An Experimental Study of Radiative Muon Decay. Brent Adam VanDevender Poquoson, VA

Transcription

1 An Experimental Study of Radiative Muon Decay Brent Adam VanDevender Poquoson, VA B.S., University of Virginia, 1996 M.A., University of Virginia, 2002 A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Physics University of Virginia January, 2006

2 Abstract Experimental measurements of the Michel parameter η can be used, along with the other Michel parameters appearing in the description of muon decays, to set limits on possible violations of the V A form of the weak interaction. All of the Michel parameters, save for η, can be measured by analyzing the ordinary muon decay µ + e + ν e ν µ. To measure η, the radiative decay µ + e + ν e ν µ γ must be observed. This work is based on more than radiative muon decays observed at the Paul Scherrer Institute meson factory using a large acceptance spectrometer. Based on these events we measure the branching ratio for the radiative decay, with the restrictions E γ > 10 MeV on the photon energy and θ > 30 on the positron/photon opening angle, to be B = [4.40±0.02 (stat.)±0.09 (syst.)] The best fit for the branching ratio is found to occur for η = ± 0.050(stat.) ± 0.034(syst.), to be compared to the V A Standard Model value η SM = 0. We interpret our result as an upper limit on the allowed value: η (68 % confidence). Combined with other measurements of η, this reduces the known upper limit to η (68 % confidence).

3 Contents 1 Introduction The Standard Model of Particle Physics Muons and The Weak Interaction Muon Decay Michel Decay: µ + e + ν e ν µ Radiative Michel Decay: µ + e + ν e ν µ γ Motivation for This Work The PIBETA Apparatus Introduction PSI Proton Cyclotron PIBETA Detector πe1 Beam Line Thin Tracking Detectors Calorimeter Electronics Triggers Front-End Computer Efficiency Data Analysis Software Calorimeter Clumps Track Finding Algorithm Michel Decay Analysis Introduction Event Selection Kinematic Cuts Time Structure of Muon Decays Results Branching Ratio Michel Parameter ρ Conclusions i

4 ii 4 Radiative Michel Decay Analysis Introduction Strategy Branching Ratio Parameter Optimization Event Selection Time Window Time Coincidence Kinematic Cuts Results The Parameters η and ρ Branching Ratio A The Functions f i (x, y, θ) 104 B Radiative Michel Decay Event Statistics 106

5 List of Figures 1.1 The differential decay rate for µ + e + ν e ν µ The Standard Model contribution to the µ + e + ν e ν µ γ branching ratio The sensitivity of µ + e + ν e ν µ γ to the parameter η The sensitivity of µ + e + ν e ν µ γ to the parameter ρ The accelerator facilities at PSI The PIBETA detector in cross-section parallel to the beam direction The PIBETA calorimeter in relief The PIBETA target and tracking detectors in cross-section perpendicular to the beam Event signal pileup in the target The effect of target pileup on the measured energy spectrum Calibrated energy spectra for each target Muon decay vertex distributions The calibrated energy deposited in the PV hodoscope for one-arm lowthreshold trigger events Tracking detector efficiencies shown to be independent of particle energy The spectrum of positrons with 40 < E CsI < 76 MeV in the one-arm low-threshold trigger The spectrum of positrons with 0 < E CsI < 60 MeV in the one-arm low-threshold trigger Energy deposited in the CsI veto crystals The individual ingredients of a pion stop signal A sketch of the beam trigger logic The angular separation between wire-chamber tracks and calorimeter clumps The identification of particles based on E PV vs. E PV + E CsI The relative difference in the µ + e + ν e ν µ decay rate when ρ = ρ SM ± The calculated time spectrum of muon decays The measured time spectrum of muon decays iii

6 4.1 A graphical definition of thrown and detected cuts Muon gate fraction cancellation in the nine-piece target data set Muon gate fraction cancellation in the one-piece target data set Radiative muon decay event timing signal-to-background χ 2 (η, ρ) for the nine-piece target data set χ 2 (η, ρ) for the one-piece target data set χ 2 (η, ρ SM ) for the nine-piece target data set χ 2 (η, ρ SM ) for the one-piece target data set The simulated opening angle distribution of misidentified nonradiative decay events The radiative Michel decay kinematic spectra for the nine-piece target data set The radiative Michel decay kinematic spectra for the one-piece target data set iv

7 List of Tables 1.1 Muon decay modes Experimental limits on the coupling constants g γ αβ The primary decay modes registered by the PIBETA experiment Scale factors for calorimeter energy calibration Hardware prescaling factors Members of a clump data structure Members of a track data structure Various time scales involved in the PIBETA experiment Tracking efficiencies and prescale factors µ + e + ν e ν µ results for the nine-piece target data set µ + e + ν e ν µ results for the one-piece target data set Time windows within which muon decay events are accepted Statistics for the normalizing, nonradiative decay µ + e + ν e ν µ Optimal values of η and ρ Gain factor correction for photons Results for the radiative muon decay branching ratio B.1 Radiative Michel decay total event statistics B.2 Event statistics for the nine-piece target data set in the first bin of cos θ.107 B.3 Event statistics for the nine-piece target data set in the second bin of cos θ B.4 Event statistics for the nine-piece target data set in the third bin of cos θ B.5 Event statistics for the one-piece target data set in the first bin of cos θ.109 B.6 Event statistics for the one-piece target data set in the second bin of cos θ B.7 Event statistics for the one-piece target data set in the third bin of cos θ.110 v

8 Chapter 1 Introduction 1.1 The Standard Model of Particle Physics The Standard Model of Particle Physics is one of the great triumphs of modern science. It is a powerful theory of the fundamental laws of nature supported by extensive experimental evidence. Nearly all of its predictions have been fulfilled and no discordant measurements have yet withstood scientific scrutiny. Nevertheless, the particle physics community is currently at a stalemate with the Standard Model. In spite of its many successes and the absence of any apparent shortcomings, the Standard Model is clearly not the ultimate theory which we seek. It is not truly fundamental as there is structure in the Standard Model which is not understood. The situation is similar to that of the Periodic Table of the Elements in the time of Mendeleev. The Periodic Table was (and is) a powerful organizational tool 1

9 Chapter 1: Introduction 2 for the known elements. It facilitated the prediction of several new elements which were subsequently discovered and found to have the expected properties. However, the organization into rows and columns was just a mnemonic device that arranged elements according to their observed properties. Only after the advent of quantum mechanics and the discovery that atoms are actually composite objects could the structure of the Periodic Table be understood. In that case, Quantum Mechanics provided the more basic understanding. In the case of the Standard Model, it is unclear where the answers lie. Current experiments push technology to its limits to search for shortcomings of the Standard Model that could indicate the direction our inquiries should take. This work describes one of those experiments. 1.2 Muons and The Weak Interaction According to our current understanding, there are four fundamental interactions which occur in nature: electromagnetic (EM), weak nuclear, strong nuclear (or simply weak and strong), and gravitational. These completely describe the behavior of the fundamental particles: the leptons, quarks and gauge bosons. The gauge bosons mediate the interactions between the quarks and leptons and even among themselves. The Standard Model encompasses the electromagnetic, weak and strong interactions. It does not describe gravity, which in any case is of negligible strength compared to the other interactions at the microscopic scale.

10 Chapter 1: Introduction 3 In this work, we will be concerned only with the weak interaction. All fundamentals particles interact weakly. In principle, we could therefore use any particle we wished to do our experiment. However, any experiment involving hadrons, which are composed of quarks, would be complicated by the presence of strong interactions, for which effects are very difficult to account. It is most convenient to use leptons, which are impervious to strong interactions to very good approximation. The additional presence of electromagnetic interactions introduces no great complications, as these are very well understood. The ideal lepton is the muon. It is heavy enough to decay into lighter leptons (electrons and neutrinos) and photons but not heavy enough to decay into hadrons. The lightest hadron, the pion, is heavier than the muon and such decays are therefore prohibited by the conservation of energy. The muon entered particle physics history in 1937 when Neddermeyer and Anderson unwittingly discovered it in cosmic rays [27]. At the time it was believed to be the pion, which had been predicted by Yukawa in 1935 [33] as the mediator of the strong nuclear force. A decade later however, experiments demonstrated that the new particle did not participate in the strong interaction and therefore could not be Yukawa s pion [5]. The discovery of a new and unexpected particle caused Rabi famously to exclaim Who ordered that?. The notion that the newly discovered muon was simply a heavy electron was also discounted by the low rate of the decay mode µ + e + γ, which was found to be B < 10 1 in 1948 [19]. The continuous energy spectrum of the electrons from muon decay was established the same year, indicating

11 Chapter 1: Introduction 4 that a muon decayed into an electron and two neutral particles [31]. The discovery of parity violation [32] prompted Feynman and Gell-Mann to suggest that the weak interaction proceeded via the exchange of charged intermediate vector bosons [14]. This mechanism however, predicted a branching ratio for µ + e + γ larger than the known upper-limit [12]. This led several authors to hypothesize that the neutrino which coupled to the muon was different from that coupled to the electron thus forbidding the decay µ + e + γ [28, 30] (now known to be B < [2]) and furthermore to the idea that neutrinos were massless fermions with only one possible spin state [23]. An experiment at Brookhaven National Laboratory [7] verified the two neutrino hypothesis a few years later, implying that lepton flavors were conserved separately. Present experiments take advantage of the muon s indifference to strong interactions and the relative ease with which they are produced at modern facilities to make very clean and precise measurements. It is hoped that these measurements will eventually reveal the inevitable signal of the theory which underlies the Standard Model. Reference [22] gives a comprehensive review of these experiments and their prospects.

12 Chapter 1: Introduction 5 Table 1.1: Muon decay modes. decay mode branching ratio reference µ + e + ν e ν µ 100 % µ + e + ν e ν µ γ (E γ > 10 MeV) (1.4 ± 0.4) % [6] µ + e + ν e ν µ e + e (3.4 ± 0.4) 10 5 [1] µ + e + γ < [2] Muon Decay As noted above, muons decay into electrons and neutrinos and possibly also a photon, which may internally convert to an electron/positron pair: µ + e + ν e ν µ (γ). (1.1) Table 1.1 lists the decay modes and measured values of their branching ratios, or upper limits on the branching ratio in the case of unobserved modes. We shall use positively charged muons (µ + ) in our notation throughout this work. The corresponding decays of negatively charged muons are related by charge conjugation, which is known to be a very good symmetry of nature. Reference [11] discusses current limits on possible violations of charge symmetry. The dominant process, µ + e + ν e ν µ, also referred to as the Michel decay, is the fate of all muons. Technically, this decay also includes µ + e + ν e ν µ γ, as is implied

13 Chapter 1: Introduction 6 by the photon in parentheses in (1.1). The vast majority of the latter decays involve a photon of very low energy emitted collinearly with the positron. These decays are thus indistinguishable from the former decay for all practical purposes. There is a significant probably however, that the decay is accompanied by a hard photon emitted at a large angle with respect to the positron. We shall treat this decay separately below (section 1.2.3). In any case, the term Michel decay refers to the process (1.1) with photons of any energy and will be denoted by µ + e + ν e ν µ in order to distinguish it from the case of µ + e + ν e ν µ γ, with an explicit hard photon. The most generic four-fermion point interaction Hamiltonian describing muon decay assumes only Lorentz invariance, local (i.e., derivative free) interactions and lepton-number conservation. The point interaction permits several equivalent Hamiltonians, related to each other via Fierz transformations, which differ in the way the fermions are grouped together. We choose the charge-exchange order, with fields of definite handedness, for which the matrix element is given by [13] M = 4 G F 2 γ=s,v,t α,β=r,l g γ αβ e α Ôγ (ν e ) n (ν µ ) m Ôγ µ β, (1.2) where G F is the Fermi coupling constant. The labels α and β denote left- (L) or right-handed (R) chirality of the positron and muon respectively. The chiralities of the neutrinos, labeled m and n, are uniquely determined for each combination of α, β and γ. The label γ distinguishes all of the interactions Ôγ allowed by Lorentz invariance: scalar (S), vector (V ) and tensor (T ). These names indicate the behavior of

14 Chapter 1: Introduction 7 e α Ôγ (ν e ) n and (ν µ ) m Ôγ µ β under Lorentz transformations and parity inversions. The explicit forms of the operators are Ô S = 1 Ô V = γ µ (1.3) Ô T = iσ µν i 2 {γ µ, γ ν }, where the γ µ s are the usual Dirac matrices satisfying the anticommutation relations {γ µ, γ ν } = g µν (1.4) and g µν is the metric tensor. There are ten complex coupling constants g γ αβ in Equation (1.2). One might naively expect twelve, but the terms involving g T LL and gt RR vanish identically due to the algebra of the associated currents. The constants are subject to the normalization condition [13] ( n S g S RR 2 + gll S 2 + grl S 2 + glr S 2) ( + n V g V RR 2 + gll V 2 + grl V 2 + glr V 2) (1.5) ( + n T g T RL 2 + glr T 2) = 1, where n S = 1, n 4 V = 1 and n T = 3. Physically, n γ g γ αβ 2 represents the relative probability that a β-handed muon will decay into an α-handed electron via the interaction Ô γ. There is no a priori reason to expect that any of these couplings vanish. However, all experimental tests are consistent with a weak interaction which has only V

15 Chapter 1: Introduction 8 Table 1.2: Experimental limits on the coupling constants g γ αβ, derived from various muon decay experiments [11]. All numbers represent a 90% confidence level and are upper limits, unless specifically noted otherwise. The maximum values allowed by definition are 2, 1 and 1/ 3 for S, V and T, respectively. g γ αβ S V T LL > LR RL RR coupling between left-handed muons and left-handed electrons. This fact is built into the Standard Model by setting g V LL = 1 (1.6) with all other coupling constants vanishing, as they must according to Equation (1.5). It is important to understand that although this action is consistent with experimental evidence, experimental uncertainties still allow the possibility of small but non-zero values of the other constants. The current experimental limits are given in Table 1.2. Reference [13] gives a comprehensive review of the experiments which led to those limits.

16 Chapter 1: Introduction Michel Decay: µ + e + ν e ν µ Beginning from the matrix element (1.2), one can arrive at the differential decay rate for Michel decay [13]: d 2 Γ dx d(cos θ) = m [ µ 4π W eµg F x 2 x 2 0 [F IS (x) + P µ + cos θf AS (x)] 1 + P e +(x, θ) ˆζ ] (1.7) where W eµ = (m 2 µ + m 2 e)/(2m µ ), x = E e +/W eµ and x 0 = m e /W eµ. Here, E e + is the energy of the positron and m µ and m e are the masses of the muon and positron, respectively. The range of allowed positron energies is m e E e + W eµ, or equivalently, x 0 x 1. The variable θ is the angle between the muon polarization P µ and the positron momentum and ˆζ is the unit vector in the direction of the positron spin polarization with respect to an arbitrary direction. Pe + is the polarization of the positron along the direction of its momentum. The functions F IS and F AS are the isotropic and anisotropic parts of the positron energy spectrum. They are given by: F IS (x) = x(1 x) ρ(4x2 3x x 2 0) + ηx 0 (1 x), (1.8) and F AS (x) = 1 3 ξ x 2 x 2 0 (1 x + 23 [ ( )]) δ 4x x (1.9) The parameters ρ, η, ξ and δ are called the Michel parameters [25]. The expression for the differential decay rate (1.7) can be simplified for the case where polarizations are not observed. Averaging over the possible polarizations in-

17 Chapter 1: Introduction 10 volves integrating cos θ over the antisymmetric interval 1 cos θ 1: 1 1 cos θ d(cos θ) = 0. (1.10) Thus, the term in (1.7) involving F AS vanishes. An analogous argument leads to the vanishing of the term P e (x, θ) ˆζ. We shall also neglect the last term in F IS since x 0 is small (x 0 = ) and furthermore the parameter η is measured to be within 1 % of its Standard Model value, η SM = 0 [22]. With these modifications we get the final form of the differential decay rate for µ + e + ν e ν µ when no polarizations are observed: dγ dx = m [ µ π W eµg F x 2 x 2 0 x(1 x) + 2 ] 9 ρ(4x2 3x x 2 0). (1.11) The relative rate is plotted in Fig The appropriate, simplified decay rate (1.11) is explicitly dependent on the Michel parameter ρ. This parameter is related to the coupling constants in Equation (1.2): ρ = [ g V LR 2 + g V RL g T LR g T RL 2 + Re(g S RLg T RL + g S LRg T LR) ]. (1.12) Recalling the Standard model prescription (1.6) and the normalization condition (1.5), one easily obtains the Standard Model value of ρ: ρ SM = 3 4. (1.13) A recent experiment has resulted in a very precise determination [26]: ρ = ± , (1.14)

18 Chapter 1: Introduction 11 Figure 1.1: The differential decay rate (1.11) for µ + e + ν e ν µ. The variable x = 2E e /m µ is the e + energy in dimensionless units. in agreement with the Standard Model prediction. A measurement in contradiction to the theory would imply scalar, vector and/or tensor coupling between left-handed muons and right-handed electrons or vise-versa, although one would not be able to distinguish exactly which couplings were present on the basis of this measurement alone. However, a corroborating measurement such as (1.14) is itself insufficient grounds to rule out the possibility of deviation from the accepted V A interaction, even in the idealized case of an exact measurement with no uncertainties. Inspection of Equation (1.12) reveals that any arbitrary values of gll S, gs RR, gv RR and gv LL can still result in ρ = 3. 4

19 Chapter 1: Introduction Radiative Michel Decay: µ + e + ν e ν µ γ The measurement of any individual Michel parameter is generally insufficient to determine the complete Lorentz structure of the weak interaction [13], as discussed above for the case of ρ. To build knowledge of the interaction then, we need to measure additional parameters. The parameter ρ exhausts the possibilities for the Michel decay in the case where polarizations are not observed. Fortunately, we may still proceed so long as we can observe photons by analyzing the radiative Michel decay µ + e + ν e ν µ γ, where the hard photon is explicitly observed as a particle distinct from the positron. Approximately 1.2% of all muon decays are accompanied by a photon with energy E > 10 MeV [21, 6]. The presence of the additional photon provides more access to the parameters of the weak interaction and, since the photon couples electromagnetically to either the muon or the positron, it introduces no new uncertainties, as discussed above. This situation is analogous to the use of deep inelastic scattering of electrons from nuclei to study the strong interaction. There, the electron couples to the hadronic constituents of nuclear matter predominantly through electromagnetic interactions and therefore allows the study of the strongly interacting quarks and gluons without introducing extraneous uncertainties. The spectrum of the radiative Michel decay has been treated by several authors [20, 17, 9]. The differential branching ratio for the case where no polarizations

20 Chapter 1: Introduction 13 are observed can be written as follows [10]: where d 3 B(x, y, θ) dx dy 2π d(cos θ) = f 1(x, y, θ) + ηf 2 (x, y, θ) + (1 4 3 ρ)f 3(x, y, θ) (1.15) x = 2E e +, y = 2E γ, cos θ = ˆp e + ˆp γ (1.16) m µ m µ and each function f i is a polynomials in x, y and = 1 β cos θ with β = p e + /E e +. Appendix A gives the explicit forms of the functions f i. Energy and momentum conservation are enforced by the inequality 2(x + y 1). (1.17) xy The parameter ρ is the same as that which occurs in the Michel decay positron energy spectrum (1.11). The parameter η is a new Michel parameter, observable only in the radiative decay. Like the other Michel parameters, it is related to the coupling constants in (1.2): η = ( grl V 2 + glr V 2) + 1 ( g S 8 LR + 2gLR T 2 + grl S + 2gRL T 2) + 2 ( glr T 2 + grl T 2). (1.18) Recalling the normalization condition (1.5) and the Standard Model prescription (1.6), we see that η is a positive semidefinite number with the nominal value η SM = 0. (1.19) The most precise measurement of η to date agrees with the Standard Model [10]: η = ± (1.20)

21 Chapter 1: Introduction 14 This result can be interpreted as an upper limit on the allowed value of η: η (68 % confidence). (1.21) Section 4.2 provides the details of this interpretation. Any deviation of η from the nominal value η SM, would imply deviation from a pure V A weak interaction. We note however, that as with measurements of ρ, corroborating measurements of η are not sufficient to establish the V A form as (1.18) and (1.19) can be satisfied for arbitrary values of gll V, gv RR, gs LL, and gs RR. If η = 0 and ρ = 3 as dictated by the Standard Model, then only f 4 1 contributes to the spectrum (1.15). Figure 1.2 shows that the most probable radiative decay has a low energy photon emitted at a small angle with respect to the positron as noted above. The greatest sensitivity to the actual physical values of η and ρ occurs in regions of kinematic phase space for which f 2 /f 1 and f 3 /f 1 are maximized, respectively. Figures 1.3 and 1.4 demonstrate that these ratios are significant in large regions of the phase space, though Figure 1.2 reminds us that these regions are relatively sparsely populated. 1.3 Motivation for This Work The PIBETA project is an ongoing series of experiments at the Paul Scherrer Institute in Villigen, Switzerland. The primary goal of the experiment was to make a precise

22 Chapter 1: Introduction 15 Figure 1.2: f 1 (x, y, θ) for various values of θ. In the Standard Model with η = 0 and ρ = 3 4, f 1 is the sole contribution to the differential branching ratio (1.15) of the µ + e + ν e ν µ γ decay.

23 Chapter 1: Introduction 16 Figure 1.3: f 2 /f 1 for various values of θ. f 2 /f 1 is a measure of the sensitivity of Equation (1.15) to the value of η.

24 Chapter 1: Introduction 17 Figure 1.4: f 3 /f 1 for various values of θ. f 3 /f 1 is a measure of the sensitivity of Equation (1.15) to the value of ρ.

25 Chapter 1: Introduction 18 measurement of the branching ratio of pion beta decay [29], π + π 0 e + ν e. (1.22) However, several other pion decay modes were measured [16] in parallel with the the mode (1.22) as well as the muon decay modes discussed above. The primary decays are summarized in Table 1.3. The reason for this methodology is twofold. On one hand it increases the return of physics results relative to the investment in the experiment. Most important though, is that it provides independent internal calibrations of the detector response over the broadest possible kinematic range. One of the great challenges of experimental science is the elimination of systematic errors. The PIBETA methodology allows for analysis to be validated by verifying internal results for well understood and precisely measured reactions (e.g., µ + e + ν e ν µ or π + e + ν e ) against external results. This lends confidence to results obtained for the primary modes of interest (e.g., µ + e + ν e ν µ γ, π + e + ν e γ and π + π 0 e + ν e ) so that if any unexpected phenomena are revealed, it is unlikely that they can be ascribed to mere systematic experimental errors. This work presents the analysis of the muon decays listed in Table 1.3 and discussed above, based on data taken by the PIBETA experiment from May through August of The Michel decay spectrum is well understood theoretically and has been very precisely measured [26, 18]. Therefore, we will use it to validate our analysis tools, particularly the simulation of the detector response and the calibration of

26 Chapter 1: Introduction 19 Table 1.3: The principle decay modes registered by the PIBETA experiment. Note that it is not meaningful to assign exact numbers to the radiative decay modes in the absence of kinematic constraints on the spectrum of the photon. decay mode branching ratio µ + e + ν e ν µ 100 % µ + e + ν e ν µ γ 1 % π + e + ν e π + e + ν e γ π + π 0 e + ν e the experimental data (chapter 2). When we are satisfied that our analysis is sound we shall then progress to the radiative Michel decay. The formal condition to be met for satisfaction is the extraction of values for the branching ratio B µ + e + ν eν µ and the Michel parameter ρ which are consistent with the best external measurements [11, 26] (Chapter 3). We shall then lay out our strategy for analysis of µ + e + ν e ν µ γ events and present the results of this analysis (Chapter 4). Our main goal is to extract the Michel parameters η and ρ from the analysis of µ + e + ν e ν µ γ events, and to measure the branching ratio B µ + e + ν eν µγ for the largest possible region of phase space.

27 Chapter 2 The PIBETA Apparatus 2.1 Introduction This chapter describes the PIBETA detector hardware and data analysis software alongside the simulation of the detector response. The hardware and software are described with details sufficient to understand the analysis in subsequent chapters. Complete details of the PIBETA detector are published in Reference [15]. 2.2 PSI Proton Cyclotron Figure 2.1 shows the layout of the PSI accelerator facilities. The cyclotron accelerates an approximately ma proton beam to an energy of 590 MeV. The accelerator operates at a frequency of MHz producing proton pulses 1 ns wide and sepa- 20

28 Chapter 2: The PIBETA Apparatus 21 rated by ns. The primary proton beam is transported to two target stations which produce pions and muons. These products are transported along secondary beam lines to the experimental areas. The PIBETA detector is operated in the πe1 experimental area which is designed for intense low-energy pion beams with good momentum resolution. The πe1 beam line can deliver a pion beam with a maximum momentum of 280 MeV/c, a full-width-half-maximum momentum resolution of less than 2 % and an accepted production solid angle of 32 msr. 2.3 PIBETA Detector Figure 2.2 shows a sketch of the PIBETA detector in a cross-sectional plane through the beam axis. Figure 2.3 is a relief of the CsI calorimeter and figure 2.4 is a crosssectional view, perpendicular to the beam, of the thin tracking detectors. This section describes the main elements shown in those figures: beam line components, multiwire-proportional-chambers (MWPC1 and MWPC2), plastic-veto hodoscope (PV) and the segmented, pure-csi calorimeter (CsI) πe1 Beam Line Positively charged pions entering the πe1 experimental area are first registered in a 3 mm thick active beam counter (BC). Immediately downstream of BC is a passive lead brick collimator (PC) with a 7 mm pinhole aperture. Pions which clear the colli-

29 Chapter 2: The PIBETA Apparatus 22 Figure 2.1: The layout of the accelerator facilities in building WEHA at PSI. The PIBETA detector operates in the πe1 experimental area.

30 Chapter 2: The PIBETA Apparatus 23 pure CsI PV BC π + beam AC1 AC2 AD AT MWPC1 MWPC2 10 cm Figure 2.2: A cross-sectional view of the PIBETA detector showing the main elements described in Section 2.3. Figure 2.3: The 240-element pure-csi calorimeter. It covers 3π sr of solid angle. The two openings allow for the beam to enter the detector and for maintenance access to the interior components.

31 Chapter 2: The PIBETA Apparatus 24 Active TGT MWPC-1 MWPC-2 PV array Figure 2.4: The stopping target and cylindrical tracking detectors shown in a cross-sectional plane perpendicular to the beam. mator are slowed in a 40 mm thick active degrader (AD) and ultimately come to rest in the active target (AT) positioned in the center of the detector. The active qualifier indicates that these elements are constructed of plastic scintillator and that their light output is detected and recorded with the rest of the data. Discriminated signals from the beam line elements are fundamental ingredients of the various electronic triggers described below (Section 2.4.1). Two different stopping targets were used in the apparatus during the course of the Summer 2004 run. Figure 2.4 shows the first of these. Note the segmentation of that target into nine individual pieces. The segmentation aids in the reconstruction of the

32 Chapter 2: The PIBETA Apparatus 25 pion stopping distribution. This reconstruction is well understood based on studies of past runs ( ) [24], so it was decided that the last weeks of the 2004 run would benefit from the improved light collection properties of a solid one-piece target. The one-piece target is of precisely the same dimensions as the nine-piece target, only without the segmentation. Because the target is such a crucial part of the detector, we choose to analyze the data from 2004 as two distinct sets, corresponding to the nine-piece target and the one-piece target. The target is the most complicated element of the detector in terms of extracting data and simulating its performance. This is due to the very high signal rate there. The target bears the total event rate of the experiment (roughly 100 khz), whereas individual elements of the calorimeter for instance, bear less than 0.5% due to the calorimeter s articulated structure. Furthermore, virtually every event in the target consists of three distinct parts: the beam pion coming to rest, the subsequent decay π + µ + ν µ which results in the muon coming to rest and finally, the decay of the muon. All of these events deposit energy in the target and their light output and subsequent photomultiplier signals pileup on each other. The situation is illustrated in Figure 2.5. We want to reconstruct the energy of the muon decay products. We therefore want the target signal most nearly in coincidence with a calorimeter shower and corresponding wire-chamber hits. Section 2.5 gives details about the algorithm that associates calorimeter showers with wire-chamber hits. The first step in extracting the

33 Chapter 2: The PIBETA Apparatus 26 Figure 2.5: A signal from the target captured by the event display program. The first peak corresponds to the stopping pion. The pion decays via π + µ + ν µ and the resulting muon comes to a stop producing the small shoulder on the pion stop signal. The muon then decays and the resulting positron causes the final peak as it exits the target. The final positron signal rides on the combined tails of two other signals. energy deposited in the target is to subtract the pedestal energy. The pedestal energy is just the peak of the energy distribution recorded for random trigger events which are not correlated with pions stopping in the target (see Section below). The peak of the energy distribution in each run is then placed at the mean peak position, obtained by averaging over all runs, by simply subtracting the difference between the peak and the average. The peak fluctuates slightly due to low-frequency noise on the signal lines. After these steps, the pion- and muon-stop pileup can be subtracted. Figure 2.6 shows how the energy in the target depends on the time between the pion

34 Chapter 2: The PIBETA Apparatus 27 Figure 2.6: The effect of pileup in the target is clearly visible in this plot of the energy left there by decay positrons in one-arm low-threshold trigger events as a function of time. The events preceding the pion stop at t = 0 are pileup free so we subtract the time dependent energy from events and then add back the constant mean of t < 0 events, denoted by the dashed line. stop and the muon decay (taken from the corresponding calorimeter shower). The increased energy at time t > 0 is due to the tail of the pion stop at t = 0. The energy for events at t < 0 is pileup free since these events precede the pion stop. The time dependence is removed from the target energy by subtracting the energy versus time distribution shown in Figure 2.6 and then adding back the constant mean value of the energy for events at t < 0, denoted in the figure with a dotted line. The last step is simply to multiply by a calibrating factor which converts ADC channels to MeV such

35 Chapter 2: The PIBETA Apparatus 28 that the peak of the experimental distribution is aligned with that of the simulated distribution. The simulated target energy is best matched to data by smearing it with photoelectron statistics and then adding pedestal noise: and subsequently E sim tgt E sim tgt E sim tgt 1 + r σ=1 ne sim tgt, (2.1) E sim tgt + r σ r u, (2.2) where r σ is a Gaussian distributed random number with variance σ 2, 0 r u 1 is a uniformly distributed random number and n is the number of photoelectrons per MeV of energy deposited. The uniformly distributed random number in the pedestal noise simulates the increasing uncertainty in the energy for events closer to the pion stop. Both targets are found to generate 80 photoelectrons/mev and the pedestal noise is σ = 1.55 MeV for the nine-piece target and σ = 2.00 MeV for the one-piece target. All of the above discussion implicitly assumes that the simulation uses the correct distribution of decay vertices. Otherwise, the energy spectra shown in Figure 2.7 would be skewed because particles would exit the target along longer or shorter paths. Figure 2.8, showing the decay vertex distribution inferred from wire-chamber tracks, confirms that the stopping distribution used in the simulation is indeed correct.

36 Chapter 2: The PIBETA Apparatus 29 Figure 2.7: The calibrated energy deposited in the stopping targets for one-arm low-threshold trigger events.

37 Chapter 2: The PIBETA Apparatus 30 Figure 2.8: The decay vertex distributions inferred from wire chamber tracks for one-arm low-threshold trigger events. The vertex (x 0, y 0, z 0 ) is taken to be the point on the wire-chamber track closest to the z-axis.

38 Chapter 2: The PIBETA Apparatus Thin Tracking Detectors: MWPC1, MWPC2 and PV Two cylindrical multi-wire proportional chambers, MWPC1 and MWPC2 precisely track charged decay products. They are highly efficient (greater than 95%) and stable at rates of up to 10 7 minimum-ionizing particles per second. Each chamber has one anode wire plane along the z-direction and two cathode strip planes in a stereoscopic geometry. The resolution with which the chambers can track charged particles is simulated by simultaneously matching the distributions of reconstructed decay vertices and the angular separation of wire chamber tracks and their corresponding calorimeter clumps. Figures 2.8 and 2.16 demonstrate the agreement between the simulation and the data. These figures will be discussed in more detail after we have elaborated on the software reconstruction of tracks in Section 2.5. The best match between the data and the simulation is found when both chambers have the coordinates of the simulated track hit smeared by the amounts x = y = 1.6 mm, z = 1.0 mm. (2.3) This implies an angular resolution of approximately 1 and is consistent with the resolution found independently from an analysis of cosmic muon events [15]. The plastic-veto hodoscope (PV) surrounds the MWPCs. It is composed of twenty individual staves of plastic scintillator, fitted together to form a cylinder covering the entire 2π azimuthal angle. Its length is such that any particle emanating from the stopping target and arriving in the calorimeter must also traverse the active volume

39 Chapter 2: The PIBETA Apparatus 32 of the hodoscope. The scintillator pieces are supported by a very thin, cylindrical carbon fiber shell ( r = 1 mm = radiation lengths). The light output of each stave is registered by photomultiplier tubes on each end (denoted ±z). The energy deposited in an individual piece is taken to be the geometric mean of the calibrated energy registered at each end separately: E exp PV = E +z E z. (2.4) PV energy deposition is reproduced in simulation by smearing the raw (simulated) energy deposition with finite photoelectron statistics and applying a gain factor: ( ) EPV sim EPV sim G + r σ, (2.5) E sim PV where G is the gain factor and r σ is a Gaussian distributed random number with variance σ 2. The parameters G and σ were found for all 20 PV elements individually by minimizing the difference between the recorded and simulated spectra. Figure 2.9 shows the cumulative result (i.e., all 20 elements together) for minimum-ionizing positrons. These same positrons can be used to measure the efficiency with which each of the tracking detectors registers minimum-ionizing charged particle hits. Here we describe the computation of the inner-chamber efficiency ɛ MWPC1. The computation of the outer-chamber and hodoscope efficiencies is analogous. The efficiency is the ratio of the number of events where the positron registered in all possible tracking detectors, including MWPC1, and the calorimeter, to the number of events where it registered

40 Chapter 2: The PIBETA Apparatus 33 Figure 2.9: The calibrated energy deposited in the PV hodoscope for one-arm low-threshold trigger events. in the other detectors regardless of whether it also registered in MWPC1: ɛ MWPC1 = N(MWPC1 MWPC2 PV CsI). (2.6) N(MWPC2 PV CsI) The efficiencies of the other tracking detectors are computed in the same way with trivial transpositions of the symbols in Equation (2.6): ɛ MWPC2 = ɛ P V = N(MWPC1 MWPC2 PV CsI), (2.7) N(MWPC1 PV CsI) N(MWPC1 MWPC2 PV CsI). (2.8) N(MWPC1 MWPC2 CsI) These numbers are computed for each run individually where there are roughly charged tracks registering under the one-arm low-threshold trigger. The statistical

41 Chapter 2: The PIBETA Apparatus 34 uncertainty in each efficiency computed for an individual run is therefore δɛ i 2 0.5%. (2.9) The statistical uncertainty in the total tracking efficiency of all three detectors together is therefore δɛ tot 3 0.5% 1%. (2.10) We can use the efficiencies to weight the number of observed events and arrive at the number of events which actually occurred (i.e., the number we would have observed in the ideal case of 100 % efficiency). If ɛ is any generic efficiency and we observe N obs events, then the number of events N act which actually occurred is N act = N obs. (2.11) ɛ In order to account properly for inefficiencies with such a simple formula we must be certain that ɛ is truly a constant rather than a function of some other observed variable, such as particle energy. We expect this to be the case since all positrons used in the analysis have energies E 5 MeV and are therefore minimum ionizing (E 1 MeV). That is, they all deposit the same amount of energy per thickness of material traversed in the thin tracking detectors and thus have equal chances of detection. Figure 2.10 confirms this assertion. The efficiencies defined by Equations ( ) represent only the instrumental efficiencies of the detectors and their attendant electronics. The physical source of

42 Chapter 2: The PIBETA Apparatus 35 Figure 2.10: The detection efficiencies of the thin tracking detectors MWPC1, MWPC2 and PV as a function of the energy in the calorimeter for a typical run. The dashed line represents the efficiencies computed via (2.6), (2.7) and (2.8) for this particular run. Note that any energy dependence is statistically insignificant.

43 Chapter 2: The PIBETA Apparatus 36 this inefficiency is ascribed primarily to discriminator units. Other sources of tracking inefficiency, namely physical processes which involve the disappearance of the particle before it reaches the calorimeter, and the software algorithm which reconstructs tracks from the topology of detector hits (Section 2.5), are included in the simulation and therefore accounted for in the detector acceptance Calorimeter The heart of the PIBETA detector is the electromagnetic shower calorimeter. Its active volume is composed of pure CsI crystal, segmented into 240 individual pieces fitted tightly together to cover π sr of solid angle. The inner radius of the assembly is 26 cm and its active volume is 22 cm thick, corresponding to 12 radiation lengths (X CsI 0 = 1.85 cm). Each individual crystal is painted with a wavelength-shifting lacquer and wrapped in aluminized mylar to improve its light collection properties and to contain showers within individual crystals to the greatest extent possible. The main volume is composed of 200 pentagonal and hexagonal pyramids with half hexagons used to finish the pattern at the edges. The beam openings themselves are each bordered by 20 trapezoidal pyramids. These latter crystals allow for the vetoing of events where some of the shower energy is likely to have spilled out of the main body of the calorimeter. The crystal shapes can be seen in Figure 2.3. The simulation of calorimeter showers is handled by the standard GEANT3 [3] software package. The only custom adaptations required are to smear the energies to

44 Chapter 2: The PIBETA Apparatus 37 simulate photoelectron statistics and pedestal noise associated with the photomultiplier tubes and apply an overall energy scale factor g, which simulates software gain factors used in the data analysis software. In principle we should apply this factor to the data, but for practical reasons we choose instead to apply it to the simulation: ECsI sim Esim CsI g. (2.12) The factor g is found by minimizing the difference between the simulated and recorded energy spectra of positron showers which initiate one-arm low-threshold triggers. The dominant source of such events is the Michel decay of the muon, µ + e + ν e ν µ. However, there is a small background of positrons from the pion decay π + e + ν e, as can be seen in Figure The high energy edge of the muon decay spectrum overlaps with the low energy tail of the monoenergetic pion decay spectrum. Although the pion decay contribution will be negligible for our later purposes, satisfactory fits cannot be obtained in this instance if the simulated spectrum neglects the pion decay background. The simulated positron energy spectrum is formed by generating the muon and pion decays independently and then combining them, allowing the relative normalization of the pion decay spectrum to be a free parameter. The overall gain factor g is also a free parameter. The fit is performed over the range 40 < E CsI < 76 MeV as shown in Figure The results are shown in Table 2.1. Note that the fits were also performed over the range 10 < E CsI < 76 MeV with no significant difference in the results for g. Neglecting the pion decay contribution results in a significant

45 Chapter 2: The PIBETA Apparatus 38 Figure 2.11: The overall calibration between data and simulation was found by simultaneously matching the µ + e + ν e ν µ edge and the π + e + ν e peak. inflation of the uncertainty in g. The agreement between the simulation and the data in Figure 2.11 also implicitly confirms the simulation of photoelectron statistics and pedestal noise discussed above. Too much noise or too few photoelectrons would result in a more gently sloping edge and a broader peak. direction would sharpen the edge and constrict the peak. An error in the other Figure 2.12 shows the energy spectrum over the full range of energies of interest to our muon decay study. The veto crystals which line the openings of the calorimeter are simulated sepa-

46 Chapter 2: The PIBETA Apparatus 39 Figure 2.12: The energy deposited in the CsI calorimeter for one-arm low-threshold trigger events. Virtually all of these events are positrons from the Michel decay µ + e + ν e ν µ. rately from the other crystals, since they were operated at higher voltages. Figure 2.13 shows the spectrum of total energy deposited in these veto crystals. 2.4 Electronics An event is recorded to the data set when it satisfies the criteria that create a highlevel trigger. The triggers vary in complexity. The most basic triggers are simply discriminated versions of analog pulses, which indicate whether the voltage in a particular channel has exceeded a predetermined discriminator threshold. More complex

47 Chapter 2: The PIBETA Apparatus 40 Table 2.1: Energy scale factors g which appear in Equation (2.12) and the relative π + e + ν e normalization N π for the two data sets. data set g N π nine-piece target ± ± one-piece target ± ± Figure 2.13: The energy deposited in the CsI veto crystals for one-arm low-threshold trigger events. The gap is due to a software threshold which zeroes any channel that reports E < 0.8 MeV.

48 Chapter 2: The PIBETA Apparatus 41 triggers are constructed from logical combinations of these discriminated outputs and indicate various temporal coincidences and topological geometries of detected hits. The ultimate, high-level triggers are formed in turn from these lower level signals and enable the front-end computer to record data from the various channels of the detector. This section describes the formation of the triggers apropos to the analysis hereafter. We shall also discuss the efficiency with which the front-end computer logs trigger events to the data set Triggers Random Trigger A small ( mm) plastic scintillator radiation counter is placed a short distance away from the main detector, but is shielded from it by a 50 mm thick lead brick wall as well as another 500 mm of concrete. Thus, signals in the random detector are completely uncorrelated with events in the primary detector. The purpose of the counter is to trigger on ionizing cosmic rays which arrive at random intervals, about 1 2 sec 1, and subsequently to record detector signals as for any other event. This random trigger gives a sampling of the ambient electronic noise in the detector. The information gained in this way is used at the end of every production run (every few hours during normal running conditions) to compute and record ADC pedestals for every channel in the detector.