Measurement of the Radiative Muon Decay As a Test of the V − a Structure of the Weak Interactions

Measurement of the Radiative Muon Decay as a Test of the V − A Structure of the Weak Interactions

Emmanuel Munyangabe Kigali, Rwanda

B.S., National University of Rwanda, 2006

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

June, 2012 Abstract

Measurements of the radiative muon decay can be used to test the validity of the V − A form of weak interactions. All the Michel parameters can be extracted from

+ + the analysis of the ordinary muon decay µ → e νeν¯µ, with the exception of the

+ + η¯ parameter which is measured by analyzing the radiative decay µ → e νeν¯µγ. This analysis is based on more than 5.1 × 105 radiative muon decays recorded by the PIBETA experiment at the Paul Scherrer Institute (PSI), Switzerland in 2004. Based on these events, the experimental branching fraction was measured to be B = [4.365 ± 0.009 (stat.) ± 0.042 (syst.)] × 10−3. Theη ¯ parameter was extracted using the least squares method and the experimental value was found to beη ¯ = 0.006 ± 0.017 (stat.) ± 0.018 (syst.). This result is to be compared to the V − A Standard

Model valueη ¯SM = 0. Our experimental result ofη ¯ gives an upper limit of:η ¯ ≤ 0.028 (68.3 % conﬁdence), a fourfold improvement in precision over the existing world average. Contents

1 Introduction 1 1.1 The Standard Model ...... 1 1.1.1 Weak Interactions ...... 3 1.1.2 V-A Theory ...... 6 1.2 Muon decay ...... 11 + + 1.2.1 Michel Decay: µ → e νeνµ(γ) ...... 13 + + 1.2.2 Radiative Michel Decay: µ → e νeνµγ ...... 15 1.2.3 Motivation ...... 20

2 PIBETA Detector 26 2.1 Introduction ...... 26 2.2 Experimental area ...... 27 2.3 Beam-deﬁning elements ...... 28 2.3.1 Passive Collimator ...... 28 2.3.2 Active beam-deﬁning elements ...... 29 2.4 Detector Tracking System ...... 31 2.5 Calorimeter ...... 33 2.6 Cosmic muon veto detectors ...... 34 2.7 PIBETA Triggering System ...... 34 2.7.1 Introduction ...... 34 2.7.2 Random Triggers ...... 35 2.7.3 Beam Triggers ...... 35 2.7.4 Calorimeter Triggers ...... 36

3 Event Reconstruction 40 3.1 Introduction ...... 40 3.2 Data Analysis Software ...... 41 3.2.1 Tracking System algorithm ...... 41 3.2.2 Clump Algorithm ...... 42 3.3 Particle Identiﬁcation ...... 45 3.4 Monte Carlo Simulation ...... 49

i ii

3.5 Data Calibration ...... 51 3.5.1 Introduction ...... 51 3.5.2 Energy Calibration (ADC Calibration) ...... 51 3.5.3 Time Calibration (TDC Calibration) ...... 53

4 Data Analysis 56 4.1 Branching Fraction ...... 56 4.1.1 Introduction ...... 56 4.1.2 Muon decay time distribution ...... 58 4.1.3 Non-Radiative Muon Decay ...... 65 4.1.4 Radiative Michel decay ...... 67 4.1.5 Background Signal ...... 68 4.1.6 Kinematic cuts ...... 71 4.1.7 Systematic errors ...... 75 4.1.8 Results (branching fraction) ...... 76 4.2 Extraction ofη ¯ and ρ Parameters ...... 80 4.2.1 Systematic error ...... 87 4.3 Conclusions ...... 98

A The functions fi(x, y, θ) 100

B The Least Squares Method 103 List of Figures

1.1 Elementary particles and gauge bosons ...... 4 1.2 Beta decay ...... 5 ◦ 1.3 f1(x, y, θ ≥ 45 )...... 18 ◦ 1.4 f1(x, y, θ ≥ 90 )...... 18 ◦ 1.5 f1(x, y, θ ≥ 135 ) ...... 19 ◦ 1.6 f1(x, y, θ ≥ 157 ) ...... 19 ◦ 1.7 (f2/f1)(x, y, θ ≥ 45 )...... 20 ◦ 1.8 (f2/f1)(x, y, θ ≥ 90 )...... 21 ◦ 1.9 (f2/f1)(x, y, θ ≥ 135 )...... 22 ◦ 1.10 (f2/f1)(x, y, θ ≥ 157 )...... 23 ◦ 1.11 (f3/f1)(x, y, θ ≥ 45 )...... 23 ◦ 1.12 (f3/f1)(x, y, θ ≥ 90 )...... 24 ◦ 1.13 (f3/f1)(x, y, θ ≥ 135 )...... 24 ◦ 1.14 (f3/f1)(x, y, θ ≥ 157 )...... 25 2.1 Detector cross-section ...... 30 2.2 Decay chain signal ...... 32 2.3 The pion stop signal ...... 37

3.1 The primary vertex x0, y0, z0 ...... 43 3.2 CsI-MWPC angle ...... 44 3.3 Positron (Michel) θ ...... 45 3.4 Positron Michel) φ ...... 46 3.5 Positron θ ...... 46 3.6 Photon φ ...... 47 3.7 Photon θ ...... 47 3.8 Positron φ ...... 48 3.9 Positron energy in CsI calorimeter for 1-arm low trigger events. . . . 52 3.10 Positron energy in plastic veto (PV) for 1-arm low trigger events. . . 53 3.11 Walk-correction eﬀect on the energy and time of PV. The straight horizontal line indicate the removal of energy-time dependence in the observed TDC values...... 55

iii iv

3.12 Walk-correction eﬀect on the energy and time of CsI calorimeter. The straight horizontal line indicate the removal of energy-time dependence in the observed TDC values in the energy range of 10-53 MeV. . . . . 55

4.1 Time coincidence of positron and photon in the CsI calorimeter. . . . 62 4.2 The time distribution of Michel events ...... 63 4.3 The time distribution of radiative Michel events. This is measured as 1 e+ γ the time diﬀerence ( 2 (tCsI + tCsI) − tDeg)...... 64 4.4 The time ratio before selection ...... 64 4.5 The time ratio after selection ...... 65 4.6 The positron energy ...... 72 4.7 The photon energy ...... 73 4.8 The opening angle ...... 74 4.9 The positron energy cut stability ...... 76 4.10 The photon energy cut stability ...... 77 4.11 The signal window cut stability ...... 77 4.12 The opening angle cut stability ...... 78 4.13 The χ2 as a function of ρ from analysis of non-radiative Michel decays 82 ◦ ◦ 4.14 (f2/f1)(x, y, 150 < θ ≤ 160 ) ...... 83 ◦ ◦ 4.15 (f2/f1)(x, y, 160 < θ ≤ 170 ) ...... 84 ◦ ◦ 4.16 (f2/f1)(x, y, 170 < θ ≤ 180 ) ...... 85 ◦ ◦ 4.17 (f2/f1)(x, y, 157 < θ ≤ 180 ) ...... 86 4.18 The χ2 as a function of photon energy scale andη ¯ for 170◦ < θ ≤ 180◦ bin...... 89 2 ◦ ◦ 4.19 The χ as a function ofη ¯ for 170 < θ ≤ 180 bin, with Cγ = 0.994. . 89 4.20 The χ2 as a function of photon energy scale andη ¯ for 160◦ < θ ≤ 170◦ bin...... 90 2 ◦ ◦ 4.21 The χ as a function ofη ¯ for 160 < θ ≤ 170 bin, with Cγ = 0.994. . 90 4.22 The χ2 as a function of photon energy scale andη ¯ for 160◦ < θ ≤ 180◦ bin...... 91 2 ◦ ◦ 4.23 The χ as a function ofη ¯ for 160 < θ ≤ 180 bin, with Cγ = 0.994. . 91 4.24 The χ2 as a function of photon energy scale andη ¯ for 150◦ < θ ≤ 160◦ bin...... 92 2 ◦ ◦ 4.25 The χ as a function ofη ¯ for 150 < θ ≤ 160 bin, with Cγ = 0.997. . 92 4.26 The χ2 as a function of photon energy scale for 160◦ < θ ≤ 180◦ bin, withη ¯ = 0.006...... 93 List of Tables

1.1 Range and lifetimes ...... 6 1.2 Weak isospin ...... 6 1.3 Muon decay modes ...... 12

2.1 Trigger pre-scaling factors ...... 39

4.1 Pibeta time scales ...... 59 4.2 Michel cuts ...... 66 4.3 Event generator level cuts...... 71 4.4 Event reconstruction cuts ...... 72 4.5 Systematic errors (largest accessible phase space ...... 78 4.6 Experimental branching fraction ...... 79 4.7 Values ofη ¯ ...... 88 4.8 Systematic errors of the region 0.5 < x ≤ 0.75, 0.28 < y ≤ 0.48 . . . . 93 4.9 Systematic errors of the region 0.5 < x ≤ 0.75 and 0.5 < y ≤ 0.75 . . 94 4.10 Systematic errors of the region 0.28 < x ≤ 0.48 and 0.5 < y ≤ 0.75 . . 94 4.11 Systematic errors of the region 0.28 < x ≤ 0.48 and 0.28 < y ≤ 0.48 . 95 4.12 Statistics for bin 170◦ < θ ≤ 180◦ ...... 95 4.13 Statistics for bin 160◦ < θ ≤ 170◦ ...... 96 4.14 Statistics for bin 160◦ < θ ≤ 180◦ ...... 96 4.15 Statistics for bin 150◦ < θ ≤ 160◦ ...... 96 4.16 Systematic errorsη ¯ parameter ...... 97

v Chapter 1

Introduction

1.1 The Standard Model

The Standard Model of particle physics [1] is a theory that describes the behavior of subatomic particles. This behavior is manifested by the interactions between these particles. So far, there are four known fundamental interactions; electromagnetic, strong, weak and gravity. The Standard Model theory was developed throughout the mid to late 20th century, the current formulation was ﬁnalized in the mid-1970s upon experimental conﬁrmation of the existence of quarks and the intermediate bosons (W and Z). Because of its success in explaining a wide variety of experimental results, the theory is sometimes regarded as a theory of almost everything.

Yet, the theory fails to incorporate the physics of general relativity, such as grav- itation and dark energy or dark matter particles that are deduced from cosmological

1 Chapter 1: Introduction 2 evidence. Nevertheless, the Standard Model theory is an important tool to explain particle physics.

The Standard Model includes 12 elementary particles of spin-1/2 known as fermions and their corresponding anti-particles. The fermions of the Standard Model are clas- siﬁed according to how they interact. Some interact strongly, weakly, and electromagnetically (quarks), while others never interact via strong interactions (leptons).

There are six quarks (up, down, charm, strange, top, and bottom) and six leptons

(electron, electron neutrino, muon, muon neutrino, tau, tau neutrino), grouped into three generations as shown in Fig 1.1.

The main diﬀerence between quarks and leptons is that quarks carry strong color charge, therefore, only quarks can interact via strong interaction. Quarks bound to one another form color-neutral hadrons. Hadrons are of two types: mesons (composed of a valence quark and anti-quark) and baryons (composed of three valence quarks). Quarks can also interact with other fermions via electromagnetic and weak interaction, because they carry electric charge and weak isospin, respectively.

Leptons and their corresponding neutrinos do not cary color charge and hence do not interact strongly. Furthermore, the three neutrino ﬂavors and their corresponding anti-neutrinos cannot interact electromagnetically as they do not carry electric charge and hence their interaction is only through the weak force. However, charged leptons interact both electromagnetically and weakly.

Each member of a generation has greater mass than the corresponding particles Chapter 1: Introduction 3 of lower generations. The ﬁrst generation of charged particles (up, down quarks and electron) being the lightest particles, are kinematically forbiden to decay into other particles. This is the reason why all ordinary (baryonic) matter is made of such particles. Speciﬁcally, all atoms consist of electrons orbiting atomic nuclei, ultimately constituted of up and down quarks. Second and third generation charged particles, on the other hand, decay with very short half-lives, and are observed only in very high-energy collisions. Neutrinos of all generations also do not decay, and even though they pervade the whole universe, they are hard to detect because they rarely interact with baryonic matter. The Figure 1.1 shows the elementary particles and the bosons that mediate the interactions.

1.1.1 Weak Interactions

This work focuses on testing the V − A structure of weak interactions [2], therefore

I will give an overview of weak interactions and give a brief theory about V − A structure of weak interactions.

Firstly, weak interactions aﬀect all known fermions; quarks and leptons interact weakly by the exchange (emission or absorption) of W ± and Z0 bosons. There are two types of weak interactions: charged and neutral interactions. W ± bosons participate in the charged interactions while Z0 bosons participate in the neutral interactions.

The best known example of this weak force is the neutron beta decay, where an electron, proton and a neutrino are emitted as shown in Figure 1.2 Chapter 1: Introduction 4

Figure 1.1: Elementary particles and the gauge bosons that mediate the interactions between them

The Z and W bosons masses are very heavy mW ∼ 80 GeV and mZ0 ∼ 90 GeV; this large mass accounts for the very short range of the weak interaction and long lifetimes for all particles decaying via weak interactions. The range of interaction is inversely proportional to the mass of the gauge boson that mediates the interaction, while the life time is inversely proportional to the decay rate. The relative decay time for strong, electromagnetic and weak interactions is shown in Table 1.1 Chapter 1: Introduction 5

The weak interaction has another unique property − namely quark ﬂavor changing where quarks can swap their ﬂavors (i.e, changing from one type of quark into another). In addition, it is the only interaction that violates parity (P)-symmetry.

Weak interactions introduce a term called weak isospin (T3) which is a property

(quantum number) of all particles that governs how particles interact in the weak interaction. As a comparison, weak isospin is to the weak interaction what electric charge is to the electromagnetism, and to what color charge is to the strong inter-

1 action. Elementary particles have weak isospin values ± 2 . For example, up-type

1 quarks (u, c, t) have T3 = + 2 and always transform into down-type quarks (d, s, b),

1 which have T3 = − 2 . During this transformation, a quark never decays weakly into

a quark of the same T3. As is the case with electric charge, these two possible values

are equal, except for sign, as shown in Table 1.2.

Figure 1.2: Feynman diagram for beta decay of a neutron into a proton, electron and electron anti-neutrino, via an intermediate W gauge boson. Chapter 1: Introduction 6

Table 1.1: Comparison of interactions in terms of energy scales, range and lifetimes. The strong interaction is asymptotically free as quarks are essentially free to move when very close to each other and essentially impossible to separate as they move apart.

Interaction Mediator Strength Range Lifetime(s) EM photons(γ) 1/137 long 10−20 Weak W ±,Z 10−5 short 10−8 Strong Gluons(G) 1 short 10−23 Gravity gravitons(g) 10−41 long long

Table 1.2: Weak isospin for fermions. In any decay through a weak interaction, weak isospin is conserved: the sum of the weak isospin numbers of the particles exiting a reaction equals the sum of the weak isospin numbers of the particles entering that reaction

Weak Isospin First Generation Second Generation Third Generation

1 + 2 Up Charm Top 1 − 2 Down Strange Bottom 1 − 2 Electron Muon Tau 1 + 2 Electron neutrino Muon Neutrino Tau Neutrino

1.1.2 V-A Theory

Parity Violation

In physics, a parity transformation (also called parity inversion) is the ﬂip of the sign of spatial coordinates. The parity operator Pˆ corresponds to a discrete transformation x → −x, y → −y etc. Chapter 1: Introduction 7

Under the parity transformation, vectors change sign while axial-vectors are un- changed. For example, if ~r is a vector and L~ is an axial-vector then:

P~rˆ = −~r and PˆL~ = L~

For a very long time, parity symmetry was considered to be conserved in all particle interactions before experimental evidence showed that weak interaction do not conserve parity. This was demonstrated by C.S.Wu in 1957 [3] in an experiment on Beta decay of Cobalt-60 nuclei:

60 60 ∗ − Co → Ni + e +ν ¯e (1.1)

She observed that electrons were emitted preferentially in the direction opposite to an applied magnetic ﬁeld. This was contrary to parity conservation, because if parity was conserved, then equal number of electrons would be emitted in the directions along and opposite to the applied magnetic ﬁeld. This demonstrated that parity is violated in the weak interaction.

V − A Structure of the Weak Interaction

The most general matrix element for the decay that embodies kinematics and spin- dependent interactions [4] is given by:

X γ ˆγ ˆγ M = gαβheα|O |(νe)ih(νµ)|O |µβi. (1.2) γ Chapter 1: Introduction 8

There are only 5 possible combinations of two spinors and the gamma matrices that are Lorentz invariant, called bilinear covariants:

Scalar: ΨΨ (1 component)

Pseudo-Scalar: Ψγ5Ψ (1 component)

Vector: ΨγµΨ (4 components)

Axial-Vector: Ψγµγ5Ψ (4 components)

Tensor: Ψ(γµγν − γνγµ)Ψ (6 components)

In total we get 16 elements of a general 4 × 4 matrix which correspond to the decomposition into Lorentz invariant combinations. The most general form of the interaction between a fermion and a boson is a linear combination of bilinear covariants. From experimental evidence, the form for weak interaction is found to be Vector minus Vector-Axial (V − A).

The parity violation of weak interactions is cause by the helicity structure of the weak interaction. The charged current(W ±) weak vertex is given by:

−ig 1 √ w γµ(1 − γ5), (1.3) 2 2

and since 1 γµ(1 − γ5), (1.4) 2 Chapter 1: Introduction 9 projects out left-handed chiral particle states then:

1 Ψ γµ(1 − γ5)Ψ = ΨγµΨ , (1.5) 2 L and by writing

Ψ = ΨR + ΨL (1.6) and as we know from QED [4] that

µ ΨRγ ΨL = 0, (1.7)

hence 1 Ψ γµ(1 − γ5)Ψ = Ψ γµΨ , (1.8) 2 L L which means that only the left-handed chiral components of particle spinors and right- handed chiral components of anti-particle spinors participate in charged current weak interactions. Moreover, at very high energy (E m), the left-handed chiral compo-

nents are helicity eigenstates, and this property gives the added information that in

the ultra-relativistic limit only left-handed particles and right-handed anti-particles

participate in charged current weak interactions. It is this helicity dependence of the

weak interaction that results in the parity violation.

The decays of pions provide a good demonstration of the role of helicity in the Chapter 1: Introduction 10

weak interaction. Experimentally, we observe that:

− − Γ(π → e ν¯e) −4 − − = 1.23 × 10 , (1.9) Γ(π → µ νµ) which is contrary to what is expected, i.e., we should expect the decay to electrons to dominate due to increased phase space. In fact the opposite happens because electron decay channel is helicity suppressed.

This observation can be explained as follows: the pion’s spin being zero, the spins of muon and neutrino are opposite to each other and as the weak interaction only couples to right-handed chiral anti-particles and since neutrinos are almost massless, then they must be in right-hand helicity state. Momentum conservation forces the muon to be emitted in a right-hand helicity state.

From the general right-handed helicity solution to the Dirac equation [4]:

1 |−→p | 1 |−→p | u ↑= P u ↑ +P u ↑= (1 + )u + (1 − )u (1.10) R L 2 E + m R 2 E + m L

and as the right-hand chiral and helicity states are identical in the limit E m,

this means, even though only left-handed chiral particles participate in the weak

interaction, still the contribution from right-hand helicity states is not necessarily

zero. Therefore, one should expect the matrix element to be proportional to left- Chapter 1: Introduction 11 hand chiral component of right-hand helicity electron/muon spinor, that is

1 |−→p | m Mfi ∝ (1 − ) = . (1.11) 2 E + m mπ + m

Since the electron mass is much smaller than the pion mass the decay

− − π → e ν¯e, (1.12)

is heavily suppressed.

1.2 Muon decay

All elementary particles interact through weak interactions, but we chose muons to test the V − A structure of weak interaction, because muons do not interact strongly while hadrons, which are composed of quarks, are harder to test for weak interactions as they also interact strongly. Strong interaction eﬀects are generally harder to calculate. Even though the electromagnetic interaction is present in the decay of muons, the eﬀects of this additional interaction are well known, and appropriate corrections are applied and hence can be separated in the experimental results. The decays of the muon, being one of the few purely leptonic weak decays, is a an important tool for studying properties of weak interactions. Table 1.3 lists the decays and their branching fractions. Chapter 1: Introduction 12

Table 1.3: Muon decay modes and the corresponding branching fractions

Decay Branching Ratio + + µ → e νeνµ(γ) ≈ 100% + + µ → e νeνµγ(Eγ > 10MeV ) (1.4 ± 0.4)% + + + − −5 µ → e νeνµe e (3.4 ± 0.4) × 10 µ+ → e+γ < 1.2 × 10−11

Muons decay into an electron and the corresponding neutrinos and sometimes the

+ + decay is also accompanied by a photon, µ → e νeν¯µ(γ). The emission of photon is most often an emission of a soft photon where the energy of the photon is below a certain threshold and therefore is not distinguished by the experiment from the normal decay which has no photon emission. However, there is also a probability for the emission of a hard photon. Hard photon emission involves photons emitted at a large angle with respect to the positron and with energy of order of MeV or more. We distinguish two types of decays: Michel decay (photon of any energy) and radiative

Michel decay (hard photon emission decay).

The Hamiltonian describing muon decay requires Lorentz invariance and lepton- number conservation. The general form of this Hamiltonian has contributions from scalar, pseudo-scalar, tensor, vector and axial-vector transformation properties. Al- though experimental evidence of muon decay is consistent with V − A terms, there is still a probability of smaller contributions from scalar, pseudo-scalar or tensor transformations. Chapter 1: Introduction 13

The matrix element for the muon decay [2] is given by:

GF X γ ˆγ ˆγ M = 4√ gαβ[eα|O |(νe)][(νµ)|O |µβ] (1.13) 2 γ

where GF is the Fermi coupling constant while α and β denote left or right handedness of the positron and muon respectively. The label γ denote the allowed bilinear covari-

γ ants: Scalar, Vector, Pseudo-Scalar, Axial-Vector and Tensor and gαβ are coupling constants.

+ + 1.2.1 Michel Decay: µ → e νeνµ(γ)

The diﬀerential decay rate for Michel decay calculated from the matrix elements given by (1.13) is:

2 q d Γ mµ 4 2 2 2 = W G x − x (F (x) + P + cos θF (x))(1 + P˜ + (x, θ) · ζˆ) dxd(cos(θ)) 4π3 eµ F 0 IS µ AS e (1.14)

2 2 where Weµ = (mµ + me)/2mµ, x = Ee+ /Weµ and x0 = me/Weµ, mµ and me are the masses of muon and electron respectively, while Ee+ is the energy of the positron.

The allowed range of the positron energy is x0 ≤ x ≤ 1. The variable θ is the angle

~ ˆ between the muon polarization Pµ and the positron momentum; ζ is the unit vector in the direction of the positron spin polarization with respect to an arbitrary direction.

~ Pe+ is the polarization of positron along the direction of its momentum. Chapter 1: Introduction 14

The functions FIS and FAS are expressed as:

2 F (x) = x(1 − x) + ρ(4x2 − 3x − x2) + ηx (1 − x), (1.15) IS 9 0 0 and 1 q 2 q F (x) = ξ x2 − x2(1 − x + δ[4x − 3 + ( 1 − x2 − 1)]). (1.16) AS 3 0 3 0

The parameters ρ, η, ξ and δ are known as Michel parameters [5].

The expression for the diﬀerential decay can be simpliﬁed in the case of no polar-

izations and becomes:

dΓ m q 2 = µ W 4 G2 x2 − x2[x(1 − x) + ρ(4x2 − 3x − x2)] (1.17) dx 4π3 eµ F 0 9 0

In this unpolarized decay, the decay rate is dependent on only Michel parameter

ρ. The Standard Model predicted value of ρ is :

3 ρ = (1.18) SM 4

which is consistent to experimental value from Particle Data Group [6]:

ρ = 0.7503 ± 0.0004 (1.19) Chapter 1: Introduction 15

+ + 1.2.2 Radiative Michel Decay: µ → e νeνµγ

Experimentaly, the radiative muon decay is characterized by detection of a photon

accompanied by a positron. Photons being massless particles, introduces infrared

divergences and collinearity.

Infrared divergences originate from massless particles with a vanishing momen-

tum in the small energy soft limit. Physical states like, for example, a single charged

particle, are degenerate with states made by the same particle accompanied by soft

photons. This corresponds to the impossibility of distinguishing a charged particle

from the one accompanied by soft photons due to the ﬁnite resolution of any exper-

imental apparatus. An infrared divergence appears in QED when the energy of the

photon goes to zero as a factor of the form

Z 1 d 0 I = (1.20) 0

where = Eγ/E is the fraction of the energy of the photon with respect to the total available energy E for the process. On the other hand, collinearity, instead, comes from photons having a vanishing value of the relative emission angle between photons and positrons. Any experimental apparatus with ﬁnite angular resolution, cannot distinguish between them. The above mentioned restrictions make it impossible to measure total branching fraction for radiative Michel decay. Correction for the above measurement restrictions, infrared and collinearity, is accomplished by setting a lower Chapter 1: Introduction 16

energy cut-off on photon energy Eγ, and a lower cut-off on the opening angle between positron and photon respectively. The spectrum of the radiative muon decay has been calculated by several authors [7, 8]. The diffential branching fraction of radiative muon decay after integrating over positron, photon and muon polarization is given by:

d3B(x, y, θ) 4 = f (x, y, θ) +ηf ¯ (x, y, θ) + (1 − ρ)f (x, y, θ), (1.21) dxdy2πd(cos θ) 1 2 3 3 where

x = 2Ee+ /mµ, y = 2Eγ/mµ, cos θ =p ˆe+ · pˆγ ,

and fi(i = 1, 2, 3) are polynomials in x, y and ∆ = 1−β cos θ with β = |pˆe+ |/Ee+ . The deﬁnitions of fi are given in the Appendix A. Energy and momentum conservation requires: 2(x + y − 1) ∆ ≥ . (1.22) xy

Energy and angular distributions of the radiative muon decay are sensitive to the parameters ρ, andη ¯. These muon decay parameters are all related to the coupling

γ constants gαβ [9]. This additional parameter,η ¯, that is only determined from radiative Michel decay, is sensitive to deviation from a V − A structure. Chapter 1: Introduction 17

The nominal Standard Model value ofη ¯ is:

η¯SM = 0, (1.23)

which is consistent with existing experimental measurements [9, 10]. Any substan-

tial deviation from theη ¯SM value, would imply deviation from a pure V − A weak interaction. The main goal of this work is to extract experimental value ofη ¯ and also to measure the branching fraction of radiative muon in the largest possible region of phase space, keeping in mind restrictions from infrared divergence and the collinearity angle between photon and positron. The diﬀerential branching fraction (1.21) implies that whenη ¯ and ρ have Standard Model values, only f1 is involved in the calculations of branching fraction. Figures (1.3, 1.4, 1.5 and 1.6) show f1 spectra against x, y and various opening angles and it is clear that the f1 term takes on larger values when a photon is emitted at small angles with respect to the positron.

The sensitive kinematic phase space regions forη ¯ and ρ are obtained from the region where |f2/f1| and |f3/f1| are at a maximum, respectively. Figures (1.7, 1.8,

1.9 and 1.10) show that the value of |f2/f1| increases as the opening angle increases.

Therefore the region of large angles is used to extract theη ¯ parameter. Figures (1.11,

1.12, 1.13 and 1.14) show that the phase space region sensitive to ρ parameter involves

the smaller angles, since |f3/f1| decreases as the opening angle increases. In this

analysis the angles greater than 150◦ are of interest in the extraction ofη ¯ parameter. Chapter 1: Introduction 18

θ f1(x,y, > 45)

y 1 40

0.9 8.5 35 0.8

0.7 30

22.8425 22.5377 0.6 25

0.5 20 0.4 29.0038 33.5316 39.5568 15 0.3 10 0.2

31.8571 39.0735 40.86 41.5 0.1 5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 x

◦ Figure 1.3: The contours of constant f1 using events passing θ ≥ 45 , as a function of photon energy(y) and positron energy(x).

θ f1(x,y, > 90)

y 1 14 0.9 2.86 3.7

0.8 12

0.7 10 4.78333 4.767 5.5 0.6 8 0.5

0.4 6.20423 7.26471 9.06782 10 6

0.3 4 0.2

8.5 11.093 13.3448 14.9286 2 0.1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 x

◦ Figure 1.4: The contours of constant f1 using events passing θ ≥ 90 , as a function of photon energy(y) and positron energy(x). Chapter 1: Introduction 19

θ f1(x,y, > 135)

y 1 2.5

0.9 1.41667 1.08971 0.864583

0.8 2 0.7

1.67118 1.43259 1.14375 1.025 0.6 1.5 0.5

0.4 1.80789 1.83985 1.8172 1.86328 1 0.3

0.2 0.5

1.98077 2.21296 2.58462 2.61389 0.1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 x

◦ Figure 1.5: The contours of f1 using events passing θ ≥ 135 , as a function of photon energy(y) and positron energy(x).

θ f1(x,y, >157)

y 1

0.9 1.06696 0.749449 0.391807 0.175 0.8

0.7

1.25941 0.939538 0.545927 0.361842 0.6

0.5

0.4 1.17045 1.01033 0.734416 0.514151 0.3

0.2

1.01293 1.0625 0.881329 0.714286 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.6: The contours of f1 using events passing θ ≥ 157 , as a function of photon energy(y) and positron energy(x). Chapter 1: Introduction 20

|f2/f1|: θ > 45

y 1

0.9 0.104547 0.156699 0.284024 0.400504 0.8

0.7

0.11826 0.153733 0.247799 0.339219 0.6

0.5

0.4 0.127478 0.136904 0.146305 0.180819 0.3

0.2

0.116267 0.118605 0.111455 0.105378 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.7: The contours of (f2/f1) using events passing θ ≥ 45 , as a function of photon energy(y) and positron energy(x).

1.2.3 Motivation

+ + The goals of this analysis are to measure the branching fraction of the decay µ → e νeνµγ using the largest accessible region of phase space, and to determine theη ¯ parameter.

Similar analysis was done by Brent VanDevender in 2006 [10], however during that analysis, data calibration had not been fully optimized. Therefore, there was a need to do an improved analysis with a better data calibration.

Additionally, this analysis revises the selection of phase space region used for the extraction of theη ¯ parameter. In this analysis, events in the region of very large Chapter 1: Introduction 21

|f2/f1|: θ > 90

y 1

0.9 0.106588 0.157414 0.284012 0.400504 0.8

0.7

0.132817 0.168727 0.24913 0.339219 0.6

0.5

0.4 0.150603 0.163308 0.159698 0.181496 0.3

0.2

0.130884 0.134562 0.121676 0.10767 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.8: The contours of (f2/f1) using events passing θ ≥ 90 , as a function of photon energy(y) and positron energy(x).

opening angle (θ > 150◦) were used while in the previous analysis [10], a broader

opening angle (θ > 90◦) cut was used. However, in both analyses, the measurement of experimental branching applied similar region of phase space (Eγ > 10 MeV and

θ > 30◦).

An overview of PIBETA detector is described in Chapter 2. The details of data calibration and event reconstruction are given in Chapter 3, while the analysis strat- egy and results are presented in Chapter 4. Chapter 1: Introduction 22

|f2/f1|: θ > 135

y 1

0.9 0.119453 0.167769 0.283952 0.400729 0.8

0.7

0.19549 0.25334 0.277725 0.339706 0.6

0.5

0.4 0.239667 0.29627 0.240239 0.201652 0.3

0.2

0.197774 0.218363 0.17737 0.124986 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.9: The contours of (f2/f1) using events passing θ ≥ 135 , as a function of photon energy(y) and positron energy(x). Chapter 1: Introduction 23

|f2/f1|: θ > 157

y 1

0.9 0.138161 0.211931 0.30015 0.394601 0.8

0.7

0.246254 0.382459 0.405341 0.356935 0.6

0.5

0.4 0.303646 0.443363 0.425877 0.278642 0.3

0.2

0.267354 0.350335 0.312488 0.181712 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.10: The contours of (f2/f1) using events passing θ ≥ 157 , as a function of photon energy(y) and positron energy(x).

|f3/f1|: θ > 45

y 1 0.55 0.9 0.553972 0.568058 0.568112 0.546467

0.8 0.5 0.7

0.357368 0.461514 0.539516 0.557082 0.45 0.6

0.5 0.4

0.4 0.212099 0.265512 0.486618 0.575447 0.35 0.3 0.3 0.2

0.349799 0.215892 0.404364 0.585217 0.1 0.25

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.11: The contours of (f3/f1) using events passing θ ≥ 45 , as a function of photon energy(y) and positron energy(x). Chapter 1: Introduction 24

|f3/f1|: θ > 90

y 1 0.55 0.9 0.550011 0.567663 0.568112 0.546467

0.8 0.5 0.7

0.309114 0.43927 0.538819 0.557082 0.45 0.6

0.5 0.4

0.4 0.207446 0.226515 0.469152 0.575256 0.35 0.3 0.3 0.2

0.347471 0.211944 0.371733 0.584084 0.1 0.25

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.12: The contours of (f3/f1) using events passing θ ≥ 90 , as a function of photon energy(y) and positron energy(x).

|f3/f1|: θ > 135

y 1 0.55

0.9 0.5203 0.552591 0.566428 0.546467 0.5 0.8

0.7 0.45

0.211565 0.33255 0.507544 0.555775 0.6 0.4 0.5 0.35

0.4 0.200553 0.164465 0.385754 0.564641 0.3 0.3

0.2 0.25

0.324864 0.195321 0.29824 0.566164 0.1 0.2

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.13: The contours of (f3/f1) using events passing θ ≥ 135 , as a function of photon energy(y) and positron energy(x). Chapter 1: Introduction 25

|f3/f1|: θ > 157

y 1

0.9 0.488683 0.51808 0.539818 0.542157 0.5

0.8 0.45 0.7

0.173173 0.234974 0.393966 0.526689 0.4 0.6 0.35 0.5 0.3 0.4 0.188863 0.133109 0.273038 0.508976

0.3 0.25 0.2 0.2 0.301319 0.17703 0.231477 0.518251 0.1 0.15 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ Figure 1.14: The contours of (f3/f1) using events passing θ ≥ 157 , as a function of photon energy(y) and positron energy(x). Chapter 2

PIBETA Detector

2.1 Introduction

The PIBETA detector is well described in Ref [11]. In this Chapter, I will give a brief description of diﬀerent components that make up the detector and explain the tasks performed by each component. The PIBETA detector is a large solid angle nonmagnetic detector optimized for measurement of photons and positrons in the the energy range of 5-150 MeV. The main sensitive components shown and labelled in

Fig. 2.1 are:

• a passive lead collimator, PC, a thin forward beam counter, BC, two cylindrical

active collimators, AC1 and AC2, and an active degrader, AD, all made of

plastic scintillators and used for the beam deﬁnition.

26 Chapter 2: PIBETA Detector 27

• active plastic scintillator target, AT, used to stop the beam particles

• two concentric cylindrical multi-wire proportional chambers, MWPC1 and MWPC2

surrounding the active target, AT, used for tracking charged particles

• a segmented fast plastic scintillator hodoscope, PV, surrounding the MWPCs

used for particle identiﬁcation.

• a high-resolution segmented shower CsI calorimeter surrounding the target re-

gion and tracking detectors.

• a cosmic muon plastic scintillator veto counters, CV, around the entire detector

(not shown).

The PIBETA detector triggering system and its data acquistion system (DAQ) will also be brieﬂy explained.

2.2 Experimental area

The detector is situated at the Paul Scherrer Institute (PSI) in Villigen, Switzerland.

The PSI ring synchrotron accelerates protons to an energy of 590 MeV. The protons are subsequently transported to two target stations where pions and muons are generated by collision with thick carbon targets. The generated pions and muons are then transported toward multiple experimental areas through secondary beam-lines.

The accelerator operates at the frequency of 50.63 MHz producing 1 ns wide proton Chapter 2: PIBETA Detector 28

pulses separated by 19.75 ns. The PIBETA experiment was set up in the PSI πE1 experimental area whose 16 m long beam line is designed to supply 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, and a Full-Width-Half-Maximum

(FWHM) momentum resolution of < 2%. The choice of a particular beam momentum is governed by the need for good time-of-ﬂight (TOF) separation of pions, positrons, and muons between production target and the forward beam counter (BC).

2.3 Beam-deﬁning elements

2.3.1 Passive Collimator

In order to reduce positron contamination, a 4 mm thick carbon degrader is inserted in the path of the πE1 beam. The momentum-analyzed pions and positrons have diﬀerent energy losses in the carbon absorber, and are therefore spatially separated in a horizontal plane after passing through the absorber. The lead collimator PC is positioned immediately upstream of the ﬁxed beamline’s exit window. Only the beam pions pass through the collimator, while positrons stop in the collimator material, since at this point they are separated from pions by 40 mm in the horizontal plane by magnetic momentum analysis (bending).

The passive collimator is composed of two stacked lead brick blocks with individual dimensions of 250 × 250 × 50 mm3. Both pieces have a central hole with a diameter of Chapter 2: PIBETA Detector 29

50 mm and a step bore extending the hole to 70 mm. This is the last beam deﬁning element before pion beam enters the active detector counter.

2.3.2 Active beam-deﬁning elements

The forward beam counter BC, is the ﬁrst active detector counter. This counter is made of plastic scintillator material with dimensions 25 × 25 × 2 mm3. Immediately after BC, the pion beam passes through a quadrupole triplet magnetic, and is focused through two active beam collimators (AC1 and AC2). These collimators are cylindrical rings made of plastic scintillators and their importance is to suppress background signals caused by detector hits that are not associated with the pion beam.

Following the active beam collimators is an active degrader. The beam of pions arriving in central area of the detector has a relatively high momentum of about 113.4

MeV/c which needs to be reduced before pions get to the active target where they are subsequently stopped. The active degrader counter reduces the average kinetic energy of the pions from 40.3 MeV to 27.6 MeV. The active degrader is made of plastic scintillator in the shape of a truncated cone to ensure that the degrader’s downstream projection covers the whole target area (40 mm diameter), while at the same time particles entering parallel to the beam axis always traverse the same 30 mm scintillator thickness.

The last active beam-deﬁning element is the active target. This target is made of cylindrical plastic scintillator, 50 mm in length and with a 40 mm diameter. In the Chapter 2: PIBETA Detector 30

Figure 2.1: A schematic cross section of the PIBETA detector showing the main components: forward beam counter (BC), two active collimators (AC1 and AC2), active degrader (AD), active target (AT), two multi-wire proportional chambers (MWPC1 and MWPC2), plastic veto (PV) and CsI calorimeter

. Chapter 2: PIBETA Detector 31

2004 experimental runs, two types of active target were used. In early runs (50000-

51017), a 9 segments target was used, while a one-piece target was used for later runs

(52000-52471). The 9 piece target was used in previous high rate runs (1999-2001).

The lower rate 2004 run dedicated to radiative decays was better served by a one piece target with improved light collection properties. Both targets were used in the the 2004 run in order to establish consistency with the earlier runs.

The relatively high π-stop rate, around 105 s−1, makes the target the most complicated component of the Pibeta detector in terms of data analysis or simulation of the its performance. The π-stop event rate in the target is roughly 100 kHz, which is very high compared to each calorimeter cystal, which bear less than 0.5% of the π-stop event rate. Once the pion is stopped in the target, a chain of decays follows. Nearly every pion mostly decays into a muon and the muon comes to rest in the target and it also decays. Figure 2.2 shows a signal produced by a stopping pion and subsequent pion decay into a muon. In this event the muon itself travels a short distance (1.4 mm) in the target before stopping and decaying.

2.4 Detector Tracking System

The Pibeta detector tracking system is comprised of 2 multi-wire proportional chambers (MWPC1 and MWPC2) and a 20-bar plastic scintillators hodoscope (PV) [11].

The wire chambers are made of low-mass material in order to minimize photon con- Chapter 2: PIBETA Detector 32

Figure 2.2: Example of signal produced by decaying pion and subsequent muon decay. The ﬁrst peak shows the stop of pion before decaying to muon. The muon produces a shoulder on the pion stop signal (at about 47 ns). The muon then decays into a positron (at about 80 ns) that causes the ﬁnal peak as it exits the active target

versions. They are highly eﬃcient and are capable of handling a high rate of up to

107tracks/s minimum-ionizing particles (MIP). They also have good radiation hard- ness and are very stable operationally. These wire-chambers are cylindrical, each having one anode wire plane along the longitudinal direction, and two cathode strip planes in a stereoscopic geometry.

The plastic scintillator hodoscope PV is located in the interior of the calorimeter Chapter 2: PIBETA Detector 33 surrounding the two concentric wire chambers (MWPCs). The PV consists of 20 independent plastic scintillator staves that form a cylinder of 598 mm long and 132 mm radius. The PV covers the whole solid angle subtended by the calorimeter as seen from the center of target.

The main parts of PV hodoscope are: plastic scintillator staves, two light guides, and two attached photomultiplier tubes. The dimensions of individual staves are

3.175 × 41.895 × 598 mm3. They are designed in such a way that a particle from the target passes through the PV before entering the calorimeter. The PV hodoscope provides efficient charged particle identification. It is reliable at differentiating charged particles (cosmic muons, positrons and protons) from photon events. Furthermore, it provides precise timing information for charged particles.

2.5 Calorimeter

The segmented PIBETA calorimeter [11] consists 240 Cesium-Iodide (CsI) crystals,

220 of these crystals are truncated hexagonal and pentagonal pyramids covering the total solid angle of 0.77 × 4πsr, while the remaining 20 crystals cover two detector openings for the beam entry and target readouts and act as electromagnetic shower leakage vetoes. The inner radius of the calorimeter is 26 cm, and its active depth is

22 cm, corresponding to 12 CsI radiation lengths (X0 = 1.85 cm).

The calorimeter is the most important part of the detector. The charged particles, Chapter 2: PIBETA Detector 34 namely positrons, electrons and protons deposit their full energies in the calorimeter while the photons that come from radiative decay are well conﬁned in the calorimeter.

It is designed in a such way that it can handle a high event rate and it also has a good energy and time resolution to minimize the background signals. Another very important function of the calorimeter is that it gives the basis for trigger logic described in Section 2.7.4.

2.6 Cosmic muon veto detectors

The detector is shielded from background radiation in the experimental hall as well as from the cosmic ray background. The active parts of the detector are doubly protected. A lead house enclosure provides inner passive shielding. It is in turn lined on the outside with active cosmic muon veto counters.

2.7 PIBETA Triggering System

2.7.1 Introduction

The triggering system accepts events that satisfy certain predetermined conditions.

These conditions are based on energy thresholds and temporal coincidences. The triggers used in this analysis are random triggers, beam triggers, and calorimeter triggers. Chapter 2: PIBETA Detector 35

2.7.2 Random Triggers

The random triggers are made using a plastic scintillator with dimensions of 190 ×

20 × 8 mm3. That counter is placed above the electronic racks, parallel to the area

ﬂoor, about 3 m away from the main detector. The scintillator is shielded from the experimental radiation by a 50 mm thick lead brick wall as well as a 500 mm thick concrete wall. It is designed in such a way that events being triggered by this trigger are random backgrounds events which are unrelated to events from the primary detector. The main sources of these random background signals are cosmic muons and electronic noise in the active detector. The information from these random triggers is used to determine the ADC pedestals in the energy spectra. For example, the average deposited energy in each CsI crystals due to random events is 0.15 MeV.

2.7.3 Beam Triggers

The beam triggers are deﬁned by a coincidence between the beam counter BC, the active degrader AD, the active target AT, and the RF accelerator signal. The signals from the beam line elements BC, AD and AT are discriminated to produce 10 ns wide logic pulses. These pulses from beam elements along with the accelerator RF make up a pion stop signal πS, which is discriminated to be 10 ns wide:

πS = BC • AD • AT • RF. (2.1) Chapter 2: PIBETA Detector 36

This coincidence is only possible if the pion that was produced at a primary target by

a proton beam which is in coincidence with the RF, traverses the beam line elements

and stops in the active target AT. The πS signal initiates another signal called pion gate, πG, which is 180 ns long and designed to start 50 ns before πS. The events

that occur during the time when the πG is open are the only recorded events. To

suppress prompt background signals, events occuring within a few nanoseconds of the

pion stop are rejected. The main source of the prompt signals are pion absorption,

elastic and inelastic pion scattering and single-charge-exchange interactions between

pions and the stopping target material:

π+ + A → π0 + B π+ + A → p + B etc. (2.2)

These πS and πG are the basis of PIBETA trigger system. Other triggers are coin-

cidences formed of calorimeter logic signals and beam signals. Figure 2.3 shows the

typical timing of beam trigger components.

2.7.4 Calorimeter Triggers

The CsI calorimeter is composed of 240 crystals. These crystals are grouped into

60 clusters, composed of 9 crystals each. These clusters are also grouped into 10

superclusters whereby 6 adjacent and overlapping clusters form one supercluster.

The simplest calorimeter trigger is the one-arm CsI trigger made in two versions: Chapter 2: PIBETA Detector 37

Figure 2.3: The pion stop signal is deﬁned as a four-fold coincidence of beam counter (BC), active degrader (AD), active target (AT) and a 19.75 ns RF cyclotron signal. The snapshot shows signals from top to bottom; BC, AD, AT and RF

Low-Threshold (5 MeV) and High-Threshold (53 MeV). It requires the ﬁring of at least one supercluster:

CS =0+1+2+3+4+5+6+7+8+9, (2.3) Chapter 2: PIBETA Detector 38

where the numbers 0-9 represent the indices of the above mentioned superclusters.

The supercluster ﬁres whenever energy deposited in one of the constituent clusters

exceeds the threshold.

The next more complicated type of calorimeter trigger is a two-arm trigger. The

two-arm trigger requires ﬁring of two non-neighboring super-clusters, and is deﬁned

as:

CSS = 0 ◦ (2 + 3 + 5 + 6 + 9) + 1 ◦ (3 + 4 + 5 + 6 + 7) + 2 ◦ (0 + 4 + 6 + 7 + 8) + 3 ◦

(0 + 1 + 7 + 8 + 9) + 4 ◦ (1 + 2 + 5 + 8 + 9) + 5 ◦ (0 + 1 + 4 + 7 + 8) + 6 ◦ (0 + 1 + 2 +

8 + 9) + 7 ◦ (1 + 2 + 3 + 5 + 9) + 8 ◦ (2 + 3 + 4 + 5 + 6) + 9 ◦ (0 + 3 + 4 + 6 + 7).

As the two-arm trigger requires ﬁring of two superclusters, it follows that there are three possible versions of it:

• high thresholds on both superclusters: CHH SS

• low thresholds on both superclusters: CLL SS

• one high, one low thresholds: CHL SS

The above calorimeter triggers play a role in the event reconstruction for the PIBETA

+ + detector. For example, non-radiative Michel decay, µ → e νeνµ requires one-arm

L + + LT trigger (CS ) while the radiative Michel decay, µ → e νeνµγ requires the ﬁring

of two-arm LT trigger (CL ). SS

There are diﬀerent decay types recorded by the PIBETA detector. Some are

common and some are rare. In order to optimize event statistics, trigger prescaling is Chapter 2: PIBETA Detector 39

Table 2.1: The prescaling factors for diﬀerent PIBETA triggers. Notice higher prescaling factors on low-threshold triggers to suppress common muon decays which would otherwise saturate the data acquistion system and prevent recording of events from rare pion decays

Trigger Prescale Factor L CS 512 H CS 1 CLL 16 SS CHH 1 SS CLH 1 SS

+ 0 + applied during data taking and it ensures that rare events like π → π e νe are always

recorded, while common events like Michel decay are prescaled appropriately. During

data analysis, the prescaling factors are applied in the number of events counting to

determine the actual number of events that occured based on the trigger prescaling

factor. The corresponding trigger pre-scaling factors for various PIBETA triggers are

shown in Table 2.1. Chapter 3

Event Reconstruction

3.1 Introduction

This chapter describes the PIBETA data analysis software, energy and time calibrations, and the detector simulation. The data analysis software reconstructs diﬀerent decay modes. The software consists of diﬀerent parts, with each part corresponding to a component of the PIBETA detector system described in the previous chapter. In order to optimize resolution of energy and time measurements, detailed and careful calibrations were performed and I will highlight the procedure that was used. The

GEANT package simulation is described in detail in Ref [12]. This chapter presents a summary of how we simulated the decay modes of interest.

40 Chapter 3: Event Reconstruction 41

3.2 Data Analysis Software

The software that is used for data analysis is written in the C language and is divided into smaller routines which describe how particles are reconstructed in the individual parts of the detector. For example, there are routines for Multi-Wire Proportional

Chamber (MWPC), a routine for Plastic Hodoscope (PV), CsI crystals etc.

The analysis software is built in such a way that a particle originating from the active target AT can be well reconstructed by following its path through different parts of the detector. The PIBETA detector identifies particles by using the tracking sytem plus energy and timing information extracted from different sub-detectors. Charged particles will leave information in the MWPC’s, and deposit energy in the PV and

CsI calorimeter. Neutral particles (photons) leave no signature in the MWPC’s, but are identiﬁed in the calorimeter.

3.2.1 Tracking System algorithm

The tracking algorithm identiﬁes charged tracks that register hits in the wire chambers. The algorithm considers hits in two wire chambers separated by an azimuthal angle of less than 30 degrees. The line through the hits is projected back to the target and forward to the struck calorimeter clump. The angle between this line and the calorimeter clump is required to be less than 13 degrees, and the energy deposited in CsI to be greater than 5 MeV. The algorithm takes the point (x0, y0, z0) shown in Chapter 3: Event Reconstruction 42

Fig. 3.1 as the decay vertex. This point is calculated by using the information from

wire chamber tracks. The point (x0, y0, z0) is taken as the point on the track closest

to the z-axis. Fig. 3.2 shows the distribution of angles between the intersection of the

wire-chamber tracks with the calorimeter face (x1, y1, z1), and the angular coordinates

of the corresponding calorimeter clump.

A comparison between simulation and measurement was performed to check if

tracks were well reconstructed. Cosine of polar angle θ and azimuthal angle of a

positron from the Michel decay are shown in Figures (3.3 and 3.4), respectively. Sim-

ilarly, for radiative Michel decay, the matching between simulation and measurement

are shown in Figures (3.5–3.8). Note that in this matching process for radiative decay,

simulated background is also shown in the histograms. Photons being neutral leave

no hits in the MWPCs or PV so they are assumed to come from the center of the

stopping distribution (x0, y0, z0). The direction cosines and calorimeter intersection

point (x1, y1, z1) are based on angles obtained from clump algorithm, explained in the next section.

3.2.2 Clump Algorithm

A clump is deﬁned as a centrally hit crystal plus 5-to-6 neighboring crystals. The central crystal is the one with the highest energy deposition. If a track intersects a crystal centrally, the particle deposits more than 90% of its energy in that crystal.

For the algorithm to accurately reconstruct the energy of an incident positron, it Chapter 3: Event Reconstruction 43

×103 MC 8000 DATA 7000 Arbitrary Units 6000

5000

4000

3000

2000

1000

0 -25 -20 -15 -10 -5 0 5 10 15 20 25 X0[mm]

×103 MC 8000 DATA 7000 Arbitrary Units 6000

5000

4000

3000

2000

1000

0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Y0[mm]

×103

4500 Data MC 4000

3500Arbitrary Units

3000

2500

2000

1500

1000

500

0 -20 -10 0 10 20 30 40 Z0 [mm]

Figure 3.1: The primary vertex x0, y0, z0

Distribution of coordnates values of the primary vertex x0, y0, z0 obtained from the wire-chambers tracks by backtracking to the point of closest approach to the z-axis.

needs to take into account the shower energy that leaks into the neighboring crystals.

The algorithm ﬁnds the centrally hit crystal and subsequently sums the energy of this Chapter 3: Event Reconstruction 44

×103 4000 MC 3500 DATA 3000 Number of Events 2500

2000

1500

1000

500

0 0 2 4 6 8 10 12 14 α CsI-MWPC(Degrees)

Figure 3.2: The angular separation between a wire-chamber track and the coordinates of the corresponding calorimeter clump

crystal and the energies in the neighboring crystals that are hit within |∆t| < 14 ns

of the central crystal. The algorithm also makes sure that two neighboring crystals

cannot be the centers of distinct clumps. The time of the clump is calculated as the

energy-weighted average of the all clump members, while clump angular coordinates

(θ, φ) are obtained using energy-weighted centers of the crystals in a clump:

Pn i=1 wiθi θCsI = Pn , (3.1) i=1 wi and similarly, Pn i=1 wiφi φCsI = Pn , (3.2) i=1 wi Chapter 3: Event Reconstruction 45

×103

4000 DATA MC 3500

3000 Number of Events 2500

2000

1500

1000

500

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 θ Cos e

Figure 3.3: Cosine of polar angle θ of a positron track with respect to the beam axis passing 1-arm trigger (Michel events).

th where θi and φi are polar and azimuthal angles of the i crystal and n is the number of crystals in the clump. The weight wi is deﬁned as:

Ei wi = a0 + ln Pn . (3.3) i=1 Ei

The simulation of calorimeter energy resolution gave as the best value of parameter a0 = 5.5.

3.3 Particle Identiﬁcation

Particle identiﬁcation is needed to distinguish protons from positrons as both may seem to be of the same type if no particle identiﬁcation criteria are applied. The separation of the positrons from protons is done through the relative amount of energy Chapter 3: Event Reconstruction 46

×103

3000

2500 Number of Events 2000 DATA 1500 MC

1000

500

0 50 100 150 200 250 300 350 Positron Trk φ

Figure 3.4: Azimuthal angle φ of a positron track passing 1-arm trigger (Michel events).

×106 2400 2200 2000 1800 1600 Signal MC 1400 1200 Background MC

1000 DATA(measured) 800 MC(signal) + MC(background) 600 400 200 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 θ Cos positron

Figure 3.5: Cosine of polar angle θ of a positron track passing 2-arm trigger (radiative Michel events). Chapter 3: Event Reconstruction 47

×106

2500

2000

Signal MC

1500 Background MC

DATA(measured)

1000 MC(signal) + MC(background)

500

0 50 100 150 200 250 300 350 Photon φ

Figure 3.6: Azimuthal angle φ of a photon passing 2-arm low trigger (radiative Michel events).

×106

2500

2000

Signal MC 1500 Background MC

DATA(measured) 1000 MC(signal) + MC(background)

500

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 θ Cos photon

Figure 3.7: Cosine of polar angle θ of a photon passing 2-arm low trigger (radiative Michel events). Chapter 3: Event Reconstruction 48

×106

2200 2000 1800 1600

Signal MC 1400 Background MC 1200 DATA(measured) 1000 MC(signal) + MC(background) 800 600 400 200

0 50 100 150 200 250 300 350 Positron φ

Figure 3.8: Azimuthal angle φ of a positron track passing 2-arm trigger (radiative Michel events).

that each particle deposits in the PV. The positrons, being minimum-ionizing particles deposit smaller amount of energy in the PV, while protons being non-relativistic, deposit greater amounts of energy in PV. Photons deposit very little (if any) energy in the PV, charged particle identiﬁcation is performed through a comparison between how much energy a particle deposited in the PV and the total energy deposited in both PV and CsI calorimeter. Two functions of energy deposited in PV (EPV) and energy in CsI (ECsI)(fγ and fe) are applied as boundaries for separating positrons, protons and photons. If energy in PV is less than fγ then the particle is identiﬁed as photon. Positrons and protons are separated by a condition of whether a particles’ energy in PV (EPV ) lies between fγ and fe or if its EPV is above fe. The functions Chapter 3: Event Reconstruction 49

fγ and fe are deﬁned as follows:

fγ = 0.2 exp(−0.007(EPV + ECsI)) (3.4)

and

fe = 2.3 exp(−0.007(EPV + ECsI)) (3.5)

Particle identiﬁcation is such that:

EPV < fγ ⇒ photon, (3.6)

fγ ≤ EPV < fe ⇒ positron, (3.7)

EPV ≥ fe ⇒ proton. (3.8)

3.4 Monte Carlo Simulation

The simulation of the PIBETA detector is done using standard GEANT3 Monte

Carlo software package [12] where both passive and active parts of the detector are deﬁned. The simulation takes into account all physics processes taking place in different detector systems. For example, the electromagnetic showers in the CsI as well as electromagnetic interactions in the active target and the tracking system are well described. Chapter 3: Event Reconstruction 50

The Monte Carlo simulation begins with deﬁning all subdetectors; beam line active detectors (upstream beam counter, active degrader and target); two cylindrical MW-

PCs, the 20 cylindrical plastic hodoscope staves and the 240-crystal CsI calorimeter.

The user can modify the simulation code according to a number of diﬀerent criteria, e.g., the run year (due to diﬀerent active targets), beam properties, or reaction/decay

final states. The user can select the final state of any particular decay channel with its relative probability defined by the reaction cross-sections and decay branching fraction. Each reaction is characterised by its own kinematics, and the event is generated by using specific event generators such as Michel or radiative Michel decays.

The simulation of energy deposition is done by simulating ADC values, and the time is simulated through TDC simulations. The simulation of ADC values and TDC hits takes into account individual detector photoelectron statististics, axial and transverse light collection nonuniformities, ADC pedestals, and electronics noise. Once the event has been generated, the simulation of the particle’s passage through the detector material is performed in such a way that the event is reconstructed using the same event reconstruction algorithm as for the measured data. Chapter 3: Event Reconstruction 51

3.5 Data Calibration

3.5.1 Introduction

Good measurement resolution is critical for a successful data analysis. In the PIBETA experiment, energy and time distributions of the detected particles are used to identify diﬀerent decay channels. Therefore, we performed a careful calibration of energy and time for the main parts of the PIBETA detector (PV and CsI calorimeter).

The entire process of calibration involved calibrating energy and time for both PV and CsI calorimeter modules. The PV is composed of 20 individual staves and there is a variation of energy and time recorded in each one. Similary, the CsI has 240 crystals, each one having its own energy and time resolution parameters. This disparity is a source of degraded resolution for the measured energy and time of any decay mode compared to an ideal detector. Therefore, to increase our resolution we performed calibration by aligning each subdetector response to one common reference point.

In addition to calibrating the PV and CsI detectors, the calibration was performed in such a way that run dependence on over all energy and time distributions was removed.

3.5.2 Energy Calibration (ADC Calibration)

The energy calibration for both CsI and PV was done by requiring the energy peak of each segment of PV or CsI to match the peak value obtained from the MC simulation. Chapter 3: Event Reconstruction 52

×103 DATA 3000 MC 2500 Number of Events 2000

1500

1000

500

0 0 10 20 30 40 50 60 Positron CsI Energy[MeV]

Figure 3.9: Positron energy in CsI calorimeter for 1-arm low trigger events.

To obtain these values, we used gaussian plus exponential functions to fit energy distributions; subsequently the fit values for each segment were applied to obtain the multiplying factor which is the ratio between the reference peak and the different peaks from each segment. These multiplying factors were then applied in the offline data analysis for each PV stave and CsI crystals. The energy distribution of the decay

+ + + π → µ → e νeν¯µ was used as a reference for both PV and CsI energy calibration.

The energy distribution peak reference values that were obtained from simulation for perpendicular PV energy and CsI energy are 0.5588 MeV and 63.29 MeV respectively.

After energy calibration, the CsI energy fractional resolution ∆E/E (∆E deﬁned as

Full-Width-Maximum-Height), decreased from 8.0% to 4.5%. Chapter 3: Event Reconstruction 53

×106 DATA 12 MC

10 Number of Events 8

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Positron Energy in PV[MeV]

Figure 3.10: Positron energy in plastic veto (PV) for 1-arm low trigger events.

3.5.3 Time Calibration (TDC Calibration)

The PV timing calibration for channels in the PV was done by using prompt trigger events and finding the time difference between each PV stave and the time zero (pion- stop time). A similar method was aplied to the CsI time calibration where we had to find the time difference between the pion-stop time and each subsequent hit in the

CsI counter. Events passing the prompt trigger were the ones used in time calibration and the measured time offsets were applied in the offline data analysis. Consequently, these offsets were applied to all other PIBETA decay modes including Michel decays.

This is done by subtracting the oﬀsets from all raw TDC values in the oﬄine data analysis. Chapter 3: Event Reconstruction 54

The ADC readouts in the PV and CsI have diﬀerent amplitudes. The smaller amplitude signal takes a longer time to rise to the ﬁxed discriminator threshold than a larger signal. Consequently, there is energy dependence of observed TDC values with higher energy signals registering earlier time while lower energy signals registering at later time. For a complete time calibration we need to perform a time walk correction in order to correct for this energy-time dependence. The correction was performed

+ + using Michel (µ → e νeν¯µ) events by ﬁtting each PV stave and each CsI channel

TDC (time) vs ADC (energy) distribution. The ﬁtting function is of the functional

form

c CTDC = TDC0 + a.(ADC − b) (3.9)

where TDC0, a, b, and c are free parameters of the ﬁt, ADC is the calibrated ADC

value proportional to the deposited energy. The resulting corrected value, CTDC

is the ﬁnal TDC value that is applied in the oﬄine analysis. Figures 3.11 and 3.12,

show the energy-time distribution before and after the walk correction procedure

is performed. Notice that after the walk-correction there is no more energy-time

dependence. This is shown in Fig 3.12 by a straight horizontal line in the energy

range of 10-53 MeV. Chapter 3: Event Reconstruction 55

Time After Walk-Correction 1 Before Calibration 0

-1

-2

-3

-4

-5

-6 0 0.5 1 1.5 2 2.5 PV Energy[MeV]

Figure 3.11: Walk-correction eﬀect on the energy and time of PV. The straight horizontal line indicate the removal of energy-time dependence in the observed TDC values.

p2 CsI Walk Correction validation Entries 8408818 5 Mean 30.63 Mean y -2.555 4 RMS 10.95 RMS y 1.766 3

1 CsI time - PV time](ns) 0

-1

-2

-3

-4

-5 0 10 20 30 40 50 60 CsI Energy [MeV]

Figure 3.12: Walk-correction eﬀect on the energy and time of CsI calorimeter. The straight horizontal line indicate the removal of energy- time dependence in the observed TDC values in the energy range of 10-53 MeV. Chapter 4

Data Analysis

4.1 Branching Fraction

4.1.1 Introduction

As described in Chapter 1, the primary objectives of this analysis are to determine the

experimental branching fraction of the radiative muon decay in the largest accessible

phase space, and to extract the Michel parameter,η ¯. This chapter explains how

the branching fraction was measured and how the Michel parameters were extracted

using both measurement and simulated data.

Although the design of the PIBETA detector is not optimized for a precise mea-

surement of the ρ parameter compared to the dedicated experiments [13], ρ parameter was extracted for the purposes of a consistency check of our data and analysis. Ex-

56 Chapter 4: Data Analysis 57

traction of the ρ parameter was performed by minimizing the chi-square diﬀerence

between the observed and simulated positron energy spectra of the non-radiative

muon decay.

The experimental branching fraction was determined by the following expression:

+ + + + + + Nµ →e νeν¯µγ Aµ →e νeν¯µ(γ) ◦ Br(µ → e νeν¯µγ)Eγ >10 MeV ,θ > 30 = · , (4.1) + + + + Nµ →e νeν¯µ(γ) Aµ →e νeν¯µγ

+ + + + where Nµ →e νeν¯µγ and Nµ →e νeν¯µ(γ) are number of observed events for radiative and

+ + + + non-radiative muon decays, respectively, Aµ →e νeν¯µ(γ) and Aµ →e νeν¯µγ are the MC

calculated acceptances for non-radiative and radiative muon decays, respectively. The

number of events for both non-radiative and radiative muon decays are determined

from measured data, while the detector acceptance for each decay channel is obtained

from detailed Monte Carlo simulations.

The Michel parameterη ¯ is extracted by minimizing the chi-square diﬀerence be-

tween the experimental branching fraction and theoretical (expected) branching frac-

tion. Theoretical branching fraction is calculated using Monte Carlo integration of

the following equation:

Z x2 Z y2 Z cos θ2 Theo 4 BR (¯η, ρ) = 2π dxdyd(cos θ) f1 +η ¯ · f2 + (1 − ρ) · f3 . (4.2) x1 y1 cos θ1 3 Chapter 4: Data Analysis 58

4.1.2 Muon decay time distribution

The decay chain that leads to a Michel decay or radiative Michel decay proceeds through the sequence of a pion stopping in the Active Target and subsequently de-

+ + caying into a muon. The muon from the decay π → µ νµ will travel about 1-2 mm before coming to rest in the target. The target radius (2 cm) is very large compared to muon travel distance and the number of decaying muons is essentially equal to the number of the stopped pions.

The parameters that are needed to describe the muon decay time [10] are: accelerator period (19.75 ns), width of pion gate, pion and muon lifetimes and the rate at which the pions are stopped in the target.

The PIBETA experiment time scales were designed based on the lifetime of the charged pion. For example, from Table 4.1, we see that the width of pion gate is wide enough for ﬁve pion lifetimes to allow the decay of almost all the stopped pions.

The situation becomes diﬀerent for the muons as their lifetime is very large, allowing muons to accumulate in the target; the decay of these piled-up muons can occur in a later pion gate with which they are not causally connected.

The rest of this section will describe in detail the time structure of the muon decays. The decay chain that leads to the muon decay following the stop of pion in the target proceeds as follows:

+ + π → µ νµ, (4.3) Chapter 4: Data Analysis 59

Table 4.1: Time scales involved in the PIBETA detector. The rate of stopping pions is around 105/sec.

Time scale value (ns)

pion lifetime (τπ) 26.02 cyclotron period (Trf ) 19.75 pion gate width 180 muon lifetime (τµ) 2197.03 Pion Stop Period (1/rπ) 10000

followed by

+ + µ → e νeν¯µ(γ). (4.4)

The probability distributions over time for these decays are given by:

exp(−t/τπ) fπ(t) = , (4.5) τπ and

exp(−t/τµ) fµ(t) = , (4.6) τµ hence the probability for sequential decay is given by;

t −t/τµ −t/τπ Z 0 0 0 (e − e ) fπ→µ→e(t) = fs(t) = fu(t − t )fπ(t )dt = , (4.7) 0 τµ − τπ

+ + + where fs(t) is the probability per unit time that the decay chain π → µ νµ, µ →

+ e νeν¯µ(γ) will happen at time t after the pion stops in the target and this probability Chapter 4: Data Analysis 60

is normalized as Z ∞ fs(t)dt = 1. (4.8) 0

As noted above, muons accumulate in the target and they can subsequently decay in a later pion gate with which they are not causally connected. Since the rate of stopping pions is comparable to the muon decay rate, we have to consider pileup eﬀects in our study of muon time structure. The probability of having a pion in a particular beam pulse is

p = rπTrf . (4.9)

Hence the rate of pileup muon decays is calculated as the sum over all beam pulses of the causal decay rate given by (4.7), weighted with the probability (4.9):

∞ X fpu(t) = pfs(t + nTrf ). (4.10) n=1

This series can be simpliﬁed by summing it as follows:

∞ ∞ X p X pf (t + nT ) = (e−(t+nTrf )/τµ − e−(t+nTrf )/τπ ), (4.11) s rf τ − τ n=1 µ π n=1 where after further summation of the series (4.11)

p e−t/τµ e−t/τπ fpu(t) = ( T /τ − T /τ ). (4.12) τµ − τπ e rf µ − 1 e rf π − 1 Chapter 4: Data Analysis 61

By ignoring the pion term (second term), which is a lot smaller compared to the

ﬁrst term and evaluating this periodic function at t = 0 we obtain

fpu(0) ≈ rπ. (4.13)

This means that the rate of muon pile-up in the target is approximately equal to the

rate at which pions are stopped in the target. In summary, the probability distribution

for all times is given by:

   fpu(t) if t < 0 f(t) = (4.14)   fpu(t) + fs(t) if t ≥ 0.

+ + Figure 4.2 shows the actual measured time structure of the decay µ → e νeν¯µ,

plotted as relative time between degrader (t = 0) and the time of CsI calorimeter

showers, while Figure 4.3 is the decay time structure for radiative muon decay which

e+ is taken as time diﬀerence between the average of CsI time of positron (tCsI) and

γ photon (tCsI) and the degrader time (tDeg), expressed as:

1 + (te + tγ ) − t . (4.15) 2 CsI CsI Deg

The time spectrum of muon decays is an important factor in branching fraction

measurement, because the number of events for non-radiative Michel decay is obtained

from the time distibution (Figure 4.2) after time window selection (explained below) Chapter 4: Data Analysis 62

CsI Time Coincidence ×103 1200

1000 ± σ = 0.71 ns N0. of Events = 518813 946 Entries/0.5 ns

800

600

400

200

0-10 -8 -6 -4 -2 0 2 4 6 8 10 ∆t = [t - tγ] (ns) e+

Figure 4.1: Time coincidence of positron and photon in the CsI calorimeter.

is applied. The number of events from the radiative decays is obtained from the ﬁt of the histogram of time coincidence of positron and photon reconstructed as per

Equation (4.16) and is shown in Figure 4.1

e+ γ ∆t = tCsI − tCsI. (4.16)

The time window is selected by dividing radiative time distribution, Figure 4.3 to non-radiative distribution, Figure 4.2 and taking a region where the ratio can be

ﬁtted by a straight horizontal line. As it can be seen from Figure 4.4, care is needed Chapter 4: Data Analysis 63

×103

1400

1200

Number of events 1000

800

600

400

200

0 -50 0 50 100 150 tposi - tdegr[ns]

Figure 4.2: The decay time distribution of Michel events. The time is the diﬀerence between CsI time (tCsI) and degrader time (tDeg).

especially at the edges of the pion gate and also at the prompt veto where there is a discrepancy between the time structure of non-radiative and radiative decay modes.

In the measurement of branching fraction, we normalize the number of radiative muon decays to the number of non-radiative muon decays. It is therefore necessary to select a time window such that there is no uncertainty in counting events due to time structure diﬀerence between the two decays.

From the ﬁt of Figure 4.5, only events in the time window of

t ∈ (−37, −6) ns ∪ (13, 144) ns, (4.17)

were selected in the determination of experimental branching fraction. Chapter 4: Data Analysis 64

45000

40000

35000 Number of Events 30000

25000

20000

15000

10000

5000

0 -50 0 50 100 150 0.5*(t + t ) - t posi phot degr

Figure 4.3: The time distribution of radiative Michel events. This is 1 e+ γ measured as the time diﬀerence ( 2 (tCsI + tCsI) − tDeg).

1.5

0.5 Rel. Time Ratio (Rad./Non-rad)

-0.5

-50 0 50 100 150 tCsI - tdeg[ns]

Figure 4.4: The ratio of radiative decay time to non-radiative muon decay time before selecting the time window. Chapter 4: Data Analysis 65

1.5

0.5 Rel. Time Ratio (Rad/Non-Rad.)

-0.5

-50 0 50 100 150 tCsI - tdeg[ns]

Figure 4.5: The ratio of radiative decay time to the non-radiative muon decay time after selecting only the range where the plot can be ﬁtted by a straight horizontal line.

4.1.3 Non-Radiative Muon Decay

Event Reconstruction

The reconstruction of non-radiative muon decay begins by taking any charged track that passes the one-arm low CsI trigger requirement. Furthermore, the track is required to have at least one hit in each wire-chamber (MWPC1 and MWPC2) and also at least one hit in the PV. The candidate events need to satisfy further requirements including positron identiﬁcation selections, requiring the energy measured to be above the CsI low-threshold (5 MeV) and also to be in the range of kinematically allowed energies of a positron coming from the decay of a muon. The background Chapter 4: Data Analysis 66

Table 4.2: Michel cuts

2 2 Positron Energy(CsI) gen. level 5 MeV < ECsI ≤ Emax = (mµ + me)/2mµ 2 2 Positron Energy(CsI) reconstructed 12 MeV < ECsI ≤ Emax = (mµ + me)/2mµ Plastic Veto(PV) Energy EPV > 0 MeV Veto CsI Veto Energy ECsI < 5 MeV Prompt Cut EPV ≤ −0.02 ∗ ECsI + 2.8

+ + −4 events from the decay π → e νe are of the order of 10 of the signal because

+ + Brπ →e νe −4 . 10 . (4.18) + + Brµ →e νeν¯µ

+ + Even though the monoenergetic positron from the decay π → e νe has a peak well above the endpoint of the energy distribution of a positron from the decay µ+ →

+ + + e νeν¯µ , there are still some events from π → e νe that are detected well below the peak due to response function of the calorimeter. Any track identiﬁed as a positron

CsI that ﬁres the one-arm trigger with energy 10 < Ee+ < 52.83 MeV was taken as a candidate for non-radiative muon decay. Additional kinematic selection applied was to reject events with more than 5 MeV of energy in the calorimeter veto crystals (221-

240), which are situated in the calorimeter perimeter of the beam openings. This cut on energy deposited in the veto crystals helps us to eliminate the events that have energy leaking from boundaries of the calorimeter. Table 4.2 summarizes the applied kinematic cuts in the selection for non-radiative muon decay. Chapter 4: Data Analysis 67

4.1.4 Radiative Michel decay

At the tree level, the radiative decay of the muon involves no complications from the strong interaction couplings or structure. This fact favors radiative muon decay in tests of the V − A structure of the weak interaction. The diﬀerential branching fraction of muon radiative decay was calculated by several authors, for example [8, 7].

From Equation (1.21), we see that the branching fraction is a function of the kinetic energy of the positron, photon energy and the opening angle between the positron and the photon.

Event Reconstruction

In the PIBETA analysis software, the radiative muon decay event is selected as follows: (1) an event that passes either low threshold trigger (one-arm trigger or two- arm trigger) condition is kept, (2) the software requires a minimum of at least two reconstructed electromagnetic showers. For better track reconstruction, the software requires (3) at least one hit in each subdetector of the tracking system (MWPC1,

MWPC2 and PV). Once the above three conditions are met, further requirements are introduced to have a well deﬁned radiative muon decay event.

If one particle is identiﬁed as a positron and another as a photon, then a calorimeter time diﬀerence between the two particles is calculated. Virtually all coincident positron and photon pairs in low threshold triggers will have a signature of radiative muon decay as shown in Figure 4.1. The event reconstruction also requires a rejec- Chapter 4: Data Analysis 68

tion of an event that contains a hard photon emmited at a small angle relative to the

positron. This kind of hard photon events are typically due to external bremstraulung

and are one of the sources of background signals described in Section 4.1.5 below.

The event reconstruction algorithm calculates the angle between the positron and

photon. This is done as follows:

~ ~ V1 · V2 1 cos θ = = 2 (v1x · v2x + v1y · v2y + v1z · v2z), (4.19) |V1||V2| R

where v1x, v1y, v1z, v2x, v2y, v2z are the particles’ trajectory intercept coordinates with the inner surface of the calorimeter and R is the inner radius of the calorimeter

(26 cm). Lastly, the angle between the calorimeter clump and the trajectory of the positron measured by the wire chamber is calculated. Once all above conditions are satisfied and the angles have been calculated, the analyzer software writes the information in an ASCII file which is converted into ROOT file for further offline analysis.

4.1.5 Background Signal

In the offline analysis, we need to study the source of background events that po- tentially contaminate the radiative muon decay process. There are two types of background events [14, 10] . The first source is the combinatoric backgrounds events which come from random photons and positrons that are coincident in the calorime- Chapter 4: Data Analysis 69 ter. As shown in Figure 4.1, they form a flat time distribution before and after the peak of ∆t = te+ − tγ. These are removed by taking only events falling inside the window

|∆t| ≤ 5 ns, (4.20) and subtracting those in the side time window

5 ns ≤ |∆t| ≤ 10 ns. (4.21)

These two windows are of equal width in the time coincidence distribution. Assuming that this combinatoric background is uniformly distributed in the whole spectrum of

∆t, the subtraction process gives an accurate estimate of the number of radiative events that are coincident in the calorimeter.

Another source of background comes from events that are non-radiative but pass the criteria of reconstructing radiative events. In order to obtain the most accurate value for the branching fraction, it is very important to determine the fraction of background signals masquerading as radiative muon decays. The source of these mis- identiﬁed events can be explained by the way showers develop in an electromagnetic shower calorimeter, such as the PIBETA detector.

Shower development in the calorimeter can give rise to background signals from non-radiative Michel decay. For example, a positron in the calorimeter can lead to a false radiative decay if a photon from its shower manages to travel in the calorimeter Chapter 4: Data Analysis 70 far enough to produce a shower in another clump. The clumping algorithm mentioned earlier will identify this new shower as a distinct clump and hence take it as radiative decay. Due to the fact that these two showers will be in close temporal coincidence, the event reconstruction procedure described above will not be able to distinguish this background signal from a radiative event. This background source takes into account all random photons that are generated inside the calorimeter, and as in ordinary bremsstrahlung, they are most likely emitted at small angles with respect to the original particle. Therefore, by applying a cut-oﬀ angle, most of these background signals are cut out. Figure 4.8 shows the applied cut where only events that register an opening angle above 45 degrees were accepted.

Even by applying the opening angle requirement, there are still some background signals that occur when a secondary shower is emitted at a very large angle with respect to the positron. The source of these background is the external bremstraulung interactions. The eﬀect of this background is estimated in simulation by determining how many events are misidentiﬁed as radiative muon decays. The correction of the order of 2.45% was then applied to the number of radiative muon decay events by taking into account the MC estimation of this background. Figures 4.6, 4.7 and 4.8 show the simulated background, measured and simulated radiative distributions for opening angle, positron energy and photon energy respectively. Chapter 4: Data Analysis 71

Table 4.3: Event generator level cuts.

Positron Energy 0.0005 ≤ x < 1 Photon Energy 0.189 ≤ y < 1 Opening Angle (θ) 30◦ ≤ θ < 180◦ 2(x+y−1) Kinematic Cut ∆ ≥ xy

4.1.6 Kinematic cuts

One goal of this work is to measure the branching fraction over the largest accessible phase space of positron energy, photon energy and the corresponding opening angle.

It is necessary to apply a lower energy cut oﬀ on photon energy to avoid the infrared divergences described in Section 1.2.2. The numbers of events for both Michel and radiative Michel decay modes were determined using the procedure described in Section 4.1.2. The experimental acceptance accounts for those events that were generated but were lost due to kinematic plus geometrical cuts (i.e., energy and opening angle cuts). The acceptance is calculated as number of simulated events that were generated and passed reconstruction cuts divided by the total number of events that passed generator level cuts

N recon A = . (4.22) N gen

Table 4.3 and Table 4.4 show the event generator level cuts and event reconstruction cuts, respectively, that were applied in the branching fraction calculation. Chapter 4: Data Analysis 72

Table 4.4: Event reconstruction cuts

Positron Energy(CsI) 0.227 > x ≤ 1 Photon Energy(CsI) 0.3789 > y ≤ 1 Opening Angle (θ) 45◦ < θ ≤ 180◦ Kinematic Cut ∆ ≥ 2(x + y − 1)/xy Plastic Veto(PV) Energy EPV > 0 MeV CsI Veto Energy ECsIVeto < 5 MeV Prompt Cut (MeV) EPV < −0.02 · ECsI + 2.8 MeV

40000 Signal MC Misid Events

35000 Signal MC + Misid. events

DATA(measured) 30000

25000

20000

15000

10000

5000

10 15 20 25 30 35 40 45 50 55 Positron Energy [MeV]

Figure 4.6: Radiative muon positron energy distribution in the calorimeter (CsI). The histogram shows distributions for the measured energy, simulated energy (signal MC), simulated background (background MC) plus the sum of signal MC and background MC. . Chapter 4: Data Analysis 73

60000 Signal MC

Misid Events 50000 Signal MC + Misid. events

DATA(measured) 40000

30000

20000

10000

0 15 20 25 30 35 40 45 50 55 Photon Energy [MeV]

Figure 4.7: Photon energy distribution in the calorimeter (CsI). The histogram shows distributions for the measured energy, simulated energy (signal MC), simulated background (background MC) plus the sum of signal MC and background MC.

. Chapter 4: Data Analysis 74

×103

Signal MC 100 Misid Events

80 Signal MC + Misid. events

DATA(measured) 60

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Cos θ

Figure 4.8: The cosine of opening angle between positron and photon. The histogram shows distributions for the measured cosine of opening angle, simulated cosine of opening angle (signal MC), simulated background (background MC) plus the sum of signal MC and background MC. Chapter 4: Data Analysis 75

4.1.7 Systematic errors

A systematic uncertainty is a possible unknown variation in a measurement, or in a quantity derived from a set of measurement, that does not randomly vary from data point to data point. It is any error that is not statistical. It represents a constant (not random) but unknown uncertainty whose size is independent of the number of events. In the process of measuring the branching fraction, main sources of systematic errors were identiﬁed to be energy calibration, subtraction of random combinatoric background signals, selection of kinematic cuts, time window selection, and the misidentiﬁed events.

The energy calibration for the photons cannot rely entirely on the calibration that was perfomed using the positron calorimeter response and therefore a small correction to the photon energy scale factor was introduced. Section 4.2.1 explains how this factor was obtained.

We also estimated the systematic errror originating from the subtraction of random coincidences (ﬂat temporal background) in Figure 4.1. This was done by varying the width of signal events under the gaussian peak and the corresponding width of

ﬂat background. After allowing a variation of one standard deviation (0.71 ns), the diﬀerence between the value of branching fractions (before and after width change) was taken as a systematic error.

The systematic errors originating from selection of kinematic cuts (energy and Chapter 4: Data Analysis 76 ) -3 4.42 Br(10

4.41

4.4

4.39

4.38

4.37

12 12.5 13 13.5 14 14.5 15 15.5 16 Ee+ [MeV]

Figure 4.9: The variation of the branching fraction due to diﬀerent positron energy threshold selection cuts

opening angle thresholds) were estimated by selecting diﬀerent cuts around the nominal cuts showed in the Tables (4.4 and 4.2). The variation on branching fraction from this procedure of varying selection cuts were taken as systematics. Figures 4.11 and

4.12 show how the branching varies when a diﬀerent threshold energy cut is applied is the event selection.

All of the estimated systematic errors are shown in Table 4.5.

4.1.8 Results (branching fraction)

The experimental branching fraction was determined using the kinematic cuts as described in Section 4.1. The numbers of events for both radiative and non-radiative decays were obtained using the procedure described in Section 4.1.2 and their respec- Chapter 4: Data Analysis 77 ) -3 4.51 Br(10

4.5

4.49

4.48

4.47

4.46

4.45 18 20 22 24 26 28 Eγ[MeV]

Figure 4.10: The variation of the branching fraction due to diﬀerent photon energy threshold selection cuts ) -3 4.375 Br(10 4.37

4.365

4.36

4.355

4.35

4.345

2 2.5 3 3.5 4 4.5 5 ∆t Cut [ns]

Figure 4.11: The variation of the branching fraction due to diﬀerent signal window selection Chapter 4: Data Analysis 78 ) -3

Br(10 4.38

4.37

4.36

4.35

4.34

42 44 46 48 50 52 54 Opening angle Cut [Deg]

Figure 4.12: The variation of the branching fraction due to diﬀerent opening angles

Table 4.5: Systematic errors associated with the measured experimental branching fraction in the region of phase space spanned by (0.0005 < x ≤ 1 , 0.189 < y ≤ 1, 30◦ < θ ≤ 180◦)

Quantity Syst. Error(%) Photon energy calibration 0.73 Background subtraction 0.14 Positron energy threshold 0.26 Positron energy calibration 0.18 Photon energy threshold 0.41 Cosine opening angle 0.29 Time window selection 0.12 Misid. events 0.03 Total Syst. Error 0.96 Chapter 4: Data Analysis 79

Table 4.6: The experimental results for the branching fraction of radiative muon decay in the region of phase space deﬁned by cuts in Tables 4.3, 4.4 and 4.2

. Quantity Value obs N + + (real plus misid. events) 518813 ± 946 µ →e νeν¯µγ misid. N + + misidentiﬁed events 12738 ± 113 µ →e νeν¯µγ obs N + + Michel events 37094400 ± 6090 µ →e νeν¯µ + + Aµ →e νeν¯µγ (Accept. radiative Michel) 0.0693 ± 0.0001 + + Aµ →e νeν¯µ (Accept. Michel) 0.7102 ± 0.00032 BExp (misid. events excl.) (4.365 ± 0.009(stat.) ± 0.042(syst.)) × 10−3

tive acceptance were calculated using Monte Carlo events. The statistical errors on number of events were estimated using standard Poisson distribution statistics [15].

The uncertainty on detector acceptance follows the binomial distribution [15] and is calculated as follows: p σA = A(1 − A)/N , (4.23)

where A is the calculated acceptance and N is the total number of generated events.

The theoretical branching fraction calculated using Monte Carlo integration in the

region of phase space given by Table 4.3 with the Standard Model values ofη ¯ and ρ

is:

BTheo = (4.342 ± 0.005) × 10−3, (4.24) Chapter 4: Data Analysis 80

which is close to our measured experimental value:

BExp = [4.365 ± 0.009 (stat.) ± 0.042 (syst.)] × 10−3. (4.25)

4.2 Extraction of η¯ and ρ Parameters

In the Standard Model,η ¯ ≡ 0 and ρ ≡ 0.75 and therefore from Equation (1.21), only

function f1 is involved in the calculation of the branching fraction. These values of

Michel parameters are consistent with V − A structure of weak interactions. To test

the validity of V − A structure of weak interaction, we extract experimental values

of bothη ¯ and ρ parameters. Any statistically signiﬁcant deviation from Standard

Model values would indicate that weak interaction is not V − A pure and there could

be contributions from scalar, pseudo-scalar or tensor transformations.

The parameter ρ was extracted using non-radiative muon decay events, µ+ → eνeν¯µ. From Equation (1.17), we see that the decay rate is dependent on ρ parameter.

The simulated positron energy spectrum is dependent on the ρ parameter through the weight Equation (1.17) which is assigned to every simulated event. The Method of least squares was used as a criterium and the experimental value of ρ parameter

was taken as the value that minimizes the χ2 diﬀerence between experimental and Chapter 4: Data Analysis 81 simulated positron CsI energy distributions:

N 2 X (yi − λi(ρ)) χ2(ρ) = , (4.26) σ2 i=1 i

2 where yi is the number of entries in bin i (measured), σi is the variance of the number of entries in bin i, λi = E[yi ] (simulated). The number of entries predicted in bin i, and the function that is ﬁtted on measured positron energy histogram is

λi(ρ), obtained from simulated positron energy distribution. Figure 4.13 shows the statistical χ2 as a function of ρ parameter. We obtained:

ρ = 0.7581 ± 0.0045, (4.27)

which is consistent with the current PDG value:

ρ = 0.7503 ± 0.0004 (PDG). (4.28)

Since our experiment was not optimized for a measurement of ρ, in the rest of this work ρ will be set to its SM value. The parameterη ¯ is deduced from the radiative

+ + muon decay sample µ → e νeν¯µγ. From Equation (1.21), we see thatη ¯ will only contribute to the branching fraction calculation if its value is diﬀerent from the Stan- dard Model value (¯ηSM = 0). From Figures 4.14, 4.15 and 4.16, the region of phase space most sensitive toη ¯ was determined to be large opening angle region, for mod- Chapter 4: Data Analysis 82

χ2 vs ρ 2 χ 180 ρ ± 160 = 0.7581 0.0045

140

120

100

0 0.72 0.74 0.76 0.78 0.8 0.82 ρ

Figure 4.13: The χ2 as a function of ρ from analysis of non-radiative Michel decays

erate values of x and y. We chose events in the region of 150 ◦ ≤ θ ≤ 180 ◦ and also in the region deﬁned by the following kinematic cuts on positron and photon energies:

0.25 < x < 0.85 and 0.189 < y < 0.8. The least squares statistical method [15] was applied in the determination of experimental value of theη ¯ parameter. The

η¯ value was taken as the one that minimizes the χ2 diﬀerence between theoretical and experimental branching fraction calculated inside the above mentioned region of phase space. The region of phase space for calculation of experimental branching fraction was split into diﬀerent parts to verify the sensitivity ofη ¯ paramater in different regions. The region was split into bins of 150 ◦ ≤ θ ≤ 160 ◦, 160 ◦ ≤ θ ≤ 170 ◦, Chapter 4: Data Analysis 83

° ° |f2/f1|: θ [150 ,160 ]

y 1

0.9 0.125024 0.159389 0.248811 0.43907 0.4 0.8 0.35 0.7

0.212907 0.282154 0.253333 0.293119 0.6 0.3 0.5 0.25 0.4 0.263077 0.337378 0.251229 0.158329

0.3 0.2 0.2 0.215762 0.238272 0.171783 0.10838 0.15 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ ◦ Figure 4.14: (f2/f1)(x, y, 150 < θ ≤ 160 )

170 ◦ ≤ θ ≤ 180 ◦ and 160 ◦ ≤ θ ≤ 180 ◦. As expected, the greatest sensitivity toη ¯ is for opening angles greater than 160 ◦. The result from the bin 150 ◦ ≤ θ ≤ 160 ◦ shows that sensitivity to theη ¯ parameter diminishes for lower angles.

When χ2 diﬀerence between theoretical and experimental branching fraction is calculated, the ρ parameter value was ﬁxed at 0.75 whileη ¯ is varied around its Stan- dard Model value. The value ofη ¯ that minimizes χ2 as explained below is taken as our experimental value.

The value of theoretical branching fraction (Btheo) was calculated by using Monte

Carlo integration: Chapter 4: Data Analysis 84

° ° |f2/f1|: θ[160 ,170 ]

y 1

0.9 0.137664 0.208531 0.279084 0.367726 0.4 0.8

0.7 0.35 0.247036 0.383426 0.404634 0.328612 0.6 0.3 0.5

0.4 0.305369 0.447383 0.428395 0.257539 0.25 0.3

0.2 0.2

0.270073 0.347863 0.303267 0.160863 0.1 0.15 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ ◦ Figure 4.15: (f2/f1)(x, y, 160 < θ ≤ 170 )

Btheo = F (x, y, θ)dV, (4.29)

where

4 F (x, y, θ) = f (x, y, θ) +ηf ¯ (x, y, θ) + (1 − ρ)f (x, y, θ), (4.30) 1 2 3 3 and

dV = dxdyd(cos θ). (4.31) Chapter 4: Data Analysis 85

° ° |f2/f1|: θ[170 ,180 ]

y 1 0.55

0.9 0.14713 0.261525 0.383471 0.419825 0.5 0.8 0.45 0.7

0.267073 0.451008 0.540158 0.461801 0.6 0.4

0.5 0.35 0.4 0.326315 0.510713 0.5609 0.43747 0.3 0.3 0.25 0.2 0.298534 0.448825 0.459294 0.324561 0.2 0.1

0 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ ◦ Figure 4.16: (f2/f1)(x, y, 170 < θ ≤ 180 )

F (x, y, θ) is a function that needs to be calculated in a region of phase space spanned by dV.

Brieﬂy, for a function of one variable, the Monte Carlo integration method works like this:

(i) Pick N randomly distributed points x1, x2, x3, ...... , xN in the interval [a, b].

(ii) Determine the average value of the function

N 1 X hfi = f(x ), (4.32) N i i=1 Chapter 4: Data Analysis 86

|f2/f1|: θ > 157

y 1

0.9 0.138161 0.211931 0.30015 0.394601 0.8

0.7

0.246254 0.382459 0.405341 0.356935 0.6

0.5

0.4 0.303646 0.443363 0.425877 0.278642 0.3

0.2

0.267354 0.350335 0.312488 0.181712 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

◦ ◦ Figure 4.17: (f2/f1)(x, y, 157 < θ ≤ 180 )

(iii) Compute the approximation to the integral

Z b f(x)dx ≈ (b − a) ∗ hfi, (4.33) a

(iv) An estimate for the error is

r hfi2 − hf2i Error ≈ (b − a) , (4.34) N Chapter 4: Data Analysis 87 where N 1 X hf 2i = f 2(x ). (4.35) N i i=1

Finally the χ2 diﬀerence is calculated as

N Exp X (B (x, y, θ, η¯) − BTheo(x, y, θ, η¯))2 χ2(¯η) = i (4.36) σ2(BExp) + σ2(BTheo) i=1 i i

Exp Theo where Bi is experimental branching fraction, B (¯η) is a calculated theoretical

Exp branching fraction and σi(B ) is a statistical error for each branching fraction,

Exp Bi .

4.2.1 Systematic error

The systematic errors associated with each branching fraction used in the calculation of the χ2 were estimated in a similar way as described in Section 4.1.7. The systematic error from photon energy calibration was estimated after ﬁnding the energy scale factor and apply its error as a systematic (Figure 4.26). As the measured experimental branching fraction depends on the applied photon energy scale, this energy scale factor in turn inﬂuences the value ofη ¯.

The minimization required the variation of both energy scale factor andη ¯ and calculating the corresponding χ2 (Equation 4.36). The minimization was performed for each of the four chosen bins of opening angle. The analysis showed that adding Chapter 4: Data Analysis 88

Table 4.7: The values ofη ¯ obtained from diﬀerent regions of phase space as showed in Tables 4.12, 4.13 and 4.15.

Angle range η¯ value 170◦ − 180◦ 0.012 ± 0.017 160◦ − 170◦ 0.002 ± 0.017 160◦ − 180◦ 0.006 ± 0.017 150◦ − 160◦ 0.042 ± 0.031

events with θ < 160◦ signiﬁcantly dilutes the sensitivity toη ¯ paramter. Figures (4.18,

4.20, 4.22 and 4.24) show the χ2 as a function of photon energy scale and theη ¯ parameter. Theη ¯ parameter values were extracted after ﬁxing the photon energy scale factor at a value that gives the minimum χ2. For large angles (160◦ < θ ≤ 180◦) the factor was found to be:

Cγ = 0.994 ± 0.003, (4.37) while for the bin of 150◦ < θ ≤ 160◦ the factor obtained was:

Cγ = 0.997 ± 0.003. (4.38)

The extractedη ¯ parameter values are shown in Figures (4.19, 4.21, 4.23 and 4.25).

Due to higher event statistics the results from the bin of 160◦ < θ ≤ 180◦ were taken as our best results.

Tables (4.8, 4.9, 4.10, and 4.11) show the systematic errors associated with the Chapter 4: Data Analysis 89

1.002 16.325 6.817 14.14 72.92

0.998 13.93 1.73 2.11 16.96 Photon Energy Scale 0.996

0.994 23.14 3.154 1.469 9.722

0.992

0.99 84.74 15.893 6.302 12.957

0.988 -0.04 -0.02 0 0.02 0.04 η

Figure 4.18: The χ2 as a function of photon energy scale andη ¯ for 170◦ < θ ≤ 180◦ bin. 2 χ 25 η = 0.012 ± 0.017

χ2/NDF = 0.48 15

-5-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 η

Figure 4.19: The χ2 as a function ofη ¯ for 170◦ < θ ≤ 180◦ bin, with Cγ = 0.994. Chapter 4: Data Analysis 90

1.002 8.32 9.495 20.01 92.08

0.998 7.583 1.522 3.49 23.37 Photon Energy Scale 0.996

0.994 15.564 2.002 1.481 13.009

0.992

0.99 71.75 13.488 5.789 14.612

0.988 -0.04 -0.02 0 0.02 0.04 η

Figure 4.20: The χ2 as a function of photon energy scale andη ¯ for 160◦ < θ ≤ 170◦ bin. 2

χ 20 η = 0.002 ± 0.017 15

χ2/NDF = 0.49 10

-5

-10 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 η

Figure 4.21: The χ2 as a function ofη ¯ for 160◦ < θ ≤ 170◦ bin, with Cγ = 0.994. Chapter 4: Data Analysis 91

1.002 11.408 8.37 15.387 67.599

0.998 9.285 2.417 3.45 18.328 Photon Energy Scale 0.996

0.994 15.92 3.03 2.37 11.401

0.992

0.99 63.29 13.764 7.898 12.668

0.988 -0.04 -0.02 0 0.02 0.04 η

Figure 4.22: The χ2 as a function of photon energy scale andη ¯ for 160◦ < θ ≤ 180◦ bin. 2

χ 18

16 η = 0.006 ± 0.017 14

12 χ2/NDF = 0.8 10

2 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 η

Figure 4.23: The χ2 as a function ofη ¯ for 160◦ < θ ≤ 180◦ bin, with Cγ = 0.994. Chapter 4: Data Analysis 92

1.006

16.35 13.62 11.26 11.38 13.3 17 22.47 29.74 1.004

1.002

13.75 10.31 7.79 6.17 5.47 5.68 6.79 8.82 1

Photon Energy Scale 0.998

19.79 15.1 11.33 8.48 6.55 5.55 5.47 6.31 0.996

0.994

24.77 19.48 15.11 11.68 9.19 7.64 7.02 7.33 0.992

0.99

0.988 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 η

Figure 4.24: The χ2 as a function of photon energy scale andη ¯ for 150◦ < θ ≤ 160◦ bin. 2 χ 20

18 η = 0.042 ± 0.031

12 χ2/NDF = 1.4 10

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 η

Figure 4.25: The χ2 as a function ofη ¯ for 150◦ < θ ≤ 160◦ bin, with Cγ = 0.997. Chapter 4: Data Analysis 93

12 Cγ = 0.994 ± 0.003

0.985 0.99 0.995 1 1.005

Figure 4.26: The χ2 as a function of photon energy scale for 160◦ < θ ≤ 180◦ bin, withη ¯ = 0.006..

Table 4.8: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by (0.5 < x ≤ 0.75 , 0.28 < y ≤ 0.48, and 160◦ < θ ≤ 180◦)

Quantity Syst. Error(%) Photon energy calibration 0.75 Background subtraction 0.15 Positron energy calibration 0.24 Cosine opening angle 0.32 Time window selection 0.14 Misid. events 0.04 Total Syst. Error 0.87 Chapter 4: Data Analysis 94

Table 4.9: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by (0.5 < x ≤ 0.75, 0.5 < y ≤ 0.75, and 160◦ < θ ≤ 180◦).

Quantity Syst. Error(%) Photon energy calibration 0.72 Background subtraction 0.18 Positron energy calibration 0.25 Cosine opening angle 0.28 Time window selection 0.12 Misid. events 0.03 Total Syst. Error 0.84

Table 4.10: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by (0.28 < x ≤ 0.48, 0.5 < y ≤ 0.75, and 160◦ < θ ≤ 180◦).

Quantity Syst. Error(%) Photon energy calibration 0.69 Background subtraction 0.16 Positron energy calibration 0.23 Cosine opening angle 0.33 Time window selection 0.15 Misid. events 0.03 Total Syst. Error 0.83

branching fractions used to extract theη ¯ parameter. These errors in turn introduced a systematic uncertainty in the measurement of theη ¯ parameter. Table 4.16 shows the details of the contribution from each source of uncertainty, with photon energy calibration being a dominant source. Chapter 4: Data Analysis 95

Table 4.11: Systematic errors associated with the measured experimental branching fraction in the the region of phase space spanned by (0.28 < x ≤ 0.48, 0.28 < y ≤ 0.48, and 160◦ < θ ≤ 180◦).

Quantity Syst. Error(%) Photon energy calibration 0.76 Background subtraction 0.17 Positron energy calibration 0.24 Cosine opening angle 0.32 Time window selection 0.12 Misid. events 0.03 Total Syst. Error 0.88

Table 4.12: Event statistics for the experimental branching fractions used inη ¯ extraction in the bin of 170◦ < θ ≤ 180◦. Calculated theoretical branching fraction in the region of phase space covered by (0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 170◦ < θ ≤ 180◦) is 2.41 × 10−6.

x Range y Range Br. Fraction (10−6) No. of Events 0.28 < x ≤ 0.48 0.28 < y ≤ 0.48 2.42 ± 0.04(stat.) 564 ± 29 0.28 < x ≤ 0.48 0.5 < y ≤ 0.75 2.39 ± 0.05(stat.) 581 ± 33 0.5 < x ≤ 0.75 0.28 < y ≤ 0.48 2.43 ± 0.04(stat.) 389 ± 36 0.5 < x ≤ 0.75 0.5 < y ≤ 0.75 2.46 ± 0.04(stat.) 384 ± 41

The obtained experimental result for parameterη ¯ extracted using statistical error

(shown in Figure 4.23) is:

η¯ = 0.006 ± 0.017, (4.39)

Our ﬁnal experimental result is:

η¯ = 0.006 ± 0.017 (stat.) ± 0.018 (syst.). (4.40) Chapter 4: Data Analysis 96

Table 4.13: Event statistics for the experimental branching fractions used inη ¯ extraction in the bin of 160◦ < θ ≤ 170◦. Calculated theoretical branching fraction in the region of phase space covered by (0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 160◦ < θ ≤ 170◦) is 8.17 × 10−6.

x Range y Range Br. Fraction (10−6) No. of Events 0.28 < x ≤ 0.48 0.28 < y ≤ 0.48 8.04 ± 0.01 1840 ± 52 0.28 < x ≤ 0.48 0.5 < y ≤ 0.75 8.17 ± 0.08 1924 ± 62 0.5 < x ≤ 0.75 0.28 < y ≤ 0.48 8.29 ± 0.02 1301 ± 60 0.5 < x ≤ 0.75 0.5 < y ≤ 0.75 8.14 ± 0.01 1424 ± 72

Table 4.14: Event statistics for the experimental branching fractions used inη ¯ extraction in the bin of 160◦ < θ ≤ 180◦. Calculated theoretical branching fraction in the region of phase space covered by (0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 160◦ < θ ≤ 180◦) is 1.02 × 10−5.

x Range y Range Br. Fraction (10−5) No. of Events 0.28 < x ≤ 0.48 0.28 < y ≤ 0.48 1.03 ± 0.03 2416 ± 62 0.28 < x ≤ 0.48 0.5 < y ≤ 0.75 1.01 ± 0.03 2494 ± 72 0.5 < x ≤ 0.75 0.28 < y ≤ 0.48 1.06 ± 0.01 1698 ± 66 0.5 < x ≤ 0.75 0.5 < y ≤ 0.75 1.04 ± 0.01 1816 ± 81

Table 4.15: Event statistics for the experimental branching fractions used inη ¯ extraction in the bin of 150◦ < θ ≤ 160◦. Calculated theoretical branching fraction in the region of phase space covered by (0.25 < x ≤ 0.85 , 0.189 < y ≤ 0.8, 150◦ < θ ≤ 160◦) is 1.59 × 10−5.

x Range y Range Br. Fraction (10−5) No. of Events 0.28 < x ≤ 0.48 0.28 < y ≤ 0.48 1.57 ± 0.05 3484 ± 73 0.28 < x ≤ 0.48 0.5 < y ≤ 0.75 1.58 ± 0.02 3371 ± 76 0.5 < x ≤ 0.75 0.28 < y ≤ 0.48 1.55 ± 0.03 3030 ± 84 0.5 < x ≤ 0.75 0.5 < y ≤ 0.75 1.64 ± 0.03 2732 ± 88 Chapter 4: Data Analysis 97

Table 4.16: Systematic errors associated with the extraction ofη ¯ parameter.

Quantity Syst. Error(δη¯) Photon energy calibration 0.016 Background subtraction 0.003 Positron energy calibration 0.005 Cosine opening angle 0.006 Time window selection 0.002 Misid. events 0.0003 Total Syst. Error 0.018

Standard statistical method [15] of setting upper limit value for a parameter was applied to calculate the upper limit of theη ¯ parameter. In this process we used the fact thatη ¯ parameter is by its deﬁnition expected to have a value between zero and one [5]. The upper limitη ¯max is thus determined by

h 2 i R η¯max (¯η−η¯0) ) 0 exp − 2σ2 C.L = η¯0 (4.41) h 2 i R 1 (¯η−η¯0) 0 exp − 2σ2 η¯0 where C.L is the conﬁdence level which can be set as 0.683, 0.95 or 0.99.

The application of Equation (4.41) on the experimental result (4.40), sets the upper limit value of theη ¯ to be:

η¯ ≤ 0.028 (68.3% conﬁdence). (4.42) Chapter 4: Data Analysis 98

The combined uncertainty in this work, δη¯ = 0.025, is about four times lower than that in the best previous measurement [9]. Our new 68% CL upper limit is four times lower than the previous average [6].

4.3 Conclusions

The experimental results obtained in this work can be compared to previous measurements [10] and [9]. The present experimental branching fraction is consistent with values obtained in Ref [10] which is not surprising as similar kinematic cuts were applied in both analyses.

The value of Michel parameter ρ obtained in this analysis is consistent with the one obtained in [10], but the precision ofη ¯ values determination is signiﬁcantly improved in this work. The improvement is caused by these factors: (a) a more precise calibration of data was implemented, (b) a larger data set was used, including runs skipped in the previous analysis, and (c) the phase space region for theη ¯ ﬁts was restricted to the most sensitive subset. This analysis used only events that were reconstructed in large opening angle (160◦ < θ ≤ 180◦) were used, in [10] the cut

on the opening angle was lowered to include all events that pass 90◦ < θ ≤ 180◦.

From Figures (4.14, 4.15, 4.16, and 4.17), we see that the most sensitive region forη ¯ extraction is in the large opening angles. In the end, the gain in sensitivity oﬀset the lower event statistics in the present, more restrictive region of phase spaces. Chapter 4: Data Analysis 99

The experimental result forη ¯ obtained from this analysis is still consistent with the

Standard Model value. This result, together with other experimental determinations of Michel parameters, will be included in a global analysis in order to determine limits to deviations from V − A structure of weak interaction. Appendix A

The functions fi(x, y, θ)

α f (x, y, θ) = (1 − R )n0 (A.1) 1 16π2y Dalitz V

α f (x, y, θ) = (1 − R )(2n0 − 2n0 + n0 ) (A.2) 2 16π2y Dalitz S V T

α f (x, y, θ) = (1 − R )(2n0 + 2n0 − n0 ) (A.3) 2 16π2y Dalitz S V T

2α ymµ 19 RDalitz = [ln( ) − ] (A.4) 3π 2me 12

+ − RDalitz is the probability that the emitted photon converts into e e . The other

100 Appendix A: The Functions fi(x, y, θ) 101

o o o variables nV , nS, nT are deﬁned as follows:

o 2 −2 nV = 4(1 − β )[2∆ x(x + y)(2(2 + y) − 3),

∆−1x2y(3 − 4(x + y)) + x3y2], (A.5)

−1 V V V 2 V +∆ G−1 + Go + ∆G1 + ∆ G2 ,

0 0 0 2 −1 2 2nS − 2nV + nT = −8xy [∆ x(1 − β + 2(1 − y − 2x) + ∆x(1 + y)], (A.6)

0 0 0 −1 2 2 3 2nS + 2nV − nT = 4(∆ 2[y (3 − 4y) + 6xy(1 − 2y) + 2x (3 − 8y) − 8x ]

+2x[−y(3 − y − 6y2) − x(3 − 5y − 10y2) + 4x2(1 + 2y)]

+∆x2y[1.5(2 − 3y − 4y2) − 2x(4 + 3y)] (A.7)

+∆2x3y2(2 + y) − (1 − β2)[∆−22x(x + y)(3 − 4(x + y))

+∆−1x2y(2(4x + 5y) − 3) − 2x3y2]),

V 2 2 3 G−1 = 8[y (3 − 2y) + 6xy(1 − y) + 2x (3 − 4y) − 4x ], (A.8)

V 2 2 2 3 Go = 8[−xy(3 − y − y ) − x (3 − y − 4y ) + 2x (1 − 2y)], (A.9)

V 2 2 3 G1 = 2[x y(6 − 5y − 2y ) − 2x y(4 + 3y)], (A.10) Appendix A: The Functions fi(x, y, θ) 102

V 3 G2 = 2x y2(2 + y), (A.11) Appendix B

The Least Squares Method

This Appendix will outline the method of Least Squares that is used in estimating parameters, given a set of experimental data. Data set will consist of a group of points xi, y(xi) and σi, where xi is the independent variable, y(xi) is the dependent variable, and σi is the standard deviation of y(xi). The subscript i denotes a particular element of N data points. This data set is to be ﬁt to an equation f(xi, a) which is a function of the independent variable and the the vector a of parameters to be evaluated. The

ﬁt is to be performed such that the dependent variable y(xi) can be approximated by the ﬁtting function evaluated at the corresponding independent variable xi , and the parameter values having the maximum likelihoof of being correct: that is

y(xi) u f(xi, a). (B.1)

103 Appendix B: The Least Squares Method 104

Least Squares analysis is comprised of a group of numerical procedures that can be used to evaluate the optimal values of the parameters in vector a for the experimental data. In general, Least Squares procedure consist of an algorithm that uses an initial approximation vector of parameters to be used in determining which one of them has the optimal value. It is iterative process which is continued until the vector of parameters converges to value such that the weighted sum of the squares of the diﬀerences between the ﬁtted function and the experimental data weighted by the inverse of variances is minimal.

The values of parameters are taken as those that minimize the quantity

N 2 X y(xi) − f(xi, a) χ2(a) = . (B.2) σi i=1

To demonstrate that the Least Squares method is appropriate and will yield the parameters a having the maximum likelihood of being correct, several interrelated assumptions must be made. We must assume: (1) that all of the experimental uncertainty can be attributed to the dependent variables y(xi), (2) that the experimental uncertainties of the data can be described by a Gaussian distribution, (3) that no systematic errors exist in the data, (4) that the functional form f(xi, a) is correct,

(5) that there are enough data points to provide a good sampling of the experimental uncertainties, and (6) that the observations (data points) are independent of each other. Appendix B: The Least Squares Method 105

Least Squares method is also connected to another method of parameter estimation called the Maximum Likelihood. This can be shown as follows: by assuming

(1) that the experimental uncertainties of the data are all in the dependent variables y(xi), (2) that the experimental uncertainties of data follow a Gaussian distribution,

(3) that no systematic errors exist, and (4) that the ﬁtting function is correct, then the N measurements of y(xi) can as well be taken as a single measurement of an

N-dimensional random vector, whose joint p.d.f is the product of N Gaussians,

N 2 Y 1 −(y(xi) − f(xi, a)) P (a; x ; y(x ); σ2) = exp (B.3) i i i p 2 2σ2 i=1 2πσi i

If we take the logarithm of the joint p.d.f we get a log-likelihood function,

N 2 1 X y(xi) − f(xi, a) logL(a) = − (B.4) 2 σi i

This is maximized by ﬁnding the values of parameters a that minimize the quantity

N 2 X y(xi) − f(xi, a) χ2(a) = . (B.5) σi i=1 Bibliography

[1] D.A. Greenwood W.N. Cottingham. An Introduction to the Standard Model of

Particle Physics. Cambridge, 2003.

[2] B. Muller W. Greiner. Gauge Theory of Weak Interactions. Springer, fourth

edition, 2009.

[3] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson. Exper-

imental Test of Parity Conservation in Beta Decay. Phys. Rev., 105:1413–1414,

1957.

[4] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field

Theory. Addison-Wesley, 2002.

[5] L. Michel. Interaction Between Four Half Spin Particles and the Decay of the

Mu Meson. Proc. Phys. Soc., A63:514–531, 1950.

[6] K. Nakamura et al.(Particle Data Group). 2011 Review of Particle Physics.

Journal of Physics, G37:1, 2011.

106 bibliography 107

[7] C. Fronsdal and H. Uberall.¨ µ-Meson Decay with Inner Bremsstrahlung. Physical

Review, 113:654–657, January 1959.

[8] S. G. Eckstein and R. H. Pratt. Radiative Muon Decay. Annals of Physics,

8:297–309, 1959.

[9] W. Eichenberger, R. Engfer, and A. Van Der Schaaf. Measurement of the Pa-

rameterη ¯ in the Radiative Decay of the Muon as a Test of the V-A Structure of

the Weak Interaction. Nucl. Phys., A412:523–533, 1984.

[10] B. A. VanDevender. An Experimental Study of Radiative Muon Decay. PhD

thesis, University of Virginia, Charlottesville, Virginia, 2006.

[11] Frleˇzet al. Design, Commissioning and Performance of the PIBETA detector at

PSI. Nucl. Instrum. Meth., pages 300–347, 2004.

[12] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, and P. Zanarini. GEANT

3.21. CERN, Geneva, 1994.

[13] J. R. Musser et al. Measurement of the Michel Parameter ρ in Muon Decay.

Phys. Rev. Lett., 94:101805, 2005.

[14] Charles A. Rey. Inner bremsstrahlung in muon decay. Phys. Rev., 135:1215–1224,

1964.

[15] Glen Cowan. Statistical Data Analysis. Oxford, 1998.