c© 2007 by Daniel Brian Chitwood. All rights reserved.

A MEASUREMENT OF THE MEAN LIFE OF THE POSITIVE MUON TO A PRECISION OF11 PARTS PER MILLION

BY

DANIEL BRIAN CHITWOOD

B.S., University of Missouri - Rolla, 1999M.S., University of Illinois at Urbana-Champaign, 2001

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2007

Urbana, Illinois

Abstract

A MEASUREMENT OF THE MEAN LIFE OF THE POSITIVE MUON TO A PRECISION OF11 PARTS PER MILLION

Daniel Brian Chitwood, Ph.D.Department of Physics

University of Illinois at Urbana-Champaign, 2007Professor David W. Hertzog, Advisor

The mean life of the positive muon has been measured to a precision of 11 ppm using a low-energy,

pulsed muon beam stopped in a ferromagnetic target, which was surrounded by a scintillator detector array.

The result, τµ = 2.197013(24) µs, is in excellent agreement with the previous world average. The new world

average τµ = 2.197019(21) µs determines the Fermi constant GF = 1.166371(6)× 10−5 GeV−2 (5 ppm).Described in this thesis are the motivating concepts, experimental procedure, data analysis, and systematic

error analysis for this measurement.

iii

This thesis is dedicated to

my loving wife Jessica

and to my children

Kieontai, Malia, and Caden.

iv

Acknowledgments

I would like to acknowledge the assistance of the entire MuLan collaboration:

T.I. Banks, M.J. Barnes, S. Battu, R.M. Carey, S. Cheekatmalla, S.M. Clayton, J. Crnkovic, K.M. Crowe,

P.T. Debevec, S. Dhamija, W. Earle, A. Gafarov, K. Giovanetti, T.P. Gorringe, F.E. Gray, M. Hance,

D.W. Hertzog, M.F. Hare, P. Kammel, B. Kiburg, J. Kunkle, B. Lauss, I. Logashenko, K.R. Lynch, R. McN-

abb, J.P. Miller, F. Mulhauser, C.J.G. Onderwater, C.S. Özben, Q. Peng, C.C. Polly, S. Rath, B.L. Roberts,

V. Tishchenko, G.D. Wait, J. Wasserman, D.M. Webber, P. Winter, and P.A. Żołnierczuk

I would like to give a special tribute to my research advisor, David Hertzog, and to the precision muon

physics group at the University of Illinois: Steven Clayton, Jason Crnkovic, Paul Debevec, Andrea Esler,

Fred Gray, Peter Kammel, Brendan Kiburg, Sara Knaack, Josh Kunkle, Ron McNabb, Gerco Onderwater,

Cenap Özben, Chris Polly, and David Webber. Without their assistance, insight, and communication the

experiment and analysis would have never been completed.

My family deserves great recognition for supporting me as a graduate student. My wife, Jessica, always

pushed me to continue in my efforts while raising our three children, Kieontai, Malia, and Caden. Without

her patience and quiet forcefulness, this work would have taken much longer. My children are my inspira-

tion, with all my work dedicated to them. It is for their success that I succeed. My parents, Elizabeth and

Dennis Chitwood, always had confidence in me and helped me to believe in myself; that I can accomplish

anything I set my mind to.

My tuition, stipend, and travel expenses were supported by the National Science Foundation under

contracts NSF-PHY-00-72044, NSF-PHY-02-44889, and NSF-PHY-06-01067. I am also grateful to the

Felix T. Adler fellowship committee for awarding me with that fellowship honor in 2005.

v

Table of Contents

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Theoretical extraction of the Fermi coupling constant . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Errors on Gµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Comparison of Gµ to other determinations of GF . . . . . . . . . . . . . . . . . . . . . . . 92.3 Comparison of τµ+ to τµ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Chapter 3 Review of pertinent muon physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Muon production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Muonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Differential decay spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 µSR effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.3 Initial polarization reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 4 Methods for measuring τµ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 DC method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 TRIUMF experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 FAST experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Pulsed method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1 Saclay experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 RIKEN-RAL experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Results of previous τµ+ measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 5 MuLan method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1 Systematic concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 µSR effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.1.2 Dead time effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1.3 Time stability of the background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vi

Chapter 6 Experimental overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.1 Accelerator facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 πE3 beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Kicker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.4 Entrance muon detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.5 SRIM simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6 Muon stopping targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7 Positron detector system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.8 Electronic components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.8.1 Discriminators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.8.2 Time to digital convertors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.8.3 Clock system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.9 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 7 Data production and histogramming . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.1 Event definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.2 Histogram details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2.1 Multiplicity correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.2.2 Dead time corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Chapter 8 Fit results and consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.1 Fit function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708.2 Global fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.2.1 Fit residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.2.2 Fit range consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.3 Run conditions and data subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.3.1 Run by run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3.2 Target and field orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.3.3 Discriminator threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3.4 Kicker settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3.5 EMC counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3.6 Detector subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8.4 Analysis choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.4.1 Data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.4.2 Rebinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.4.3 Artificial dead time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 9 Systematic error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.1 Extinction stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.1.1 Kicker voltage stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.1.2 Extinction factor change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.1.3 Calculation of systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.2 Errant muon stops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.2.1 Muons missing the targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2.2 Simulation effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.3 Dead time correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.4 Gain stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

vii

9.4.1 Evaluating ∆N/N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.4.2 Gain stability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.5 Systematic errors associated with the MTDC . . . . . . . . . . . . . . . . . . . . . . . . . 1099.5.1 Response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.5.2 Repeated events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9.6 Timing shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139.7 Afterpulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 10 Final results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendix A Muonium lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Appendix B Dephasing calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Appendix C Errant muon simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.1 SRIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.2 GEANT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Author’s biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

viii

Chapter 1

Introduction

The standard electroweak model describes the interaction of quarks and leptons via the electromagnetic

and weak forces [1]. As with any theory, there are certain input parameters, derived from measurable

observables, needed to allow predictive calculations. The input parameters needed for the electroweak

theory include

1. the masses of the leptons, quarks, and Higgs boson (not yet determined),

2. the Cabibbo-Kobayashi-Maskawa matrix elements for quark mixing,

3. the Maki-Nakagawa-Sakata matrix elements for neutrino mixing,

4. and three additional values dictating the strength of the interactions and relative masses of the weak

gauge bosons.

The determination of the values for the first three groups will not be discussed here. For the third group, it

is natural to choose the values that can be measured or derived with the highest precision. The three most

precisely known parameters are the electromagnetic coupling constant, α, the mass of the neutral Z boson,

MZ , and the Fermi coupling constant, GF .

The value of α is one of the most precisely determined parameters in physics. Obtained from a mea-

surement of the electron magnetic moment and quantum electrodynamics (QED) calculations, the value of

α is given as [2]

α−1 = 137.035 999 710 (96) [0.70 ppb]. (1.1)

Despite the name “fine-structure constant,” α is not constant, but depends on the energy scale of the

interaction—the coupling “runs.” For energies above a few hundred MeV, uncertainties arise from the

hadronic contributions to the vacuum polarization. Thus, both the central value and uncertainty are depen-

1

dent on energy. For example, at the energy scale of MZ [1],

α−1(MZ) = 127.918 (18) [140 ppm]. (1.2)

The LEP1 Electroweak Working Group combined the results of four measurements from the ALEPH,

DELPHI, L3, and OPAL experiments to obtain the current world average value for MZ [3]:

MZ = 91.1876 (21) GeV [23 ppm]. (1.3)

This precision is remarkable when compared to estimates, before LEP was turned on, expecting at best a

∼ 550 ppm measurement of MZ [4]. The theoretical cross section and measurements near the Z-pole can beseen in Fig. 1.1(a), and the comparison of the four determinations of MZ are shown in Fig. 1.1(b).

(a) (b)

Figure 1.1: (a) “The hadronic cross-section as a function of center-of-mass energy. The solid line is theprediction of the SM, and the points are the experimental measurement. Also indicated are the energy rangesof various e+e− accelerators. The cross-sections have been corrected for the effects of photon radiation.” [3](b) Measurements of MZ for the four experiments of the LEP Electroweak Working Group. The averageis obtained by taking into account the correlated errors between the measurements. Figures are from theElectroweak Working Group [3].

The determination of GF is best accomplished through the measurement of the muon lifetime, τµ. A

series of previous measurements of τµ+ [5, 6, 7, 8], performed in the 1970s and 1980s, had produced a

combined relative uncertainty on the lifetime of 18 ppm, contributing 9 ppm to the uncertainty on GF . At

1Large Electron-Positron Collider at CERN.

2

the time, GF had a total relative error of 17 ppm, with the dominant error (15 ppm) coming from the second-

order radiative corrections in the extraction of GF from τµ. In 1999, the completion of these calculations [9,

10, 11] resulted in a reduction of the error on GF from theoretical considerations to 0.2 ppm, giving the

Fermi coupling constant as

GF = 1.16639 (1)×10−5 GeV−2 [9 ppm]. (1.4)

The uncertainty is dominated solely from the error on the muon lifetime. Fig. 1.2 shows the relative error

on GF as a function of time since 1962. The last experimental improvement came over 20 years ago.

Year1960 1970 1980 1990 2000 2010

[p

pm

]F

Rel

ativ

e E

rro

r o

n G

1

10

210

Measurementreported hereTheoretical uncertainty

MuLan Goal

Figure 1.2: The relative error on GF as a function of year from 1962 to the present. The solid line indicatesthe relative error and the points indicate the reduction of the error from measurements of the muon lifetime.The dashed line is the contribution to the error from theoretical uncertainties, with the drop at 1999 from thecompletion of the second-order radiative corrections. The current measurement reported here is indicatedin the year 2007 and the MuLan goal of 0.5 ppm is indicated for the year 2010.

Given the reduction of the theoretical uncertainty and the precise measurements of other electroweak

parameters, especially α and MZ , it is worth improving the precision of GF . The MuLan experiment is

designed to improve the precision of the muon lifetime to a relative error of 1 ppm, and once again put

the theoretical and experimental uncertainties associated with the determination of GF at nearly the same

level. The experiment, taking place at the Paul Scherrer Institut in Villigan Switzerland, is ongoing. This

dissertation reports an intermediate measurement of the positive muon lifetime with a precision of 11 ppm,

3

reducing the relative error on GF to 5 ppm. The measurement reported here constitutes the results from the

R04 data set, collected during a six week period in the fall of 2004.

The MuLan experiment is expected to produce at least two more precision measurements of the muon

lifetime. In 2006 the R06 data set was collected; the first data set having a fully operational kicker and

waveform digitizers. The expected precision is ∼ 1 ppm. Currently planned is a R07 data set to be taken inthe summer of 2007 with the goal of another ∼ 1 ppm measurement on a quartz target, known for forminga large fraction of muonium.

4

Chapter 2

Motivations

2.1 Theoretical extraction of the Fermi coupling constant

Presented in this section is the historical context for the use of GF as well as its derivation from τµ. This will

include a discussion of the relevance of the weak charged exchange boson, the necessity of radiative calcu-

lations, and the relationship between GF and the weak coupling constant, gw. In addition, the uncertainties,

both experimental and theoretical, in determining GF are discussed and quantified.

2.1.1 Historical overview

Prior to the discovery of the muon in 1936 [12, 13] the formalism for calculating the muon lifetime already

existed from the beta decay theory developed by Enrico Fermi [14], and known as the local four-fermion

interaction theory1. As applied to muon decay, the theory assumes that the muon decays through the direct

interaction with a positron and two neutrinos at a single point, as illustrated in Fig. 2.1a and represented by

the expression: µ+ → e+ + νe + ν̄µ. Assuming the vector minus axial vector (V-A) interaction of the weakforce, the lifetime of the muon, defined as the inverse total decay rate, for this interaction is simply

1τµ≡ Γµ = Γ0µ

(1+ f

(m2em2µ

)), (2.1)

where

Γ0µ =G2µm

5µ

192π3, (2.2)

and

f (x) =−8x−12x2 lnx+8x3− x4 . (2.3)1This does not imply that the muon was anticipated prior to its discovery, just that the local four-fermion weak interaction had

already been formulated

5

+µ

µν eν +e

µG

(a)

+µ

µν

+W

eν +e

Wg

Wg

(b)

Figure 2.1: Feynman diagrams for muon decay. (a) The local four-fermion model as described by Fermi.(b) Utilizing the W+ boson as the charged weak propagator.

In the above equations, the mass of the electron and muon are given by me and mµ, respectively, and the

strength of the interaction is represented by the coupling constant Gµ. It is implicitly assumed that the mass

of the neutrinos are sufficiently small such that the phase space consideration only involves the electron

mass and is accounted for in f (m2e/m2µ). Within the framework of the standard model, the coupling constant

Gµ is a universal constant defining the strength of all weak interactions, and as such it is better known as the

Fermi coupling constant, GF .

In the modern description the weak interaction is not a local four-fermion interaction, but rather takes

place via the propagation of an exchange boson. In the case of positive muon decay this is through the

charged weak boson, W+, as illustrated in Fig. 2.1(b). The non-zero mass, MW of the W boson contributes

to the matrix element, modifying the decay rate. But the effect is suppressed by the large difference between

mµ and MW . The correction to the decay rate is given by

Γµ → Γµ(

1+35

m2µM2W

), (2.4)

resulting in a 1 ppm correction. Using this description the strength of the interaction is given by the weak

6

coupling constant gw, which is related to Gµ by

Gµ√2

=g2w

8M2W. (2.5)

2.1.2 Radiative corrections

As early as the 1950’s, it was realized that the discrepancy between the observed muon lifetime and the

value calculated using Eq. (2.1) required the inclusion of radiative corrections [15, 16]. The lowest-order

radiative corrections are dominated by virtual photon interactions as illustrated in Fig. 2.2. To quantify these

+µ

µν eν +e

µG

γ

(a)

Figure 2.2: An example of a Feynman diagram showing a radiative correction.

additional corrections, it is useful to revert back to the local interaction model [9] and define the relationship

between τµ and Gµ as

Γµ = Γ0µ (1+∆q) , (2.6)

where

∆q =∞

∑i=0

∆q(i) (2.7)

contains all QED corrections and the index i refers to the power of the renormalized electromagnetic cou-

pling constant, αr, contained in each correction term ∆q(i) [9].

The zeroth-order term in Eq. (2.7) is just the phase space term from Eq. (2.1) plus additional terms if

7

the neutrino masses are retained as non-zero parameters. The first-order QED corrections have long been

known [15, 16]. The second-order corrections have recently been calculated [9], greatly reducing the

uncertainty on ∆q.

Eq. (2.6) is written in the local four-fermion model, neglecting effects from the W boson. In this frame-

work, all W boson modifications to the decay rate, as in Eq. (2.4), are removed. Instead effects arising from

the W boson are contained in a modification of Eq. (2.5), by the inclusion of a correction term ∆r:

Gµ√2

=g2w

8M2W(1+∆r). (2.8)

Similar to the expansion of ∆q, ∆r can be expanded in powers of αr. Recent theoretical work has completed

the determination of ∆r to O(α2) [17].

2.1.3 Errors on Gµ

When using Eq. (2.6) to compute Gµ the relative error can be expressed as

δGµGµ

=

√(12

δτµτµ

)2+

(52

δmµmµ

)2+

(12

δ∆q∆q

)2, (2.9)

where the errors are assumed to be independent and δ∆q/∆q represents the relative error from theoretical

uncertainties to the radiative corrections.

The best known of these quantities is the muon mass given in atomic mass units as [1]

mµ = 0.1134289264 (30) u [26 ppb]. (2.10)

The uncertainty in the conversion factor from atomic mass units to MeV,

1 u = 931.494043(80) MeV [86 ppb], (2.11)

is larger than the measured uncertainty on the mass itself. Thus, the uncertainty in the determination of Gµ

involving the muon mass is dominated by the conversion factor and results in a 0.22 ppm relative error.

Prior to the completion of the calculations of ∆q(2) in 1999 [9], δ∆q contributed the dominant error in

Eq. (2.9), giving a relative uncertainty on Gµ of 15 ppm. Currently, the theoretical uncertainty, δ∆q, in

8

determining Gµ is on the order of a few tenths of a ppm and is comparable to the uncertainty from the muon

mass. The remaining uncertainty to ∆q comes from three sources. The first, and smallest, is an additional

hadronic contributions at the 10−8. The second is uncalculated two-loop QED corrections that scale as

α/2x ln2 x, where x = m2e/m2µ, and conservatively contributes a relative uncertainty of 1.7×10−7. The finaltheoretical uncertainty comes from the sum of all unaccounted three-loop QED corrections, which have

been estimated to be at the 1.4×10−7 level.The improvements on ∆q leaves the muon lifetime as the largest contributing factor to the uncertainty

on Gµ. Prior to this work, the world average for the positive muon lifetime was

τµoldworld average = 2197.03 (4) ns [18 ppm] [1]. (2.12)

It was obtained from four precision measurements [5, 6, 7, 8], the most precise having a relative error of

27 ppm [5]. The error from these measurements contributed a relative uncertainty of 9 ppm to Gµ, more

than an order of magnitude larger than the other sources of uncertainty. The δτ/τ = 11 ppm result from this

work reduces the overall relative uncertainty on Gµ to 5 ppm.

While Eq. (2.9) is formally correct, there is an additional uncertainty contained in δ∆q/∆q, which is

attributed to the uncertainty on the mass of the muon neutrino, mνµ . This error comes from the phase

space calculation, which introduces an additional term of −8 m2νµ

m2µto ∆q(0). The best direct upper limit from

pion decay gives mνµ < 170 keV(C.L. = 0.9) [18], corresponding to a 10 ppm shift of Gµ from the case

of mνµ = 0. On the other hand, results from astrophysics constrains the total mass of all neutrinos to be

less than 1.8 eV [19], implying at most a 10−15 shift of Gµ. Given the astrophysical observations, it seems

reasonable to ignore the mass of the neutrinos and calculate Gµ assuming mνµ ∼ 0.

2.2 Comparison of Gµ to other determinations of GF

Care has been taken to avoid labeling Gµ as GF . The point is that GF is the assumed “universal” scale of

the interaction for any weak mechanism, while Gµ is the extracted value based on the muon decay process.

Other determinations of GF can be compared to Gµ to verify the universality of the weak interaction and in

some cases constrain theories that are beyond the standard model. Marciano [20] reviewed some of these

other evaluations of GF and the following are some of his ideas for comparing Gµ to other determinations

9

of GF .

A natural test is to compare the decay of the tau lepton (τ→ lνν(γ), l = e,µ) to obtain a tau based Fermicoupling constant, Gτl . By comparing the overall decay rate with the lepton branching ratios and mass of

tau, the coupling constants have been evaluated as

Gτe = 1.1666(28)×10−5 GeV−2 [2400 ppm] (2.13)

Gτµ = 1.1679(28)×10−5 GeV−2 [2400 ppm] (2.14)

The uncertainty of these values are significantly larger than Gµ, but as Marciano points out, they confirm

lepton universality of the weak interaction at the ±0.2% level. The agreement between the values can beused to constrain certain new physics models such as the mixing angle involving a heavy fourth generation

neutrino.

Alternatively, instead of a direct derivation of GF from a single measured quantity, other observables,

such as sin2 θW , MW , and mτ, along with the relevant radiative corrections can be combined to give in-

dependent determinations of GF . The details of several of these calculations can be found in Marciano’s

work [20]. Currently, the combinations presented have errors that are more than 200 times larger than the

error on Gµ from this work. The errors are dominated by uncertainties in the masses of the tau and Higgs

boson, the later being presently undiscovered. Future measurements of these masses will lead to better eval-

uations of GF . These comparisons, assuming consistency, will serve to further constrain some new physics

models such as fourth generation fermions and technicolor models.

2.3 Comparison of τµ+ to τµ−

According to the standard model, there should be no difference between the lifetime of free negative and

positive muons. A discrepancy between the two, would imply a CPT violation. For muons stopped in matter,

the two observed decay rates are not expected to be identical. In fact, this difference can be used to obtain

information about other physical parameters. The remainder of this section discusses the extraction of the

nucleon’s induced pseudoscalar form factor, gP, from the difference in lifetimes between the oppositely

charged muons.

The negative muon can form a Coulombic bound state with a positively charged nucleus. The simplest

10

case is muonic hydrogen, where a negative muon replaces the electron on a single proton nucleus. In this

bound state, the free decay rate is affected by nuclear muon capture,

p+µ−→ νµ +n (En = 5.2 MeV). (2.15)

This process has a characteristic rate, ΛS, for capture from the singlet hyperfine state, resulting in a second

disappearance mechanism for the muon. The muon capture reduces the observed negative muon lifetime:

1τµ−

= Λµ− = (Λfree +ΛS) , (2.16)

where Λfree = 1/τµ+ . Experimentally, there are additional modifications to τµ− resulting from a proton rich

environment, including a relativistic correction because the muon is in a bound system [21] and a difference

in the capture rate for muons in the molecular pµp state [22, 23, 24]. Since these are peripheral topics, they

are ignored here.

Within the standard model the matrix element for the muon capture rate can be factorized into leptonic

and hadronic V −A components:

M =−iGFVud√

2u(pνµ)γα(1− γ5)u(pµ)u(pn)[V α−Aα]u(pp). (2.17)

If second-class currents are ignored, the hadronic components can be written in terms of the four nucleon

form factors gV , gA, gM , and gP:

Vα = gV (q2)γα +

igM(q2)

2MNσαβqβ (2.18)

Aα = gA(q2)γαγ5 +

gP(q2)

mµqαγ5. (2.19)

In the above, Vud is the Kobayashi-Maskawa up-down matrix element, γα and σαβ = 12 [γα,γβ] are the Dirac

gamma matrices, and q is the momentum transfer. Because this is a monoenergetic process, q2 is fixed at

q2 =−0.88m2µ. With the exception of gP, most of the form factors are experimentally determined quite well

11

at this momentum transfer:

gV = 0.9755(5), gM = 3.5821(25),

gA = 1.245(4), gP = 8(4).(2.20)

These values can be combined with the appropriate coefficients to give their effect on the capture rate. These

coefficients, to first-order, and the resulting relative error on ΛS for each form factor can be expressed as

δΛSΛS

=

0.47 δgVgV = 0.024%, 0.15δgMgM

= 0.01%,

1.57 δgAgA = 0.5%, 0.18δgPgP

= 9%.

(2.21)

Thus, the dominant error on the theoretical extraction of ΛS is from gP, or conversely, a measurement of

ΛS provides a mechanism for determining gP. The MuCap collaboration [25] is currently preforming the

measurement of τµ− in a 10 bar protium vessel. The combination of their first result–the Ph.D. thesis of

UIUC student Steve Clayton—and the measurement discussed here, produce a value for the capture rate of

ΛS = (725.1±17.5) s−1. (2.22)

The extracted gP is then 7.3±1.1 [26]. Their final goal is a 7% relative uncertainty on gP.

12

Chapter 3

Review of pertinent muon physics

3.1 Muon production

The decay of the positive pi meson (pion) is the source of muons for the MuLan experiment and can be

expressed as π+ → l+ + νl , where l = e or µ. Based on phase space considerations, the pion decay to apositron, π+ → e+ + νe, would appear to be favored over the muon decay mode, π+ → µ+ + νµ; howeverthis is not the case. Two-body kinematics require the decay particles be emitted anti-parallel. Angular

momentum conservation along with the left-handed helicity state of the neutrinos imply that the charged

decay particle is likewise left handed. In the chiral limit (m∼ 0), the positive charged lepton is purely righthanded, resulting in angular momentum not being conserved. The positron mass, being much less than the

muon mass, results in a chiral suppression of that decay channel by an amount (me/mµ)2 ≈ 2.3× 10−5.1

Thus, the muon decay channel is dominant. Being a two-body decay, the kinetic energy and momentum of

the muon is trivial to calculate:

Kµ = 4.12 MeV (3.1)

pµ = 29.79 MeV/c (3.2)

A pictorial representation of the both pion decay methods is shown in Fig. 3.1.

Because of the spin-zero nature of the pion, the decay muons are emitted isotropically. Hence a pion

source produces an unpolarized muon distribution. The helicity discussion implies that for any positive

muon the spin is anti-parallel to the momentum. Thus, to an observer with a small solid angle view of a

pion source, the muons are highly polarized.

This is exactly the case for muons that are guided from a pion source to an experiment using a beamline.

1The ratio of pion branching ratios, BR(π→ eν)/BR(π→ µν)≈ 1.2×10−4, is larger than implied because of the phase spacesuppression of the muon compared to the positron.

13

+π=0+πS

+e

+eS

eν

eνS

(a)

+π=0+πS

+µ

+µS

µν

µνS

(b)

Figure 3.1: Pictorial representation of pion decay. The arrow extending from the decay particle representsthe momentum direction and the arrow beneath represents the particles spin. (a) The positron channel ishighly suppressed by the chiral limit. (b) The muon decay channel is favorable by a factor of 7792 over thepositron.

The beamline, having a limited angular acceptance of the pion source, will transport muons with a high

polarization. In particular, the highest polarization, P ≈ 1 is achieved if the beamline is tuned to transportparticles with momentum near 29.8 MeV/c from muons generated by pions at rest. These muons are called

“surface muons” because they originate from pions on the surface of a production target. Surface muon

beamlines have been known to produce muon polarizations as large as 97.85% [27].

3.2 Muonium

To perform a high-precision measurement of the free muon lifetime, it is desired that the muons be nearly

stationary to avoid complications from time dilation. Typically, this means the muons are stopped in a

target where the positive muon can form a Coulombic bound state with an electron, which is analogous to

a hydrogen atom and known as a muonium atom. The fraction of muonium formation is highly dependant

on the stopping medium, with nearly 100% muonium formation in some insulators and very little in most

metals. If a significant distortion of the decay rate arises from this bound state it would be imperative

to determine the fraction of muonium (as a function of time) for the stopping target. Fortunately for this

experiment, it has been shown that the lifetime of muonium is different from that of a free muon by only

≈ 0.6 parts per billion [28]. This evaluation of the muonium lifetime was motivated by the MuLan goalof δτµ/τµ = 1 ppm and a previous evaluation [29, 30] of the muonium lifetime that suggested a 484 ppm

14

difference. The arguments for such a small difference are presented in appendix A.

3.3 Differential decay spectrum

In Sec. 2.1 the total decay rate was used, but in the context of MuLan it is worth reviewing the differential

decay rate, dΓ. In terms of the Michel parameters ρ, η, ξ, and δ the decay rate as a function of the positron

energy fraction x = Ee/Emax and the angle between positron emission and muon spin, θ, is given by

d2Γdx d cosθ

= 4Γ0x2{

3(1− x)+ 2ρ3

(4x−3)+3ηx0 1− xx +ξcosθ[

1− x+ 2δ3

(4x−3)]}

. (3.3)

The maximum positron energy is given by Emax = (m2µ + m2e)/(2mµ) ≈ mµ/2 = 52.83 MeV. The quantity

x0 ≡ me/Emax is essentially twice the ratio of the electron to muon mass. In the standard model, the Michelparameters are

ρ = 34 , η = 0,

ξ = 1, δ = 34 .(3.4)

Using these values the differential decay rate simplifies to

d2Γdxd cosθ

=Γ02

n(x)(1+a(x)cosθ) , (3.5)

where the normalized positron energy spectrum is

n(x) = 2(3−2x)x2, (3.6)

and the amplitude of the angular dependance, the asymmetry, is

a(x) =2x−13−2x . (3.7)

Fig. 3.2(a) shows the n(x) curve. The distribution indicates a range of positron energies, the most

probable being at the maximum energy. The asymmetry as a function of positron energy is shown in

Fig. 3.2b. This indicates that the asymmetry is maximal for the highest energetic positrons. Notice that

the asymmetry is negative for x < 0.5 and positive for x > 0.5. Thus, the positron is emitted preferentially

15

Positron Energy [x]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

No

rmal

ized

Dec

ay P

rob

abili

ty

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a)

Positron Energy [x]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Asy

mm

etry

-0.2

0

0.2

0.4

0.6

0.8

1

(b)

Figure 3.2: (a) Normalized decay probability, n(x), as a function of normalized positron energy. (b) Asym-metry, a(x), with respect to the spin vector as a function of normalized positron energy.

in the same direction as the muon spin when the kinetic energy of the positron is greater than 26.15 MeV,

but anti-parallel when the kinetic energy is less than that value. Integrating with energy weighting, gives an

average asymmetry of

〈a〉=∫ 1

0n(x)a(x)dx =

13. (3.8)

The polar probability plot is given in Fig. 3.3. This figure shows the decay probability contours for equal

energy positrons, with the radius indicating the relative decay probability and the angle is identical to θ.

There are several features evident from this figure. First, the highest energy (x = 1) positrons have zero

probability of decaying anti-parallel to the muon spin. Second, it is again obvious that the energy midpoint

(x = 0.5) is the only energy with isotropic decay, the lower energies decaying preferentially in the anti-

parallel direction.

To give a qualitative understanding to the asymmetry function, a few limiting cases can be reviewed.

First, consider the case of maximal positron energy. This occurs when both neutrinos are emitted in the

same direction. The neutrinos are of definite helicity states with the νµ being right handed and the νe

being left handed. The net angular momentum carried by the neutrinos is zero. This requires the spin of

the positron to match the original spin of the muon. Because the energy of the positron is much greater

than its mass, this situation approaches the chiral limit. Therefore, the highest probability of decay will

have the positron momentum in the same direction as its spin, in this case in the direction of the muon

spin, as represented in Fig. 3.4(a). Next, consider the lowest-energy positron. As the positron momentum

16

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

99.4

99.6

99.8

100

100.2

100.4

100.6

x=0.25

x=0.50

x=0.70

x=0.85

x=1.00

S

e+

θ

Figure 3.3: Isoenergetic decay probabilities shown in polar coordinates with the θ as the angle and decayprobability as the radius.

+µ

+µS

+e

+eS

µν

µνS

eν

eνS

(a)

+µ

+µS

+e

+eS

µν

µνS

eν

eνS

(b)

Figure 3.4: Pictorial representation of muon decay showing the chiral limit spin dependance. The arrowextending from the decay particle represents the momentum direction and the arrow beneath representsthe particles spin. (a) Highest positron energy limit where the direction of the positron is parallel and theneutrinos are anti-parallel to the muon spin. (b) Lowest positron energy limit where the two neutrinos areemitted anti-parallel to each other with the νµ in the direction of the muon spin. The positron is preferentiallyemitted in the same direction as the νe.

17

approaches zero, the neutrinos must emerge in opposite directions with equal energy. Because of the helicity

of the neutrinos, their net angular momentum is 1 To match the initial angular momentum of the muon, the

positron spin must then be in the direction of the νe. Again, in the chiral limit (much less justified here) the

positron momentum will be anti-parallel to the initial muon spin. This example is represented pictorially in

Fig. 3.4(b).

3.4 µSR effects

The discussion has focussed on the decay probability for a single muon. When considering an ensemble of

muons, the average polarization vector ~P must be considered instead of the spin of the single muon. The

consequence is that the cosθ term of Eq. (3.5) is replaced by PcosΘ, where P = |~P| is the magnitude of thepolarization and Θ is defined to be the angle between the polarization and decay direction.

Often the polarization is time dependent. For a positron detector at a fixed angle relative to the stopped

muons, the changing of the polarization results in a modification of the decay positron spectrum. In a

precision measurement of the lifetime, this can result in a distortion of the signal and thus an incorrect

determination of the lifetime. The methods by which the polarization can change are discussed in this

section. In general this field of study is called µSR for “Muon Spin Rotation/Relaxation” [31]. A complete

review of µSR is beyond the scope of this discussion and only a brief description with the consequences on

a precision measurement of the muon lifetime are presented.

3.4.1 Rotation

The rotational component of µSR results in a change of the polarization direction but not the magnitude.

The Larmor precession of the muon spin in a local magnetic field, B, has an angular frequency given by

ω = gµeB

2mµc. (3.9)

The gyromagnetic ratio, gµ, has been measured to the incredible precision of ≈ 0.6 ppb [32], but for thiscontext it is sufficient to approximate it by gµ = 2, which is correct to nearly one part in 800. This implies

a frequency of f = ω/2π≈ B ·135 MHz/T. A magnetic field of only 3.4 mT will produce a rotation periodequal to the muon lifetime and for the earth’s magnetic field (≈ 50 µT), the rotation period is nearly 150 µs.

18

Imagine recording the time of the decay positrons relative to the stopping time for muons in a magnetic

field using a detector at a fixed angle relative to the initial muon spin. The expected decay curve f (t) will

be modulated by the Larmor precession, complicating the extraction of the lifetime:

f (t)→ f (t)(1+aPcos(ωt +φ)) . (3.10)

The asymmetry, a, is the average asymmetry of a(x) and is dependant on the detector energy resolution. The

phase φ serves to correct for the position of the detector relative to the initial polarization and magnetic field.

Consider the effect of a 3.4 mT field, see Fig. 3.5(a). The oscillation of the number of counts significantly

distorts the lifetime spectrum in a measurable manner. A more troubling case is an unobserved oscillation

from a lower magnetic field, such as one close to the earth’s field, see Fig. 3.5(b). In that case, the oscillation

can result in a undetected shift of the measured lifetime.

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-410

-310

-210

-110

1

(a)

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-410

-310

-210

-110

1

(b)

Figure 3.5: The relative decay rate for a detector with a limited angular resolution for a muon in no magneticfield, compared to that of muon in a particular magnetic field. The black line shows the unperturbed decayprobability. (a) For a magnetic field of 3.4 mT the oscillation is obvious and detectable. (b) For a magneticfield of 100 µT the oscillation is too long to be easily detected, but nevertheless distorts the function resultingin an incorrect determination of the lifetime.

One important point is that the rotation effect can be canceled by summing the counting rates of two

identical detectors that are collinear with the muon source located at the midpoint between them. Configured

in this way, the phase and amplitudes are exactly matched so that the summed count rate returns to a

pure exponential. Fig. 3.6 illustrates this cancelation by showing the relative measured event rates for two

identical detectors oriented in this manner. Fig. 3.6(c) shows the difference, which is used to expose the

19

µSR signatures.

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-410

-310

-210

-110

1

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-410

-310

-210

-110

(a)

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-410

-310

-210

-110

1

(b)

s]µtime [0 2 4 6 8 10 12 14 16 18 20

-0.2

-0.1

0

0.1

0.2

0.3

(c)

Figure 3.6: The relative decay rates for two identical detectors having equal solid angle coverage, butopposite view points of a muon in a 7.0 mT magnetic field. (a) The two curves shown separately. (b) Thetwo curves added together, having no rotation component. (c) The two curves subtracted, maximizing therotation effect.

3.4.2 Relaxation

Relaxation usually refers to a decrease in the magnitude of the polarization, but this phenomenon also

includes methods which allow a polarization magnitude to increase with time. Relaxation is typically de-

scribed by two time constants known as T1 and T2. The reduction in polarization that is independent of

the magnetic field and is material dependant is called T1. It is assumed to be dominated by spin exchange

collisions within the medium. In an inhomogeneous field, the spins precess at different rates, reducing the

20

magnitude of the average polarization; this effect gives rise to the T2 term. The rotation effect is still present

with the frequency given by the average field. A graphical representation of both relaxation components is

shown in Fig. 3.7. Just as for the rotation component, we can consider the effect on a detector from each

s, P = 1.000µt= 0 s, P = 0.607µt= 1 s, P = 0.368µt= 2 s, P = 0.223µt= 3 s, P = 0.135µt= 4 s, P = 0.082µt= 5 s, P = 0.050µt= 6

(a)

s, P = 1.000µt= 0 s, P = 0.956µt= 1 s, P = 0.830µt= 2 s, P = 0.645µt= 3 s, P = 0.434µt= 4 s, P = 0.246µt= 5 s, P = 0.164µt= 6

(b)

Figure 3.7: The polarization as a function of time being reduced by the two relaxation terms, T1 and T2, withtime running along the x-axis between subfigures. (a) The longitudinal relaxation component, T1 results ina reduction in polarization magnitude with no change in direction. Here represented by an exponentiallydecreasing function with a 2 µs relaxation constant. (b) The magnetic field dependant relaxation component,T2, changes both the magnitude of polarization with the rotation given by the mean field. In this examplethe individual spin probabilities (shown with black arrows) of 5 muons sum to give the polarization vector(shown by the red arrow).

of these relaxation components. In general, the functional form for the relaxation is unknown and thus the

magnitude of polarization is replaced by an explicitly time-dependant version P → P(t) and the frequencyof oscillation is given by the mean frequency ω→ 〈ω〉 to give a more general form

f (t)→ f (t) [1+aP(t)cos(〈ω〉 t +φ)] . (3.11)

3.4.3 Initial polarization reduction

In addition to the change of polarization with time, the initial polarization is often reduced from unity when

the muon comes to rest in a material. The exact mechanism of depolarization is unknown, but it is believed to

be a consequence of the muon passing through multiple local fields while slowing to rest and it is associated

with the repeated transition into and out of the muonium state. A list of a few materials and the residual

muon polarization is given in table 3.1.

21

Material Residual PolarizationGraphite 1.00±0.03Silicon 1.10±0.05Aluminium 0.91±0.04Polystyrene 0.31±0.04Sulfur 0.06±0.05

Table 3.1: Measured residual muon polarization after coming to rest in various materials. The data arenormalized to the observed asymmetry in graphite and hence can be greater than 1 [33].

The muonium atom reduces the polarization for several reasons. First, the precession frequency is

almost 103 times larger than the free muon. Second, the precession direction is opposite to that of the free

muon. Third, transitions to the triplet hyperfine state effectively remove half the polarization.

22

Chapter 4

Methods for measuring τµ+

Traditionally there have been two very different methods used to measure the muon lifetime. The first is

the DC method utilizing a continuous beam of incoming muons, and the second is the pulsed method with

a periodic muon source. Reviewed in this chapter are two experiments, an older and a recent measurement,

for each of these methods, with a discussion of the

1. experimental method and apparatus,

2. effective muon collection rate,

3. limiting systematic effects, and

4. results or current status.

4.1 DC method

The DC method utilizes a continuous incoming muon beam and requires the association of each recorded

decay positron to the muon from which it originated. Thus it is necessary to record the arrival time of

each muon in addition to the decay time. If more than one muon is present, the combinatorics artificially

increase the background and reduce the sensitivity of the experiment. This hurdle can be circumvented in

two ways. The first is to run at a beam rate that is low enough to have, on average, the interarrival time

between muons longer than the desired decay measurement period after a muon arrival. The second is to

have a segmented and active target so that multiple decays can be registered with minimal muon pileup in

one detector segment.

In the DC method, the effective measuring rate is determined by the stochastic beam rate, R, and the

measuring period TM. To avoid the confusion resulting from multiple muons in a target, data are rejected

if another muon arrival is recorded within ±TM of any other muon. We can then calculate the fraction of

23

accepted events by first noting that the normalized probability of having two particles separated by a time

∆t is given by

P(∆t) = Re−R∆t . (4.1)

The fraction of events for which the next muon arrives after a fixed time TM is computed as

f =∫ ∞

TMP(t)dt = e−RTM . (4.2)

The fraction of events for which the previous muon came before the measurement period is identical to the

value of f above. Thus, the effective rate is reduced to

Reff = R f 2 = Re−2RTM . (4.3)

Differentiating this with respect to R allows for the determination of the maximum effective rate

Rmaxeff =1

2TMe−1 for R =

12TM

. (4.4)

The calculation of the effective rate assumes all decay positrons are detected. In any real experiment, this

value is reduced by the energy dependance and solid angle coverage of the detector. For example, suppose

TM = 22 µs≈ 10× τµ+ , then the optimal beam rate is 22.7 kHz with a maximum effective collection rate of8.4 kHz. Using any other beam rate will produce a slower data collection rate. If the detector is incapable

of measuring positrons with energies below 10.5 MeV, x = 10.5 MeV/52.3 MeV = 0.2, the effective rate is

reduced by the multiplicative factor

ε =∫ 1

0.2n(x)dx = 0.986. (4.5)

4.1.1 TRIUMF experiment

In 1981 a measurement of the positive muon lifetime was carried out at the “TRI-University Meson Facility”

(TRIUMF) [5, 34]. The time of the decay positron relative to the stopping of a pion in a water-filled container

was measured. The use of a stopped pion beam is desired because the isotropic decay of pions nullifies the

spin dependance inherent in muon decay, resulting in an isotropic positron distribution in the detector.

Furthermore, the pion lifetime, τπ ≈ 26 ns, is sufficiently short so that delaying the recording of positrons

24

by 350 ns ≈ 11.5 τπ after a recorded pion stop ensures that any detected positron originated from a muon.Pions with momenta between 150 and 170 MeV/c, and a rate of 5 to 30 kHz, were transported into the water

tank. The incoming particles were recorded by two plastic scintillation counters. The Čerenkov radiation

produced by the decay positron in the water was registered by two photomultiplier tubes (PMT). The water

was doped with a waveshifter to transition the Čerenkov light to a frequency more suitable to the PMTs

and also to reduce any directionality associated with the decay positron. A diagram of the experimental

apparatus can be seen in Fig. 4.1. Given TM = 20 µs used in this experiment, the optimal beam rate was

Figure 4.1: The experimental setup for the TRIUMF muon lifetime measurement. Beam particles weredetected by scintillator counters S1 and S2. Čerenkov radiation from decay positrons in the water werewavelength shifted and registered by the photomultiplier tubes L and R (Figure from Giovanetti et al. [5]).

25 kHz with an effective measurement rate of ∼ 9.2 kHz. They reported the beam rate as varying between5 and 30 kHz as a result of accelerator fluctuations. The combination of energy resolution and solid angle

coverage reduced the effective rate by about 5 %.

This experiment was ultimately limited by statistics, but several potential systematic effects were also

25

investigated. These included time varying background, µSR effects from beam muons, and dead time effects.

All of these effects were found to be negligible compared to the statistical uncertainty.

The final value for the TRIUMF experiment was

τµ+ = 2196.95 (6) ns [27 ppm]. (4.6)

4.1.2 FAST experiment

A variation on the DC experiment has been approved at PSI [35], based on an active segmented target that

allows the effective rate to be increased by a factor proportional to the number of independent stopping loca-

tions. The FAST method is to use a diffuse pion beam passed through a wedge shaped degrader to stop pions

at varying depths in a plastic scintillator fiber array. The scintillator array is made of 4 mm× 4 mm fibersimplanted with waveshifters and readout with position sensitive photomultiplier tubes. The experimental

design, illustrating the estimated resolution, is shown in Fig. 4.2. The use of the pion stopping gradient

(a) (b)

Figure 4.2: The FAST detector showing the wedged degrader and scintillator array. (Figure from FASTcollaboration [35])

.

along with position sensitive recording stops and decays is estimated to allow an average of 30 pion stops

with a 30 µs measurement period at any given time. Thus, the optimal beam rate can be determined from

Eq. (4.4) with the multiplicative factor of 30 to be R = 500 kHz. This beam rate and measurement length

then imply an effective collection rate of 184 kHz, which is 20 times faster than the TRIUMF experiment.

The combination of detector resolution and solid angle are not expected to significantly impact the collection

26

rate.

The FAST collaboration has investigated several possible systematic errors including effects from µSR ,

time dependant detector efficiency, time dependant backgrounds, and dead time. They have concluded that

all known effects are below their stated statistical goal of 1 ppm [36]. They are currently analyzing data and

it is hoped they will present a comparable measurement to the one reported here shortly.

4.2 Pulsed method

An alternative method for measuring the muon lifetime is through a pulsed beam having a fixed duration

and periodicity, which is used to implant multiple muons into a target during a short accumulation period,

TA. The decay times of the positrons are measured as a function of time after the accumulation period during

a measurement period, TM. By definition, the time between beam bursts, TB, is greater than or equal to the

sum of TA and TM. The pulsed beam method is inherently different from the DC method because of the

desire for more than one muon in the target at a time to increase the effective rate. This method is also

known as the “radioactive” method because it resembles a radioactive source with the number of muons

decreasing during the measurement time. A benefit over the DC method is the potential for higher effective

rates with the simultaneous measurement of τµ+ from multiple muons. The largest undesired consequence

is a pileup effect where multiple muons decay close enough in time that only one positron is detected.

In the pulsed method, the calculation of the effective rate is qualitatively different than for the DC

method and is determined from TA, TM, and TB. During the accumulation period, the number of muons in

the target is increasing from the constant incoming beam rate R, but also decreasing from muon decay. The

number of muons in the target can be expressed in the same manner as a charging capacitor:

dN(t)dt

= R−N(t)/τ. (4.7)

This has the familiar solution of

N(t) = Rτ(1− e−t/τ). (4.8)

For the measurement period, there is no source term and therefore the number of muons in the target simply

27

decreases with time:dN(t)

dt=−N(t)/τ. (4.9)

This has the simple solution of

N(t) = N0e−t/τ, (4.10)

where N0 = Rτ(1− e−TA/τ) guarantees a match of the amplitude at the end of the accumulation period.Eqs. (4.8) and (4.10) can be combined to give the total number of muons in the target as a function of time

as:

N(t) =

N0exp(TA/τ)−exp(−t/τ)

exp(TA/τ)−1 for −TA ≥ t < 0N0 exp(−t/τ) for 0 < t ≤ TM

, (4.11)

The effective measuring rate is given by the number of decays during the measurement period divided by

the cycle period:

Reff = Rτ(1− e−TA/τ)(1− e−TM/τ)/(TB. (4.12)

Once again, the energy dependance and solid angle coverage of the detector will reduce this rate from

the nominal value. As an example, consider TA = 5 µs, TM = 22 µs, and TB = TA + TM. Then, Reff =

0.073R, which requires an instantaneous beam rate of 126 kHz to match the collection rate of the TRIUMF

experiment.

It is important to emphasize that Reff is proportional to the beam rate. Unlike the DC measurements,

where the optimal beam rate is determined to avoid muon pileup, here the optimal rate is unrestricted.

4.2.1 Saclay experiment

The 1981 Bardin et al. [6] experiment used the Saclay linear accelerator to deposit pions into a sulfur target

during a beam burst of 3 µs every ∼ 333 µs, a 1 % duty cycle. The pions promptly decayed to muons andthe positrons from the muon decays were detected by a nearly cylindrical array of six scintillator telescopes.

The use of sulfur was motivated by an order of magnitude reduction in initial muon polarization. The

experimental setup is shown in Fig. 4.3. A measurement period of 65 µs≈ 29× τµ+ was used to histogramthe positron times relative to the end of the beam burst. Even with this very long measurement time, the

experiment sat idle for more than 80% of the time because of the low 3 kHz repetition rate of the accelerator.

After accounting for the solid angle coverage, Ω = 0.75, the effective rate for this experiment was Reff =

28

Figure 4.3: The experimental setup for the Bardin et al. muon lifetime measurement. Shown in the lateralview is the lead collimators, sulfur target, and plastic scintillator telescopes. The axial view shows the sulfurtarget surrounded by the six scintillator telescopes. (Figure from Bardin et al. [6])

3.7× 10−3R. They ran at two beam rates of 0.49 MHz and 2.45 MHz, giving effective measurement ratesof 1.8 kHz and 9.0 kHz, quite similar to the TRIUMF experiment.

Pileup was the largest systematic issue for this experiment. Because of the electronics available at the

time, only the first decay positron in a measurement period could be recorded per scintillator tile. Thus,

subsequent decays were missed, causing the measured lifetime to be too low. The effect of these missed

events was minimized by creating a circuit that rejected fills where more than one hit was detected during

the gate. Unfortunately, the dead time associated with the electronics prohibited identifying all fills with

multiple hits and therefore a rate dependance remained. The effect can be seen in Fig. 4.4 where the mea-

sured muon lifetime as a function of fit start time is shown. Since the probability of missing an additional

event is proportional to the instantaneous rate squared, the measured value for the lifetime converges to the

true value with a characteristic lifetime given by τµ+ . By extrapolating their results as a function of fit start

time, the Saclay experiment claimed a value of

τµ+ = 2197.078 (73) ns [33 ppm]. (4.13)

4.2.2 RIKEN-RAL experiment

A much more recent measurement has taken place at the RIKEN-RAL Muon facility [37]. This measure-

ment uses a high-intensity muon source with a repetition rate of only 50 Hz. They utilize a surface muon

beam with nearly 100% polarization. A paramagnetic holmium target, with a polarization relaxation con-

stant of 500 ns, is used to stop and depolarize the muons. Two sets of multi wire proportional chambers

(MWPC), placed symmetrically around the stopping target, create a segmented positron detector with 192

29

Figure 4.4: Results from the Bardin et al. experiment showing the measured lifetime as a function of fit starttime. The trend towards longer lifetimes is a consequence of positron pileup. (Figure from Bardin et al. [6])

individual elements. A diagram of the experimental setup is shown in Fig. 4.5. Each pulse delivers between

Figure 4.5: The setup of the RIKEN-RAL measurement of the muon lifetime. During each fill 104 to 105

muons are stopped in a holmium target and the decay times of the emitted positrons are registered using twosets of MWPC. (Figure from D. Tomono[37])

104 and 105 muons into the target region over a short 320 ns double muon burst. These high initial rates

result in maximal effective rates between 50 kHz and 500 kHz. But the MWPCs have a solid angle coverage

of Ω≈ 6%, reducing the effective rate substantially to at most 30 kHz.Unlike the Saclay measurement, the times of all hits during a measurement period can be recorded.

Thus, instead of rejecting measuring periods with multiple events in any given detector element, all events

30

Year Reported Measurement Value [ns] Uncertainty [ns] Reference1962 2203 2 [38]1963 2202 3 [39]1963 2197 2 [40]1972 2200.26 0.81 [41]1973 2197.3 0.3 [8]1974 2197.11 0.08 [7]1984 2196.95 0.06 [5]1984 2197.08 0.07 [6]

Old Average 2197.03 0.04 [1]2007 2197.013 0.024

New Average 2197.019 0.021

Table 4.1: List of previous precision lifetime values and uncertainties. The world average prior to this workis listed, as well as the new measurement from this work an the new world average.

can be recorded and used in the analysis. A rate dependent systematic error, associated with missed positrons

because of detector dead time, is the dominant systematic error. This error is dependent on the fit start time,

tstart, decreasing with an exp(−tstart/τ) form (more on this topic can be found in Sec. 9.3). Additionally,the statistical error increases with fit start time with an exp(tstart/τ) form. By varying the fit start time, an

optimal start time can be obtained that minimizes the overall error. For the RIKEN-RAL experiment, the

optimal fit start time is almost 1.6× τµ after the beam burst. The error associated with the dead time is thecurrent limiting systematic error.

At this time, the RIKEN-RAL experiment has released a preliminary (unpublished) lifetime value of

τµ+ = 2197.01 (11) ns [51 ppm]. (4.14)

They claim to be statistically limited and hope to obtain a 10 ppm measurement, which will be a direct

confirmation of the measurement presented here.

4.3 Results of previous τµ+ measurements

Earlier τµ+ measurements typically used the DC method. The previous measurements, including the ones

described above, are summarized in table 4.1. The uncertainty on τµ+ as a function of year is shown in

Fig. 4.6.

31

Year1960 1970 1980 1990 2000 2010

[n

s]+ µτ δ

-110

1

Figure 4.6: The uncertainty on the muon lifetime as a function of year including the improvement based onthe results reported here.

32

Chapter 5

MuLan method

The MuLan experiment is a pulsed experiment taking place in a DC surface muon beam. The time structure

is artificially imposed through the use of an electric kicker, with the ability to adjust the length of the

accumulation and measurement periods. A judicious choice of these periods results in efficient use of the

beam, and with TB ≡ TA + TM , there is no idle time between cycles, thereby improving the collection ratecompared to the pulsed beam method, where idle times greater than 99% occur.

When the cycle period is equal to TA + TM, the effective rate Eq. (4.12) for the pulsed method has a

maximum of (Rmaxeff = 0.203R) for TA = TM = 1.256τ ≈ 2.76 µs. However, the electric kicker is limited toa cycle frequency of 75 kHz, almost 2.5 times lower than the ideal frequency. The desire to observe the

background and a larger fraction of the decay curve resulted in the choice of TM = 22 µs ≈ 10τ, resultingin a best effective rate of 0.0734R for TA = 2.61τ = 5.7 µs. In practice, the R04 data set was taken with

TA = 5 µs, insignificantly reducing the effective rate to 0.0731R. The typical beam rate was 2 MHz, and

the detection efficiency is 64%, resulting in a data collection rate of 93500 events/s. In the R06 data set, a

higher beam rate of 10 MHz implies more than 46500 events/s. A histogram showing the positron count as

a function of time in the kicker cycle is seen in Fig. 5.1.

5.1 Systematic concerns

The MuLan experiment was designed to measure τµ+ with a relative uncertainty of 1 ppm, through the

detection of more than 1012 muon decays, and thus systematic errors must be kept below 1 ppm. The

systematic issues that influenced the design of the experiment are reviewed in this section. These and other

systematic errors related to the measurement presented here are discussed in more depth in chapter 9.

33

Time relative to kicker transition [ns]-5000 0 5000 10000 15000 20000

Co

un

ts p

er 4

2 n

s

610

710

810

Kic

ker

Tra

nsi

tio

n

Background Level

MMeasurement Period, T

AAccumulation Period, T

Figure 5.1: Sample of data from the MuLan experiment illustrating the accumulation and measurementperiods. The background is dominated by the incomplete extinction of the beam.

5.1.1 µSR effects

A polarization change from µSR effects can result in a distortion of the decay time distribution for a detector

viewing a subsection of the solid angle, potentially resulting in an incorrect determination of τµ+ . Since,

a surface muon beam with nearly 100% polarization was used, this problem was a major concern in the

design of the experiment, resulting in three different efforts to reduce the effect. First, the polarization is

reduced by a dephasing mechanism during the accumulation period. Second, the target is chosen to reduce

the polarization at the start of the measurement period. Third, the detectors are arranged symmetrically

around the stopping target.

In the R04 data set, two different target assemblies were utilized. The first was a material called

ArnokromeTMIII (AK-3) [42], a ferromagnetic metal with a high internal magnetic field of about 400 mT.

The second was a sulfur disc with a mean applied magnetic field of 13 mT. These assemblies are discussed

in more detail in Sec. 6.6, but for now only the magnetic field is relevant. The fields were arranged to be

perpendicular to the initial polarization vector so that maximal precession amplitude is achieved. Similar to

the T2 relaxation component of µSR from field inhomogeneity, the stochastic arrival of the muons in a fixed

magnetic field reduces the polarization during the accumulation period. This effect, known as dephasing,

results in a reduction of the polarization given by the expression (see appendix B for the details of this

34

calculation):

P(TA) = P0

√(cosh

(TAτ)− cos(TAω)

)csch2

(TA2τ

)

2τ2ω2 +2. (5.1)

Using this formula, the polarization magnitude at the beginning of the measurement period can easily be

calculated. For P0 = 1, the polarization as a function the accumulation length is shown in Fig. 5.2 for the

two magnetic fields. With the 5 µs accumulation period the depolarization factor for the sulfur and AK-3

1 2 3 4 5 6

0.005

0.01

0.05

0.1

0.5

1

|P|

Time [microcsecond]

TA

Time [microsecond]

(a)

1 2 3 4 5 6

0.005

0.01

0.05

0.1

0.5

1

|P|

Time [microcsecond]

TA

Time [microsecond]

(b)

Figure 5.2: The magnitude of polarization as a function of accumulation length. The reduction is a resultof dephasing in a fixed magnetic field. (a) Dephasing for a 13 mT field as found in the sulfur target. (b)Dephasing for a 400 mT field as found internally in the AK-3 target.

targets was 0.045 and 0.0013, respectively.

The material-dependant reduction of polarization is also used to reduce the initial polarization. The

sulfur target was motivated by the known depolarization in this material. A measurement, performed by us,

showed that relative to a silver target, a polarization preserving material, sulfur has a residual polarization of

AS/AAg ≈ 9 percent, see Fig. 5.3. On the other hand, the AK-3 target, being metallic and resisting muoniumformation, is not expected to cause any reduction to the initial polarization.

A symmetric detector, discussed in Sec. 6.7, also helps to further reduce polarization effects. For each

detector element, another complementary detector element is positioned oppositely with respect to the tar-

get. Summing two identical detectors that are exactly point symmetric in this manner, results in the exact

canceling of any time dependance to the polarization. The matching of detector efficiencies and precise

positioning greatly reduce the magnitude of polarization changes. The relative efficiency is estimated to

further reduce the effective polarization by a factor of 2×10−2.

35

Figure 5.3: Asymmetry plot showing the reduction in initial polarization of sulfur compared to Silver. Theasymmetry is computed as A = F−BF+B where F and B refer to the count rates of detectors on opposite sides ofthe stopping target.

The combination of these three factors serves to reduce the effective initial polarization to a value of

8×10−5 for the sulfur target and less than 3×10−5 for the AK-3 target. Additional polarization reductionmay come from the relaxation terms T1 and T2 during the accumulation period. The effect on the lifetime

from a residual polarization is expected to be less than 0.5 ppm.

5.1.2 Dead time effects

Dead time effects refer to the consequences of missing events because of dead time associated with the

detector system. Typically, these effects are known as “pileup”, since they are associated with events that

pileup with one or more being uncounted. Consider the simple case of “double” pileup where only one

event is uncounted1. The probability per unit time of detecting a single hit in any detector is given by the

probability distribution

P(t) = N(e−tτ +X) (5.2)

where X = BN is the ratio of the background level to the counts at t = 0, which can be determined empirically

for each detector. The values are constrained by normalizing the distribution to 1 over the measurement

period, ∫ Tmeasure0

P(t)dt = 1. (5.3)

1Higher order pileup, with more than one unaccounted event, is a straight forward extension of the shown formulas.

36

The probability per unit time to have a second event following a first at time t within a dead time given by

∆T is simply

P2(t) =∫ t+∆T

tP(t)2dt

=τN2

2(1− e− 2∆Tτ )e− 2tτ +

2N2Xτ(1− e− ∆Tτ )e− tτ +

N2X2∆T

≈ N2∆T(

e−2tτ +2Xe−

tτ +X2

), (5.4)

giving the time distribution of missed events. The second and third terms have no affect on the extraction

of the lifetime because they have the same functional form as the single decay. The first term, falling as

exp(−2t/τ), will increase the extracted lifetime if it is not included in the fit function or if the histogramis not corrected to account for pileup. Pileup is dependant on the amplitude, N, and dead time, ∆T . By

reducing these terms, the systematic error can be reduced.

The pileup effect is graphically illustrated in Fig. 5.4. The first figure shows the lifetime exponential

and a pileup exponential having a characteristic constant 1/2 that of the lifetime. The second figure shows

the lifetime exponential along with the histogram generated by subtracting the pileup exponential. At early

times, there is a larger deficit of counts than at later times, resulting in the fit to this distribution giving a

lifetime that is longer than the true lifetime.

The amplitude N is proportional to the instantaneous rate at the beginning of the measurement period,

and should be as large as possible to increase the data collection rate. By splitting the detector into m indi-

vidual elements, the coefficient in the pileup probability is reduced by a factor of m−2, while the amplitude

for the sum of all detectors remains the same. In this experiment the detector system is divided into 170

individual detector elements, reducing the pileup factor from that of a single detector by a factor of 28900.

5.1.3 Time stability of the background

A systematic error arises if the background has a time dependance. Previous pulsed experiments had time

varying backgrounds as a result of neutron beta decay (and other long-lived nuclear states). These particles,

which were produced during the burst period, have lifetimes that are much longer than TM, creating the

37

s]µtime [0 1000 2000 3000 4000 5000

Co

un

ts [

au]

-210

-110

1

Lifetime Exponential

Pileup Exponential

(a)

s]µtime [0 1000 2000 3000 4000 5000

Co

un

ts [

au]

-110

1

Lifetime exponential

Pileup affected histogram

(b)

Figure 5.4: The effect of pileup is shown with (a) giving example lifetime and pileup exponentials and (b)showing the lifetime exponential and pileup affected histograms. The reduced counts at early times resultsin the measured lifetime being longer than the true lifetime.

observation of a background level that decreases with time. If this were to occur and not explicitly included

in the fit function, the measured lifetime would be lower than the real lifetime. In MuLan , the cycle

time is independent of the cyclotron time, and therefore particles with long lifetimes have no effect on the

measurement.

The greater concern for the MuLan experiment is a time varying background associated with kicker

instabilities. The background is directly proportional to the kicker voltage. To minimize this effect, the

kicker was designed to have a quick switching mechanism, with a time constant less than 45 ns and a

variation over the measurement period of less than 3.6 V on the applied 25 kV.

38

Chapter 6

Experimental overview

6.1 Accelerator facility

The MuLan experiment takes place at the Paul Scherrer Institut (PSI) [43] utilizing muons produced from

the proton accelerator facility. Protons are accelerated using a Cockcroft-Walton accelerator with energies

of 870 keV into an injection cyclotron producing a ”high intensity, high quality” beam of 72 MeV protons.

The protons are further accelerated to 590 MeV in the primary ring cyclotron operating at 50.63 MHz with

a current of up to 2 mA. The proton beam is collided with two carbon production targets with designa-

tions of M (for mince, French for thin) and E (for epaisse, French for thick) with lengths of 5 mm and

40 mm respectively. Charged particles produced at the production targets are magnetically channeled to

seven experimental areas, two utilizing the M target and five utilizing the E target. The proton accelerator

components are shown in Fig. 6.1, the production targets are shown in Fig. 6.2, and the layout of the proton

facility is shown in Fig. 6.3.

6.2 πE3 beamline

The πE3 beamline, as shown in Fig. 6.4, accepts muons produced at the E production target. The muons

are gathered by a quadrupole triplet and then guided 5 m above the production position by two 60◦ dipoles,

operating in opposing directions. Between the dipoles, a system of slits and quadrupole doublets constrain

the momentum phase space, resulting in a relative momentum acceptance of∆p

FWHMp ≈ 2%. Following the

second dipole, a quadrupole doublet produces a nearly parallel beam for more than 2 m, allowing the muons

to pass first through a velocity selecting−→E ×−→B separator and then through an electrostatic kicker. After the

kicker, two sets of quadrupole triplets and a slit are used to focus and collimate the muon beam to a 1.2 cm

by 0.5 cm spot at the center of the experimental area.

39

(a) (b)

(c)

Figure 6.1: Images of the PSI proton accelerator facility: (a) the 870 keV Cockcroft-Walton accelerator, (b)the 72 MeV injector cyclotron, and (c) the 590 MeV primary ring cyclotron. (Figures from PSI [43].)

A measurement of the muon flux at the target as a function of the beam momentum was performed by us

and the results are shown in Fig. 6.5. It is clear from this figure that the maximum rate occurs near the muon

momentum of 28 MeV/c. The experiment was operated at a slightly higher momentum of 28.8 MeV/c to

reduce the number of muons that stopped between the evacuated beamline and the muon stopping target, a

distance of about 40 cm.

6.3 Kicker

The electrostatic kicker [44] is one of the crucial elements of the beamline. This device, developed explicitly

for the MuLan experiment, creates the artificially pulsed time structure from the continuous muon beam. The

40

(a) (b)

Figure 6.2: The production targets in the proton beamline: (a) the M target and (b) the E target. Each targetis rotated at 1 Hz. (Figures from PSI [43].)

kicker consists of two pairs of electrode plates, biased to produce a potential difference of up to VK = 25 kV,

with a virtual ground at the midplane. Each plate is 75 cm long and 20 cm wide. The electric field created

by the 15 cm separation between each pair deflects the muons over the 1.5 m long region. The plates are

charged to high voltage, or discharged to ground, in less than 45 ns through the use of four MOSFET based

modulators operating in a push-pull mode. During the accumulation period, the kicker is uncharged allowing

unimpeded muon delivery to the experimental area. The measurement period is initiated by the charging of

the kicker with the muons deflected into the downstream quadrupoles and slits.

During the R04 run period, two of the modulator stacks malfunctioned. This was overcome by short-

circuiting each of the two pairs and driving them with only two modulators at a slightly reduced potential

difference of VK = 22.7 kV. The consequence is a longer transition time of 60 ns. With a fully operational

kicker the extinction factor, ε, defined as the ratio of the beam rate between the accumulation and measure-

ment periods, is measured as εfull = 800, but with the reduced kicker voltage, this dropped to ε≤ 390.

6.4 Entrance muon detector

At the exit of the beamline is a 76 µm mylar window, followed by a 5 cm gap of air, and a multi-wire

proportional chamber (MWPC). The MWPC, known as the entrance muon chamber (EMC), records the

41

Figure 6.3: A map showing the layout of the proton accelerator facility. Highlighted by circles are 1. theCockcroft-Walton accelerator, 2. the injector cyclotron, 3. the primary cyclotron, 4. the M target 5. the Etarget, and 6. the πE3 experimental area. (Figure from PSI [43].)

Figure 6.4: A diagram of the πE3 beamline showing the path of muons from the E target to the experimentalarea.

beam spot and number of muons during the accumulation and measurement periods. The 39 cm gap between

the EMC and stopping target is filled with helium at 1 atm, which is contained in a 37 µm thick nylon bag.

The helium, having a density ≈ 10% of air, is used to reduce the number of muon stops in this region.

42

]cMeVmomentum [

24 25 26 27 28 29 300

0.01

0.02

0.03

0.04

Cou

nts

[arb

itrar

y un

its]

Figure 6.5: Relative flux of muons measured at the MuLan target as a function of beamline momentum.

The EMC is designed to measure both the horizontal and vertical beam position while minimizing muon

scatter and stops prior to the target. The anode wires are 15 µm thick tungsten. The 25 µm thick, aluminized

mylar, exterior windows function both as the amplification gas container and cathode planes. A 12.5 µm

thick cathode plane separates the two wire planes. The wire planes are comprised of 128 wires with 2 mm

pitch, read out in pairs, to give a spatial resolution of 4 mm. The intrinsic dead time per wire was measured

to be 100 ns. The EMC is efficient at detecting 98% of muons and about 8% of positrons, with no ability

to distinguish between the two. A typical beam profile is shown in Fig. 6.6(a,b, and c), and an image of the

EMC is shown in Fig. 6.6(d).

A permanent magnet array, placed at the exit of the EMC, produces a central magnetic field of 11 mT.

The purpose is to precess and dephase the polarization of muons that stop in the EMC materials. Without

this magnet array, any muons stopping in the EMC would have been exposed to the earth’s magnetic field

resulting in a potential distortion of the lifetime curves. The magnet array with the measured magnetic field

is shown in Fig. 6.7

6.5 SRIM simulation

The simulation package “Stopping and Range of Ions in Matter” (SRIM) [45] uses Monte Carlo techniques

to simulate the interaction of charged particles passing through matter. This tool is used to simulate the

muons as they pass from the beamline exit window to the stopping target. Muons stopping in this region

are of concern since the magnetic field is not well defined and the detector is not symmetric at this location.

43

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

y [m

m]

-40

-30

-20

-10

0

10

20

30

40

200

400

600

800

1000

610×

(a)

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

Co

un

ts p

er 2

mm

0

2

4

6

8

10

12

14

16

18

910×

rms = 4.6 mm

(b)

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

Co

un

ts p

er 2

mm

0

1000

2000

3000

4000

5000

6000

610×

rms = 12.2 mm

(c) (d)

Figure 6.6: (a) Beam profile as measured by the EMC with the color indicating the number of particles at agiven location. (b) The beam profile in the vertical direction with an RMS of 4.6 mm. (c) The beam profilein the horizontal direction with an RMS of 12.2 mm. (d) Figure of the entrance muon chamber (EMC) withthe anode wires and cathode planes removed.

Using SRIM, the relative number of muon stops in each of the materials is obtained, see Fig. 6.8, with

≈ 3× 10−4 stopping in the EMC, ≈ 1× 10−4 stopping in the helium (dominated by backscatter off thetarget), ≈ 4×10−5 backscattering into the beamline, and the remainder stopping in the MuLan target. Theeffects of non-target-stopped muons are given in the systematic chapter, Sec. 9.2.

6.6 Muon stopping targets

Two stopping targets are used in the experiment, a sulfur target and a ferromagnetic sheet of

ArnokromeTMIII (AK-3) [42]. Either targ