IMPROVED SEARCH FOR CP-VIOLATION IN ORTHO-POSITRONIUM DECAY

By

Tom-Erik Haugen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2024

ABSTRACT

The observed baryon asymmetry of the universe requires combined Charge Parity (CP) violation,

likely more than the Standard Model contains. Simultaneously, the scarcity of antimatter makes

direct tests of CP-violation difficult to perform in a laboratory, and many tests search for time-

reversal violation (T ) and use combined CPT -symmetry to equate this with CP-violation. Direct

searches involving matter and antimatter are robust and insensitive to false signals like final state

interactions.

Positronium is a bound state of an electron and a positron, and can be copiously produced in

a laboratory. This motivates the design and construction of a dedicated detector array to search

for CP-violation in the 3-𝛾 decay of ortho-positronium, with a target sensitivity of 10−5 for the

measured asymmetry, a factor of 10 improvement on current limits.

The experiment will require tensor polarized positronium, which can be achieved by utilizing

a magnetic field. The sensitivity target will require high statistics and large detector acceptances.

To these ends the array will feature three rings of 𝛾-detectors with 16 crystals in each ring. The

detector array and all readout electronics will be constructed to fit within the warm bore of the FRIB

Positron Polarimeter magnet. The detector geometry and placement were studied in Monte-Carlo

simulations in order to optimize the array for this specific experiment. This allowed optimization

of detector size, shape, array geometry, and energy and multiplicity cuts.

Extensive tests of the crystal shape and geometry were performed, these characterized and

removed a geometric light collection distortion. A test stand for optimization of positronium

formation was constructed. Tests of multiple powders showed that using chunks of silica-aerogel

could achieve a lifetime of 135 ns, and up to 40% formation fraction. This was then placed in

a three crystal demonstrator to prototype the online DAQ. The three crystal and start detector

combination was able to remove backgrounds and extract the continuous 2-D energy distribution

of ortho-positronium decay.

The direct comparison of measured count asymmetries with theory motivated quantities (La-

grangian parameters, mixing coefficients, etc.)

is non-trivial, and cannot be done in a model-

independent way. A detailed discussion of removal of detector acceptances and efficiencies is

presented in the context of a specific model. This further required extension of the theory analysis

to incorporate the effects of a static magnetic field, which induces non-trivial time dynamics of

the different angular distributions. This clarifies some inconsistencies in the literature on the time

dependence of the asymmetry in a magnetic field. The detector array is under construction and

will be able to reach the target sensitivity with 35 days of continuous runtime.

ACKNOWLEDGEMENTS

I would like to thank my parents, Jan-Petter and Elizabeth, for being very encouraging of pursuing

an academic career, and instilling a deep curiosity about all things. I would also like to thank my

amazing grandparents, Tom, Barbara, Frank, and Evelyn. They, along with my parents, always

encouraged me to learn as much as possible, approach all discussions critically, and maintain a

sense of humor throughout.

I learned so much during my time at Michigan State, in large part because of my amazing

advisor Oscar Naviliat-Cuncic. The breadth and depth of his knowledge about physics is inspiring,

and working with him was always very instructive and fun. He encouraged me to really dig through

the details of the experiment and the underlying theory, and he would regularly drop what he was

doing to talk through details in two hour or longer impromptu meetings.

During my time at Michigan State I had the opportunity to work with many amazing faculty. It

was fantastic getting to work with Paul Voytas and Elizabeth George from Wittenberg University.

They were always available to help, and really pushed me to be careful and meticulous in my work.

Many faculty went above and beyond for what is expected in a graduate education, but especially

Jaideep Singh. He went far out of his way to ensure that I was part of a research community,

and that I was safe and supported during the pandemic. He and the rest of my committee: Greg

Severin, Joey Huston, and Kirtimaan Mohan had large positive impacts on my project, and were

deeply encouraging of me as a scientist. I would further like to thank the professors that had a large

impact on my time at MSU: Andreas von Manteuffel, Andrea Shindler, Scott Pratt, Remco Zegers,

Paul Gueye, and Xavier Fleéchard during my time at LPC-Caen. Lastly, I would like to thank the

members of my research group over the years: Shloka, Xueying, Max, Mohamad, and Romain.

And, of course, Kim Crosslan for helping me navigate the program.

I would like to thank some of my educators, advisors, and mentors from before my time at

MSU. Those are: David Latimer, Rachel Pepper, Betsy Kirkpatrick, Ms. Foster, and Ms. Rohmer.

I learned so much from all of them, and they each had a large impact on who I became as a scientist

and academic. I would like to further thank both Brahim Mustafa and David Brown. They both

iv

helped bridge my undergraduate education towards a future career. Without either of their help, I

would not have started graduate school.

The support of my family has helped me throughout this process. I am lucky to have three

amazing sisters: Erin, Rebecca, and Penryn. My extended family have all had a large on me as

a person: Alec, Alex, Barbara, Carson, Dave, Dutch, Elizabeth, Emily, Jan, Julia, Kalei, Kanoa,

Merethe, Momi, Nikoline, Ola, Scott, Sebastian, Shelby, Sloane, Sofie, and Suzy.

I’m very lucky to have friends that always believed in me, especially the UPS boys: Theo, Jens,

and Garrett, and my Michigan family: Patti, Adam, Brean, Adam, Brynn, Aryn, Morgan, and Nick.

I would also like to thank: Todd, Carlo, Gayané, Marissa, Daniel, Kat, Patrick, Jennet, Jordan,

Thomas, Cami, Corey, Alyssa, Ava, Jack, Jason, Tamas, Carl, Isaac, Kyle, Swapna, Joe, and Tawfik.

v

LIST OF ABBREVIATIONS .

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ix

TABLE OF CONTENTS

CHAPTER 1

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PHYSICS OF CP-VIOLATION . . . . . . . . . . . . . . . . . . . . . .
1.1 Overview .
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1.2 Discrete spacetime symmetries . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Observed discrete symmetry violations . . . . . . . . . . . . . . . . . . . . . .
1.4 Cosmological considerations . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Experimental evidence and searches for CP-violation . . . . . . . . . . . . . .

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POSITRONIUM DECAY AND CP-VIOLATION THEREIN . . . . . .
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2.1 Overview of positronium physics and history . . . . . . . . . . . . . . . . . .
2.2 Positronium discovery .
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2.3 Overview of 3-photon kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Searches for fundamental symmetery violations in positronium . . . . . . . . . 14
2.5 University of Michigan CP-violation search . . . . . . . . . . . . . . . . . . . 15
2.6 University of Tokyo search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8 Spin-1 positronium mixed state . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9
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2.10 Optimal B-field for tensor polarization . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Open question about time evolution of signal . . . . . . . . . . . . . . . . . . . 29
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2.12 The experiment

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INITIAL DESIGN AND SIMULATIONS . . . . . . . . . . . . . . . . . 33
3.1 Extracting the symmetry violating term . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Planned experiment .
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Initial positronium event generator . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4
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3.5 Geometric Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1 Overview .
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4.2 Scintillation detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3
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4.4 Single crystal design and simulation . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 First prototype and observed distortion . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Final crystal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Initial tests . .

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POSITRONIUM FORMATION . . . . . . . . . . . . . . . . . . . . .
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5.1 Overview .
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5.2 Ps formation and Positron annihilation lifetime spectroscopy . . . . . . . . . . 60
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5.3 Ps test stand construction and testing . . . . . . . . . . . . . . . . . . . . . .
. 63
5.4 Time spectrum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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vi

Initial results

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DESIGN AND SIMULATION OF INNER MODULE .
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CHAPTER 9

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9.3 Extraction of count asymmetry terms .
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NEW PHYSICS IN 3-𝛾 DECAY OF POSITRONIUM .
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CHAPTER 10

10.1 Detector efficiencies inducing false asymmetries .
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10.6 Energy cuts and 2-𝛾 dilution .
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CHAPTER 11

SUMMARY AND FUTURE WORK .

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APPENDIX A

POSITRONIUM SPIN STATE FOR SPIN-0 AND SPIN-1 .

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MATRIX ELEMENT AND 3-𝛾 PHASE SPACE .

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viii

LIST OF ABBREVIATIONS

MSU

Michigan State University

FRIB

Facility for Rare Isotope Beams

DAQ

Data Acquisition System

QED

Quantum Electrodynamics

MIT

p-Ps

o-Ps

Massachussets Institute of Technology

para-positronium

ortho-positronium

FWHM Full Width at Half Maximum

FoM

Figure of Merit

WU

Wittenburg University

SiPM

Silicon Photomultiplier

PMT

Photomultiplier Tube

FASTER Fast Acquisition SysTem for nuclEar Research

QDC

Charge to Digital Converter

GEANT GEometry ANd Tracking

EGS

LPC

Electron Gamma Shower

Labratoire de Physique Corpusculaire

ADC

Analog to Digital Converter

CFD

Constant Fraction Discriminator

MSPS

Mega Samples Per Second

ix

CHAPTER 1

PHYSICS OF CP-VIOLATION

1.1 Overview

In this chapter we give an overview of discrete spacetime symmetries and their relation to anti-

matter, fundamentally connected through the CPT -theorem of relativistic quantum field theories.

This includes a brief review of the cosmological observations that lead to the widely held belief that

there must be more CP-violation than currently accomadated in the Standard Model. Following

this is an overview of the history and observation of discrete symmetry violations, culminating in

a discussion of CP-violation. This finally motivates neutral mixed matter-antimatter systems as

particularly clean systems to search for CP-violation.

As the positron and positronium are both historically important for the conceptual develop-

ment of quantum field theory, and early precision tests of Quantum Electrodynamics (QED), the

discussion begins with a general conceptual overview of antimatter and its natural connection to

spacetime symmetries. This will follow the discussions in Refs. [1, 2] but highlighting the role of

discrete symmetries.

1.2 Discrete spacetime symmetries

The current most precise framework to describe particle interactions is using relativistic quantum

field theory. This describes the interactions of quantized fields in a flat spacetime. The structure of

spacetime is taken to be flat Minkowski space following from special relativity. The finiteness of

the speed of light means that for any event (x, 𝑡) the whole of spacetime can factor into 5 distinct

regions, illustrated in Figure 1.1. These are the timelike future, timelike past, lightlike future,

lightlike past, and spacelike separation. Since no signal can travel faster than the speed of light, no

two events that are spacelike separated can affect each other.

The five regions of spacetime are invariant under rotations and boosts. This means all inertial

reference frames agree on the time order of events that are timelike separated. The concept of

cause and effect ("causality") can be applied within the past and future lightcones. However, any

signal that can reach a spacelike separated event (and therefore travel faster than light) will violate

1

Figure 1.1 Decomposition of Minkowski spacetime into past and future lightcones, and spacelike
separated region. Image from Ref. [3]

causality. Different observers can disagree on which event was first and which was second (or

which was the cause and which was the effect).

Beyond rotations and boosts there are two more spacetime transformations that warrant consid-

eration. The three spatial axes have an intrinsic handedness (right or left), a right handed spacetime

cannot be mapped into a left handed one through any rotations or boosts, but instead requires a dis-

crete transformation termed "Parity" or P. Similarly, the future and past lightcone are individually

Lorentz invariant, but they can be mapped between each other through "time-reversal" or T .

Now consider a single particle state defined on Minkowski space, it is specified by an energy,

momentum, helicity (or chirality), and internal quantum numbers. A state of a single particle

of species 𝑎 can be considered as |𝑎, p, 𝜎⟩, where p is the momentum and 𝜎 is chirality or

helicity. Further, consider that this particle has a conserved quantum number, like electric charge,

ˆ𝑄 |𝑎, p, 𝜎⟩ = 𝑞𝑎 |𝑎, p, 𝜎⟩.

Study of the causal structure of relativistic single particle quantum mechanics quickly runs into

trouble. The "propagator" gives the amplitude for a particle to travel from position 𝑥𝑖 to 𝑥 𝑓 in time

2

𝑡 𝑓 − 𝑡𝑖,

𝐾 (𝑥 𝑓 , 𝑡 𝑓 ; 𝑥𝑖, 𝑡𝑖) = ⟨𝑥 𝑓 , 𝑡 𝑓 |𝑥𝑖, 𝑡𝑖⟩ = ⟨𝑥 𝑓 |𝑒−𝑖𝐻 (𝑡 𝑓 −𝑡𝑖)/ℏ|𝑥𝑖⟩

(1.1)

For the single particle theory to respect causality this should vanish at spacelike separations

(ensuring no particle can travel faster than light). Direct computation using the relativistic energy
momentum dispersion relations 𝐸 = +√︁p2 + 𝑚2 gives a non-vanishing amplitude at spacelike
separations, Δ𝑥2 − 𝑐2Δ𝑡2 > 0. Instead the propagator falls off like 𝑒−Δ𝑥/𝜆𝐶 for Δ𝑥 >> 𝑐Δ𝑡, where

𝜆𝐶 is the Compton wavelength for the particle [1, 2]. This means that relativistic single particle

quantum mechanics cannot respect causality on the scale of the Compton wavelength of the particle.

For spacelike separations the order of the two events is dependent on the reference frame of the

observer. Instead of abandoning quantum mechanics or special relativity, the solution comes from

abandoning a single particle description and instead considering only systems where the particle

number can change. So it is in-fact inappropriate, for spacelike separations, to only consider a

particle traveling from (𝑥𝑖, 𝑡𝑖) → (𝑥 𝑓 , 𝑡 𝑓 ), without also considering a separate particle traveling

from (𝑥 𝑓 , 𝑡 𝑓 ) → (𝑥𝑖, 𝑡𝑖).

This gives an exact cancellation for propagation across the lightcone when we restrict to only

considering observables constructed out of sums of creation operators that create a particle of

charge 𝑞 and helicity 𝜎 with destruction operators that annihilates a particle of charge −𝑞 and

helicity −𝜎. This operator is called a "quantum field".

In effect this construction requires that a particle with charge and helicity {𝑞, 𝜎} propagating

over (Δ𝑥, Δ𝑡) must have a partner with charge and helicity {−𝑞, −𝜎} such that it interferes when

travelling over (−Δ𝑥, −Δ𝑡) across the lightcone. This partner is called the antimatter partner for the

original particle. This construction demonstrates the connection between spacetime symmetries

and charge conjugation. The two spacetime intervals, (Δ𝑥, Δ𝑡) and (−Δ𝑥, −Δ𝑡), are related by a

combined parity-time reversal operation PT . Antimatter conceptually arises as a method to ensure

causality in a relativistic quantum theory [1, 4, 5], at least in the context of a particle interpretation.

It requires most generally that there is a symmetric particle state with opposite charge and opposite

helicity. The mapping between these states is anti-unitary and termed Θ [4]. If the theory admits

3

an operator P and T , then Θ can be considered as a combined CPT -transformation of the particle

state. But this should really be taken as defining the operator C for the theories that admit a P and

T .

This construction highlights the deep connection between antimatter and spacetime geometry,

which appears accidental when presenting each discrete operation individually. It also highlights

CPT as a deep underpinning of the conceptual framework for describing modern physics.

1.3 Observed discrete symmetry violations

All three discrete symmetries C, P, and T were assumed to be individually respected. Ex-

perimental observation shows that electromagnetism and strong interactions do respect these sym-

metries. Increasingly precise studies of particle properties in the 1950’s called these ideas into

question. In response to the "𝜃-𝜏" puzzle, Lee and Yang proposed that parity may be violated in

weak interactions [6]. They formulated an extension of the Fermi theory of weak interactions that

allowed for the violation of parity.

This was discovered in the famous Madame Wu experiment in 1957 [7], by the observation that

𝛽 particles were preferentially emitted opposite to the direction of the spin of a 60Co nucleus. This

demonstrated that P was violated in weak interactions [8].

These experiments did not provide information on possible combined charge-parity symmetry,

shortened to CP. Such a test in nuclei would require measuring the angular distribution of 𝛽−

particles from a 60Co nucleus and compare with the angular distribution of 𝛽+ particles from an

anti-60Co nucleus [4]. Direct tests of CP-violation prove difficult due to the lack of availability of

antimatter.

Nevertheless, CP-violation has been observed in neutral meson physics, first by Fitch and

Cronin in neutral kaons in 1964 [9]. The CP-violation in the current Standard Model is very small,

entirely contained in the 3×3 Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix [10],

although there is increasing evidence of a similar mechanism in the neutrino mixing matrix as well

[11]. There is good reason to expect that there are as of yet unobserved sources of CP-violation.

4

1.4 Cosmological considerations

The observed baryon asymmetry of the universe quantifies the imbalance between matter and

antimatter at cosmological scales. It is defined as,

𝜂 =

𝑛𝐵 − ¯𝑛𝐵
𝑛𝛾

(1.2)

where 𝑛𝐵 is the Baryon density, ¯𝑛𝐵 is the anti-baryon density, and 𝑛𝛾 is the photon density. Current

measurements show this number to be very small 𝜂 = 10−10 [12].

In 1967 Andrei Sakharov

identified a set of three conditions that are sufficient (though not strictly necessary) to generate a

baryon asymmetry [12, 13],

1. thermal inequilibrium

2. baryon number violation

3. C- and CP-violation

the first two ingredients are mostly straightforward, the universe must not be in equilibrium, or

else as many particles would be becoming antiparticles as the other way around. Similarly, for a

net change in baryon number, baryon number must be violated. Understanding the necessity of

CP-violation is more nuanced.

It might seem that C-violation alone is enough to generate an imbalance in matter versus

antimatter, however this is not the case. Consider a particle 𝑋 that decays to 𝑞𝑞, and its antiparticle

¯𝑋 that decays to ¯𝑞 ¯𝑞, the conservation of C implies that Γ(𝑋 → 𝑞𝑞) = Γ( ¯𝑋 → ¯𝑞 ¯𝑞). If the universe

starts with an equal number of 𝑋 and ¯𝑋, then the rate of change of baryon number,

d𝐵
d𝑡

∝ Γ(𝑋 → 𝑞𝑞) − Γ( ¯𝑋 → ¯𝑞 ¯𝑞)

(1.3)

is equal to zero unless C-symmetry is violated. Consider the fact that particles with spin come

with a handedness, and that left handed particles are transformed into right handed particles under

parity. Now under CP the particles are mapped such that 𝑋 → ¯𝑋, 𝑞 𝐿 → ¯𝑞𝑅, 𝑞𝑅 → ¯𝑞 𝐿, etc.

C-violation means that Γ(𝑋 → 𝑞 𝐿𝑞 𝐿) ≠ Γ( ¯𝑋 → ¯𝑞 𝐿 ¯𝑞 𝐿). However, CP-symmetry means that

5

Γ(𝑋 → 𝑞 𝐿𝑞 𝐿) = Γ( ¯𝑋 → ¯𝑞𝑅 ¯𝑞𝑅) and Γ(𝑋 → 𝑞𝑅𝑞𝑅) = Γ( ¯𝑋 → ¯𝑞 𝐿 ¯𝑞 𝐿). Now if the universe starts

with the same amount of 𝑋 and ¯𝑋, it can generate a net change in baryon number proportional to,
(cid:19)

(cid:18)

(cid:19)

(cid:18)

∝

Γ(𝑋 → 𝑞 𝐿𝑞 𝐿) + Γ(𝑋 → 𝑞𝑅𝑞𝑅)

−

Γ( ¯𝑋 → ¯𝑞 𝐿 ¯𝑞 𝐿) + Γ( ¯𝑋 → ¯𝑞𝑅 ¯𝑞𝑅)

(1.4)

d𝐵
d𝑡

which is only non-zero with full CP-violation, not just C-violation. If all 𝑋 decay to 𝑞𝑞 and all

¯𝑋 decay to ¯𝑞 ¯𝑞, then once all 𝑋 and ¯𝑋’s have decayed the baryon number would return to zero.

Further, CPT requires that the lifetimes of 𝑋 and ¯𝑋 must be equal, so there must be competing

decay channels as well, Γ(𝑋 → 𝑌 ) and Γ( ¯𝑋 → ¯𝑌 ). So long as 𝑌 has a different baryon number

than 𝑞𝑞, then a non-zero baryon number can be generated [12, 14].

1.5 Experimental evidence and searches for CP-violation

CP-violation was discovered in the neutral kaon system in which the kaon and anti-kaon mix.

Further studies have shown CP-violation in B-mesons beyond 5𝜎 as well [10]. These are induced

by the complex phase of the CKM matrix, and indeed the third generation of quarks was predicted

by Koboyashi and Maskawa to explain the CP-violation observed in kaons [15].

Permanent electric dipole moments (EDMs) are highly sensitive to CP-violating physics.

Decades of searches for neutron EDMs have tightly constrained possible values of 𝜃𝑄𝐶𝐷 [10].

Similar searches for electron EDMs have constrained the possible electric dipole moment of the

electron to 10−29 𝑒cm [16].

There is a distinction between two types of CP-violation searches. One searches for T -

odd properties and through CPT -symmetry relates this to CP-violation. This includes EDM

searches, and the D-coefficient in 𝛽-decay. T -violation can be mimicked in decaying systems by

processes that respect T [17, 18]. In this way, equating CP-violation with signatures of T -violation

in decaying systems are technically model dependent interpretations (even when requiring strict

CPT -conservation).

Searches for CP-violation that use neutral systems such as kaons, B-mesons, or 𝑍-bosons are

termed as "clean signals" of CP-violation, in that they cannot be mimicked in a theory that respects

CP. This means that such signals cannot be induced by higher order radiative corrections, or

absorptive processes like final state interactions, unless those processes themselves violate CP-

6

symmetry [18, 19, 20]. While the interpretation of such signals is very clean, the experiments

are hard to perform because of the rarity of such systems limiting the statistical reach of such

experiments.

This motivates the use of mixed matter-antimatter bound state searches that utilize systems that

can be produced in large quantities, and have moderately long lifetimes. Positronium is one such

system.

7

CHAPTER 2

POSITRONIUM DECAY AND CP-VIOLATION THEREIN

In this chapter we give a brief historical overview of the discovery of the positron, and of the

bound state positronium. This includes a discussion of some early experimental work that utilized

magnetic fields to study the different spin states of positronium. We focus on two previous searches

for CP-violation in the angular distribution of the photons from positronium decay, and give a

detailed overview of the theoretical description of these searches.

2.1 Overview of positronium physics and history

Antimatter is a generic prediction of relativistic quantum mechanics, historically motivated

by the "negative energy" solutions to the Dirac equation. These solutions could not be discarded

without sacrificing a Hilbert space interpretation of the solutions. The details of the discussions

and theories, along with the resolution are given in Ref. [21].

Dirac’s theory generically predicted "positive electrons" or positrons. The positron was quickly

discovered in 1932 by Anderson, though it is interesting that positron tracks had been appearing

in many cloud chambers for the better part of a decade by that point, but were not identified as

such [21]. Figure 2.1 shows one of the cloud chamber images from Anderson’s studies. There

was a reluctance to equate these positive electrons with the positron from Dirac’s theory. The

Dirac theory of electrons and positrons was not widely accepted until absorption lengths of the

2.6 MeV 𝛾-ray from 208Tl could not be reproduced without including the pair creation that high

energy 𝛾’s undergo in matter [21]. In the 1930’s this was the highest energy 𝛾 source available,

and pair creation, which begins to dominate over photo-absorption at higher energies [22], is not

even possible below 1 MeV.

As the positron is simply an electron but with positive charge, there is a Coulomb attraction

between the two particles. This allows for possible bound states between the two in analogy with

the Hydrogen atom. Wheeler proposed that there could be an electron-positron bound state [24].

He coined the term "polyelectrons," and worked out some atomic and molecular properties of the

system. A similar proposal by Ruark coined the term "positronium" [25] . These initial works

8

Figure 2.1 Image of positron track in a cloud chamber from Ref. [23]. The cloud chamber has a
magnetic field coming out of the page. The positron enters from the bottom left, and its curvature
corresponds to a positive charge.

suggested looking at stellar spectra for the spectral lines that would correspond to positronium,

and not the method of directly producing it in the laboratory. Wheeler’s work estimated speeds

of positron thermalization, and lifetimes of spin-0 ground state positronium.

In 1939 Ore and

Powell calculated the lifetime of spin-1 positronium and the energy distribution of the photons

from unpolarized spin-1 positronium decay [26].

2.1.1 Symmetry properties of positronium and decay selection rules

Positronium has well defined behavior under C and P.

Its eigenvalues correspond to C =

(−1) 𝐿+𝑆, and P = (−1) 𝐿+1, where 𝐿 is the orbital momentum, and 𝑆 is the spin of the bound state

[1, 27]. As this bound state is mixed matter-antimatter it is mapped back to itself under charge

conjugation, unlike charged particles like electrons or protons.

The photon also has a well defined charge conjugation eigenvalue of −1. This follows from

Maxwell’s equations [27]. This gives the selection rule that dominates the positronium lifetimes.

When C is a good symmetry, spin-0 positronium can only decay to an even number of photons and

spin-1 can only decay to an odd number.

Positronium has two nearly degenerate ground states separated by an 841 𝜇eV hyperfine interval.

The ground state spin-0 positronium is referred to as singlet positronium or "para-positronium"

(p-Ps). Due to the selection rules above it can only decay to an even number of photons, primarily

9

2. It has a lifetime of 125 ps. Spin-1 positronium is referred to as triplet positronium, or "ortho-

positronium" (o-Ps). Due to the C-selection rules it must decay to an odd number of photons,

primarily 3. This increases the lifetime of ortho-positronium to 142 ns, over one thousand times

longer than that of para-positronium. Quite generally a spin-1 boson cannot decay to two photons

due to the Bose-statistics of the photons [28, 29]. This means that ortho-positronium cannot decay

to two photons in vacuum for any theory that conserves angular momentum.

2.2 Positronium discovery

Positronium was discovered by Martin Duetsch at MIT in 1951 [30]. The group was studying

positron thermalization and annihilation times in different gasses. They had containers filled with

gasses (O2 and N2), and would bombard them with positrons. They observed a clear drop in time

of the amount of annihilation events, not matching the expected 10’s of ns lifetime of thermalized

positrons in gasses. There was twice as much quenching in Oxygen compared to Nitrogen as

well. After multiple experiments Deutsch realized that if some electrons form positronium in the

gasses, then that positronium scatters off a gas molecule in such a way that there is a spin flip, the

atom would be in a short lifetime state and decay. Then if N2 and O2 had similar formation rates,

the discrepancy in observed time spectra would be explained by the increased interaction strength

between positronium and O2 causing more spin flips. To test this he added nitric oxide gas and

measured the subsequent proportional quenching of the longer lifetime state [30, 31, 32].

2.2.1 Early studies of positronium in a magnetic field

After discovering positronium the MIT group performed further studies of the system. Deutsch

performed the calculations of the magnetic mixing of ortho- and para-positronium states that

occur in a magnetic field. Since the electron and positron have opposite magnetic moments, the

|𝑠 = 1, 𝑚 = 1⟩ and |𝑠 = 1, 𝑚 = −1⟩ states have no net magnetic moment, whereas both 𝑚 = 0 states

do have net magnetic moments. This means in a B-field the |𝑠 = 0, 𝑚 = 0⟩ and |𝑠 = 1, 𝑚 = 0⟩ states

mix. We call the resulting states "pseudo-singlet" and "pseudo-triplet" states, corresponding to

shorter and longer lifetimes respectively [33].

Deutsch and Dulit performed coincidence measurements with detectors arranged in a plane

10

perpendicular to the magnetic field and indeed observed a quenching in the counts. They assigned

a large uncertainty of 15% to the quoted value [33].

The lifetime of triplet positronium as calculated by Ore and Powell [26] followed the standard

procedure of averaging the initial spin states, and summing over final states. Drisko later observed

that this loses some experimentally testable predictions [34]. He drew attention to dependence

of the angular distribution of final state photons on the initial spin state of the positronium. This

intuitively follows from conservation of angular momentum, if the initial positronium carries

angular momentum along some axis, the final state of three photons must as well, and therefore

likely has a non-trivial angular distribution.

Drisko calculated the angular distortions that occur in QED [34]. This gave the prediction that

in the plane perpendicular to the quantization axis, 1/2 of the decays occur from the 𝑚 = 0 state, as

opposed to 1/3 as expected by counting statistics.

This led to experiments by Wheatley and Halliday to measure the possible angular dependence

to the quenching of counts in a magnetic field [35]. The group measured the coincidence counts for

two of the three photons from ortho-positronium decay. The authors utilized a permanent magnet

and adjusted the field strength by changing the position. A similar experiment was performed

by Hughes, Marder, and Wu [36]. They recorded single counts in the plane perpendicular to the

magnetic field axis. They applied a tuneable field using an electromagnet that they powered with

two submarine batteries. Both experiments saw a clear quenching that matched the predictions by

Drisko. The results from Ref. [36] are shown in Figure 2.2, and show that the data only agrees

with the theoretical curve after correcting for the angular dependence.

This non-trivial effect was neglected in the initial positronium experiments in magnetic fields,

but was smaller than their large uncertainty estimates. Further experiments that studied the angular

distribution of photons from positronium decay searched for more complex distributions. This

requires a more thorough discussion of ortho-positronium decay kinematics.

11

Figure 2.2 Observed quenching of counts in the plane perpendicular to the magnetic field. Ex-
perimental data does not agree with the theoretical prediction unless corrected for the angular
distribution that occurs in a magnetic-field. From Ref. [36].

2.3 Overview of 3-photon kinematics

The three photons from positronium decay have a continuous energy distribution given by the

Ore-Powell distribution [26]. This distribution is shown in Figure 2.3b. The photons are ordered

by their energies, 𝜔1 > 𝜔2 > 𝜔3, where 𝜔𝑎 = |k𝑎 | is the energy of photon 𝑎. The kinetic energy

of the positronium is negligible compared to the energies of the photons, so it is safe to assume

k1 + k2 + k3 ≈ 0. Momentum conservation requires that the three photons are co-planar. Further the

event can be fully described by only observing two of the photons, as the third is entirely constrained

kinematically. Figure 2.3c informs the needed detector placement. For instance, measuring ˆk1 and

ˆk2 in coincidence requires an angle between the two detectors of 120◦ − 180◦. The distribution

peaks around 160◦.

Figure 2.4 shows the geometry of a decay event. The two photon momenta are confined

to a plane. The normal to the decay plane is determined by these momenta, and taken to be

ˆn = ˆk1 × ˆk2/| ˆk1 × ˆk2| or any cyclic permutation of indices. Figure 2.4 also includes some axis

defined by the positronium state, either the vector polarization or (a component of) the tensor

polarization. It is represented with ˆs to match existing literature.

If the vector polarization and tensor polarization lie on the same axis then their values are given

by the state populations as,

𝑠𝑧 =

𝑁+ − 𝑁−
𝑁+ + 𝑁0 + 𝑁−

12

(2.1)

(a)

(b)

(c)

Figure 2.3 (a) Three photons ordered by their energies |k1| > |k2| > |k3|, and respective angles
between the photons.
(b) Energy distribution of events for each photon, and summed together
reproducing the Ore-Powell distribution. (c) Distribution of angles between photons, 𝜓12 shows
the optimal detector placement to see the two photons in coincidence.

𝑃2 =

𝑁+ − 2𝑁0 + 𝑁−
𝑁+ + 𝑁0 + 𝑁−

(2.2)

where 𝑁𝑖 is the population of the state with 𝑚 = 𝑖 along the axis. We call these the polarization

and the aligment of the state. We reserve "unpolarized positronium" to refer to a positronium state

with neither vector nor tensor polarization.

The charge conjugation properties of the decay are completely fixed by the positronium initial

state and the 3-𝛾 final state. This means the observation of parity violation in spin-1 positronium

decaying to 3-𝛾 immediately indicates combined CP-violation.

Similar arguments have been applied to argue that an observation of a P-even "T -odd" angular

correlation would therefore imply full CPT -violation. This claim is incorrect [19], because

T -violation can be mimicked in decaying systems by absorptive processes such as final state

interactions [4, 17]. Such interactions cannot induce a CP-violating signature in this decay unless

13

Figure 2.4 ˆk1, ˆk2, and ˆk3 are the 3 linearly independent vectors describing the 3-𝛾 final state. ˆs is
the "quantization axis" for the positronium atom, indicative of some axis defined by the positronium
state, either by a spin polarization or a tensor polarization.

they also violate CP-symmetry themselves [18].

2.4 Searches for fundamental symmetery violations in positronium

Further studies of positronium were performed in the 80’s and 90’s by a group at University

of Michigan. This group proposed a set of measurements in positronium that would be sensitive

to new physics.

In Ref.

[37] Arbic, Hatamian, Skalsey, Van House, and Zheng proposed and

performed a search for an asymmetry in events with the 3-𝛾 decay plane aligned with the initial

positronium polarization, or anti-aligned. This would not be induced in the Standard Model except

at higher orders due to photon-photon scattering. They performed this search with two sets of

NaI detectors read out in coincidence. They utilized parity violation in nuclear 𝛽-decay to have

polarized positrons that formed vector polarized positronium. They could change the direction of

the decay plane by switching which pair of detectors they read out. Utilizing this simple setup they

demonstrated the feasibility of such studies. They state that such an asymmetry would be induced

by a term in the angular distribution of the form 𝐶𝐶𝑃𝑇 ˆs · ( ˆk1 × ˆk2), coining the term 𝐶𝐶𝑃𝑇 as the

"coefficient" of the "angular correlation" ˆs · ˆn. This language matches similar studies in nuclear

𝛽-decay, and was suggested by Natchmann. At the end of this work they identified a few other

angular distributions that would be indicative of symmetry violations. Of interest for this current

work is the "correlation," (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2) which is odd under parity and would be indicative of

14

CP-violation. They stated that since ˆs enters this correlation twice it would require tensor polarized

positronium in-lieu of vector polarized positronium [37].

In conjunction with this work, Bernreuther, Löw, Ma, and Natchmann worked through the

general theory of 3-𝛾 decay of spin-1 positronium [19]. They performed a general tensor de-

composition to represent the possible angular distributions by irreducible tensors constructed from

kinematic vectors each multiplied by a form factor. They then analyzed term by term what sym-

metries each form factor would respect or violate. They identified four terms that could only be

induced by CP violation, two requiring vector polarized positronium, and two requiring tensor

polarized positronium, one of which corresponds to the term referred to as (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2)

above. In Ref. [19] this is written as 𝑠𝑖 𝑗 ( ˆk1𝑖 ˆn 𝑗 + ˆk1 𝑗 ˆn𝑖), where 𝑠𝑖 𝑗 is the tensor polarization. This

highlights the tensorial nature of this quantity, and avoids the possible misconception of equating

the vector polarization squared with the tensor polarization, as the two are independent quantities.

To simplify the discussion throughout this work we will use the following notation when

discussing unit vectors,

ˆk𝑎 = 𝜅𝑎𝑥 ˆx + 𝜅𝑎𝑦 ˆy + 𝜅𝑎𝑧 ˆz

n = ˆn = 𝑛𝑥 ˆx + 𝑛𝑦 ˆy + 𝑛𝑧 ˆz

(2.3)

(2.4)

where 𝑎 = 1, 2, 3. We take n to always be a normalized unit vector, otherwise we write ( ˆk1 × ˆk2).

For a tensor polarization along the ˆ𝑧-axis this correlation can be reweritten as 𝜅1𝑧𝑛𝑧 which is an

unambiguous analogue of (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2), although the two differ by the magnitude of ˆk1 × ˆk2.

To date there have been two searches for this tensor term that would indicate CP-violation.

2.5 University of Michigan CP-violation search

The first search for the quantity (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2) was performed at the University of Michigan

in 1991 [38]. They utilized the same gamma detector setup used in their previous search for the

correlation ˆs · ˆn [37]. Their setup is shown in Figure 2.5. It featured the following components:

1. 68Ga positron emitting 𝛽 source,

2. Plastic scintillator to detect 𝛽 emission,

15

(a)

(b)

Figure 2.5 Diagram of the detector setup used for the University of Michigan measurement [38].
Both top-down and side view of the setup.

3. Magnesium Oxide (MgO) powder for positronium formation,

4. Permanent magnet centered on the positronium source,

5. Three NaI 𝛾-detectors.

The detectors had 9.5% FWHM energy resolution at 511 keV, and 3.9 ns FWHM time resolution

between two NaI detectors (for 511-511 coincidences). The setup featured a dedicated highest

energy photon detector, and two second highest energy detectors at 145◦ from the highest energy

detector. These were placed such that the quantity (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2) is positive for one pair of

detectors, and negative for the other pair. CP-violation would manifest as an asymmetry in counts

between the two pairs of detectors.

For further control of the systematics they measured in two time windows. The addition of a

magnetic field reduces the lifetime of the 𝑚 = 0 state, meaning more 𝑚 = 0 states decay at an

earlier time. They state that the alignment of the decaying positronium atom within a time window

will flip sign; the early window is mostly negative from the 𝑚 = 0 decays, and the late window is

mostly positive from the 𝑚 = ±1 decays. The authors state that they can change the sign of the

signal both geometrically, and between two time windows.

They measured a count asymmetry of 𝐴𝑀𝑖𝑐ℎ = −0.0004 ± 0.0010. They searched for a term

𝐶𝐶𝑃 (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2), and they defined an “analyzing power" as 𝐴 = 𝐶𝐶𝑃/𝑆𝑎𝑛. This gives their

analyzing power as 𝑆𝑎𝑛 = 1
2

𝑃2⟨cos(2𝜃)sin(𝜓12)cos(𝜙)⟩, where 𝑃2 is the alignment, and 𝜃, 𝜓, and

16

𝜙 are given in Figure 2.4. They estimated their analyzing power as 𝑆𝑎𝑛 = 0.072 ± 0.015, giving a

final value for 𝐶𝐶𝑃 = −0.0056 ± 0.015.

2.6 University of Tokyo search

A subsequent search was performed at the University of Tokyo in 2010 [39]. They used a similar

mechanism to form positronium, induce an aligment, and measure a count asymmetry between

pairs. They state that the number of events can be written as,

𝑁 = 𝑁0 [1 + 𝐶𝐶𝑃 (ˆs · ˆk1)ˆs · ( ˆk1 × ˆk2)]𝑒−𝑡/𝜏

(2.5)

where we have slightly changed the notation of unit vectors to match the rest of this discussion. The

array featured sets of detectors (each with two possible pairings) placed on a turntable as shown in

Figure 2.6.

Figure 2.6 Setup for the University of Tokyo search [39]. The four detectors can be combined into
two sets of three detectors to compare with the University of Michigan setup.

Positronium was formed using a 1 MBq 22Na source for their 𝛽+, and silica aerogel. The 𝛾-

detectors were LYSO crystals with 11.7% energy resolution FWHM at 511 keV, and 1.2 ns timing

resolution FWHM for two 511 keV photons.

If there was a term (ˆs· ˆk1)ˆs· ( ˆk1 × ˆk2) it would create a modulation of counts that goes as cos(𝜙),

where 𝜙 is given in Figure 2.4, which also corresponds to the angle the turntable has rotated. They

observed no signal and quote a final value of 𝐶𝐶𝑃 = 0.0013 ± 0.0021(stat) ± 0.0006(syst). They did

17

not measure in two time windows with flipping alignment, instead they used a 0.49 T permanent

magnet and selected events between 50-130 ns after the 𝛽-detection. The experiment collected

data for six months and was statistically limited. The major systematic uncertainties were from the

stepper motor that rotated the setup.

2.7 Theoretical description

The authors of Ref. [19] investigated the general form of the 3-𝛾 final state distribution from

spin-1 ortho-positronium. For the discussion in this chapter, there is no implied summation over

repeated indices. They call the mapping of the ortho-positronium state to the 3-𝛾 distribution

the "decay matrix." They performed a tensor decomposition on this matrix and identified the 9

independent terms, and what each terms discrete symmetry properties are. Before summarizing

this work, we will take one step back to highlight the connection between the decay matrix elements

and the time evolution of the positronium states. This will be key to clarifying future discussions

in this work.

A pure quantum state |𝑖⟩ can decay to a variety of final states (for example states with a different

number of photons, or different angular distributions of photons). The partial decay rate for this

state to a given final state | 𝑓𝑎⟩ is given as Γ𝑖→ 𝑓𝑎 = ⟨𝑖| 𝒯 | 𝑓𝑎⟩ ⟨ 𝑓𝑎 | 𝒯 |𝑖⟩, where 𝒯 is the "transition

matrix," or more standardly the S-matrix [1]. The decay width of the state is given by the sum of
the partial decays to all final states, Γ𝑖 = (cid:205) 𝑓𝑎 Γ𝑖→ 𝑓𝑎. Assuming the initial state diagonalizes the

Hamiltonian, then the state will have the following time evolution,

|𝑖; 𝑡⟩ = √︁𝑁𝑖𝑒−𝑖𝑡𝜔𝑖− 1

2

𝑡Γ𝑖 |𝑖; 𝑡 = 0⟩

(2.6)

which reproduces the exponential decay expected for the total number, 𝑁𝑖 (𝑡) = ⟨𝑖, 𝑡|𝑖, 𝑡⟩ = 𝑁𝑖𝑒−𝑡Γ𝑖 .

Also consider the population of the final state 𝑓𝑎 as 𝑁 𝑓𝑎 (𝑡). The decay rate of state 𝑖, and the rate

of population of state 𝑓𝑎 are,

(cid:164)𝑁𝑖 (𝑡) = −𝑁𝑖Γ𝑖𝑒−𝑡Γ𝑖

(cid:164)𝑁 𝑓𝑎 (𝑡) = 𝑁𝑖Γ𝑖→ 𝑓𝑎 𝑒−𝑡Γ𝑖

18

(2.7)

(2.8)

(cid:164)𝑁 𝑓𝑎 (𝑡) = − (cid:164)𝑁𝑖 (𝑡), meaning all decaying states decay to some final state. Now consider

Note that (cid:205) 𝑓𝑎
the rate for some statistical mixture of initial states. For instance in spin-1 𝑚 = +1, 0, −1, with

initial populations 𝑁+, 𝑁0, 𝑁−. The final state distribution is,

(cid:164)𝑁 𝑓𝑎 (𝑡) = 𝑁+Γ+1→ 𝑓𝑎 𝑒−𝑡Γ+ + 𝑁0Γ0→ 𝑓𝑎 𝑒−𝑡Γ0 + 𝑁−Γ−1→ 𝑓𝑎 𝑒−𝑡Γ−

(cid:164)𝑁 𝑓𝑎 (𝑡) = 𝑁+𝐵𝑅+1→ 𝑓𝑎 Γ+𝑒−𝑡Γ+ + 𝑁0𝐵𝑅0→ 𝑓𝑎 Γ0𝑒−𝑡Γ0 + 𝑁−𝐵𝑅−1→ 𝑓𝑎 Γ−𝑒−𝑡Γ−

(2.9)

(2.10)

where 𝐵𝑅𝑖→ 𝑓𝑎 = Γ𝑖→ 𝑓𝑎/Γ𝑖. This can be simplified by invoking rotational invariance, which requires

Γ+ = Γ0 = Γ− = Γ. It does not imply that partial decay rates are equal only the summed final rate

[4]. Now consider the final state as a 3 photon state with momenta {k1, k2, k3},

(cid:164)𝑁 (k1, k2, k3) = (𝑁+𝑅++(k1, k2, k3) + 𝑁00𝑅00(k1, k2, k3) + 𝑁−𝑅−−(k1, k2, k3)) Γ𝑒−𝑡Γ

(2.11)

where we have introduced 𝑅𝑖𝑖 (k1, k2, k3) = Γ−1| ⟨3𝛾(k1, k2, k3)| 𝒯 |𝑃𝑠, 𝑖⟩ |2 = 𝐵𝑅𝑖 (k1, k2, k3).

This illustrates that the initial positronium state can be mapped into a distribution of final state

photons, and to do so it is useful to introduce some sort of "normalized partial decay rate" that is

really just the branching ratio for that initial state to that final state. Following Ref. [19] we refer to

this as "the decay matrix." The state populations can be represented in terms of a polarization and

an alignment,

(cid:164)𝑁 (k1, k2, k3) = ( 𝐴(k1, k2, k3) + 𝑠𝑧 𝐵𝑧 (k1, k2, k3) + 𝑃2𝐶00(k1, k2, k3)) Γ𝑒−𝑡Γ

(2.12)

where 𝐴 = 𝑅++ + 𝑅00 + 𝑅−−, 𝐵𝑧 = 𝑅++ − 𝑅−−, and 𝐶00 = 𝑅++ − 2𝑅00 + 𝑅−−. This highlights the

dependence of the angular distribution on the polarization (and alignment) of the initial positronium

atom. As described here, the formalism is too limiting and cannot accommodate many states of

practical importance. To do so we need to use the density matrix as a description of a "mixed"

quantum state.

2.8 Spin-1 positronium mixed state

A normalized spin-1 system requires 8 independent numbers to be fully described. These

can be taken as the 3 components of the vector polarization, and the 5 components of the tensor

19

polarization (a symmetric traceless tensor) [40]. In the Cartesian basis, the density matrix takes

the simple form,

𝜌𝑖 𝑗 =

𝛿𝑖 𝑗 +

1
3

1
2𝑖

∑︁

𝑘

𝜖𝑖 𝑗 𝑘 𝑠𝑘 − 𝑠𝑖 𝑗

(2.13)

where 𝑠𝑖 is the 𝑖th component of the vector polarization and 𝑠𝑖 𝑗 is the 𝑖 𝑗th component of the tensor

polarization. This is described in more detail in Appendix A.

2.9

3-𝛾 phase space

The final state is described by the massless 3-body Lorentz-invariant phase space for 3 identical

particles.

In Ref.

[19] this is written as dΓ 𝑓 , but to avoid confusion in future discussions, we

will call this d 𝑓 3𝛾. The form of the 3-body phase space is worked out in detail in many standard

texts, for example Ref. [2]. In the center of momentum frame the phase space measure takes the

following form,

d 𝑓 3𝛾 =

3
(cid:214)

𝑗=1

d3𝑘 𝑗
2𝜔 𝑗

𝜃 (𝜔1 − 𝜔2)𝜃 (𝜔2 − 𝜔3)
(2𝜋)5

𝛿

(cid:32) 3
∑︁

𝑖=1

(cid:33)

𝜔𝑖 − 𝑚𝑃𝑠

𝛿(3)

(cid:32) 3
∑︁

(cid:33)

k𝑖

𝑖=1

(2.14)

where 𝑚𝑃𝑠 is the rest mass of the initial positronium, 𝜔 𝑗 = |k𝑖 | is the energy of the photon, and

𝜃 (𝑥 − 𝑦) is the Heaviside step function that imposes the energy ordering 𝜔1 > 𝜔2 > 𝜔3. The

ordering of the photons is a restriction of the phase space so we do not overcount final states. This

is ultimately a convention. This is discussed more in Appendix B.

In general any 3-body decay is described by 5 variables. This follows from the 4-momentum of

each particle, minus the energy-momentum relation for each particle, total momentum conservation,

and total energy conservation. This gets 12 − 3 − 3 − 1 = 5 independent variables. We can choose

these to be the energies of two photons, 𝜔1 and 𝜔2, the direction of the normal to the decay plane

ˆn(𝜃𝑛, 𝜙𝑛), and an azimuthal rotation of ˆk1 within the decay plane, 𝜙. The angle between ˆk1 and ˆk2

is fixed by their energies, 𝜓12(𝜔1, 𝜔2).

2.9.1 Tensor decomposition of the decay matrix

The transition matrix maps an initial pure positronium state into a 3-photon final state. We

shorten the notation for the state from |3𝛾; k1, 𝜖𝜖𝜖 1; k2, 𝜖𝜖𝜖 2; k3, 𝜖𝜖𝜖 3⟩ = |3𝛾(k, 𝜖𝜖𝜖)⟩ to match the notation

in Ref. [19]. Take an element of the transition matrix to be ⟨3𝛾(k, 𝜖𝜖𝜖)| 𝒯 |𝑃𝑠, 𝑖⟩. The distribution

20

of final photons for a mixed initial state is given by the decay matrix, which is summed over photon

polarizations as those are not measured,

𝑅𝑖 𝑗 (k1, k2, k3) = Γ−1

𝑜−𝑃𝑠

∑︁

𝜖

⟨3𝛾(k, 𝜖𝜖𝜖)| 𝒯 |𝑃𝑠, 𝑖⟩∗ ⟨3𝛾(k, 𝜖𝜖𝜖)| 𝒯 |𝑃𝑠, 𝑗⟩

(2.15)

where Γ𝑜−𝑃𝑠 is the vacuum lifetime for ortho-positronium. This is normalized so that,

∫

1
3

d 𝑓 3𝛾 ∑︁

𝑖

𝑅𝑖𝑖 (k1, k2, k3) = 1

(2.16)

This contains all measurable information about angular distributions of the photon momenta

from an arbitrary mixed 𝐽 = 1 positronium state, assuming photon polarizations are not measured.

For an observable defined on the final state the expectation value can be calculated as,

(cid:10) ˆO (k1, k2, k3)(cid:11)

𝜌 = ˆO (k1, k2, k3)

∑︁

𝑖 𝑗

𝑅𝑖 𝑗 (k1, k2, k3) 𝜌𝑖 𝑗

(2.17)

Now perform a tensor decomposition of the decay matrix. As a 3×3 Hermitian matrix acting

on the spin-1 Hilbert space it can be decomposed into 9 form factors, one scalar, three vector, and

five tensor. These can be constructed from the three linearly independent vectors ˆk1, ˆk2, and ˆn,

𝑅𝑖 𝑗 = 𝛿𝑖 𝑗 𝑎(𝜔1, 𝜔2) +

1
𝑖

∑︁

𝑙

𝜖𝑖 𝑗𝑙B𝑙 (k1, k2, k3) − 𝐶𝑖 𝑗 (k1, k2, k3),

(2.18)

where,

and,

B = ˆk1𝑏1(𝜔1, 𝜔2) + ˆk2𝑏2(𝜔1, 𝜔2) + ˆn𝑏3(𝜔1, 𝜔2),

(2.19)

𝐶𝑖 𝑗 =(𝜅1𝑖𝜅1 𝑗 −

𝛿𝑖 𝑗 )𝑐1(𝜔1, 𝜔2)

1
3
+ (𝜅2𝑖𝜅2 𝑗 −

1
3
𝜅1𝑖𝜅2 𝑗 + 𝜅1 𝑗 𝜅2𝑖 −

(cid:18)

+

𝛿𝑖 𝑗 )𝑐2(𝜔1, 𝜔2)

(cid:19)

( ˆk1 · ˆk2)𝛿𝑖 𝑗

2
3

𝑐3(𝜔1, 𝜔2)

(2.20)

+ (𝜅1𝑖𝑛 𝑗 + 𝑛𝑖𝜅1 𝑗 )𝑐4(𝜔1, 𝜔2)

+ (𝜅2𝑖𝑛 𝑗 + 𝑛𝑖𝜅2 𝑗 )𝑐5(𝜔1, 𝜔2)

21

Finally, for some observable 𝐴 on the 3-𝛾 final state, the expectation value is,

∫

⟨𝐴⟩𝜌 =

d 𝑓 3𝛾 𝐴(k1, k2, k3)

(cid:40)

𝑎(𝜔1, 𝜔2) + ˆs · B(k1, k2, k3) +

(cid:41)

𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3)

.

(2.21)

∑︁

𝑖 𝑗

where ˆs is the vector polarization, and 𝑠𝑖 𝑗 is the full tensor polarization. This follows from
⟨𝐴⟩𝜌 = ∫ d 𝑓 3𝛾Tr( 𝐴𝑅𝜌)

The authors of Ref.

[19] identified the symmetry properties of each form factor and what

expectation values it would induce. We shorten these here, so for instance ⟨ ˆk𝑎⟩𝜌 means "if this

form factor is nonzero then there is a positronium state 𝜌 that would produce a non-zero expectation

value for one or more of the components ⟨𝜅𝑎𝑖⟩ of photon 𝑎." Similarly the tensor terms are shortened,

so the term ⟨ ˆk𝑎 ˆk𝑏⟩𝜌 really means, "if this form factor is nonzero there is a state 𝜌 that would produce

a non-zero value for ⟨𝜅𝑎𝑖𝜅𝑏 𝑗 + 𝜅𝑎 𝑗 𝜅𝑏𝑖 − 2
3

ˆk𝑎 · ˆk𝑏𝛿𝑖 𝑗 ⟩." This is quoted below,

1. 𝑎(𝜔1, 𝜔2) – The stardard Ore-Powell distribution for unpolarized ortho-positronium decay

2. 𝑏1(𝜔1, 𝜔2), 𝑏2(𝜔1, 𝜔2) – CP-violating form factors. These would induce non-zero expec-

tation value for ⟨ ˆk𝑎⟩𝜌 for any photon 𝑘 𝑎. These terms have never been searched for.

3. 𝑏3(𝜔1, 𝜔2) – This form factor has been searched for 5 times [37, 41, 42, 43, 44], however it

has erroneously been claimed in those searches to be indicative of CPT -violation. A non-

zero value would be indicative of new physics, but would give no indication of the symmetry

properties of that physics [19]. This term would induce a non-zero value for ⟨ ˆn⟩𝜌.

4. 𝑐1(𝜔1, 𝜔2), 𝑐2(𝜔1, 𝜔2), 𝑐3(𝜔1, 𝜔2) – These form factors are non-zero in QED, and are

precisely the terms measured in the early QED angular anisotropy experiments [35, 36].

They generate non-zero values for ⟨ ˆk𝑎 ˆk𝑎⟩𝜌, ⟨ ˆk𝑎 ˆk𝑏⟩𝜌, and ⟨ ˆn ˆn⟩𝜌

5. 𝑐4(𝜔1, 𝜔2), 𝑐5(𝜔1, 𝜔2) – These are indicative of CP-violation. They would generate a

non-zero expectation value for ⟨ ˆk𝑎 ˆn⟩𝜌

These functions are given explicitly in Ref. [19], and reproduced here in Appendix B. The

authors calculated 𝑎, 𝑐1, 𝑐2, and 𝑐3 from QED, and 𝑏1, 𝑏2, 𝑐4, and 𝑐5 in the context of a CP-

violating mixing of 𝑛3𝑆1 and 21𝑃1 states. Though not explicitly stated in Ref. [19] these terms

22

are not completely independent. Within each item above the functions are related due to photon

indistinguishability. The relationships are the following,

𝑎(𝜔1, 𝜔2) = 𝑎(𝜔2, 𝜔1)

𝑏1(𝜔1, 𝜔2) = 𝑏2(𝜔2, 𝜔1)

𝑏3(𝜔1, 𝜔2) = −𝑏3(𝜔2, 𝜔1)

𝑐1(𝜔1, 𝜔2) = 𝑐2(𝜔2, 𝜔1)

𝑐4(𝜔1, 𝜔2) = −𝑐5(𝜔2, 𝜔1)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

this is shown for the scalar, vector, and CP-odd tensor terms in Appendix B. 𝑐3 is also related with

𝑐1 and 𝑐2 although it is not readily apparent in this choice of coordinates.

The CP-violating quantity that has been searched for in Refs. [38, 39] is induced by a nonzero

𝑐4(𝜔1, 𝜔2) and 𝑐5(𝜔1, 𝜔2). It is important to note that there are actually two terms, and both must

be present together due to photon indistinguishability (as shown in Appendix B).

No singular number quantifying CP-violation has appeared, like the proposed 𝐶𝐶𝑃 in Refs.

[38, 39]. Instead some model dependent functions of energies, 𝑐4(𝜔1, 𝜔2) and 𝑐5(𝜔1, 𝜔2), carry the

information. It is possible that the terms 𝐶𝐶𝑃𝑇 and 𝐶𝐶𝑃 were introduced as a rough characteristic of

the scale of the sensitivity. These quantities are inherently model dependent. The structure of the

form factors that contribute (a combination of both 𝑐4 and 𝑐5) also eludes any simple factorization

into a purely geometric factor multiplied by a purely energy dependent factor.

For this reason we find the terminology "coefficients of angular correlations" to be unclear, as

they are functions whose contributions are intertwined with the detector placement and energy cuts

used in each search.

2.10 Optimal B-field for tensor polarization

Throughout this work Γ𝑝−𝑃𝑠 exclusively refers to the para-positronium lifetime in vacuum,

and Γ𝑜−𝑃𝑠 to refer to ortho-positronium lifetime in vacuum. A magnetic field can induce a time
dependent Nenote the relative spin states of the electron and positron as (cid:12)
(cid:11). These can be

(cid:12)𝑠 𝑝
𝑧 , 𝑠𝑒
𝑧

23

combined into states of definite spin in the following combinations,

|𝑆 = 0, 𝑚 = 0⟩ =

1
√

2

(|↑↓⟩ − |↓↑⟩)

|𝑆 = 1, 𝑚 = 1⟩ = |↑↑⟩

|𝑆 = 1, 𝑚 = 0⟩ =

1
√

2

(|↑↓⟩ + |↓↑⟩

|𝑆 = 1, 𝑚 = −1⟩ = |↓↓⟩

(2.27)

(2.28)

(2.29)

(2.30)

The introduction of an external B-field breaks overall rotational symmetry, but maintains azimuthal

rotational symmetry. This means in general there can be mixing between states with the same 𝑚

but different 𝐽, and level splitting between states with different 𝑚, but no mixing of states with

different 𝑚. CPT -symmetry requires that two states related by a CPT -transformation have the

same mass and lifetime [4]. The addition of a B-field does not break CPT symmetry. Under CPT

the 𝑚 = ±1 states transform as Θ |𝑃𝑠, +𝑚⟩ → 𝜁Θ |𝑃𝑠, −𝑚⟩, where 𝜁Θ is a phase. This means that

in the Standard Model (and most extensions that respect CPT ) there is no level splitting between

|𝑃𝑠, 𝑚⟩ and |𝑃𝑠, −𝑚⟩ in a magnetic field to all orders. A more common argument is that CPT

requires the electron and positron have opposite magnetic moments and therefore the net magnetic

moment of positronium cancels when the two are spins are parallel. This is a less general argument

and does not rule out complicated positronium structure effects, possible higher order interactions,

or effects of renormalization of the magnetic moment in the bound state.

Purely by symmetry arguments, positronium in a B-field can only have mixing between states

with the same 𝑚, and level shifts between states with different |𝑚|. The leading order interaction of a

positronium atom with a magnetic field is determined by the magnetic moments of the electron and

positron. The |𝑆 = 1, 𝑚 = ±1⟩ are unaffected by the field and can be disregarded in this discussion.

In the subspace of {|𝑆 = 0, 𝑚 = 0⟩ , |𝑆 = 1, 𝑚 = 1⟩} the Hamiltonian takes the form,

𝑚=0 = 𝐸0𝛿𝑖 𝑗 − 2𝜇𝐵𝑧𝜎𝑖 𝑗
𝐻𝑖 𝑗

1

− Δℎ 𝑓 𝑠𝜎𝑖 𝑗
3

(2.31)

where 𝜇 is the magnetic moment of the electron and Δℎ 𝑓 𝑠 is the hyperfine splitting of positronium.

24

We define 𝑥 = 2𝜇
Δℎ 𝑓 𝑠

|𝐵| = |𝐵|

3.63 T the stationary states are [45],

|𝜓 𝑝𝑆⟩ = cos(𝜃)|𝑆 = 0, 𝑚 = 0⟩ − sin(𝜃)|𝑆 = 1, 𝑚 = 0⟩

|𝜓 𝑝𝑇 ⟩ = sin(𝜃)|𝑆 = 0, 𝑚 = 0⟩ + cos(𝜃)|𝑆 = 1, 𝑚 = 0⟩
√︁

Δℎ 𝑓 𝑠 (1 +

1 + 𝑥2)

𝐸 𝑝𝑆 = 𝐸0 −

1
2
1
𝐸 𝑝𝑇 = 𝐸0 +
2
√︄

cos(𝜃) =

1
√

2

1 +

√

1
1 + 𝑥2

Δℎ 𝑓 𝑠 (1 +

√︁

1 + 𝑥2)

(2.32)

(2.33)

(2.34)

where 𝑝𝑆 is short for "pseudo-singlet" and 𝑝𝑇 is short of "pseudo-triplet." Finally, the singlet and

triplet states have relative lifetimes 𝜏𝑝−𝑃𝑠 = 125 ps and 𝜏𝑜−𝑃𝑠 = 142 ns. This results in the new

mixed lifetimes,

Γ𝑝𝑆 = cos2(𝜃)Γ𝑝−𝑃𝑠 + sin2(𝜃)Γ𝑜−𝑃𝑠

Γ𝑝𝑇 = sin2(𝜃)Γ𝑝−𝑃𝑠 + cos2(𝜃)Γ𝑜−𝑃𝑠

(2.35)

(2.36)

Even at modest field values of the order of 1 Tesla, there is only a small amount of mixing but a

large quenching of the lifetime. This is shown in Figure 2.7, which shows the ortho-positronium

decay rate in vacuum and in a magnetic field. This follows from the factor of 103 difference in the

lifetimes of singlet and triplet positronium. This induces a time evolution of the effective 𝑚 = 0

state relative to the 𝑚 = ±1 states. By careful selection of time cuts, the different spin states of

positronium can be separated.

The instantaneous alignment for a decay event is,

𝑃2(𝑡) =

𝑛𝑇 (𝑡) − 2𝑛𝑝𝑇 (𝑡)
𝑛𝑇 (𝑡) + 𝑛𝑝𝑇 (𝑡)

(2.37)

where 𝑛𝑖 (𝑡) = 𝑁𝑖/𝜏𝑖𝑒−𝑡/𝜏𝑖 is the instantaneous rate of decays from that state, and 𝑁𝑖 is the initial

population of that state. Averaging over a time window, not all events have the same alignment.

The average needs to be weighted by the number of decays with that alignment,

⟨𝑃2⟩ 𝜌 =

∫ 𝑃2(𝑛)d𝑛
∫ d𝑛

=

d𝑡 d𝑡

∫ 𝑃2(𝑡) d𝑛
∫ d𝑛
d𝑡 d𝑡

25

(2.38)

Figure 2.7 Time spectrum for positronium decay. The time spectrum for positronium in vacuum
is shown in red. The addition of a B-field shifts one lifetime component to be shorter, shown in
blue. At the same time, the B-field shortens the lifetime by opening up a decay channel to 2𝛾, so
the actual number of 3-𝛾 events for this component is greatly reduced (black).

with the number of decays being the infinitesimal times the rate,

⟨𝑃2⟩ 𝜌 =

(cid:0)𝑛𝑇 (𝑡) + 𝑛𝑝𝑇 (𝑡)(cid:1)(cid:17)

∫ d𝑡

(cid:16) 𝑛𝑇 (𝑡)−2𝑛 𝑝𝑇 (𝑡)
𝑛𝑇 (𝑡)+𝑛 𝑝𝑇 (𝑡)
∫ d𝑡 (cid:0)𝑛𝑇 (𝑡) + 𝑛𝑝𝑇 (𝑡)(cid:1)
𝑡1
𝑁𝑇 𝑒−𝑡/𝜏𝑇 − 2𝑁 𝑝𝑇 𝑒−𝑡/𝜏𝑝𝑇 (cid:17) (cid:12)
(cid:12)
(cid:12)
(cid:12)
𝑡2
𝑡1
(cid:12)
(cid:12)
(cid:12)
(cid:12)
𝑡2

(cid:0)𝑁𝑇 𝑒−𝑡/𝜏𝑇 + 𝑁 𝑝𝑇 𝑒−𝑡/𝜏𝑝𝑇 (cid:1)

(cid:16)

=

(2.39)

for the time window [𝑡1, 𝑡2].

However, this has not included the changing branching ratios. This new pseudo-triplet state

has some probability to decay to 2-photons and some to 3-photons. Figure 2.7 shows the ortho-

positronium decay rate in a magnetic field, and the rate of 3-𝛾 decays. There is a net quenching

of 3-𝛾 decays at early times (from the pseudo-triplet). This means increasing the field value also

decreases the statistics. The relationship is given as

𝐵𝑅(o-Ps → 3𝛾) = 𝐵𝑅3𝛾 = 𝜏𝑝𝑇 /𝜏𝑜−𝑃𝑠

(2.40)

The alignment increases for larger 𝐵, but the counts decrease. To identify an optimal field value

it is necessary to define a "Figure of Merit." Consideri the statistical sensitivity: doubling the tensor

26

0100200300400500T (ns)D4-103-102-101-10events (arb)o-Ps decay rate vacuumo-Ps decay rate (B=0.4 T) event rate (B=0.4 T)g3-polarization doubles the size of the asymmetry, but doubling the branching ratio only increases the

statistical sensitivity by a factor of

√

2. We take the Figure of Merit as,

𝐹𝑜𝑀 = ⟨𝑃2(𝐵)⟩𝜌

√︁𝑁 (𝐵)

(2.41)

Returning to Equation 2.39 shows that the Figure of Merit is defined for some time window. To

match the Michigan experiment, we identify 2 time windows with opposite tensor polarizations, a

start window of 10 ns, and a stop of 500 ns. This determines the time cut between the two windows.

The result of optimal time cut versus lifetime for these time windows is plotted in Figure 2.8.

(a)

(c)

(b)

(d)

Figure 2.8 The effect of a magnetic field on positronium. (a) The optimal timing cut between two
time windows to form populations with opposite tensor polarization (given an initial time start at
10 ns and final at 500 ns. (b) The instantaneous alignment for the population. (c) The tradeoff
with increasing tensor polarization is the decreasing branching ratio decreasing statistics. (c) The
Figure of Merit, tensor polarization weighted by square root of the counts. For the perturbed time
window there is a clear optimal value between 0.4-0.5 Tesla. For unperturbed the optimal choice is
as large a field as possible (kill the pseudo-triplet and only take triplet events with complete tensor
polarization).

27

00.10.20.30.40.50.60.70.80.91B (Tesla)20406080100 (ns)cutt00.10.20.30.40.50.60.70.80.91B (Tesla)00.10.20.30.40.50.60.70.80.9|2|P00.10.20.30.40.50.60.70.80.91B (Tesla)00.20.40.60.81)gBR(300.10.20.30.40.50.60.70.80.91B (Tesla)00.20.40.60.811.2FoMperturbedunperturbedThe Figure of Merit versus the B-field is shown in Figure 2.8d. This shows a value around

0.45 T optimizes the statistical sensitivity. This is in agreement with the field values used in the

previous experiments. The time spectroscopy related quantities from Refs. [38, 39] are tabulated

in Table 2.1. This includes the values obtained using Equation 2.39. We do not expect a perfect

reproduction of the quoted values because of corrections for systematic effects. However we are

not able to reproduce the quantities quoted in both texts.

Group
Michigan

Tokyo

𝜏𝑝𝑇 (ns)

𝜏𝑇 (ns)

𝑡0 (ns)

30
30
22.5

124
124
126

10.3
65
50

𝑡1 (ns) Calculated ⟨𝑃2⟩ Quoted ⟨𝑃2⟩
0.37
negative
positive
0.87

−0.43
0.68
0.57

64.6
270
130

Table 2.1 Summary of time-spectroscopy related quantities for Refs. [38, 39], and the results from
Equation 2.39. These values do not include any systematic corrections.

There is one systematic effect that is particularly important for this discussion and that is "the

2-𝛾 dilution." The quenching of the 𝑚 = 0 triplet state is performed by mixing with the singlet state,

allowing decay to 2-𝛾. As the vacuum decay strength for para-positronium to 2-𝛾 is a factor of one

thousand times greater than ortho-positronium to 3-𝛾, a small mixing induces a large branching

ratio to 2-𝛾 for the pseudo-triplet state. This means that the relative 2-𝛾 and 3-𝛾 branching ratios

are different for the different spin states.

The University of Michigan group considered the 2-𝛾 dilution in terms of how many 2-𝛾 events

could mimic a true 3-𝛾 decay (by Compton scattering in such a way that they passed the energy

cuts). They determined that 12.8% of the events in the first time window were due to 2-𝛾 events,

and 3.8% for the second time window. Indeed taking our calculated ⟨𝑃2⟩ for the first time window

and correcting by a 12.8% dilution recovers an averaged alignment of -0.37.

It has little to no

change on the second time window.

We can reproduce the University of Tokyo group’s value by applying the 2-𝛾 quenching correc-

28

tion to the pseudo-triplet’s contribution to the signal. They calculated the alignment as,

(cid:16)

𝑁𝑇 𝑒−𝑡/𝜏𝑇 − 2𝑁 𝑝𝑇 𝐵𝑅3𝛾𝑒−𝑡/𝜏𝑝𝑇 (cid:17) (cid:12)

(cid:11)

(cid:10)𝑃′
2

𝜌 =

(cid:0)𝑁𝑇 𝑒−𝑡/𝜏𝑇 + 𝑁 𝑝𝑇 𝐵𝑅3𝛾𝑒−𝑡/𝜏𝑝𝑇 (cid:1)

𝑡1
(cid:12)
(cid:12)
(cid:12)
𝑡2
𝑡1
(cid:12)
(cid:12)
(cid:12)
(cid:12)
𝑡2

(2.42)

where the dilution is included in terms of the state that it relates to. Our estimation produces a

value of 0.91 using this definition.

Ultimately it is not immediately clear how the alignments were calculated, or how 2-𝛾 dilutions

were accounted for. It does seem that these were treated by mutually exclusive methods by both

groups.

2.11 Open question about time evolution of signal

According to the theoretical analysis in Ref. [19] the angular anisotropies are proportional to

the tensor polarization, both the QED anisotropy (due to the form factors 𝑐1, 𝑐2, and 𝑐3) and the

possible CP-odd anisotropy (due to 𝑐4 and 𝑐5). The QED anisotropy was measured by the groups

of Wheatley and Halliday, and Hughes, Marder, and Wu in Refs. [35, 36]. These groups used a

magnetic field to quench the 𝑚 = 0 state. Similarly, the University of Michigan group used a magnet

to quench the 𝑚 = 0 state and searched for the CP-odd tensor form factor. In this experiment

the authors induced a time dependence and state that the signal changes sign between two time

windows due to the flipping of the alignment (more 𝑚 = 0 decay at early times, and more 𝑚 = ±1

at late times). The treatment of the time dependence in the Michigan experiment is inconsistent

with the treatment in the two previous experiments.

Firstly, at any given time the number of 3-𝛾 events in a magnetic field will always be less than

that in a vacuum. This can be seen quite generally as follows: take a state 𝐴 with partial width Γ𝐴𝐵

to state 𝐵. Now suppose we can turn on a new final state 𝐶 with partial width Γ𝐴𝐶. Opening new

channels will always decrease the instantaneous rate of decay to the existing channels. This can be

explicitly shown by taking the difference between the two rates of decays to the state 𝐵 for the two

conditions,

ΔΓ𝐴𝐵 (𝑡) =

(cid:16)

(Γ𝐴𝐵 + Γ𝐴𝐶)𝐵𝑅𝐴→𝐵𝑒−(Γ𝐴𝐵+Γ𝐴𝐶 )𝑡 − Γ𝐴𝐵𝑒−Γ𝐴𝐵𝑡 (cid:17)

29

(cid:18)

=

(Γ𝐴𝐵 + Γ𝐴𝐶)

Γ𝐴𝐵
Γ𝐴𝐵 + Γ𝐴𝐶

= Γ𝐴𝐵𝑒−Γ𝐴𝐵𝑡 (cid:16)

𝑒−Γ𝐴𝐶 𝑡 − 1

(cid:17)

𝑒−(Γ𝐴𝐵+Γ𝐴𝐶 )𝑡 − Γ𝐴𝐵𝑒−Γ𝐴𝐵𝑡

(cid:19)

ΔΓ𝐴𝐵 (𝑡) < 0

(2.43)

(2.44)

The takeaway is that in the presence of a magnetic field the 3-𝛾 rate will always be decreased at all

times.

According to the tensor decomposition into irreducible form factors in Ref.

[19], the QED

anisotropy and the CP-odd tensor correlation must have the same dependence on the positronium

state 𝜌, which in Ref. [19] is parameterized by the vector and tensor polarization. But as argued

in Equation 2.39, starting with an even state population in a B-field the decaying positronium will

have a tensor polarization of changing sign that averages to zero (in the end all the states decay and

we started with a uniform state population). But this is at odds with the experimental results from

the early QED anisotropy measurements. In Ref. [36], the positronium decays in a tuneable in a

magnetic field and measured the reduction of counts in the plane perpendicular to the magnetic field

axis. However the authors did not measure a time spectrum, they did not trigger on the 𝛽-emission

and record a time difference between Ps formation and decay. This means that the experiment was

only sensitive to a net anisotropy. They observed that in a B-field the net angular distribution of

3-𝛾 decays was anisotropic. It did not distort in a positive way at early times, then flip to a negative

distortion at later times such that the total decay distribution was isotropic.

The entire line of symmetry violating angular distribution searches in positronium decay was

started by the pioneering studies at University of Michigan. They identified the correlation ⟨𝑛𝑧⟩ as

a correlation that would indicate new physics. As it is a vector correlation would require polarized

positronium [37]. They identified the tensor correlation ⟨𝜅1𝑧𝑛𝑧⟩ as being indicative of CP-violation,

and as it is a tensor correlation it would require tensor polarized positronium. This statement is

unequivocally correct.

The authors of Ref. [19] worked out the general theory of angular correlations in 3-𝛾 decay

of ortho-positronium. The tensor terms in the angular distribution would be driven by the tensor

30

polarization in the positronium, which would imply for these experiments a signal that integrates

to zero over the full time spectrum. The inconsistency between the Michigan treatment and the

early QED anisotropy tests comes from the assumption of rotational invariance in the theoretical

treatment. The authors of Ref.

[19] equated the tensor term in the positronium state with the

tensor term in the final state photon distribution. However the addition of a B-field had broken

rotational invariance: total angular momentum and tensor polarization were no longer conserved

quantities in the positronium time evolution and decay. Indeed a direct consequence of rotational

invariance is that all 𝑚 states have the same lifetime and energy [4] which is certainly not the

case for positronium in a B-field. This further means that this analysis evades a simple rate-like

equation like that given in Equations 2.5 and 2.12. Any such equation cannot be correct as the

positronium state is a combination of different lifetime components. Indeed when we illustrated

the time dependence of the various angular distributions in Equations 2.9 through 2.12, we had to

invoke rotational invariance to factor out the overall exponential and re-write the time dependence

of the angular correlation in terms of the alignment.

The analysis of Ref.

[19] is directly applicable to tensor polarized positronium without an

external field. This can only be produced by impinging polarized positrons on polarized electrons

(this is calculated in Appendix A). A proper accounting of the angular correlations including

the induced time-dependence will require further in-depth analysis and will be the main focus of

Chapter 10.

For the majority of this work it is sufficient to assume that we have a tensor polarized positronium

source without entering into the details of the pseudo-triplet state and 2-𝛾 branching ratio. We

could realize this experimentally by using the B-field to quench the pseudo-triplet and choosing

a time window after the pseudo-triplet has decayed. This of course still has the broken rotational

invariance, however there is no level splitting between the |𝑆 = 1, 𝑚 = ±1⟩ states in a magnetic

field. At late times the B-field is having no effect on the dynamics. The system can be treated as

if it had a maximal positive tensor polarization. This is how we will interpret the analysis for most

of this work. Throughout we will treat the direction of alignment, and the axis of the B-field as

31

intechangeable concepts, at least insofar as their effect on the observable.

2.12 The experiment

We intend to measure the CP-odd tensor form factors 𝑐4(𝜔1, 𝜔2) and 𝑐5(𝜔1, 𝜔2) and to

achieve a 10-fold improvement in sensitivity over previous experiments. The planned experiment

is structurally similar to the Michigan experiment, with the addition of greatly increased solid angle

coverage, and replacing the permanent magnets with an electromagnet. This requires designing

and constructing a dedicated detector array from scratch.

32

CHAPTER 3

INITIAL DESIGN AND SIMULATIONS

This chapter covers the general design work in close contact with the analytic structure of the ortho-

positronium decay distribution. First we discuss extracting the signal by forming an asymmetry.

This is followed by a brief discussion of the superconducting magnet, that constitutes the main

geometric constraint for the experiment. Next we outline the basics of the geometry forming

"configurations" between sets detector pairs. We identify both a cylindrical array and spherical

array geometry. These have tradeoffs between solid angle, analyzing power, and detector efficiency

(by varying possible detector sizes). Further investigation requires defining a "Figure of Merit" to

weight sensitivity of different designs.

The experiment follows the basic structure of the University of Michigan measurement [38].

They had a triplet of detectors, one for highest energy, two for second highest, centered on a

positronium source. This is illustrated in Figure 3.1a. They oriented the detectors such that the

signal is positive or negative between the two pairs of detectors.

Now as we need very high statistics we must fit as many sets of detectors in the array as possible,

with the constraint that the magnetic field is horizontal. This leads to the conceptual design shown

in Figure 3.1b, consisting of circular rings of detectors following the cylindrical geometry of the

magnet. This will allow many detector combinations between the rings, and leads to a dramatic

increase in the number of possible detector pairs.

3.1 Extracting the symmetry violating term

The previous measurements utilized an "asymmetry" to extract the symmetry violating term

[38, 39]. They took two pairs of detectors that see some number of decay events. Call these UP

and DOWN. Assuming equal efficiencies the coincidence counts go as,

𝑁𝑈 = 𝑁 (1 + 𝑃2𝛼𝐺 𝑎𝑛)

𝑁𝐷 = 𝑁 (1 + 𝑃2𝛼(−𝐺 𝑎𝑛))

(3.1)

(3.2)

33

(a)

(b)

Figure 3.1 (a) A single "configuration," a set of three detectors taken in two pairs, sharing a highest
energy detector. (b) Rotating the configuration around the magnetic field axis allows many sets of
detectors to be placed in a ring.

where 𝑁 is the total normalization, 𝑃2 is the alignment, 𝐺 𝑎𝑛 is the "geometric analyzing power"

and purely dependent on the geometry of the event, and 𝛼 is the symmetry violating term. They

defined their analyzing power as 𝐺 𝑎𝑛 = 𝜅1𝑧 ( ˆ𝑘1 × ˆ𝑘2)𝑧 and we will mirror this analysis until Chapter

9, whose main purpose is to define an analyzing power in the context of the theoretical analysis of

Ref. [19]. The detectors were arranged so that 𝐺 𝑎𝑛 changes sign between the two configurations.

𝛼 could then be extracted by forming an asymmetry,

𝛼 =

1
𝑃2𝐺 𝑎𝑛

(cid:19)

(cid:18) 𝑁𝑈 − 𝑁𝐷
𝑁𝑈 + 𝑁𝐷

(3.3)

This cancels the total normalization, and the background cannot induce a false signal. An additive

background does increase the denominator, and therefore reduces the sensitivity.

3.2 Planned experiment

Both previous experiments were sensitive to possible asymmetries induced by 𝛾’s scattering on

their iron magnets. We will remove this effect by designing the entire experiment to fit inside the

Positron Polarimeter magnet, a superconducting magnet with maximum field value of 2 Tesla at

FRIB.

This will be the primary geometric constraint for the experiment, the diameter of the warm

bore is 22 cm. The primary goal will be to fit as many 𝛾-detector sets in the magnet as possible

34

Figure 3.2 One of the two FRIB Positron Polarimeter magnets. The detector array will be placed
in the warm bore of the magnet.

to achieve a large increase in statistical sensitivity. This will require relatively small scintillators

ideally read out directly in the magnet, mounted onto an array with many sets aimed at the central

source.

3.3 Basic geometry

Fix the B-field direction along the ˆ𝑧-axis. Following Ref. [39], define the "geometric analyzing

power" for an event as the ˆ𝑧 component of ˆk1 multiplied by the ˆ𝑧 component of the normal to the

decay plane, 𝐺 𝑎𝑛 = 𝜅1𝑧 ( ˆk1 × ˆk2)𝑧. The larger this quantity the better the sensitivity. Parameterize

the detector pairs as shown in Figure 3.3a. Hold the cylindrical radius constant, and plot 𝐺 𝑎𝑛 versus

the position along 𝑧. Define the unitless position of the ring that detects the highest energy photon

as 𝜒1 = 𝜌1/𝑧1, the ring that detects the second highest energy photon as 𝜒 − 2 = −𝜌2/𝑧2, and the

azimuthal angle between the two as Φ. This gives the geometric analyzing power as,

𝐺 𝑎𝑛 =

𝜒1sin(Φ)
√︃

(1 + 𝜒2
1

)

1 + 𝜒2
2

(3.4)

Plotting this for fixed Φ = 90◦ gives the distribution shown in Figure 3.3b. This alone does not

determine the optimal detector placement as it includes no actual information about the positronium

35

decay. Kinematically the opening angle is constrained between 120◦ and 180◦, as shown in Figure

2.3c. To investigate this we must simulate the distribution of 3 photon events and create a Figure

of Merit, which we calculate by weighting the geometric sensitivity by the statistical sensitivity.

(a)

(b)

Figure 3.3 (a) The coordinate system for the cylindrical ring geometry, where 𝜒1 = 𝑧1/𝜌1, and
𝜒2 = −𝑧2/𝜌2. (b) The geometric analyzing power for that ring placement (plotted for Φ = 𝜋/2).
This does not account for kinematics and includes non-physical regions.

3.4

Initial positronium event generator

The phase space for a three-body decay to zero mass particles is flat in the energies 𝜔1 vs 𝜔2.

The angles between the photons is entirely determined by their energies and is given as,

cos(𝜓𝑖 𝑗 ) = 1 − 2𝑚𝑒

𝜔𝑖 + 𝜔 𝑗 − 𝑚𝑒
𝜔𝑖𝜔 𝑗

(3.5)

The energy distribution for unpolarized ortho-positronium decay follows the Ore-Powell distribution

[26],

∝

1
𝜔1𝜔2𝜔3

{(1 − cos(𝜓12))2 + (cycl. perm.)}

(3.6)

This is taken as a probability distribution function, and two energies are sampled from it. We

start with ˆk1 along the ˆ𝑥-axis, and ˆn along the ˆ𝑧-axis. With these specified, the direction of ˆk2 is

fully determined, since the angle between ˆk1 and ˆk2 is dependent on the energies 𝜓12(𝜔1, 𝜔2). Now

36

we generate a random angle 𝜙 and rotate both ˆk1 and ˆk2 around ˆn by the angle 𝜙. Finally a random

direction is thrown, and the decay plane is rotated so that the normal points in that direction.

Note that the angular distribution and energy distribution are fundamentally intertwined. Spec-

ifying an opening angle between detectors inherently determines an energy range for 3 photon

events, as shown in Figure 3.4. Finite detector solid angle will make the wedge wider. For this

reason we must investigate the angular distribution and energy distribution in parallel. Detector

placement and energy dependent effects (cuts, resolution, efficiency) are considered in a combined

way.

Figure 3.4 Given the energy of two photons 𝜔1 and 𝜔2, the opening angle is entirely determined
kinematically. In red the phase space condition of 𝜔1 > 𝜔2 > 𝜔3 restricts events to this region.

3.5 Geometric Optimization

To compare different designs we must define a quantitative Figure of Merit that combines the

geometric analyzing power with the statistical weight of the number of the events,

𝐹𝑜𝑀 = 𝐺 𝑎𝑛

√

𝑁

(3.7)

It is worth stressing that Φ in Equation 3.4 is the standard cylindrical azimuthal coordinate, not the

angle between the two photons. The two angles are related but that relation depends on the distance

between the rings. The Figure of Merit is a function of three variables, so we need to make some

simplifying assumptions.

37

Firstly, we consider two symmetrically placed rings, 𝜒1 = 𝜒2 = 𝜒. The Figure of Merit as a

function of 𝜒 versus Φ is shown in Figure 3.5. This shows that the sensitivity maximizes between

130 − 165◦, with ring placement 0.3 < 𝜒 < 0.8; this corresponds to an angle between the central

ring and the outer ring of 11◦ < 𝛽 < 39◦.

Figure 3.5 Symmetric configuration, two rings symmetrically placed. The corresponding Figure
of merit is shown for the ring placement.

Now we consider a second configuration, where one ring is centered on the positronium source.

Plotting the Figure of Merit for events where the highest energy photon hits the outer ring, and the

second highest hits the central ring as a function of the placement of the outer ring gives Figure 3.6.

Comparing the two distributions shows the optimal placement of the outer ring lines up for both

configurations. This means we can consider a three ring design, with Symmetric configurations

and Asymmetric configurations.

We will proceed with the design incorporating three rings, two outer rings and one central ring.

This design has two categories of events, "Symmetric" events have both photons hit the outer ring,

and "Asymmetric" events have one photon hit an outer ring and the other hit the middle ring. The

events where ˆk1 hits the central ring has no sensitivity for the analyzing power defined in Equation

3.4. Throughout this chapter "Asymmetric events" exclusively refers to events with ˆk1 in an outer

ring.

38

Figure 3.6 Asymmetric configuration, one ring centered around the source with another offset along
the B-field axis. The corresponding Figure of Merit is shown for the ring placement.

This is still an idealized distribution.

It is easy enough to consider a "solid angle" for the

detector by drawing a box over the diagrams, but in practice there is a solid angle to consider for

both detectors. This would lead to a smearing out of the distribution shown (smearing out ˆk1 over

one detector face), then specifying detector 2 by placing a box on the diagram.

3.5.1 Cylindrical or spherical

There are two simple designs to place the detectors within the warm bore of the magnet: a

cylindrical design (mirroring the geometry of the warm bore), and a spherical design (where the

outer rings are tilted inwards). These are shown in Figure 3.7. The cylindrical design is much

simpler, makes detector mounting substantially easier, and enables the addition of simple shielding

between rings. However, the spherical design features increased solid angle for the outer detectors,

and the possibility of a wider opening angle. The tapered crystal design is ideal for the spherical

array as the angle is set so the detector takes up optimal space within its solid angle, whereas the

cylindrical only leaves room for a taper along the ˆ𝜙-direction.

The distribution of events is shown in terms of the opening angle between the two photons

in Figure 3.8. This assumes 16 crystals in a ring, and displays the events in the Symmetric

and Asymmetric configurations labeled by the azimuthal angle between the two detectors. The

39

Figure 3.7 Two possible designs, either a set of cylindrical rings with shielding between each ring,
or tilted outer rings all aimed at a central source. The tilted design allows for increased solid angle
and wider opening angle between the photons but is substantially harder to fabricate.

combined distributions are shown in Figure 3.9. The Symmetric events gain from the tilted design,

this is due to the increased solid angle for both detectors.

Figure 3.8 Distribution of all events where the highest energy photon hits the outer ring. Shown is
the angle between the two photons for each pair of detectors that see the events. The configurations
are labeled as 1) Symmetric when both photons hit outer rings, or 2) Asymmetric when the highest
energy photon hits an outer ring, and the second highest hits the inner ring. The number next to
Symmetric (Asymmetric) refers to the azimuthal angle between the two detectors).

The spherical geometry has an increased geometric acceptance, with an increase of roughly

60%. For this reason we choose to pursue the spherical design despite being more difficult to

construct.

3.5.2 Number of crystals

The warm bore has a diameter of 22 cm. The design must include realistic space for the readout

and mounting on the back-end of the crystals. We budget 9 cm from the center of the magnet to

40

100110120130140150160170180(deg)12y020406080100120140160180200counts (arb)Symmetric 157.5Symmetric 135Symmetric 112.5Asymmetric 157.5Asymmetric 135Cylindrical Array100110120130140150160170180 (deg)12y020406080100120140160180200counts (arb)Spherical ArrayFigure 3.9 Summing all the distributions in Figure 3.8 shows the increase in statistics for the
spherical design. Note that each configuration has a different sensitivity so directly summing the
counts only captures the statistical increase. It is more correct to interpret this as a 1-D projection
of the phase space seen by the detector pairs.

the back-end of the crystal (leaving 2 cm behind for mounting and readout).

The design can accommodate 12 or 16 crystals in each ring. Increasing the number of crystals

increases the angular granularity of the array, but also decreases the efficiency of the detector

themselves (in that they need to be smaller to fit more detectors in the same space). The complicated

interplay between the number of detectors in a ring, the size of the detectors, and the angle of the

tilt for the outer ring is illustrated in Figures 3.10 and 3.11.

Following the coordinates from Figure 3.10, the geometric constraint goes as,

cot(𝛼/2)cos(𝛽) − sin(𝛽) > cot(𝜋/𝑛)

(3.8)

This is plotted for the twelve crystal array and the sixteen crystal array with a radius of 6 cm in

Figure 3.12. In principle any choice beneath the blue (red) curve is valid for the sixteen (twelve)

crystal array.

Considering the possible pairings, the 16 detector configuration offers a substantial increase in

the number of pairs with kinematic sensitivity. In principle each detector has 2 sets of pairs in the

other rings for 16 detectors compared to 1 set of pairs for 12 detectors. There is a change in the

possible solid angle of the detectors (so the 12 detector array crystals sees more events). However,

the increased granularity is worthwhile in and of itself.

41

100110120130140150160170180 (deg)12y050100150200250counts (arb)Spherical ArrayCylindrical Array(a)

(b)

Figure 3.10 Diagrams illustrating the relationship between the tilt angle of the outer ring, and the
geometric constraint on the crystal size. (a) The side view, with detectors of opening angle 𝛼, and
a ring tilt of 𝛽. This gives the cylindrical radius of the outer ring as 𝜌. (b) Front view for a 5 crystal
ring. Once the edges of the crystals are touching we cannot tilt the ring farther. This defines a
minimum radius 𝜌𝑚𝑖𝑛. This gives the geometric constraint of 𝜌 > 𝜌𝑚𝑖𝑛, and is given in terms of 𝛼
and 𝛽 for an n-detector ring by Equation 3.8.

Figure 3.11 Illustration of the geometric concept in Figure 3.10, holding the crystal size constant
they get closer and closer as we tilt the outer ring, until they touch.

3.6 Summary

We will construct an array of detectors to fit inside the FRIB Positron Polarimeter magnets.

These will have 3 rings with 16 detectors in each ring. The detectors will be roughly 3 cm deep

and have a front face between 1-2 cm in width. The outer two rings will be tilted inwards towards

the central source for positronium formation.

42

Figure 3.12 The geometric constraint for the crystals based on angle of ring (if the crystal is 6 cm
from the source). For a given tilted angle between the central ring and the outer rings, the largest
size crystal is shown. In principle any choice below the blue curve is possible for 16 crystal ring,
and any below the red is possible for a 12 crystal ring.

43

CHAPTER 4

𝛾-DETECTOR PROTOTYPING

4.1 Overview

This experiment will require high statistics to reach the target sensitivity. Since the array will be

constructed from scratch to perform this specific measurement, it is worth spending time optimizing

the individual detectors before purchasing and constructing the full array. This chapter is entirely

focused on the study of the individual crystals and their response within the energy range of interest

(sub MeV photons). We first give a basic overview of scintillation detectors and photomultipliers

including the specific implementation we choose. This is followed by some simulations of single

crystal efficiency as a function of detector geometry. Once we have settled on a design we present

the initial testing of single crystal prototypes.

4.2 Scintillation detectors

In general 𝛾-detectors can be simple scintillation detectors. There is a monolithic crystal with

a high Z-value, the 𝛾 hits the crystal and creates an amount of scintillation light proportional to the

energy deposited. The scintillation light is collected by some photomultiplier and is transduced into

an electrical current that is recorded by a data acquisition system [22]. The principle is illustrated

in Figure 4.1. At the energies of interest the 𝛾-ray can either Compton scatter or be fully absorbed.

There is the possibility for multiple scatterings as well. The full energy peak is composed of all

events that deposit their full energy (either through photoabsorption, or multiple scatterings until

full absorption), and the Compton continuum is the plateau of events with a shape determined by

kinematics. An ideal detector has all events in the full energy peak, however there is both a material

dependence and an energy dependence for the relative cross-sections of these processes [22].

4.2.1 LYSO crystal

Common inorganic scintillation crystals used for 𝛾-ray spectroscopy are NaI, CsI, LaBr3. These

are usually doped with "light emitters," elements that add states in the bandgap and therefore allow

scinitllation to occur. A comparative study of these inorganic scintillators for high energy physics

experiments can be found in Ref. [46]. The University of Michigan measurement utilized NaI

44

(a)

(b)

Figure 4.1 a) Illustration of the interaction of a low energy photon with a scintillation detector. b) A
sample spectrum for a 137Cs source with a (background subtracted) LYSO crystal. The x-axis is the
digitized current output from the photomultipler. The spectrum shows a clear peak and Compton
plateau.

scintillators [38], and the University of Tokyo measurement utilized LYSO crystals [39], which

is becoming the standard in the PET industry [47, 48]. LYSO refers to Cerium doped Lutetium

Yttrium Oxyorthosilicate (Lu1.9Y0.1SiO5). These crystals have a density of 7.25 g/cm3, a decay

time of 40 ns, and a light yield of 30 photons/keV. [49]. These crystals are increasingly being

used for calorimeters in nuclear and high energy physics experiments [50, 51]. LYSO crystals

suffer from internal radioactivity from the 176Lu, creating 3.9 cps/g [52]. This gives a constant

background of singles for the detectors. The decay scheme is shown in Figure 4.2. The internal

Figure 4.2 Decay scheme for 176Lu, taken from Ref. [52]. The nucleus decays with a half life
of 3.76×1010 years to 176Hf by 𝛽 emission with an enpoint energy of 593 keV (right). This is
accompanied by 3 characteristic 𝛾-emissions at 307 keV (6+ → 4+), 202 keV (4+ → 2+), and 88
keV (2+ → 0+).

45

radioactivity is a 𝛽-decay followed by a 𝛾-cascade, and completely covers the energy region of

interest for ortho-positronium decay (less than 511 keV). Sample spectra are shown in Figure 4.3.

To recover the clean spectrum shown in Figure 4.3 we had to run twice, once with a source and

once without. Careful matching of the runtime and monitoring for gain shifts allowed us to perform

a background subtraction to remove the intrinsic radioactivity backgrounds. This was performed

for all single crystal studies presented in this chapter, unless otherwise specified. The internal

radioactivity has been extensively studied in Refs. [52, 53].

Figure 4.3 Illustration of background subtraction for a LYSO crystal read out by a silicon photo-
multiplier, using a 137Cs source. Spectra taken at WU.

4.2.2 Silicon photomultiplier

A silicon photomultipler (SiPM) is an array of single photon avalanche diodes, held at break-

down voltage. The working principle is that an incoming scintillation photon from the crystal

causes the breakdown in the photodiode and allows passage of current. The current from the

SiPM is proportional to the number of photodiodes that experienced breakdown, which is ideally

proportional to the number of scintillation photons emitted in the crystal, which finally is ideally

proportional to the energy deposited by the 𝛾. A thorough overview of silicon photomultipliers can

be found in Refs. [54, 55].

For the LYSO array, we plan to use SiPMs in-lieu of PMTs. The SiPMs are much smaller than

a PMT, they can run at a much lower voltage (around 25-30 V), and they are not affected by a

magnetic field. This will allow us to place the detector modules directly in the warm bore of the

46

020040060080010001200140016001800ADC units0123456310·counts per binsourcebackgroundsubtractedmagnet itself. The combination of LYSO crystals read out with an SiPM is well characterized and

becoming a standard technology in PET systems [47, 48].

Figure 4.4 Prototype silicon photomultipler for detector readout. This board features a 2×2 array
of 6 mm SiPMs, no active amplification, and readout connections on the opposite side of the board.

4.3

Initial tests

Initial testing of the LYSO crystals read out by SiPMs were performed in Spring 2019. We

had pairs of LYSO crystals and CsI crystals in small and large sizes, shown in Figure 4.5. The

resolution of the crystal is based on the counting statistics of the scintillation photons. CsI has a

higher light output (about 50 photons/keV) than LYSO (about 30 photons/keV), and we therefore

expect a worse resolution for LYSO. CsI has a slower primary decay time at 4 𝜇s compared to 40

ns for LYSO [53]. Further benefits of LYSO crystals are that they are non-hygroscopic [49] and

therefore do not deteriorate like CsI, and they have a substantially decreased afterglow [56].

These crystals were coupled to a PM3325-WB 2x2 array of 3 mm2 Ketek SiPMs [57]. The

SiPM was coupled with an optical gel of width 1 mm, and read out with the FASTER data acquistion

system [58]. The sides of the crystals that were not coupled to the SiPM were coated in three layers

of Teflon tape. This acted as a reflective coating to increase the light collection of the scintillation

light at the SiPM. The large CsI coupled to the SiPM and connected to a preamplifier is shown in

Figure 4.6.

The observed response for all four crystals is shown in Figure 4.7. These show the spectra

for 22Na and 60Co. The LYSO showed degraded resolution relative to the CsI detector. The large

LYSO has a resolution of 17.6% at 511 keV. This is to be compared with 11.7% for the Tokyo

experiment [39]. The Tokyo experiment used 30 mm diameter by 30 mm length crystals read out

47

CsI
10 x 10 x 15 mm3
20 x 20 x 50 mm3

LYSO
10 x 10 x 15 mm3
30 x 30 x 40 mm3

Small
Large

Figure 4.5 Two sets of scintillation crystals, large and small sizes for both CsI and LYSO crystals.
Dimensions of the crystals are listed in the table.

Figure 4.6 Early test setup. The SiPM was attached to the back of a crystal and fed through a
pre-amplifier before being read out by the DAQ system. The radioactive source was placed next to
the crystal.

by PMTs. It seems that we are far from a comparable resolution using SiPMs. However, we tested

various crystal sizes, but always used the same sized SiPM. As shown in Figures 4.5 and 4.6 the

size of the SiPM was small compared to the face of the large crystal. It is reasonable to expect an

improved resolution for a larger SiPM designed specifically for the final detectors. All tests so far

indicate that using LYSO crystals read out with SiPMs will be feasible for the final design target.

4.4 Single crystal design and simulation

The array will feature 3 rings with 16 detectors in each ring, with the detectors aimed at the

central source. The sensitivity increases as the tilt of the outer ring approaches 45◦, but the detectors

48

(a) 60Co source

(b) 22Na source
Figure 4.7 𝛾-spectra for two sources for all four crystals. The CsI showed an improved resolution
compared to the LYSO, and the large crystal showed a much cleaner response than the small.

will need to be large enough that they will have good photopeak efficiency and solid angle coverage.

The warm bore of the magnet is 22 cm diameter which constrains the widest part of the array. A

reasonable range of crystal depths is 2-5 cm, and the width of the front face for those respective

depths of at most 3.5-2.35 cm (this can be ascertained from Figure 3.12).

Further single crystal design work was carried out in concurrent simulations using Geant4

[59, 60] and EGSnrc [61]. Here we present the results from Geant4 tests. These simulations study

the response to monochromatic 511 keV 𝛾’s, emitted isotropically from a point 6 cm in front of

49

Array of 2x2 SiPMs PM3325-WBCo Source60TEH/ONC/EZ@MSUNormalized counts/bin00.20.40.60.81CsI(Tl)310 x 10 x 15 mmNormalized counts/bin00.20.40.60.81LYSO310 x 10 x 15 mmBackground SubtractedQDC units020040060080010001200310·Normalized counts/bin00.20.40.60.81CsI(Tl)320 x 20 x 50 mmQDC units020040060080010001200310·Normalized counts/bin00.20.40.60.81LYSO330 x 30 x 40 mmBackground SubtractedArray of 2x2 SiPMs PM3325-WBNa Source22TEH/ONC/EZ@MSUNormalized counts/bin00.20.40.60.8180 keV FWHM @ 511 keVCsI(Tl)310 x 10 x 15 mmNormalized counts/bin00.20.40.60.81LYSO310 x 10 x 15 mmBackground SubtractedQDC units0100200300400500600700800900310·Normalized counts/bin00.20.40.60.8178 keV FWHM @ 511 keVCsI(Tl)320 x 20 x 50 mmQDC units0100200300400500600700800900310·Normalized counts/bin00.20.40.60.8190 keV FWHM @ 511 keVLYSO330 x 30 x 40 mmBackground Subtractedthe crystal. We record the energy deposited in the crystal from the initial event (including any

secondaries that do not escape the crystal). These simulations do not include any internal optics of

the scintillation light, and therefore give no information on finite energy resolution.

These simulations are used to study three aspects of the crystal geometry: length, width, and

taper. To compare these aspects we record the total efficiency, and the photopeak efficiency. The

total efficiency is merely the "total number of counts" for a fixed number of events regardless of

the energy left in the detector (Compton or photopeak). The "photopeak efficiency" is the number

of counts in the photopeak divided by the total counts for the detector. This means that, if both

the photopeak and the Compton plateau scale by a factor of two, the detectors would have the

same photopeak efficiency. In reality the term "photopeak efficiency" is a bit of a misnomer: the

term "total energy peak efficiency" is more accurate. Increasing the size of the crystal increases

the number of photons that scatter multiple times and therefore leave all their energy by either

subsequent scatterings or subsequent absorption.

4.4.1 Crystal taper

The gamma detectors will be held in a spherical array and aimed at a central source. This means

that, for a cuboid geometry, photons coming from a point source will not see an equal crystal length

whether they hit the center of the crystal or the edge. For this reason we could benefit greatly by

using a tapered crystal design. This could greatly increase the detection efficiency across the front

face of the detector by decreasing the number of photons that "clip" the edge. The basic geometries

considered in this study are shown in Figure 4.8. An image from the simulation of a tapered crystal

is shown in Figure 4.9.

4.4.2 Crystal width

The width of the front face has a large effect on the overall acceptance of the detector. Doubling

the width of the front face quadruples the solid angle seen by the detector, and the efficiency should

scale roughly as the solid angle. The possible width will largely be determined by the geometric

constraint of fitting 16 crystals in a tilted ring inside the cylindrical magnet. In reality all of the

detectors will need mounts to hold them in place in the apparatus and this will greatly limit the size

50

Figure 4.8 The three geometries considered, a rectangular crystal, a tapered crystal, and a partially
tapered crystal.

Figure 4.9 Geant4 simulation of a monochromatic point source of 511 keV photons in front of a
tapered crystal.

of the crystals. The dependence of the efficiency on the varying crystal width is shown in Figure

4.10. This shows that the total efficiency roughly scales as the width squared.

4.4.3 Crystal length

The length of the crystal affects the detection efficiency by increasing the amount of material

the photon must pass through (increasing the likelihood of interaction). Similarly it also increases

the events with multiple scattering, as such it should increase the total efficiency and the photopeak

efficiency, although not as dramatically as increasing the width. The effect is shown in Figure 4.11.

4.4.4 Partial taper

The mockup of the detector placement for tapered crystals is shown in Figure 3.10. This has left

almost no room for mounting the detectors. We want to make room for mounting, while keeping

51

Figure 4.10 Changing width of the crystal holding its front face 6 cm from a point source of 511
keV photons. The length of the crystal is held at 3 cm. Rectangular crystal is shown in blue, tapered
(with the angle of taper matching the solid angle of the front face) shown in red, and a partially
tapered crystal with the front half tapered shown in black.

Figure 4.11 Changing the length of the crystal, holding the front face position 6 cm from a point
source of 511 keV photons. The front face is fixed at 2×2 cm2. Rectangular crystal is shown in
blue, tapered (with the angle of taper held constant) shown in red, and a partially tapered crystal
with the front half tapered shown in black.

as much material as possible, as illustrated in Figure 4.12. This can be achieved by using partially

tapered crystal. The photons that enter the detector at an angle have a longer path length to the

back of the crystal than those that enter head on (for a tapered design). Removing part of the back

region of the crystal will not dramatically affect the path length for the majority of the photons that

52

11.21.41.61.822.22.42.6crystal width (cm)123456789relative total efficiency11.21.41.61.822.22.42.6crystal width (cm)0.460.480.50.520.540.560.580.60.620.64photopeak efficiency22.533.54crystal length (cm)11.21.41.61.82relative total efficiency22.533.54crystal length (cm)0.50.520.540.560.580.60.620.64photopeak efficiencyhit the detector.

Figure 4.12 Diagram of the space gained between the crystals as we change the fraction that is
tapered.

The benefit of the partial taper is many-fold. Firstly, it maintains some of the benefit of the

tapered design versus the rectangular. Secondly, it buys substantially more mounting space at

the back of the crystal without sacrificing space at the front. The photopeak efficiency and total

efficiency for a 1.5×1.5 cm front face by 3 cm deep crystal are shown in Figure 4.13. While the

gains in efficiency might not look dramatic, the experiment will record coincident events. This

means it is more appropriate (though still rough) to consider the efficiency squared, meaning an 18%

increase in single detector efficiency roughly corresponds to a 39% increase in coincidence statistics.

Considering the photopeak efficiency we see a greater increase, roughly as 1.18∗(0.59/0.56) ≈ 1.24

for a single crystal.

4.5 First prototype and observed distortion

We settled on a design of 1.68×1.68×3 cm with the front 1.5 cm of the crystal tapered (the back

end has dimensions 2.13×2.13 cm). The first prototype was studied at WU in January 2021. This

crystal produced dramatic distortions, shown in Figure 4.14b. The spectra were taken with a 137Cs,

and the spectra showed very different responses whether the source was in front of the crystal or to

the side. There was a long high energy tail from the front, and a double peak from the side. Scans

53

Figure 4.13 Total efficiency and photopeak efficiency as the fraction of the partially tapered crystal
is varied. A value of 0.4 corresponds to the front 40% of the crystal being tapered. This is for a
crystal of 1.5×1.5 cm front face and 3 cm depth, and the angle of the taper is held constant.

(a)

(b)

Figure 4.14 (a) The first partially tapered LYSO prototype, in a 3-D printed cover. (b) Response of
the tapered LYSO prototype to a 137Cs source. Experimental spectra with the source aimed at the
front and at the side of the crystal. These results are after background subtraction. Data taken at
WU.

with a collimated source along the side of the crystal showed a clear position dependence to the

gain (and shape of the response). Two possible explanations are, a non-uniformity of the cerium

doping, or a position dependent light collection efficiency. With only a single crystal there was no

clear non-destructive test to distinguish these two possible causes. For this reason we performed

54

00.20.40.60.81proportion tapered11.21.41.61.82relative efficiency00.20.40.60.81proportion tapered0.520.5250.530.5350.540.5450.550.5550.56photopeak efficiency010002000300040005000ADC units0200400600800100012001400160018002000arbfrontsidea series of destructive tests at LPC-Caen. We started with a rectangular crystal and demonstrated

that it had a uniform well-behaved response. After the uniformity of the response was verified, we

had the crystal cut into the partially tapered geometry and measured the response. Observing a

distortion after cutting would be a clear indication that it is a geometric effect.

The experimental setup is shown in Figure 4.15a. We used a 22Na source in a Pb container

with a 1 cm diameter opening. This source was further collimated with a brass tube of 1 cm outer

diameter, 3 mm inner diameter, and 1 cm length. The crystal was read out with a SiPM and was

placed on a jack to move relative to the source. The crystal was moved in increments of 0.5 cm,

with position 0 referring to the front face, and positive displacement means towards the readout

end.

The rectangular crystal showed no distortions and a very clean spectrum, shown in Figure

4.16a. The peaks were centered and did not move relative to each other as we scanned along the

side of the crystal. The changing height of the histogram is an indication of imperfect collimation

of the source. As we scanned the crystal saw more of the cone emitted by the source. This is

illustrated in Figure 4.15b. Once the crystal was cut, but the cut sides were still unpolished, we saw

a clear shifting of the peak, shown in Figure 4.16b. The peak moved from high energy to low as

we scanned from the front face towards the silicon photomultiplier. Finally when the sides of the

crystal that were cut were polished (Figure 4.16c), any semblance of a clean response was lost. The

peaks were highly asymmetric and non-gaussian when the source was aimed at the tapered region

of the crystal.

Results for each configuration with the source aimed at the front face are shown in Figure

4.17. A simple energy calibration was applied to make the peaks lie on top of each other, and the

spectra were scaled to have equal geometric area (not integral). Figure 4.17a demonstrates that the

rectangular crystal had a clean response that was dramaticaly distorted when the crystal was cut and

polished. However, our collaborators from WU identified a solution to the problem. They painted

the crystal with TiO2 paint and were able to completely remove the effect. As illustrated in Figure

4.17b, changing the surface treatment resulted in the distortion going away and a clean response

55

(a)

(b)

Figure 4.15 A 𝛾 source was collimated and aimed at the side of the crystal. The crystal could
be moved using a raising and lowering jack. a) Image of the setup constructed at LPC-Caen. b)
Cartoon illustrating the working principle, and how imperfect collimation lead to a change in solid
angle.

being regained.

(a) Initial rectangular crystal

(b) cut crystal

Figure 4.16 Measured response of the crystal to a collimated 22Na source aimed at different positions
along the side of the crystal. Changes in height are due to imperfect collimation of the source.

(c) cut and polished crystal

The non-uniform response could be caused by a geometric focusing of the scintillation photons.

56

02004006008001000310·SiPM1_QDC105001000150020002500300035004000-0.5 cm0.0 cm0.5 cm1.0 cm1.5 cm2.0 cm2.5 cm3.0 cm3.5 cm0200400600800100012001400310·SiPM1_QDC105001000150020002500300035004000-0.5 cm0.0 cm0.5 cm1.0 cm1.5 cm2.0 cm2.5 cm3.0 cm3.5 cm0200400600800100012001400310·SiPM1_QDC105001000150020002500300035004000-0.5 cm0.0 cm0.5 cm1.0 cm1.5 cm2.0 cm2.5 cm3.0 cm3.5 cm(a) Response after cutting, and after polishing

(b) Response after partial painting, and full painting.

Figure 4.17 Response of the crystal to photons from a 22Na source aimed at the front face. All
histograms were scaled on the x-axis to overlay with a very rough energy calibration. The cut and
polished crystal did not provide a good energy calibration, so there is an arbitrariness to the scaling.
Similarly all histograms were normalized by geometric area to account for their different bin sizes.

As demonstrated in Ref. [62], in a tapered scintillation crystal when the light bounces off the edges

of the crystal the light can gain a more favorable angle at the readout face for transmission. This

leads to a position dependence to the light collection efficiency along the length of the crystal that

is absent for a rectangular crystal. Changing the surface treatment changes the reflective properties

which can remove the geometric focusing.

57

200400600800100012001400160018002000310·Energy (eV)00.20.40.60.81arbtraryrectangularcut unpolishedcut polished0500100015002000310·energy (eV)00.20.40.60.81arbunpaintednon tapered sides paintedall sides painted4.6 Final crystal results

With the benefit of painting we will not need to alter the geometry of the crystal. Therefore we

use 1.68×1.68×3 cm3 crystals where the front 1.5 cm are tapered. Each crystal is read out by a 2×2

array of 6 mm SiPMs. They are coupled with 1 mm thick optical gel to the crystal. Each is painted

with at least 3 layers of TiO2 paint, and then covered with a Tyvek cover. These sit in a custom

built crystal clamp. The crystals are shown in Figure 4.18, and their spectra are shown in Figure

4.19. We could achieve roughly 12% FWHM energy resolution for the final crystals. Production

of all 48 detectors is ongoing at present.

Figure 4.18 Three finalized 𝛾-detectors in their clamps. Two of these are with prototype SiPM
boards, and one had the finalized board (with the proper screw placement to attach the board onto
the back of the crystal).

58

(a)

(b)

(c)

Figure 4.19 Gamma peaks for all three crystals shown in Figure 4.18, with FWHM resolution at
511 keV of 12.1%, 11.5%, and 14.1% respectively. These spectra were not background subtracted,
but instead read out in coincidence with the detection of the 𝛽-emission.

59

0123456789101112310·Energy Detector 0x10 (ADCu)00.511.522.533.5310·counts (arb)0123456789101112310·Energy Detector 0x17 (ADCu)00.511.522.533.54310·counts (arb)0123456789101112310·Energy Detector 0x19 (ADCu)00.511.522.533.5310·counts (arb)CHAPTER 5

POSITRONIUM FORMATION

5.1 Overview

The final triple coincidence rate in the online experiment will be in part determined by how

many of the positrons emitted from the 𝛽 source actually form positronium. This required analysis

of different materials for positronium formation. The target is to reach is a formation fraction of at

least 40%, as achieved in Magnesium Oxide powder in Refs. [63, 64], and a lifetime of at least 125

ns as achieved in the previous experiments [38, 39]. For both of these quantities, larger values are

always better. There has been much activity recently in optimizing positronium formation in large

scale antimatter experiments [65, 66].

5.2 Ps formation and Positron annihilation lifetime spectroscopy

Positron annihilation lifetime spectroscopy is a well developed method to study material prop-

erties in Condensed Matter and in Material Science [67, 68], as well as exciting new applications in

identifying tumor tissues [69]. The time spectrum will reveal material properties about the sample.

The positrons either annihilate directly in the bulk, or form positronium. Positronium has a positive

work function in materials, so it is repelled to the voids and pores of the material.

Interactions between the positronium and the bulk can cause a spin flip so that ortho-positronium

changes to para-positronium and quickly annihilates. As such, the lifetime of the positronium atom

depends on the size of the void it exists in. Measuring the distribution of lifetimes for a given

sample reveals the relative sizes, and interconnectedness of the pores in the material [67].

The experimental spectrum is built as follows, we start with a 𝛽+ source that also emits a

de-excitation photon. We use two 𝛾-detectors, one for the de-excitation photon (START), and one

for measuring annihilation photons (STOP). Finally, we record the time of each signal and construct

the Δ𝑡 = 𝑡𝑆𝑇𝑂𝑃 − 𝑡𝑆𝑇 𝐴𝑅𝑇 spectrum of events. This is shown schematically in Figure 5.1.

Extracting the lifetimes and relative amplitudes of the different components can reveal properties

of the material, as well as the "formation fraction" of the material. For the studies presented here,

the target is purely maximizing the lifetime and the formation fraction.

60

Figure 5.1 Cartoon illustrating the working principle of the powder test stand. First a 𝛽 decay
occurs, the 𝛽 enters the powder, and there is a concurrent 1275 keV de-excitation photon detected
with the start detector. The 𝛽 can form positronium in the powder, eventually annihilating and
leaving a signal in the stop detector.

5.3 Ps test stand construction and testing

We used a 22Na source 𝛽+ emitter while testing the powders. The nuclear decay emits a

concurrent 1275 keV 𝛾 along with the 𝛽+. This 𝛾 gives the START. This source was placed

directly against powder held inside an aluminum holder. We placed another powder container on

the opposite side. However, due to the asymmetry of this source almost no 𝛽’s escape the back side

of the source (layers of plastic and the label).

These containers could be placed inside an aluminum vacuum tube. Spacers inside the vacuum

tube ensured the powder sits at the same longitudinal position inside the tube from run to run. The

vacuum tube was held in a frame that fixed its position on the table. This is illustrated in Figure

5.2.

We utilized two LaBr3 detectors. These were held in frames relative to the vacuum tube for

reproducible alignment. The detectors were cylindrical 38.1 mm diameter by 38.1 mm length,

and placed 5.5 cm from the powder. Lateral displacements of the powder were constrained by the

frames, and longitudinal displacements were controlled by the spacer inside the vacuum tube. The

largest variation was the placement of the source within the powder container, which was difficult to

61

constrain. Variations in position resulted in a negligible change in the solid angle of the detectors.

Each detector could see the start signal or a stop signal (or both). The detectors were not

back-to-back so they could not see the 511-511 coincidences unless one Compton scattered. We

recorded coincident events, the energy (in ADC units), and the timestamp of the event (with CFD

timing). The digitization was performed with NSCLDAQ using PIXIE-16 data acquisition modules

[70, 71, 72]

Figure 5.2 Positronium test stand. A vacuum tube connected to a roughing pump, with two LaBr3
detectors aimed at a central source in the tube.

We distinguished the start and stop signals by placing energy cuts on the distribution. A data

run could take somewhere between one hour to two days depending on the activity of the source

used. For this reason we needed to perform energy calibrations to correct for gain shifts.

The 22Na source provides two gamma peaks. We automatically calibrated the spectrum by

fitting the two peaks, extracting a gain and offset and then rebuilding the spectrum in units of

energy. We applied energy cuts based on the calibrated spectrum which removed the effect of gain

drifts between runs. Two uncalibrated energy spectra with fits are shown in Figure 5.3.

We considered a START signal if the hit had between 1200 and 1350 keV (to capture the

1275 keV annihilation photon). We considered two energy windows for the STOP, one called

"continuum" between 250 keV to 490 keV, and another called "peak" between 490 keV to 530 keV.

We considered these energy windows for each detector, which gave us four separate time spectra

for any given run. In principle these were different methods to construct the same time spectrum,

62

Figure 5.3 The observed energy spectra for both detectors with a 22Na source in ADC units (ADCu).
The gain and offset were extracted with the fits of the 511 keV and 1275 keV peaks.

so extracted lifetimes and formation fractions should agree.

5.4 Time spectrum model

The time spectrum was modeled as a delta function plus some finite number of decaying

exponentials. These are convolved with a gaussian response function. This gave a gaussian prompt

peak, and a complementary error function as the lifetime component. The analytic form of this is,

𝐹𝑖 (𝑡) =

(cid:18) 𝐴𝑖
2𝜏𝑖

exp

(cid:26) 𝜎2
2𝜏2

(cid:27)(cid:19)

(cid:26)

−

exp

(cid:27) (cid:20)

𝑡 − 𝜇
𝜏𝑖

1 + erf

(cid:18) 𝑡 − 𝜇
√
2𝜎

−

√

𝜎
2𝜏𝑖

(cid:19)(cid:21)

(5.1)

where 𝐴𝑖 is the integral of the lifetime component, 𝜇 is the start time of the signal, 𝜏𝑖 is the lifetime,

and 𝜎 is the resolution.

However, actual data showed a different response.

In general we saw some small lifetime

component that was on par with the time resolution. We show a sample fit function in Figure 5.4,

with an asymmetric peak, a 5 ns lifetime component, and a 100 ns lifetime component. So a "fit"

corresponds to fitting a function 𝑓 (𝑡) = (cid:205)𝑖 𝐹𝑖 (𝑡), where all 𝐹𝑖 have the same 𝜎 and 𝜇, but different

𝐴𝑖 and 𝜏𝑖. A simple log likelihood fit can easily discriminate lifetimes that vary by a factor of 2,

however it struggles when multiple components have similar lifetimes. This is largely unimportant

for our purposes.

63

0100020003000400050006000Energy (ADCu)110210310countsFigure 5.4 Illustration of a lifetime spectrum fit function. This features an asymmetric peak with
two lifetime components. All components have the same resolution and are centered on the same
time.

5.5

Initial results

5.5.1 Aluminum tests

We carefully prepared aluminum targets to match the density of the MgO powder, both 0.3 and

0.6 g/cm3. This should have provided the most precise prompt decay measurement. We cut the

aluminum foil into small circles, then layered them within the container, shown in Figure 5.5. These

matched the target density. Initial tests seemed accurate, except we saw a high level of accidentals.

The accidental rate depends on the rate of the source as 𝑅𝑎𝑐𝑐 = 𝑅2Δ𝑇 where Δ𝑇 is the coincidence

time window. The distributions for three different source rates are shown in Figure 5.6.

Figure 5.5 Constructed aluminum target designed to match the target densities of the powders to be
tested.

At low rate levels we saw a long lifetime component in the aluminum container. This lifetime

component appeared when we cut on the low energy part of the spectrum, not on the 511 keV

peak, indicating positronium formation. For this reason we switched to a solid piece of Aluminum,

64

Figure 5.6 Data taken with cuts on continuum energy, showing the peak and the accidentals level.
This data was taken with the custom made aluminum target at density 0.3 g/cm3. At low rate we
could resolve a small long lifetime component.

with results shown in Figure 5.7. The lifetime component had dissappeared, so we could have

been seeing positronium forming on the surfaces between the thin foil layers. This shows a clean

Figure 5.7 Time spectrum for a solid piece of aluminum. Two spectra were constructed depending
on which detector is treated as start and which is stop. Note the clear lifetime component with 2 ns
lifetime, indicative of positronium formation in plastic.

response for both directions the spectrum was built, and clearly displays a lifetime component of

around 2 ns. This is to be expected for positrons forming positronium in plastic. The 22Na source

was deposited on a thin mylar foil, which was further held in a thin plastic layer (with a window so

65

200-100-0100200300time (ns)1-10110210310410counts (arb)11.5 kcps3.4 kcps1.7 kcps80-60-40-20-020406080100time (ns)110210310410counts (arb)the 𝛽’s escape). This plastic component is likely from the positronium forming in that plastic of

the source itself, and will be present for all tests in this chapter.

5.5.2 MgO powder

We replaced the aluminum with powder and immediately saw a long lifetime component, shown

in Figure 5.8. The magnesium oxide powder was 35 nm grain size and compressed to a density of

0.3 g/cm3. Initial fits returned a long lifetime component of 66±6 ns. The amplitude of the long

Figure 5.8 Data with unprepared Magnesium Oxide powder. The powder spectrum shows a (small)
long lifetime component compared to data taken with a solid aluminum block.

lifetime component was small, so there was not much positronium formation. Also shown is the

spectrum when pumped with a roughing pump for 20 hours. Holding the powder under vacuum

did indeed increase the lifetime with the fit returning a long lifetime of 98±10 ns, but the integral of

the long lifetime component was not changed. The result of fitting the MgO powder under vacuum

is shown in Figure 5.9. Translating the amplitudes of each lifetime component into a "formation

fraction" is non-trivial and requires some modeling to extract the relevant information.

5.6 Extracting lifetime and formation fraction

The positron travels through a series of materials after emission. In each material, it can do one

of three things: annihilate, form positronium, or pass through the material. Each of these options

affect both the time and energy distribution of the final gamma distribution.

66

40-20-020406080100120140time (ns)110210310counts (arb)AluminumMgOMgO PumpedFigure 5.9 Example of a fit of a lifetime spectrum. There is clearly a sharp peak, a lifetime component
around 2 ns (plastic), and a long lifetime (o-Ps). The lifetimes and the relative amplitudes are
extracted for each component.

5.6.1 Positronium state population

In order to model the positronium state populations, we make some simplifications. We consider

only 2 materials:

1. "Plastic" - The source itself is in a plastic holder, but we treat all the materials of the source

(plastic, mylar foil, paper label) to be the "plastic" material.

2. "Powder" - This is the target we want to study, an aluminum container filled with powder. This

model does not include the positrons that annihilate against the aluminum of the container.

The model is illustrated in Figure 5.10. Starting with with 𝑁𝛽 𝛽-decays, some fraction stop in plastic,

𝑓 𝑝. The remaining fraction 1 − 𝑓 𝑝 reach the powder target. Finally, in each of these materials there

is some probability for the positrons to form positronium. We call this the positronium formation
fraction 𝑓𝑃𝑠. There is a formation fraction in the plastic 𝑓 𝑝

𝑃𝑠, and the powder 𝑓𝑃𝑠. The ultimate target
is to measure 𝑓𝑃𝑠 for the given powder, everything that follows is attempting to extract this quantity.

One quarter of the positronium forms para-positronium and three quarters form ortho-positronium.

Para-positronium has a lifetime of only 125 ps and is indistinguishable from the direct annihilation.

This gives the following five distributions:

𝑁𝑎𝑛𝑛ℎ𝑖𝑙𝑎𝑡𝑒 = 𝑁𝛽 ( 𝑓 𝑝 (1 − 𝑓 𝑝

𝑃𝑠) + (1 − 𝑓 𝑝) (1 − 𝑓𝑃𝑠))

(5.2)

67

Figure 5.10 Model for positronium lifetime distribution. Positrons are either stopped in plastic of
the powder. Both materials have a chance for direct annihilation, or formation of positronium. 1/4
of the formed positronium has a lifetime 125 ps and is indistinguishable from the prompt events,
the other 3/4 forms ortho-positronium with a characteristic lifetime.

𝑁 𝑝𝑙𝑎𝑠𝑡𝑖𝑐

𝑝−𝑃𝑠 =

𝑁 𝑝𝑙𝑎𝑠𝑡𝑖𝑐

𝑜−𝑃𝑠 =

𝑁 𝑝𝑜𝑤𝑑𝑒𝑟

𝑝−𝑃𝑠 =

𝑁 𝑝𝑜𝑤𝑑𝑒𝑟

𝑜−𝑃𝑠 =

1
4
3
4
1
4
3
4

𝑁𝛽 𝑓 𝑝 𝑓 𝑝
𝑃𝑠

𝑁𝛽 𝑓 𝑝 𝑓 𝑝
𝑃𝑠

𝑁𝛽 (1 − 𝑓 𝑝) 𝑓 𝑝
𝑃𝑠

𝑁𝛽 (1 − 𝑓 𝑝) 𝑓 𝑝
𝑃𝑠

(5.3)

(5.4)

(5.5)

(5.6)

Direct annihilation and para-positronium decays have a distinct time signature (a delta function),

and the various ortho-positronium decays have distinct lifetimes for each material. This means that

these populations can in principle be separated by their various time-dependencies.

𝑓 𝑝 𝑓 𝑝

𝑃𝑠 + (1 − 𝑓 𝑝) (1 − 𝑓𝑃𝑠) +

(1 − 𝑓 𝑝) 𝑓𝑃𝑠

(cid:19)

𝛿(𝑡)

1
4

𝑓 𝑝 (1 − 𝑓 𝑝

𝑃𝑠) +

1
4
𝑃𝑠𝑒−𝑡/𝜏𝑝𝑙𝑎𝑠𝑡𝑖𝑐

𝑁𝛽 𝑓 𝑝 𝑓 𝑝

𝑁 𝑝𝑟𝑜𝑚 𝑝𝑡 (𝑡) = 𝑁𝛽

(cid:18)

𝑁 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 (𝑡) =

𝑁 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 (𝑡) =

3
4
3
4

𝑁𝛽 (1 − 𝑓 𝑝) 𝑓𝑃𝑠𝑒−𝑡/𝜏𝑝𝑜𝑤𝑑𝑒𝑟

(5.7)

(5.8)

(5.9)

Where 𝑁 (𝑡) is the state population at time 𝑡 (allowing the liberty of calling "direct annhilation" a

"state"). This model assumes that all ortho-positronium forms in the ground state.

68

Figure 5.11 Starting with the lifetime distributions (which in-principle we can separate), the Ps
decay events are mapped into 2-𝛾 and 3-𝛾 decays dependent on the relative branching ratios (which
are material dependent). In principle all 5 of these are distinguishable because they have different
time-dependence and different energy distributions. However finite detection efficiency will limit
the ability to completely distinguish 2-𝛾 from 3-𝛾 events.

5.6.2 Positronium decay

The angular distribution of the decay products is taken to be isotropic. The 2-𝛾 decay results

in two 511 keV photons emitted back-to-back. The 3-𝛾 decay results in three photons in a plane

with a continuous energy distribution from 0-511 keV. This means that the relative sensitivity to

the decay channels depends on the energy cuts. Gating on the 511’s biases the results to see less

o-Ps decay.

Ortho-positronium in a material has a more complicated time evolution than in vacuum. As-

suming a simple uniform material, there are two decay channels for the ortho-positronium. It can

either decay to 3-𝛾, or it can annihilate due to interaction with the surrounding material. The
𝑝𝑖𝑐𝑘𝑜 𝑓 𝑓
lifetime in the 𝑖th material, Γ𝑖, depends on the decay width due to "pickoff annihilation," Γ
𝑖

.

This gives two competing decay widths:

𝑝𝑖𝑐𝑘𝑜 𝑓 𝑓
Γ𝑖 = Γ𝑜−𝑃𝑠 + Γ
𝑖
1
Γ𝑖

𝜏𝑖 =

We assume that all pickoff annihilation goes to 2-𝛾. This gives the branching ratio of o-Ps

𝐵𝑅(𝑜 − 𝑃𝑠 → 3𝛾) =

Γ𝑜−𝑃𝑠
Γ𝑖

=

𝐵𝑅(𝑜 − 𝑃𝑠 → 2𝛾) = 1 −

𝜏𝑖
𝜏𝑜−𝑃𝑠

𝜏𝑖
𝜏𝑜−𝑃𝑠

69

(5.10)

(5.11)

(5.12)

(5.13)

where 𝜏𝑜−𝑃𝑠 = 142 ns, the vacuum lifetime for ortho-positronium. For plastic with a lifetime of
𝜏𝑃 = 2.25 ns the positronium has 𝐵𝑅(3𝛾) = 2.25
142 = 1.58%. Whereas for the unprepared powder

with a lifetime 65 ns, the positronium has 𝐵𝑅(3𝛾) = 45.8%.

This means that for each "state" the 𝛽+ can end in, the model must account for the final state

photons it leads to. The energy distribution for a 2-𝛾 decay is a simple delta function energy at

511 keV. For the 3-𝛾 decays it is a continuous energy distribution with complicated angles between

each photon. The lifetimes are measured when we perform the fits, so the branching ratios are

also already being measuring. This must be accounted for to extract the correct state populations.

To fully disentangle these effects and extract a formation fraction, we need to model the detector

response and how the energy cuts bias different final states.

5.6.3 Detector Response

The final measurement (a set of hits with a timestamp and a reconstructed energy) is complicated

by the detector response. We assume that there is a single efficiency for 2-𝛾 decays and for 3-𝛾

decays. They are functions of the energy cuts, but not of time, stopping position, or stopping

material, etc. This is an oversimplification, spreading of the source will affect the geometric

acceptance, different materials will scatter photons at different rates, etc.

Now we consider the ratio 𝜖3𝛾/𝜖2𝛾 of the detection efficiencies for a given energy range. The

detectors are more efficient for lower energies, so this would raise the relative efficiency to a higher

number. But Compton scattering also maps some 2-𝛾 events into the energy cuts.

As defined above, the detection efficiency 𝜖 𝑝𝑒𝑎𝑘
counts in the 𝑝𝑒𝑎𝑘 energy window." Similarly for 𝜖 𝑐𝑜𝑛𝑡

2𝛾 means "for 𝑁 2-𝛾 decays, there are 𝑁𝜖 𝑝𝑒𝑎𝑘
2𝛾
3𝛾 . These values can be

2𝛾 , 𝜖 𝑝𝑒𝑎𝑘
3𝛾

, and 𝜖 𝑐𝑜𝑛𝑡

estimated in a simple Monte-Carlo. This was performed by David-Michael Peterson (MSU) and

shown in Figures 5.12 and 5.13. We place a LaBr3 cylinder 5.5 cm from a point source and fire a

definite number of 2-𝛾 and 3-𝛾 events. We ran a second simulation with the distance increased to

6.5 cm. This should cover the range of values for the distance between the source and the detector

and give a stronger understanding of how sensitive we are to mismodeling this setup. The number

of events within an energy window is an estimate of the detection efficiency. Ultimately the relative

70

efficiencies 𝜖3𝛾/𝜖2𝛾 enter the calculation of the formation fraction, so we are less sensitive to effects

like offset of the source. The results are given in Table 5.1.

Figure 5.12 Example of a three photon event occuring in front of the LaBr3 detector. Of course,
this event leaves no signal in the crystal.

Figure 5.13 Simulated energy deposited in a LaBr3 crystal from a point source of positronium.
Blue shows the continuous 3-𝛾 events, and red the back-to-back 2-𝛾 events. The photopeak at 511
goes well off the plot. The two energy windows are shown in green and purple. This includes no
smearing for finite detector resolution.

5.6.4 Final distribution

All of the above effects are combined and give an analytic expression for the amplitude of the

lifetime component 𝐴𝑖 in terms of the state populations.

𝐴𝑝𝑟𝑜𝑚 𝑝𝑡 = 𝜖2𝛾 (𝑁𝑎𝑛𝑛𝑖ℎ𝑖𝑙𝑎𝑡𝑒 + 𝑁 𝑝𝑙𝑎𝑠𝑡𝑖𝑐

𝑝−𝑃𝑠 + 𝑁 𝑝𝑜𝑤𝑑𝑒𝑟
𝑓 𝑝
𝑃𝑠) + (1 − 𝑓 𝑝) (1 −

𝑜−𝑃𝑠

)

= 𝑁𝛽𝜖2𝛾 ( 𝑓 𝑝 (1 −

3
4

𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐 =

𝑁𝛽𝜖2𝛾 𝑓 𝑝 𝑓 𝑝

𝑃𝑠

3
4

𝑓𝑃𝑠))

3
4

(5.14)

(5.15)

71

𝜖 𝑐𝑜𝑛𝑡
2𝛾
𝜖 𝑐𝑜𝑛𝑡
3𝛾
2𝛾 /𝜖 𝑐𝑜𝑛𝑡
𝜖 𝑐𝑜𝑛𝑡
3𝛾
𝜖 𝑝𝑒𝑎𝑘
2𝛾
𝜖 𝑝𝑒𝑎𝑘
3𝛾
𝜖 𝑝𝑒𝑎𝑘
2𝛾

/𝜖 𝑝𝑒𝑎𝑘
3𝛾

Centered

Offset
0.50% 0.333%
2.56% 1.88%
0.185
0.195

1.57% 1.18%

0.232% 0.167%

6.77

7.07

Table 5.1 Detection efficiency for a single LaBr3 detector for a source located 5.5 cm from the front
face, and for a source located at 6.5 cm from the front face.

𝐴𝑝𝑜𝑤𝑑𝑒𝑟 =

3
4

𝑁𝛽 (𝐵𝑅2𝛾𝜖2𝛾 + 𝐵𝑅3𝛾𝜖3𝛾) (1 − 𝑓 𝑝) 𝑓𝑃𝑠

(5.16)

which includes the simplifying assumption that all ortho-positronium in plastic decays to two

photons. Now, in principle, we directly measured 𝐴𝑖 from the fits, we measured the lifetimes (and

therefore the branching ratios), and we have simulated the detection efficiencies. We cannot extract
𝑓𝑃𝑠 directly from these quantities. This is because 𝑓 𝑝 and 𝑓 𝑝

𝑃𝑠 are fully correlated (varying the

number of positrons that stop in plastic versus the number that form positronium in the plastic

affects the counts in the same way). From this data we can extract,

𝑓 𝑝 𝑓 𝑝

𝑃𝑠 =

(1 − 𝑓 𝑝) 𝑓𝑃𝑠 =

¯𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐
¯𝐴𝑝𝑟𝑜𝑚 𝑝𝑡 + ¯𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐 + ¯𝐴𝑝𝑜𝑤𝑑𝑒𝑟
¯𝐴𝑝𝑜𝑤𝑑𝑒𝑟
¯𝐴𝑝𝑟𝑜𝑚 𝑝𝑡 + ¯𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐 + ¯𝐴𝑝𝑜𝑤𝑑𝑒𝑟

3
4

3
4

(5.17)

(5.18)

where ¯𝐴𝑖 is the amplitude corrected for the efficiency (or efficiencies weighted by branching ratios).

These quantities cannot be disentangled without running some dedicated tests to determine either

the stopping fraction in plastic, or the formation fraction in plastic.

5.6.5 Study of plastic

If all of the positrons stop in plastic then Equation 5.17 could be used to directly extract the

formation fraction for plastic. Since all decays in this scenario would be 2-𝛾 decays, the amplitudes

would not need to be corrected for detection efficiency as well. The formation fraction in plastic

would be,

𝑓 𝑃
𝑃𝑠 =

𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐
𝐴𝑝𝑟𝑜𝑚 𝑝𝑡 + 𝐴𝑝𝑙𝑎𝑠𝑡𝑖𝑐

4
3

72

(5.19)

We measured this by taking standard cling wrap and layering it over an aluminum block. The

plastic was held flat on a table while the block was flipped over and over to apply multiple layers

at once. This plastic was pulled tight to avoid rippling and air pockets within the plastic. This is

shown in Figure 5.14. The aluminum block wrapped in plastic was placed on top of the source to

not affect the source position and relative efficiencies between runs (although these should cancel

run by run).

Figure 5.14 Small aluminum block carefully layered in standard cling wrap. The plastic was
held tight across the table while the block was flipped over repeatedly to try to avoid rippling.
Nevertheless some small ripples did occur.

The results are shown in Figure 5.15. The spectrum was built using the continuum energy

cuts, and the results are shown for both detector combinations (using one as START and the other

as STOP or vice-versa). The errors were taken from the fit results, but were overestimated as the

amplitude for the two fit components were negatively correlated.

Assuming that the formation fraction in the plastic did not change, and only the number of

positrons in plastic changed, then adding more layers should approach the limit 𝑓 𝑝 → 1. This

means the data in Figure 5.15 should asymptote to the value of 𝑓 𝑃
𝑃𝑠.

These tests gave a formation fraction for plastic between 25-33%. The values found in literature

give 58% [73], citing Ref. [74] as making this claim. However this appears to be a misprint as

Ref. [74] claims a value of 28 ± 3% [74] in agreement with the measurement presented here. The

73

Figure 5.15 Extrapolating 𝑓𝑝 𝑓𝑃𝑠 from the plastic data as adding more and more layers of plastic.
Red and black are are the results depending on which detector we call "START" and "STOP." Note
that errors are from the fit results and do not include correlated errors.

formation fraction for plastic as defined in this dataset was not purely plastic, but included the effect

of the mylar foil, paper label, etc. This also assumed a single formation fraction for the different

kinds of plastic.

Taking the formation fraction in plastic as 28% and using the results from the previous MgO

powder tests shown in Figure 5.8, we extract a "stopping fraction" 𝑓 𝑝 = 0.6. This is reasonable as

half of the positrons are entirely stopped in the back half of the source.

5.7 Powder tests

Initial tests of 35 nm MgO powder at 0.3 g/cm3 density featured a very small amplitude long

lifetime component and a lifetime around 70 ns, as shown in Figure 5.8. The chamber was connected

to a roughing pump and the pump ran overnight. In principle the vacuum pump removes moisture

from the powder, and oxygen in the voids, and should therefore increase the lifetime. Indeed, for

the MgO powder, we saw the lifetime increased to 90 ns under vacuum.

Previous experiments that used MgO powder prepared the powder in a vacuum oven to fully

desiccate it. The powder was then kept under vacuum or flushed with dry air. Recent experiments

have increasingly been using chunks of silica aerogel (SiO2) for positronium formation. We studied

both "unprepared" 35 nm MgO powder at 0.3 g/cm3 density and "unprepared" SiO2 powder at 1200

74

01020304050Layers of plastic0.150.20.250.30.35psf*f𝜇m grain sizes powders at 0.1 g/cm3 density. The time spectra are shown in Figure 5.16.

Figure 5.16 Lifetime spectrum for MgO powder and SiO2 powder, showing a substantial increase
in the long lifetime component.

We saw a substantial improvement in the SiO2 powder without having to place it in an oven.

We produced the spectra using two sets of energy cuts for the stop signal. We considered the

"peak cut" between 490-520 keV, and the continuum cut between 250-490 keV. These should have

different efficiencies for 2-𝛾 and 3-𝛾 events. The results for the same powder in air and under

vacuum are shown in Figures 5.17a and 5.17b respectively, with the observed time spectra for the

peak energy cut (red) and continuum energy cut (blue). The two datasets did not have the same

total normalization. Before pumping we saw a lifetime of 72 ± 0.8 ns, after pumping we saw a long

lifetime of 132 ± 2.8 ns (compared with 142 ns theoretical lifetime for o-Ps). The disappearance of

the long lifetime counts for the peak cut energy region was the direct effect of the relative branching

ratios dependence on the lifetime. With 72 ns lifetime 50% of the ortho-positronium decayed to 2

photons, but at 132 ns only 7% of the ortho-positronium decayed to 2 photons, resulting in almost

no 511 keV events in the long lifetime component.

The relative detection efficiencies can also be estimated from the data in Figure 5.17. For a

lifetime of 72 ns the branching ratios are nearly 0.5 for both channels. This means that the relative

amplitudes of the peak counts versus the continuum cuts should give an estimate of the relative

75

100-0100200300400500time (ns)110210310410counts (arb)MgO powder powder2SiO(a) SiO2 in air

(b) SiO2 in vacuum

Figure 5.17 The increase in lifetime when the powder was under vacuum. This also shows the
change to the branching ratios for 2-𝛾 to 3-𝛾 decays. The "peak" cut primarily saw 511 keV back-
to-back photons, and the "continuum" cut primarily saw the continuous 3-𝛾 distribution. Clearly
the number of events in the peak for the long lifetime component was dramatically reduced when
the powder was under vacuum.

detection efficiency of 2-𝛾/3-𝛾 events that can be compared to the Monte-Carlo results. The ratio

of the amplitudes of the long lifetime components should give,
0.5𝜖 𝑝𝑒𝑎𝑘
+ 0.5𝜖 𝑝𝑒𝑎𝑘
2𝛾
3𝛾
0.5𝜖 𝑐𝑜𝑛𝑡
2𝛾 + 0.5𝜖 𝑐𝑜𝑛𝑡
3𝛾

𝐴𝑝𝑒𝑎𝑘
𝑙𝑜𝑛𝑔
𝐴𝑐𝑜𝑛𝑡
𝑙𝑜𝑛𝑔

=

(5.20)

The Monte-Carlo simulation in Table 5.1 give relative efficiencies of 0.59 for the centered

source, and 0.61 for the offset source. This can be compared with the relative amplitudes in Figure

5.17, which give 0.6. This somewhat obscures the effect of offsetting the source, as that varies the

relative efficiencies in opposite directions which cancels when the efficiencies are summed in this

way.

The extracted formation fractions for the MgO powder, SiO2 in air, and SiO2 in vacuum are

shown in Table 5.2. The results are quoted for both sets of efficiencies that were simulated. The

different estimated efficiencies in Table 5.2 change the extracted formation fractions by a 10%

relative shift. There is a large formation fraction for SiO2 powder under vacuum, at around 50%,

exceeding the target formation fraction.

With these results we should be able to achieve the source rate needed to reach the target final

statistics. We achieved a lifetime of 132 ns, compared to 125 ns for the Michigan measurement

[38], and 126 ns for the Tokyo experiment [39]. The use of SiO2 powder has removed the necessity

76

100-50-050100150200250300350400time (ns)110210310410counts (arb)continuum cutpeak cut100-50-050100150200250300350400time (ns)110210310counts (arb)continuum cutpeak cutMgO
SiO2
SiO2 vacuum

𝑓𝑃𝑠 (%)
11.4
41.3
52.5

𝑓𝑃𝑠 varied 𝜖 (%)
10.4
38.1
48.0

Table 5.2 Measured formation fraction using results from fits and correcting for relative efficiencies.
All cuts are on the continuum energy. The two columns correspond to using the efficiencies in each
column of Table 5.1.

of disassembling the apparatus to re-prepare the powder (by baking it in a vacuum desiccator), this

will be a large benefit over the course of the experiment.

77

CHAPTER 6

DESIGN AND SIMULATION OF INNER MODULE

6.1 Overview

The inner module consists of the 𝛽+ source, the start detectors, and the powder for positronium

formation. The size and geometry of the start detector itself affects how many 𝛽’s reach the powder,

and where they stop in the powder. This directly affects the spatial distribution of the positronium

atoms. A broader spatial distribution will change the acceptances for both 2-𝛾 and 3-𝛾 events

in the experiment, and means the positronium will have a broader distribution of lifetimes due

to inhomogeneities in the magnetic field. These effects were studied for different combinations

of start detectors and powders in dedicated Monte-Carlo simulations. In this chapter we present

the results from the Geant4 simulations that were primarily performed by David-Michael Peterson

(MSU). These effets were also studied by Paul A. Voytas and Elizabeth A. George (WU).

6.2 Start detector and powder formation

There are two common methods to build the lifetime spectrum of positronium, 𝛾-𝛾 coincidence,

and 𝛽-𝛾 coincidence. Throughout Chapter 5 we investigated the lifetime spectrum built between

two 𝛾 hits. The actual experiment will directly measure the 𝛽 from the 𝛽-decay. This will give a

much higher trigger rate for the start signals, and it will allow the use of a source that does not emit

𝛾’s (which can induce accidentals in the measurement).

"Organic" scintillators are commonly used for charged particles, herein referred to as plastic

scintillators. The basic principle is the same as an inorganic scintillator, but the mechanism of

scintillation and sensitivity to different particles is different [22]. An optimal start detector will

detect the 𝛽’s, but not stop them from reaching the powder. Ideally it will have as large of a solid

angle coverage as possible, and will cover the powder so that no 𝛽 can reach the powder without

passing through the scintillator.

We will utilize a thin plastic scintillator for the detector. It needs to be thick enough to generate

enough scintillation light to trigger on, but also thin enough that it will not stop the 𝛽’s. This is

non-trivial as the 𝛽’s below 1 MeV do not penetrate very far through plastic (to be shown below).

78

Even a very thin layer of plastic can stop a substantial fraction of the 𝛽’s, and will dramatically

reduce the number that reach the powder to form positronium. Further, different sources have

different endpoint energies, and the higher the 𝛽 energy the more likely it is to punch through the

detector, as illustrated in Figure 6.1a.

This will all sit in the middle of the detector array, so it must be low mass, and operate in a

magnetic field. The plan for the online experiment is to couple the light guide to a PMT outside

the magnet warm bore using optical fibers. For the tests performed throughout this work we simply

read the lightguide with SiPMs.

(a)

(b)

Figure 6.1 (a) Cartoon of the design, a 𝛽 source emits positrons that travel through a plastic
scintillator and stop in powder to form positronium.
(b) A realization of this idea expanded to
show each individual piece, a lightguide (square with cylinder cut out) to couple the scintillator to
a readout, the scintillator with a layer of Al foil on each side, and finally the powder. This would
be one side of the combined start detector and powder module, with a symmetrically placed setup
on the opposite side of the source.

For these studies we use a square plexiglass lightguide with a circular cut through the center.

The scintillator itself is an Eljen-212 scintillator [75]. There are two thicknesses, 0.5 mm and 0.15

mm. These have dramatically different amounts of scintillation light and stopping fractions. Note

that unlike with the 𝛾 detectors, the coating of the scintillator can’t be neglected when considering

particle propagation. Two layers of aluminum foil on the scintillator dramatically increases the

width of the scintillator and stop a large fraction of the positrons that would otherwise make

it through. At the same time, with no coating we do not get enough scintillation light to the

79

photomultipliers.

Figure 6.2 One design for the light guide (with the scintillator in the center). This features the 6
mm thick light guide and the 0.5 mm scintillator.

As the plastic scintillator will need to be read out in the magnetic field we currently utilize

SiPMs. These will not be used in the final experiment for three reasons, 1) they break the symmetry

of the setup, 2) they add too much matter between the powder and the 𝛾-detectors, and 3) they

cannot handle the high rate expected for the source.

6.3 Simulation and tracking

This simulation records the initial energy and direction of the 𝛽+, its stopping position, as well

as the ID of the object it stopped in. In principle we can therefore study, for a given number of 𝛽

emissions what percentage stop in the powder (and what stop in other materials as well). We can

then see where the 𝛽 stopped within that specific material. In practice we only look at this for the

powder.

6.4

𝛽-primary generator

We study two 𝛽+ sources, 22Na, and 68Ga. The resulting spectra for these sources are generated

using the BetaShape program [76, 77]. These are high precision spectra calculations that account

for competing electron capture processes. As shown in Figure 6.3, the gallium source has a much

80

Figure 6.3 Energy distribution for 𝛽+ coming from 22Na (blue) and 68Ga (red), generated using the
BetaShape program [76, 77].

higher end-point energy. This will translate to more positrons punching through the start detector,

but also to a much larger spreading of the source throughout the powder.

Unless specified, all results shown until the very end of this chapter are for a point source of

radioactivity. All positrons are emitted isotropically in the forward half-sphere (+ ˆ𝑧).

6.5 Specification of geometry

The simulations estimate the stopping positions of the initial 𝛽’s. Secondaries and any material

that the 𝛽’s cannot reach are unimportant. The inner module must include 4 pieces,

1. the lightguide,

2. the scintillator,

3. the wrapping for the scintillator,

4. the powder.

This does not include the 𝛽-source itself, or the aluminum container that holds the powder.

These will both have an effect on stopping position. The source itself is more important as it can

stop positrons that backscatter off the scintillator and reach the powder in the opposite direction.

The aluminum container is beyond the powder, and its main contribution would be the 𝛽’s that

81

020040060080010001200140016001800Energy (keV)00.511.522.533-10·Counts (arb)backscatter off of it and stay in the powder. Most combinations of start detector and powder stop

the positrons before they reach the exterior edge of the powder, with one notable exception. We

have also neglected the TiO2 paint that covers the lightguide. As our final interest is in the powder

(and materials between the source and the powder) this should have minimal impact.

(a)

(b)

Figure 6.4 (a) Prototype of start detector and powder container. (b) Reproduction in Geant4. Note
that the SiPMs along with the aluminum can are not included in the Geant4 construction.

Here we present the study of 3-different start-detectors, and three different kinds of powders.

The powder is always a cylinder of 2.5 cm radius and 2.3 cm length. This is always making direct

contact with the scintillator (or more accurately with the foil on the scintillator).

6.5.1 Start detector

We present 3 designs for the start detector that vary in the geometry of the lightguide, the

thickness of the scintillator, and the wrapping on the scintillator.

The first design, called the "5 mm inset square" is a thicker version of the geometry specified in

Figure 6.2, with a 0.5 mm thick scintillator, and two layers of Al foil estimated at 0.016 mm thick

each. This is an implementation of the actual physical modules we will test in Chapter 8. Notably,

this design has a 5 mm inset where the radius of the cutout changes. This means that the powder

will always be offset from the source by at least 5 mm.

The second design, called the "3 mm inset square" is identical to the first design, but now

82

matches the geometry in Figure 6.2. The thickness of the lightguide is 6 mm, meaning there is a 3

mm inset and the powder is therefore closer to the source.

The third design has not been constructed or tested, but it serves as an estimate for what an

"optimized design" might look like. It is called the "0.5 mm inset cylinder". It is a cylindrical

lightguide shown in Figure 6.5. The main benefit is that the cylindrical inset is reversed, the wider

part (2.7 cm) faces the source and the narrower part (2.5 cm) holds the powder. This allows for

optically coupling the scintillator to the resulting 2 mm inset, but also allows the scintillator to be

closer to the source. The scintillator itself is 0.15 mm thick, and is wrapped with 2 𝜇𝑚 aluminized

Mylar (2% aluminum).

Figure 6.5 The implementation of the 0.5 mm inset cylindrical start detector. This moves the
scintillator and the powder as close to the source as possible. The direction of the inset is reversed,
so that the powder sits in the narrower regions.

6.5.2 Powder

We study three kinds of powders, all three of which have been prepared, though further work

is required for quantifying positronium formation in the different powders. The powder container

is specified as a 2.5 cm radius cylinder of length 2.3 cm. The powder is assumed to have uniform

density throughout.

We model the powder from Chapter 5 as 0.1 g/cm3 SiO2 powder uniform in density. We also

have MgO powder at two densities, 0.3 g/cm3 and 0.6 g/cm3. The expected benefit of increased

powder density is making the formation region smaller, at the cost of potentially reduced lifetime

83

or positronium formation.

6.6 Stopping fraction and stopping position

Here we explore the parameter space by varying the powders while keeping the start detector

the same, then varying the start detector while keeping the powder the same. All results in this

section are for a point source of 𝛽+.

Here we quote 4 quantities for the different combinations. Firstly the stopping fraction is the

number of 𝛽’s that stopped in the powder divided by the total number of 𝛽 emissions. The larger

this number the better, as it directly relates to the final event rate. We quote the mean offset in the

𝑧-direction relative to the 𝑧 = 0 position at the source, the standard deviation in 𝑧 and in 𝜌. This

quantifies the offset of the formation region, and the general size of the spreading throughout the

powder.

In Table 6.1 the results for varying the powder is shown. All these runs are with the 5 mm inset

square start detector, and as such the powder starts at 5 mm from the source.

Powder Density (g/cm3)
SiO2
SiO2
MgO
MgO
MgO
MgO

0.1
0.1
0.3
0.3
0.6
0.6

Source
22Na
68Ga
22Na
68Ga
22Na
68Ga

Stops in Powder (%)
10.6
40.3
10.6
74.0
10.6
73.3

⟨𝑧⟩ (mm) Δ𝑧 (mm) Δ𝜌 (mm)
1.85
6.17
0.62
4.43
0.31
2.22

7.17
14.28
5.75
10.77
5.39
7.83

1.84
6.14
0.62
3.79
0.32
1.89

Table 6.1 The percentage of 𝛽+ that stop in the powder, their mean depth, and the standard deviation
of the depth and radial coordinate. All results run with the 0.5 mm thick scintillator in the 5 mm
inset square start detector.

Clearly the gallium source has almost 7 times as many 𝛽+ survive the start detector and reach

the powder compared to the sodium source. For the low density SiO2 powder with a gallium source

we see that almost 41% of the 𝛽+ that reach the powder do not get stopped in the powder and

escape. The offset along the 𝑧-axis is dominated by the size of the inset in the start detector (5

mm) for all combinations except low density SiO2 with the gallium source. Increasing the density

dramatically reduces the size of the stopping distribution, more than 3 times smaller in 𝑧 and six

84

times smaller in 𝜌. Figure 6.6 shows the distribution of stopping positions for the SiO2 powder and

the 0.6 g/cm3 MgO. The 𝜌-axis is scaled by 1/𝜌 to account for the cylindrical geometry, meaning

this corresponds to the distribution seen in a "slice" through the powder.

(a) 22Na in SiO2 at 0.1 g/cm3

(b) 68Ga in SiO2 at 0.1 g/cm3

(c) 22Na in MgO at 0.6 g/cm3

(d) 68Ga in MgO at 0.6 g/cm3

Figure 6.6 Stopping positions for two 𝛽 sources in two different powders, using the same start
detector design. The 22Na source in 0.6 g/cm3 MgO is substantially more localized than all other
combinations, and is entirely dominated by the geometry of the start detector (the 5 mm inset).

Next we consider changing the geometry of the start detector while using the 0.6 g/cm3 MgO

powder. These results are shown in Table 6.2. Changing the size of the inset dramatically changes

⟨𝑧⟩, as expected from moving the powder closer to the source. However, it does not decrease the

spread in 𝜌, as the powder moves closer to the source more and more 𝛽’s emitted at an angle will

85

051015202530z (mm)051015202530 (mm)r0100020003000400050006000700080009000051015202530z (mm)051015202530 (mm)r050010001500200025003000051015202530z (mm)051015202530 (mm)r0100002000030000400005000060000700008000090000051015202530z (mm)051015202530 (mm)r05000100001500020000250003000035000reach the source. In the final design, switching to the 0.15 mm scintillator means dramatically more

angled 𝛽’s survive the start detector and reach the powder.

Design
5 mm inset
5 mm inset
3 mm inset
3 mm inset
0.5 mm inset
0.5 mm inset

Source
22Na
68Ga
22Na
68Ga
22Na
68Ga

Stops in Powder (%)
10.6
74.4
10.6
74.4
52.5
87.3

⟨𝑧⟩ (mm) Δ𝑧 (mm) Δ𝜌 (mm)
0.31
2.22
0.31
2.22
0.42
2.35

0.32
1.89
0.44
2.03
0.89
1.96

5.39
7.83
3.63
6.07
0.96
3.56

Table 6.2 The percentage of 𝛽+ that stop in the powder, their mean depth, and the standard deviation
of the depth and radial coordinate. All results run with 0.6 g/cm3 MgO powder.

Comparing the currently implemented design (5 mm inset square with 0.5 mm thick scintillator

and 0.1 g/cm3 powder), with a more optimized design (0.5 mm inset cylinder with 0.15 mm thick

scintillator and 0.6 g/cm3 powder) we see a dramatic improvement to almost all recorded quantities.

These correspond to the first two lines in Table 6.1, and the last two lines in Table 6.2. Considering

the 22Na source specifically, the more optimal design has about 5 times as many positrons survive

the start detector and reach the powder. This directly translates to a 5 times higher event rate in the

final experiment. The mean 𝑧-position is about 1 mm, decreased by a factor of 7. We will have

two start detectors and two powder containers, one in + ˆ𝑧 and on in − ˆ𝑧. This means the two powder

containers with the two positronium distributions will be about 2 mm apart compared to 14 mm

apart. Finally the spreading in 𝑧 is reduced by a factor of 4, and in 𝜌 by a factor of 2.

The spreading of the positronium position has two separate effects, 1) sampling a wider range

of field values and therefore broadening the distribution of pseudo-triplet lifetimes and directions

of tensor polarization, and 2) changing the geometry of the 2-𝛾 and 3-𝛾 events the detector pairs

see.

6.7 Estimation of pseudo-triplet lifetime

The magnetic field map was provided by the manufacturer in 0.25 inch steps. These are linearly

interpolated to provide a more fine grained map of the field. The field values used are shown

in Figure 6.7. We track the ˆ𝑧-component of the B-field, and the ˆ𝜌-component. The lifetime is

86

determined by the absolute value of the field, the direction of the alignment is determined by the

field direction.

(a) B𝑧

(b) B𝜌

Figure 6.7 Field maps for the ˆ𝑧-component and ˆ𝜌-component of the magnetic field. Values are
normalized to the "nominal" value at the center of the field.

We take the stopping positions from the earlier section and for each 𝛽 that stops in the powder

we record the two field components at its position. The distributions for the 5 mm inset square

light guide with 0.1 g/cm3 SiO2 powder are shown in Figure 6.8. This configuration is sampling

the broadest range of field values.

We need to consider how the field value affects the lifetime (as discussed in Chapter 3), and how

the direction of the field affects the value of 𝑃2. The B-field induces an alignment along its axis.

The detector geometry is sensitive to distortions induced by alignment along the ˆ𝑧 axis. Relating

the direction of induced alignment 𝑃′

2, to the direction along the ˆ𝑧-axis goes as,

𝑃′
2 = 𝑃2

(cid:18) 1
2

−

1
2

(cid:19)

cos(2𝜃)

(6.1)

where 𝜃 is the angle between the axes. This follows from 𝑃2 being a component of a (traceless)

symmetric second rank tensor. In reality for both sources the range of values of cos(2𝜃) is miniscule,

with over 90% of events having less than a 0.00001% reduction.

87

200-150-100-50-050100150200z (mm)020406080100 (mm)r0.20.40.60.811.2200-150-100-50-050100150200z (mm)020406080100 (mm)r0.4-0.3-0.2-0.1-00.10.20.30.4(a) B𝑧 for forward 𝛽+

(b) B𝜌 for forwards 𝛽+

Figure 6.8 Firing 𝛽+ in a forwards aimed half-sphere for 22Na (blue) and 68Ga (red). The distribution
of field values are plotted separated between the 𝛽+ that scatter forwards and those that backscatter
into the second powder container. Field values are normalized to the "nominal" value at the center
of the magnet.

The magnitude of the field affects the lifetime. We take the "nominal field value" (the value

a the center of the magnet) as 0.4 T. We show the normalized distributions of lifetimes for two

configurations, the currently implemented design, and the more optimized design with higher

density MgO powder in Figure 6.9. For 22Na we get a sharply peaked distribution of lifetimes that

all fall within a few hundred ps range. The 68Ga shows more spreading than the 22Na, but almost

the full distribution still falls within 1 ns, with the majority of events within a 500 ps time window.

As such, for all combinations of powders and start detectors studied here we expect that the

spreading of the source will have a minimal effect on the tensor polarization. The magnet is

homogeneous enough that the variation of field directions has little effect on the alignment. The

variation in the lifetimes is also small enough to fall below the expected resolution of the electronics.

The connection between the lifetime and the size of the signal will be the main topic of Chapter 10.

6.8 Results for three designs

Before concluding this discussion we present full results for three finalized designs. One of

which is already constructed, one which could be reasonably constructed and tested with current

88

0.980.9850.990.99511.0051.01|0/|BzB020406080100120140310·Counts0.015-0.01-0.005-00.0050.010.015|0/|BrB020406080100310· Counts(a) Current start detector, SiO2 at 0.1 g/cm3

(b) Optimized start detector, MgO at 0.6 g/cm3

Figure 6.9 Distribution of lifetimes for 22Na (blue) and 68Ga (red) for the two start detector designs
studied. Clearly the galium source is sampling a wider range of field values, but the majority of all
the distributions fall within a 1 ns range.

prototypes, and one which is far more optimized. We label these as Design (A), Design (B), and

Design (C). The results for these designs will be used in later chapters.

All results presented so far are missing one large effect, the size of the radioactivity for realistic

sources is not point-like, but is instead on the millimeter scale, and therefore will have a large effect

when compared with ⟨𝜌⟩ values estimated above. For this reason we include an estimation of the

spread position of the radioactivity as an input for the following results. We model the radioactivity

as having a gaussian profile in the x-y plane, a reasonable estimate would be a 𝜎 = 1.5 mm,

although depending on the manufacturing method for the source it could be larger than this, and

likely will not have a gaussian profile.

Design thickness of PVT thickness of coating

(A)
(B)
(C)

0.5 mm
0.5 mm
0.15 mm

16 𝜇m
16 𝜇m
2 𝜇m

distance to powder
5 mm
3 mm
0.5 mm

powder density
0.1 g/cm3
0.3 g/cm3
0.6 g/cm3

Table 6.3 Summary of the main properties of the 3 Designs. We will study the stopping distribution
for these designs going forwards.

89

31.631.83232.232.432.632.833 (ns)t00.010.020.030.040.050.060.07counts (arb)31.631.83232.232.432.632.833 (ns)t00.050.10.150.20.250.3counts (arb)6.8.1 Design (A)

This is an implementation of the actual start detector prototype we have tested. This is the 5

mm inset start detector with 0.5 mm width scintillator with foil on both sides, shown in Figure 6.2.

The powder container is 0.1 g/cm3 SiO2 powder. We do not have plans to use this setup in the final

experiment (or in the magnet at all), so we do not enter into estimating the lifetime distribution.

The stopping position distribution is given in Figure 6.10, and the number of positrons stopping in

the powder, and their spreading is given in Table 6.4.

(a) 22Na

(b) 68Ga

Figure 6.10 Stopping position for Design (A), representing the currently implemented start detector
and powder combination.

Source
22Na
68Ga

Stopping fraction (%) Δ𝑧 (mm) Δ𝜌 (mm)

10.6
43.9

1.85
6.17

1.95
6.06

Table 6.4 Source spreading for Design (A). This reflects the currently implemented start detector
and powder container.

6.8.2 Design (B)

This is an improved implementation that could reasonably and quickly be used in further tests.

This is the 3 mm inset start detector with 0.5 mm width scintillator with foil on both sides. The

powder container is 0.3 g/cm3 MgO powder. The stopping position distribution is given in Figure

6.11, and the number of positrons stopping in powder, and their spreading is given in Table 6.5.

90

051015202530z (mm)051015202530 (mm)r02004006008001000051015202530z (mm)051015202530 (mm)r050100150200250300350400(a) 22Na

(b) 68Ga

(c) pseudo-triplet lifetime

Figure 6.11 (a), (b) Stopping position for Design (B), representing a more optimized combination
of start detector and powder using existing prototypes. (c) The pseudo-triplet lifetime distribution
in 0.4 T field for 22Na (blue) and 68Ga (red).

Source
22Na
68Ga

Stopping fraction (%) Δ𝑧 (mm) Δ𝜌 (mm)

10.6
74.9

0.62
4.44

1.14
3.82

Table 6.5 Source spreading for Design (B). This design is more optimized than (A), and could be
implemented with current equipment.

6.8.3 Design (C)

This is the optimized design for the start detector presented above. This also uses MgO powder

at a higher density of 0.6 g/cm3. This is an achievable design, but would require further prototyping

and testing work to be performed. The stopping position distribution is given in Figure 6.12, and

the number of positrons stopping in powder, and their spreading is given in Table 6.6.

91

051015202530z (mm)051015202530 (mm)r050010001500200025003000051015202530z (mm)051015202530 (mm)r0200400600800100012001400160031.631.83232.232.432.632.833 (ns)t00.020.040.060.080.10.120.14counts (arb)(a) 22Na

(b) 68Ga

(c) pseudo-triplet lifetime

Figure 6.12 (a), (b) Stopping position for Design (C), representing an optimized design to maximize
the amount of positrons reaching the powder, and minize their spreading through the powder. (c)
The pseudo-triplet lifetime distribution in 0.4 T field for 22Na (blue) and 68Ga (red).

Source
22Na
68Ga

Stopping fraction (%) Δ𝑧 (mm) Δ𝜌 (mm)

54.5
87.6

0.42
2.35

1.25
2.05

Table 6.6 Source spreading for Design (C). This represents a more optimized design that greatly
diminishes the spreading of the source.

92

051015202530z (mm)051015202530 (mm)r0500010000150002000025000051015202530z (mm)051015202530 (mm)r05001000150020002500300035004000450031.631.83232.232.432.632.833 (ns)t00.020.040.060.080.10.120.140.160.180.2counts (arb)CHAPTER 7

SIMULATION OF 𝛾-DETECTOR ARRAY

We have established the conceptual outline of the experiment, and identified a general structure

for the array of 𝛾-detectors and the optimized 𝛽-trigger module. In this chapter we cover the more

detailed design and optimization of the array. This extends the analytic work done in Chapter 3, to

a full Monte-Carlo simulation in Geant4. These studies include optimizing the angle of the outer

rings, determining the geometric analyzing power of each configuration, estimating the effect of

finite energy resolution and spreading of the source. These simulations study both 3-𝛾 events and

2-𝛾 backgrounds. Finally we consider adding shielding between the rings in the array.

The simulations in this chapter were performed using Geant4. Much of this work was studied

in parallel by Paul A. Voytas and Elizabeth A. George (WU) using EGSnrc [61].

7.1 Primary event generator

This chapter is concerned with the 2-𝛾 and 3-𝛾 events and their interplay with the array geometry.

The first half of this section is concerned with positronium decay at a point exactly in the center

of the detector array. The latter half of this chapter discusses the effect of spreading of the source.

The event generator is described below.

7.1.1

2-𝛾 events

These are very simple, a random vector ˆk is thrown, then the event consists of two photons with

511 keV. One in the + ˆk-direction, and the other in the − ˆk-direction.

7.1.2

3-𝛾 events

At the beginning of the run a vector polarization and a tensor polarization are specified for the

positronium atom. The matrix element is created including the Ore-Powell distribution, 𝑎(𝜔1, 𝜔2),
and the tensor term (cid:205)𝑖 𝑗 𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3). We do not consider the vector term s · B(k1, k2, k3)

in this work. The matrix element is normalized to have a maximum value of 1 (in reality this is

slightly more complicated due to the divergence mentioned in Appendix B). This chapter only

presents simulations using the isotropic distribution.

A complete description of the 3-𝛾 phase space was given in Chapter 2. The phase space is flat

93

in the energies, (𝜔1, 𝜔2). It is flat for two angles, the azimuthal coordinate for the decay plane 𝜙,

and the rotation of the momenta vectors within the decay plane, 𝜙12. Then it has the standard sin(𝜃)

distribution for the polar spherical coordinate. This is worked out in Ref. [2]. For generation of

events the matrix element (squared) is treated as a probability distribution on the phase space. A

random 𝜔𝑎 and 𝜔𝑏 values are sampled from the kinematically allowed region, and a third energy is

determined by 𝜔𝑐 = 2𝑚𝑒 − 𝜔𝑎 − 𝜔𝑏. These are then ordered by their energies 𝜔1 > 𝜔2 > 𝜔3. This

is tantamount to restricting the generation to the red triangle in Figure 7.1. The three photons are

initialized with ˆk1 along ˆ𝑥, and ˆn along ˆ𝑧. Finally, a random isotropic unit vector is sampled and

assigned to ˆn. The three momenta are then collectively rotated by randomly sampled Euler angles.

Finally the matrix element for this primary event is evaluated, and a random number between

0 and 1 is thrown. If that number is larger than the matrix element then the event is discarded and

a new event is generated. If the number is less than the matrix element then the event is used in the

simulation.

Figure 7.1 The kinematically allowed regions for two of the three energies from 3-𝛾 decay. Each
triangle corresponds to a different ordering of photons. Due to photon indistinguishability all
considerations can be restricted to a single triangle. The red triangle is where 𝜔𝑎 > 𝜔𝑏 >
2𝑚𝑒 − 𝜔𝑎 − 𝜔𝑏.

94

7.2 Detector array and configurations

The experiment will measure coincidences in pairs of detectors. Each pair will have a partner

pair that is its image under parity (combined with rotation). Call this set of three detectors a

"configuration". Each configuration has one dedicated "highest energy detector", and two "second

highest energy" detectors. Two configurations are illustrated in Figure 7.2. Take the highest energy

detector marked in green. This forms a configuration with two detectors marked in red, and a

second configuration with two detectors marked in blue. As these two configurations have different

opening angles, and therefore different energy ranges (see Figure 3.4) they select different parts of

the phase space. This induces geometric structure onto the distributions measured in (𝜔1,𝜔2). For

this chapter we refer to the 2-D energy plane restricted to the range 𝜔1 > 𝜔2 > 𝜔3 as the "phase

space" (in fact it is a 2-D projection of the full 5-D phase space).

Figure 7.2 Array of detectors with no mounting. Two configurations are highlighted, the Symmetric
157.5◦ between the green and red detectors, and the Symmetric 135◦ between the green and blue
detectors.

The following configurations have sensitivity for the tensor term 𝜅1𝑧 ( ˆk1 × ˆk2)𝑧,

1. Symmetric

95

Figure 7.3 Coordinates of the full array. Detector ID is specified by two digits in hexadecimal. The
first digit is the ring number, increasing for − ˆ𝑧. The second digit is the ID of the detector within the
ring. Detectors in different rings, but with the same index within a ring are referred to as a column,
for example column 1 is the set of detectors {0x01,0x11,0x21}.

a) 157.5◦

b) 135◦

c) 112.5◦

2. Asymmetric

a) 157.5◦

b) 135◦

c) 112.5◦

96

(a) Detectors involved in "Symmetric events"

(b) Detectors involved in "Asymmetric events"

(c) Detectors involved in "Asymmetric with ˆk1 in center ring"
Figure 7.4 The various classes of events. The arrow starts at the detector that ˆk1 hit and points
towards the detector that ˆk2 hit. Within a subfigure (Symmetric, Asymmetric, etc.) arrows with
the same color have the same analyzing power, dashed (full) line corresponds to negative (positive)
value for analyzing power.

97

where the angle refers to the cylindrical azimuthal angle between detectors, which is not equal to

the opening angle between the photons in the decay plane. "Symmetric" refers to both hits in the

outer rings, "Asymmetric" refers to events where the highest energy photon is in an outer ring, and

the second highest is in the middle ring. Any detector in the outer rings can serve as the "highest

energy detector". So there are two sets of independent Symmetric configurations with the highest

energy in the upper ring (ring 0), and in the lower ring (ring 2).

The flipped Asymmetric events, where the highest energy photon is in the middle ring, have

no sensitivity for the tensor term. This is because these events have ˆk1 perpendicular to the ˆ𝑧-axis.

This is only valid for the 𝑐4 form factor, this conclusion is no longer valid when including the effect

of 𝑐5. For this chapter we will be following the design and analysis only considering 𝑐4 and more

closely mirroring the previous experiments in Refs. [38, 39].

7.2.1 Detector numbering

We introduce labels for all 48 detectors by simply numbering them in hexadecimal. The first

digit corresponds to ring number, and the second to detector number. This is illustrated in Figure

7.3.

An events refers to an ordered pair of detectors IDs ( ˆk1, ˆk2). Detector 0x00 forms configurations

with the following sets of detectors,

1. Symmetric

a) 157.5◦ – {(0x00,0x27),(0x00,0x29)}

b) 135◦ – {(0x00,0x26),(0x00,0x2a)}

c) 112.5◦ – {(0x00,0x25),(0x00,0x2b)}

2. Asymmetric

a) 157.5◦ – {(0x00,0x17),(0x00,0x19)}

b) 135◦ – {(0x00,0x16),(0x00,0x1a)}

c) 112.5◦ – {(0x00,0x15),(0x00,0x1b)}

98

This is illustrated in Figure 7.4. Due to azimuthal symmetry we only consider events where the

highest energy photon was in "column 0", for this chapter we only consider events where one of

the photons hit detector 0x00. All of these events have a pair with the same sensitivity where the

highest energy photon hit the lower ring. This is illustrated in Figure 7.4.

7.3 Outline of analysis

A specified detector configuration has a set of coincident hits, with the recorded energy of each

hit. An example is shown in Figure 7.5. This shows the energy deposited in detector 0x00 on the

y-axis, and energy deposited in detector 0x27 (0x29) on the x-axis for the left (right) plot. An

asymmetry is formed between coincidences in detectors (0x00,0x27) versus (0x00,0x29). Some

energy cuts are specified on this distribution which correspond to integrating all the counts within

the 2-D cuts. For this chapter we merely draw a red triangle for the phase space of the 3-𝛾 event,

where 𝜔1 > 𝜔2 > 𝜔3.

Figure 7.5 2-D energy distribution for coincident hits in the "Symmetric 157.5◦" configuration. An
count asymmetry is measured between the counts in the red triangle on the left versus the right.

Now following Refs. [38, 39], the analyzing power for the signal is 𝜅1𝑧 ( ˆk1 × ˆk2)𝑧. For the

Monte-Carlo simulations this can be directly calculated from the generated kinematic vectors. The

averaged geometric analyzing power for the distribution in Figure 7.5 is shown in Figure 7.6, but

99

where each entry is weighted by the geometric analyzing power, then normalized by the number of

events in that bin. This illustrates which events have a positive and negative value of 𝜅1𝑧 ( ˆk1 × ˆk2)𝑧.

A configuration 𝑎 has three quantities of interest,

1. Counts – 𝑁 𝑎

2. Average Geometric Analyzing power – 𝐺 𝑎

𝑎𝑛 = (1/𝑁 𝑎) (cid:205)𝑖 𝐺 𝑎

𝑎𝑛,𝑖

3. Figure of Merit –

√︃(cid:205)𝑁

𝑖 (𝐺 𝑎

𝑎𝑛,𝑖)2

Figure 7.6 The same events as in Figure 7.5, but where each bin has been weighted by the averaged
geometric analyzing power. Note many bins outside the main kinematic region are affected by the
low number of counts in that region.

The average geometric analyzing power can be discerned from Figure 7.6 by integrating within

the energy cuts, then dividing by the counts within those cuts (calculated from integrating Figure

7.5). The Figure of Merit, as described in Chapter 3, corresponds to the statistical sensitivity for

those kinematic events. This is shown in Figure 7.7.

7.4 Coincidence cuts

Investigation of the observed distributions shows some oddities. In particular, in Figure 7.8,

there appears to be spurious structures in the energy distribution. There is an apparent dip along

the 𝐸1 = 𝐸2 line, but only for some configurations and not others.

100

Figure 7.7 "Figure of Merit" for the same data shown in Figure 7.5. This corresponds to the sum of
the analyzing power weighted by the square root of the number of events, and is indicative of the
statistical sensitivity.

Figure 7.8 Symmetric events at 157.5◦ (left) and Symmetric events at 135◦ (right) with an exclusive
2 hit coincidence condition. Note the visible structure within the band, most clearly visible in the
135◦ pair as a dip on the E1=E2 diagonal. There are possibly six similar dips in the 157◦ pair. The
quoted number of counts corresponds to the number within the phase space triangle, not drawn
here so as to not cover any structure.

101

00.10.20.30.40.50.6Energy Detector 0x27 (MeV)00.10.20.30.40.50.6Energy Detector 0x00 (MeV)010203040506070=18038(cid:176)157.5UPN00.10.20.30.40.50.6Energy Detector 0x26 (MeV)00.10.20.30.40.50.6Energy Detector 0x00 (MeV)01020304050=6983(cid:176)135UPNConsider the geometry of the event as the energy of the photons changes. The three photons lie

in a plane and their momentum sums to zero. Geometrically, ˆk3 is confined to lie between − ˆk1 and

− ˆk1 − ˆk2, as illustrated in Figure 7.9. ˆk3 = − 1√
2

( ˆk1 + ˆk2) as 𝜔2 → 𝜔1 (stated otherwise, momentum

conservation means k3 = −k1 − k2, when |k1| = |k2| this can be translated to their unit vectors).

Reconsider Figure 7.8, the vectors ˆk1 and ˆk2 are fixed to hit the two detectors, as the energies

change ˆk3 moves through the plane. The plane is defined by the highest energy detector, second

(a) 𝜔1 ≠ 𝜔2 ≠ 𝜔3

(b) 𝜔1 ≈ 𝜔2

Figure 7.9 Distribution of photons within the decay plane, with ˆk1 in red, ˆk2 in blue, and ˆk3 in
(a) The general range that ˆk3 can lie in, constrained between the two dashed lines.
green.
(b)
When 𝜔2 → 𝜔1 then 𝜓23 → 𝜓13 by momentum conservation. The bounds of the region for ˆk3 are
perpendicular to ˆk1 + ˆk2 for 𝜔1 = 𝜔2, and the reflection of ˆk2 over ˆk1 for 𝜔2 = (1/2)𝜔1

highest energy detector, and the origin (where the decay occured). This is shown graphically for

Symmetric 157.5◦ events and 135◦ events in Figure 7.10. The lowest energy photon lies in the

plane and falls somewhere between the reflection of k1 and the intersection of the decay plane and

the x-y plane (z=0). When 𝜔2 → 𝜔1, then the lowest energy photon is constrained to the lie in

the x-y plane (which coincides with the central ring of detectors). For the 135◦ events it is aimed

directly at detector 0x16 and therefore all 3 photons are detected. For the 157.5◦ pair it lies between

detectors 0x15 and 0x16, but as ˆk3 sweeps out an arc in this plane as 𝜔2 → 𝜔1 it can cross through

detector 0x16 when 𝜔2 ≲ 𝜔1.

The analysis above applies cuts based on the number of hits, specifically a strict two hit

102

(a) Symmetric 157.5◦

(b) Symmetric 135◦

Figure 7.10 Taking the event in the decay plane from Figure 7.9 and embedding it into the detector
array shows how certain configurations restrict ˆk3 to hit specific detectors in an energy dependent
way.

coincidence is imposed. This restriction is removing perfectly good events where all three photons

are detected. This is inherently related to the geometry of the event, and through the argument

above, directly related to the 2-D energy distribution of the observed photons. So restricting to

strictly two hits results in dips in the 2-D energy distribution. Updating the coincidence condition

to allow for events with three hits (in specific conditions) removed much of this structure. This is

shown in Figure 7.11. This is most notable for the 135◦ pair and increases the counts by 42%, the

two dips closest to the diagonal for the 157.5◦ are also reduced but only result in an 11% increase

in counts.

The specific three hit coincidence condition is, each hit in a separate ring, their summed energy

is near 1022 keV, and that no back-to-back detectors were hit. Looking at the 157.5◦ configuration

in Figure 7.11, there are still some dips. Reflecting on the geometry, there are still configurations

where the third photon crosses through the lower ring. These are rejected by the current coincidence

conditions. This coincidence condition is used for the rest of this work.

103

Figure 7.11 Loosening the restriction of only 2-hits by allowing a third hit in the middle ring
removes the structures visible in Figure 7.8. The counts increase by about 11% for the 157.5◦ pair,
and 42% for the 135◦ pair respectively.

7.5 Optimization of geometric sensitivity

At this point the geometry is largely determined based on the discussions in Chapter 3. The

main undetermined quantity is the angle of the tilt of the outer ring. Before discussing this it is

worth surveying the sensitivities of the various configurations first, and how they can be combined

into a net Figure of Merit.

7.5.1 Sensitivity of each configuration

The counts for all six configurations are shown in Figure 7.12 when holding the outer rings

tilted at 30◦ . The geometric analyzing powers are shown in Figure 7.13. Finally the Figures of

Merit are shown in Figure 7.14. These are summarized in Table 7.1 where these quantities are

integrated over the energy cuts, taken to be the phase space triangle.

Each of these configurations select different parts of the phase space. The Figures of Merit are

combined into a aggregate Figure of Merit by the square root of the sum of squares of the individual

detector configurations.

104

00.10.20.30.40.50.6Energy Detector 0x27 (MeV)00.10.20.30.40.50.6Energy Detector 0x00 (MeV)0102030405060708090=22381(cid:176)157.5UPN00.10.20.30.40.50.6Energy Detector 0x26 (MeV)00.10.20.30.40.50.6Energy Detector 0x00 (MeV)01020304050=10050(cid:176)135UPNFigure 7.12 Coincidence counts for all 6 configurations considered here, all with ˆk1 hitting detector
0x00. These correspond to the plots shown in Figure 7.5.

105

Figure 7.13 Geometric analyzing power for all 6 configurations, corresponding to the information
shown in Figure 7.6.

106

Figure 7.14 Figure of Merit (FoM) for all six configurations, corresponding to the information
shown in Figure 7.7.

107

Configuration Counts
22381
S157.5
10050
S135
1753
S112.5
12651
A157.5
3552
A135
34
A125.5

G𝑎𝑛
0.125
0.242
0.294
0.144
0.258
0.240

FoM
19.92
24.83
12.93
17.25
15.87
1.67

Table 7.1 Relative counts, geometric analyzing powder, and Figure of Merit for each configuration
considered in this chapter. This is with the tilt of the rings fixed at 30◦.

7.5.2 Outer ring tilt

As defined the geometric analyzing power is largely dependent upon the opening angle between

the photons. Increasing the angle of the outer ring up to 45◦ has 3 features: 1) it increases the

opening angle between the photons, 2) it makes 𝜅1𝑧 larger, and 3) it makes 𝑛𝑧 larger. The results of

changing the tilt of the outer ring are given for an angle of tilt between 20◦ to 32.5◦ in Table 7.2.

The results are presented for the Symmetric configurations and the Asymmetric separately.

Outer ring tilt (◦) N𝑆𝑌 𝑀 G𝑆𝑌 𝑀
𝑎𝑛
0.126
20
0.139
22.5
0.150
25
0.160
27.5
0.168
30
0.175
32.5

26338
28495
30245
32082
34184
36336

FoM𝑆𝑌 𝑀 N𝐴𝑆𝑌 𝑀 G𝐴𝑆𝑌 𝑀
𝑎𝑛
0.127
23791
0.141
21693
0.153
20335
0.163
18290
0.169
16238
0.176
14499

22.76
25.97
28.79
31.62
34.36
36.86

FoM𝐴𝑆𝑌 𝑀
21.72
22.72
23.72
23.96
23.50
23.14

Table 7.2 Effect of increasing the outer ring tilt. The summed counts, averaged geometric analyzing
power, and cumulative Figure of Merit are quoted for the Symmetric and Asymmetric events
separately.

The Figure of Merit merely increases as the angle increases. Having a larger tilt makes the

support structure much more complicated, but it appears vital to reaching the sensitivity needed.

We opt for a tilt at 30◦.

7.6 Estimation of final statistical sensitivity

Now we can estimate a statistical sensitivity for the planned experiment to compare with the

previous searches. For the purposes of this chapter we are considering only using the unperturbed

triplet lifetime state reducing the statistics by only considering two of the three states. The statistical

108

sensitivity from an asymmetry measurement should go as Δ𝐶𝐶𝑃 = 1/(𝐺 𝑎𝑛

√

2𝑁) = 1/(

√

2𝐹𝑜𝑀)

using the aggregate Figure of Merit defined as the square root of the sum of squares of each

configurations Figure of Merit. The simulations shown so far do not correspond to the full planned

statistics, but instead to 108 ortho-positronium decays. Calculating the cumulative Figure of Merit

for this level of statistics gives 𝐹𝑜𝑀 = 41.66.

If we plan to run for 35 continuous days with

a 1.85 MBq source with 50% of the positrons surviving the start scintillator, then 50% forming

positronium in the powder, 1/2 in 𝑚 = ±1 ortho-positronium, then this simulation corresponds to

1/7000 of the full planned statistics. This translates to Δ𝐶𝐶𝑃 = 2 × 10−4 a factor of 10 times higher

statistical sensitivity than the previous search. This is if we assume the same tensor polarization

as the Tokyo experiments [39]. This level of increased sensitivity will also require a reduction in

systematic uncertainties by a factor of 3. The primary systematics for the Tokyo experiment were

from the stepper motor that rotated the setup. This is completely removed in the present array,

and replacing the permanent magnets with an electromagnet will allow careful characterization of

many systematic effects.

7.7 Spreading of the source, finite energy resolution, and 2-𝛾 backgrounds

7.7.1

2-𝛾 events

Some portion of the positronium decays through 2-𝛾 annihilation, resulting in two back-to-back

511 keV photons. One photon could Compton scatter off of inactive material and hit a detector

in a configuration we record. Currently this simulation has no inactive material, and the final

experiment will have all material between the source and the detector made from low Z material,

which will minimally scatter the photons.

Including finite spreading of the source makes the detector configurations geometrically sensi-

tive to back-to-back photons. This is illustrated in Figure 7.15 for the central ring detectors, drawn

roughly to scale. Spreading of the source means the decay is no longer at the origin, and if it spreads

far enough then a straight line can connect two detectors and the positronium decay position. For

the 157.5◦ pairs, highlighted in darker blue, The configuration is sensitive to 2-𝛾 decays once the

decay occurs within the region banded by the two red or purple lines. Also drawn are circles of

109

radii 1 cm and 2 cm. Further complicating issues, the combinations shown in this figure do not

represent any of the configurations in Figure 7.4. The combinations sensitive to the signal are those

between detector rings, which have an even sharper opening angle than that shown.

Figure 7.15 Pairs of detectors in the central ring, highlighting the regions where 2−𝛾 decays are
visible (between the red lines or the purple lines). Some scale is given for the central circles, to
be compared with the stopping position simulations in Chapter 6. Note this is the geometry for
coincidences within the central ring, which we do not consider here. For coincidences between the
outer ring the angle between detectors is closer to 180◦, and the two bands are even closer.

7.7.2 Finite source spread

The positronium source is not point-like. The spreading of the source is dependent on the 𝛽

source used. The 68Ga source produces substantially broader spreading of the positronium source

compared to 22Na.

To estimate the effect of spreading we directly take the stopping distributions from Chapter

6. The 2-D histogram of stopping positions is used as an input to be sampled as a probability

distribution function. One limitation is that the 2-D histogram is taken as a distribution for 𝑧 and

𝜌, and a random 𝜙 is generated, as well as randomly choosing whether 𝑧 > 0 or 𝑧 < 0. All inputs

110

generate an azimuthally symmetric source with mirror symmetry between positive and negative

𝑧. The caveat to this exception is that the entire source distribution can be shifted to an arbitrary

position, so the origin for the distribution can be shifted away from the origin of the detector array.

The 2-D energy distributions for the 3-𝛾 and 2-𝛾 events for the Symmetric 157.5◦ configuration

for a point source, a 22Na source, and a 68Ga source are shown in Figure 7.16 for Design (B), and in

Figure 7.17 for Design (C). The dilution due to 2-𝛾 is substantially worse for the Symmetric 157.5◦

configuration than for any other configuration (this is the closest to back-to-back). There are far

more back-to-back photons when spreading of the source is included, however these events leave

a very distinct energy pattern in the detectors. The spreading of the source increases the amount

of phase space selected, but it does not lead to misidentification of events, the energy recorded is

still the energy of the photon (up to the effect of Compton scattering and X-ray escapes). It is an

interesting coincidence that the corner of phase space where all 3 photons have the same energy

(𝜔1 = 𝜔2 = (2/3)𝑚𝑒) is exactly equal to the "double Compton shoulder" for two 511 keV photons.

The 2-𝛾 distribution is still separable with energy cuts. The coincidence counts for each

configuration for both sources in Design (B) and (C) is given in Table 7.3. The spreading generally

reduces the number of events in the configurations that select large regions of phase space (157.5◦),

but increase the number of events for the configurations that are only sensitive to a small corner of

phase space (112.5◦).

𝑝𝑡𝑠𝑟𝑐
N
3𝛾
22412
10122
1711
12651
3552
34

(B) N𝑁𝑎
3𝛾
22056
9254
1769
13072
3860
91

(B) N𝐺𝑎
3𝛾
20252
7989
1571
1300
4305
346

(C) N𝑁𝑎
3𝛾
22526
9982
1804
12679
3663
66

(C) N𝐺𝑎
3𝛾
22274
9327
1827
13051
3921
132

S157.5
S135
S112.5
A157.5
A135
A112.5

Table 7.3 3-𝛾 coincidences for each configuration including the spreading of the source. Design
(C) shows the least amount of spreading of the positronium.

111

Figure 7.16 Point source (left), 22Na (middle), and 68Ga (right) coincidence events for 3-𝛾 decays
(top) and 2-𝛾 decays (bottom) using the Design (B) for the start detector and powder. Note the
spreading of the source merely makes more of the phase space visible, but does not map events into
or out of the phase space (with the exception of the included effect of Compton scattering).

7.7.3 Finite energy resolution

The detectors will have roughly 12% FWHM resoltuion at 511 keV and a 1/

√

𝐸 scaling. More

specifically we take the function,

𝜎(𝐸)/𝐸 = (cid:0)0.0433(cid:1)/√︁𝐸 (MeV)

(7.1)

as extracted from data that will be presented in Chapter 8. For any event with a given deposited

energy, a random number gaussian distributed around 𝐸 with width 𝜎𝐸 is sampled. This is

illustrated in Figure 7.18. The Symmetric 157.5◦ configuration’s 3-𝛾 distribution for a point source

is shown in Figure 7.19. The effect on the measured distributions, for Design (B) are estimated in

Figure 7.20, and for Design (C) in Figure 7.21. The smearing of the 3-𝛾 events is not particularly

nefarious, it simply moves events around inside of the wide energy cuts. The primary concern

is when it leads to a misidentification of which photon had highest energy and which had second

112

Figure 7.17 Point source (left), 22Na (middle), and 68Ga (right) coincidence events for 3-𝛾 decays
(top) and 2-𝛾 decays (bottom) using the Design (C) for the start detector and powder. Note the
spreading of the source merely makes more of the phase space visible, but does not map events into
or out of the phase space (with the exception of the included effect of Compton scattering).

highest. By intuition this might seem like a disastrous thing to happen, however inspection of the

tensor term shows that flipping ˆk1 ↔ ˆk2 does not change the sign of the signal for Symmetric

events. This is because flipping the two photon labels flips the normal of the decay plane, but it also

flips 𝜅1𝑧. It is clear that there is no energy cut that removes all 2-𝛾 events without also removing

good 3-𝛾 events.

7.7.4 Dilution for various start detector designs

Now the effects of the combined radioactive source, start detector geometry, powder density,

and energy resolution can be combined. The reduction in counts for all configurations for these

combinations is given in Table 7.4. There is a reduction in overall counts on the scale of 10-20%.

This reduces the statistical sensitivity of the experiment and will require slightly longer runtime.

For the Symmetric 157.5◦ configuration the 3-𝛾 and 2-𝛾 counts are given in Table 7.5. These are

113

Figure 7.18 Estimation of the response function for the final crystals. These values are taken from
data in Chapter 8. The width of the peak is taken for the 202, 307, 511, and 1275 keV peaks and
𝐸. The right plot demonstrates the Gaussian response with the corresponding width at
fit with 1/
each energy.

√

Figure 7.19 Coincidence distribution for Symmetric 157.5◦ pair of detectors (and a point source).
This shows the same event distribution with (right) and without (left) applying the response function
from Figure 7.18.

114

Figure 7.20 Same events as shown in Figure 7.16 for Design (B), with the response function from
Figure 7.18 applied event by event.

the number of coincidences in the energy cuts of the "phase space triangle", for 108 3-𝛾 decays and

108 2-𝛾 decays.

N

𝑝𝑡𝑠𝑟𝑐,𝜎∞
3𝛾
22412
10122
1711
12651
3552
34

𝑝𝑡𝑠𝑟𝑐
N
3𝛾
18549
9239
1948
11783
3368
224

S157.5
S135
S112.5
A157.5
A135
A112.5

(B) N𝑁𝑎
3𝛾
17923
8386
1965
12035
3742
342

(B) N𝐺𝑎
3𝛾
16821
7292
1849
11937
4205
608

(C) N𝑁𝑎
3𝛾
18531
9123
2000
11724
3437
285

(C) N𝐺𝑎
3𝛾
18223
8512
2027
12046
3724
348

Table 7.4 Combined effect of source spreading and finite resolution on the number of 3-𝛾 coinci-
dences observed for each coincidence configuration. This is presented for both radioactive sources
and both inner module designs.

7.8

Inter-ring shielding

The array will feature a lot of detectors, and each detector sees a lot of detectors. Nearly 90%

of the pseudo-triplet decays to two photons, and a large fraction of those Compton scatter and only

115

Figure 7.21 Same events as shown in Figure 7.17 for Design (C), with the response function from
Figure 7.18 applied event by event.

Design
point source, infinite resolution
point source, finite resolution
Design (B) finite resolution
Design (B) finite resolution
Design (C) finite resolution
Design (C) finite resolution

source
ptsrc
ptsrc
22Na
68Ga
22Na
68Ga

N3𝛾
22412
18549
17923
16821
18531
18223

N2𝛾
–
26
485
2733
641
2031

Table 7.5 Detection efficiency for 2-𝛾 and 3-𝛾 events for the 157.5 ◦ pair of detectors in all three
configurations.

116

deposit part of their energy. This leads to many events where a photon scatters to its neighbor

and yields hits with lower energy. There is no room to add shields between detectors in a ring,

however it might be feasible to add shielding between the rings, and this could substantially reduce

the amount of Compton coincidences. Consider a "wedge" shield specified as in Figure 7.22. This

forms a circular wedge between the rings.

Figure 7.22 Geometry of the shield, showing a radial slide through the shield.
placeholder).

(currently a

Two shields are shown in Figure 7.23, with 𝛽0 = 9◦, 𝛽1 = 23◦, and two different inner radii (𝑟0).

These shields need to extend inwards beyond the detector face to truly block the detectors from

each other.

(a)

(b)

Figure 7.23 Array with shielding between the rings. The left shows shielding with an inner radius
of 4 cm, and the right shows inner radius of 6 cm.

117

Firstly, consider the dependence on the material for the shielding. The shielding should block

photons scattered off the detectors. At the same time adding shielding adds events that scatter off

the shield and hit the detectors. Figure 7.24 shows the energy deposited in the shield for 511 keV

photons fired isotropically in the array. This features the two likely shielding options, lead and

brass, and also includes silver simply because it lies between the two options in terms of atomic

number and density. The amount of Compton scatter versus photopeak events is highly dependent

on the material, quickly reducing as the atomic number is decreased.

Figure 7.24 Energy deposited in the shield by isotropic 511 keV photons. Brass is shown in red,
Silver in pink, and lead in blue. Note the relative heights of the Compton continuum versus the
photopeak. Lead will provide the best shielding as it is produces almost an order of magnitude less
scattering.

There are a few ways to slice up the data to glean insights into this design. We record events

with 2 hits, and create a matrix where the x-index corresponds to the first object hit, and the y-index

to the second object hit. All hits within a ring are summed, and the two shields are treated as one

object. This is shown for the detector array with no shielding in Figure 7.25. The diagonal elements

mean a photon scattered from one detector to another within the same ring. An off-diagonal means

it scattered from one ring to another.

Figure 7.26 corresponds to the array with shielding. This shows a decrease in events scattering

between rings. For a shield of inner radius 6 cm (in line with the front face of the detectors)

scattering from outer rings to inner rings is reduced from 1700 events to 300 events with a shield.

Extending the shield inwards to 4 cm radial distance this number dramatically drops again down to

118

Figure 7.25 Quantifying the scattering between detector rings for isotropic 511 keV photons. The
x-axis is the first object hit and the y-axis is the second object hit.

about 45 events.

The amount of photons that scatter off the shielding and hit a detector far outweigh any reduction

of scatters between crystals. For the inner ring has an additional 8700 events where the photon

scattered off the shield, and for the smaller radius shield this number goes up to 10200. In effect

this adds a huge amount of background, but only minimally reduces the coincidence hits between

detectors. This is worse than it seems, for two neighboring crystals in the final experiment both hits

are recorded and this does not look like a signal. The shielding is not an active detector element,

so for hits scattering off the shielding the events cannot be simply vetoed. This amounts to a large

distortion to the spectrum to fix a problem that could largely be removed by coincidence logic.

(a) 𝑟0 =6 cm

(b) 𝑟0 =4 cm

Figure 7.26 Scattering between detector rings and shielding for lead shielding. The color scaling
is on the same scale as Figure 7.25.

Now consider some more finely grained information. Take a slice in 𝜙 centered on three

detectors, column 0 (0x00, 0x10, and 0x20). The observed spectra for these detectors and the

neighboring crystals in columns 1 and 2, with and without the shielding is shown for a lead shield

matching the design described above in Figure 7.27, extending to 𝑟0 =6 cm inner radius. The

119

spectra in blue are with no shielding, and in red is with shielding. This has no trigger on number

of hits so it includes multiple scatterings.

Figure 7.27 Energy spectra of column 0, 1, and 2 detectors for 511 keV singles restricted to
azimuthal angles centered on column 0. Shown for no shielding (blue) and Pb shielding extending
to 𝑟0 =6 cm radius (red).

There is little to no benefit looking at the nearest neighbor, it seems to reduce the low energy

hits, but increase the high energy. There is a clear reduction in hits to the crystals two away.

However, compare the y-axis scales shows that any effects on neighboring crystals is very small

compared to the increase in counts in the central crystals.

This was the pattern throughout the full parameter space. In general, making the shield thinner

and going to a smaller radius reduces the spurious scatterings. The closest design to something

that would be considered beneficial (reduction in scattering events relative to no shielding) was

when the shielding was made of lead, angled at 𝛽0 = 15◦ and 𝛽1 = 17◦, and extended all the way to

𝑟0 =1.5 cm radius. The results are shown in Figure 7.28. This unrealistic design has a front edge

that is less than a mm wide.

120

Figure 7.28 Energy spectra of column 0, 1, and 2 detectors for 511 keV singles restricted to azimuthal
angles centered on column 0. Shown for no shielding (blue) very thin Pb shield extending down to
𝑟0 =1.5 cm from the origin (red).

Brass shielding, which is far easier to produce, was purely detrimental in all configurations

tested. We chose to not pursue shielding between rings.

It seems to almost exclusively be

detrimental, adding spurious counts while reducing very little of the backgrounds that matter.

121

CHAPTER 8

FIRST PROTOTYPE TESTING

8.1 Overview

In this chapter we present the design of the support structure that will hold the array of detectors

in the magnet. This is followed by the design and printing of a support for the start detector and

powder that will hold the prototype inner module in the center of the detector array.

This is followed by tests with three finalized crystals mounted on the central support ring and

read out with the start detector in the center of the ring. We studied the timing properties of the

system, the various coincidences observed, and the positronium physics observed.

8.2 Gamma array support structure

The 𝛾-detectors will all be placed in custom designed frames that both mount the crystal onto

the frame, and hold the SiPM on the back of the crystal. This piece must handle the tilt of the outer

rings, as such there are two designs of frames. The frames with an angle of 60◦ are labeled the

"outer frames", and those with a 90◦ angle are the "central frames". An outer frame is shown in

Figure 8.1 The frame features a base that directly attaches to the support structure, and a "clamp"

(a)

(b)

Figure 8.1 Frame for mounting the detector onto the support structure. It features a "clamp" that
screws onto a back frame that holds the crystal onto the frame. (a) The combined frame and crystal
in the CAD file. (b) A finalized crystal in the constructed frame.

that screws onto the base. The clamp holds the crystal onto the base from about halfway along the

length of the crystal. This ensures that there will be little material between the front face of the

122

detector and the central source. The back of the frame has a set of holes for the SiPM to screw

directly into.

The frames attach onto an aluminum ring, with sixteen detectors attached on one side. Figure

8.2 shows two outer ring crystals attached onto the support. The tilted outer ring crystals sit very

close to each other.

Figure 8.2 Two final LYSO crystals in the tilted frames attached to the support ring. The inner
clamp of the frames are nearly touching. The final LYSO crystals will be covered by a Tyvek foil
over the TiO2 paint not included in this image.

The main support structure consists of two aluminum rings, each with 16 tilted detectors

attached on the outer edges. The central detectors can be attached to the inner side of either ring.

This design is shown in Figure 8.3.

Finally, this three ring structure is connected by four support legs onto outer rings that attach

to the outer sides of the magnet. The full model for the support structure with all crystals is shown

in Figure 8.4. The manufactured structure without crystals (or crystal frames) is shown installed in

the warm bore of one of the FRIB Positron Polarimeter magnets in Figure 8.5.

The diameter of the warm bore of the magnet is about 22 cm. Accounting for the mounting and

the crystals that are 30 mm in length leaves about a 12 cm diameter cylinder for the entire inner

detector support structure.

123

Figure 8.3 Cross section of detector array, showing all three rings of detectors mounted on the
central aluminum support. Each central detector can attach to either the upper or lower ring, but
here they are all attached to the lower ring.

Figure 8.4 Full array support structure. The central aluminum frame is attached to two large rings
that will attach to the outer frame of the magnet.

8.3 Start detector support structure

The start detector and the powder need to be held in the center of the array. The position of the

inner module should be adjustable to moderately high precision to avoid any offsets of the source.

The powder and start detector are on the scale of 2-3 cm radially, with 6 cm radius to the crystals.

124

Figure 8.5 Aluminum support structure installed in one of the Positron Polarimeter magnets. All
48 detectors will be attached ot the two central rings.

There is still the possibility of manufacturing a smaller start detector and powder container (at the

cost of losing some 𝛽’s that could have formed positronium), but for now we proceed with the

current prototypes that we have on hand.

Two students, Vimbainashe Chado (MSU) and David-Michael Peterson (MSU), worked on

designing a support structure to hold the inner module. this must be non-magnetic, and induce as

little scattering as possible. The natural solution is to 3-D print a support structure that can attach

to the main support frame. 3-D printers mainly print in polylactide (PLA) plastic which is low Z

and will induce minimal scattering in the final experiment.

The support accommodates the current start detector read out with SiPMs. This breaks the

azimuthal symmetry of the structure and will not be the design for the final experiment. The

structure is shown in Figure 8.6, and in the full array support in Figure 8.7. The piece highlighted

in dark blue holds the start detector, and places enough pressure onto the SiPMs to couple them to

the sides of the start detector. This piece will be used for future tests to be performed in the magnet.

8.4 Demonstrator

We constructed a demonstrator using the 3 finalized 𝛾-detectors and the start detector and

powder specified in Design (A). In principle this is all that is needed for the experiment, as the

125

Figure 8.6 Prototype design for the inner module support structure for initial tests in the magnet.
This will hold the powder and start detectors centered in the detector array.

Figure 8.7 Model of the inner module support placed in the full detector array. Not featured are
the screws that will attach the outer ring of the inner support structure to the legs of the 𝛾-detector
support structure.

count asymmetries will always use independent sets of 3 detectors. The prototype is shown in

Figure 8.8. The three crystals correspond to a configuration with an opening angle of 157.5◦. This

is not a final configuration that will be included in the asymmetry, as all the detectors were in

the middle ring. Following Figure 7.3 the detectors are labeled starting at the top of the ring and

126

proceeding counterclockwise as 0x10, 0x17, and 0x19. The start detector and powder were placed

in the center of the ring. There was no mechanism to carefully fine tune the position of the source

and powder. This is visibly evident in Figure 8.8. The entire setup was raised 18 inches off the

table by foam supports which dramatically reduced backscatter of photons from the wooden table.

Figure 8.8 Demonstrator with three finalized crystals and a start detector read out by SiPMs.
The detectors are ordered following Chapter 7, starting with 0x10 at the top, 0x17, and 0x19 in
counterclockwise order. Note the asymmetric placement of the start detector and powder. Similarly
the source itself was not perfectly centered beneath the start detector.

The two sides of the start detector were summed and amplified before going to the DAQ.

This amplified the range of the ADC bits used and gave increased timing resolution [72]. No

preamplification was used on the LYSO crystals, which were digitized directly.

The signals were digitized using 250 MHz PIXIE-16 modules with NSCLDAQ [71, 72]. The

digitization used CFD timing on the signals to get improved timing from just recording timestamps.

The timing parameters for the fast filter and the CFD were tuned for the plastic and the LYSO

individually.

Most of the results in this chapter used a 22Na source beneath the start detector. At the very

end of the chapter we will present the results for a 68Ga source. As such, almost all plots have

coincidences between the 𝛽+ annihilation and the 1275 keV photon. This added structure helped to

characterize the system, but is inherent to the sodium source and should be absent when we using

127

a gallium source.

8.5 Timing properties

The system has two timing properties that are important. These are the time resolution between

a hit in the start detector and a hit in a LYSO crystal, and also the timing between the LYSO’s. The

start signal is in the plastic and the stop in a LYSO. The timing between the two LYSO crystals is

useful for rejection of backgrounds. The final experiment will record triple coincidences between

two LYSO crystals and the start detector. Sharper timing resolution between LYSO crystals will

give a stronger ability to separate true coincidences from accidentals.

All timing properties presented in this section were built with an exclusive 2 hit coincidence

condition. When using a 22Na source there were 3 classes of coincidences between the 𝛾-detectors,

1) 1275 keV de-excitation photon with an annihilation photon, 2) two of the annihilation photons,

3) inherent radioactivity coincidence.

All 6 coincidence time spectra are shown in Figures 8.9 and 8.10. The timing between two

LYSO crystals was more or less straightforward. The timing between the plastic and the LYSO was

more difficult. The plastic signals were very noisy, and because the SiPM was slow the start detector

experienced pileup. This ultimately required a moderately high energy cut on the start detector to

remove improperly digitized events. Some structure persisted in the time spectrum shown in Figure

8.10. The combined results for the time resolution are quoted in Table 8.1. We achieved a timing

resolution of roughly 3.2 ns FWHM between the plastic and LYSO, and 4.3 ns between two LYSO

crystals. This data was taken with the 0.1 g/cm3 SiO2 powder, and a long lifetime component is

visible in Figure 8.10.

start
–

0x10
3.13
–

0x17
3.20
4.14
–

0x19
3.39
4.47
4.38
–

start
0x10
0x17
0x19

Table 8.1 FWHM time resolution between each pair of detectors in ns. Further optimization is
needed to balance faster timing versus proper digitization of energy.

A more thorough analysis must be performed that more carefully studies the timing properties

128

(a)

(b)

(c)

Figure 8.9 Timing between each pair of LYSO detectors giving between 4.1 and 4.5 ns FWHM for
each pair. These events had an exclusive 2 hit coincidence condition, and the y-axis is log-scale

versus the reconstructed energies. Ultimately the final experiment will use a much faster start

detector by replacing the SiPMs by PMTs, so a careful optimization of the timing was not pursued

at this time.

8.6 Crystal coincidences

The studies of single LYSO detectors in Chapter 4 required a "background subtraction", that is

running without a source to remove the counts from internal radioactivity of the 176Lu (Figure 4.2).

These counts are expected to be removed in the online experiment by a coincidence condition with

the start detector.

The 2-D energy distributions for coincidences between LYSO crystals is shown in Figure 8.11.

All these were coincidences between the LYSO crystals that were not in time with the plastic start

129

80-60-40-20-020406080 (ns)0x10 - t0x17t310410510counts (arb)80-60-40-20-020406080 (ns)0x10 - t0x19t310410510counts (arb)80-60-40-20-020406080 (ns)0x19 - t0x17t310410510counts (arb)(a)

(b)

(c)

Figure 8.10 Time spectrum when an event had only two hits, one in the start detector, and one
in a LYSO crystal. The y-axis is log-scale. There was some structure before the peak and some
structure around 130 ns after the peak.

detector. Some of these were from the 22Na source which emits a 1275 keV photon. However the

structure was dominated by the 3 peaks of 176Lu below 511 keV, at 307, 202, and 88 keV 𝛾-rays.

This structure in the coincidence spectrum occurs when one or more of the 𝛾’s from the 176Lu

decay escape the crystal and hit another LYSO crystal. For this reason there were substantially

more of these events for the two crystals that were closer to each other (0x17 and 0x19).

Looking at the 1-D spectra for these detectors (by projecting the plots down onto the x-axis

or y-axis) would show the structure dominated by the continuous 𝛽 distribution. However, this

projection throws away a lot of info. We can make better informed cuts on the 2-D distributions.

The maximum 𝛾 is at 307 keV, there was a hit with energy above that in one crystal means the 𝛽

from the intrinsic radioactivity had to be in that crystal, and the 𝛾 in the other. Some flaws in this

130

300-250-200-150-100-50-050100150 (ns)start - t0x10t410510610counts (arb)300-250-200-150-100-50-050100150 (ns)start - t0x17t410510610counts (arb)300-250-200-150-100-50-050100150 (ns)start - t0x19t410510610counts (arb)(a)

(b)

(c)

Figure 8.11 2-D energy distribution from LYSO coincidences with an exclusive 2-hit coincidence
(a) and (b) correspond to crystals on opposite sides of the ring and show clear 22Na
trigger.
coincidence spectra, and three 𝛾 lines below 511 keV, along with continuous spectra from the
(c) The two crystals were much closer, and the three 𝛾 lines are more
internal radioactivity.
pronounced than the 511 keV and 1275 keV from the 22Na source.

argument are that both the 202 and 307 keV could be in the same crystal, but this would require

both to escape the original crystal and then hit the same crystal. This occurs, but is not as likely as

the high energy signal being from the 𝛽. We divide into 3 regions illustrated in Figure 8.12. These

separate the structure from the 511-1275 keV coincidences from the sodium source.

Plotting the distribution in each of these regions produces the spectra shown in Figure 8.13.

131

0123456789101112310·Energy Detector 0x17 (ADCu)0123456789101112310·Energy Detector 0x10 (ADCu)0501001502002503000123456789101112310·Energy Detector 0x19 (ADCu)0123456789101112310·Energy Detector 0x10 (ADCu)0501001502002503000123456789101112310·Energy Detector 0x19 (ADCu)0123456789101112310·Energy Detector 0x17 (ADCu)050010001500200025003000These cuts on the coincidence 2-D energy distribution demonstrated an immensely cleaner internal

radioactivity spectra. This set of cuts almost entirely removed the continuous 𝛽 distribution above

307 keV. The 202 and 307 keV peaks from this data were two of the peaks used to extract the

resolution used in Figure 7.18.

Figure 8.12 Same data shown in Figure 8.11c, now plotted log scale clearly showing more structures.
Three energy windows are displayed on the x-axis and y-axis corresponding to cutting between the
307 keV peak and the 511 keV peak, isolating the 511 keV peak, and cutting above the 511 keV
peak.

Imposing a timing coincidence with the start detector should remove the internal radioactivity

events. The measured spectra in each LYSO crystal in coincidence with the start detector is shown

in Figure 8.14. This recovered a clean 22Na spectrum from the crystals without performing a

background subtraction. These two peaks served as the other two data points used in Figure 7.18

for the energy resolution. These are also the "final 𝛾-detector spectra" shown at the end of Chapter

4. The two coincidence spectra for detectors 0x17 and 0x19 are shown in Figure 8.15. This shows

the internal radioactivity events summed from windows A and C in black, and the coincidence

spectra with the start detector (scaled down by a factor of 16) in blue. This data was from the

same run, simply using different coincidence conditions. Note the unfortunate overlap between the

132

(a)

(b)

Figure 8.13 The measured spectra for the internal radioactivity coincidences. Including no cuts on
the 2-D distribution, and each of the three windows shown in Figure 8.12.

Compton shoulder for the 511 keV peak (340 keV) and the backscatter peak (170 keV) with two of

the 𝛾 peaks from 176Lu at 307 keV and 202 keV.

(a)

(b)

(c)

Figure 8.14 The observed 22Na spectrum in each individual LYSO crystal when a coincidence
condition was applied with the plastic detector.

8.7 Triple coincidences

Next we studied the timing properties with a triple coincidence. We required one hit in the

plastic, and two LYSO hits. The time spectra for the (0x10,0x17) and (0x10,0x19) pairs are shown

in Figure 8.16.

In comparison with the exclusive 2 hit data, shown in Figure 8.10, the triple

coincidence data had: a lower level of accidentals, a more prominent long lifetime component,

133

0123456789101112310·Energy Detector 0x17 (ADCu)024681012310·counts (arb)No cutsWindow AWindow BWindow C0123456789101112310·Energy Detector 0x19 (ADCu)0246810310·counts (arb)No cutsWindow AWindow BWindow C0123456789101112310·Energy Detector 0x10 (ADCu)00.511.522.533.5310·counts (arb)0123456789101112310·Energy Detector 0x17 (ADCu)00.511.522.533.54310·counts (arb)0123456789101112310·Energy Detector 0x19 (ADCu)00.511.522.533.5310·counts (arb)(a)

(b)

Figure 8.15 Two spectra from the same run, blue is the LYSO crystal in coincidence with the start
detector (scaled down by a factor of 16), black is the sum of the first and third window cuts from
Figure 8.13. These are shown for detector 0x17 and 0x19 but not 0x10 as that detector was much
farther away and did not get much statistics for the internal coincidences.

and clear structure before the peak. The detector placement was positioned to favor seeing 3-𝛾

decay and therefore made the long lifetime component look more prominent than the simple double

coincidence condition.

(a)

(b)

Figure 8.16 Time spectra with a 3-hit coincidence, the start detector and two LYSO crystals. This
spectrum displays a more pronounced long lifetime component compared to the 2-hit coincidence
spectra.

The data was cut into two time windows, one on the peak, and one on the long lifetime. This

gave two very distinct energy distributions. We define the "peak" window between −156 ns <

134

0123456789101112310·Energy Detector 0x17 (ADCu)00.511.522.533.54310·counts (arb)0123456789101112310·Energy Detector 0x19 (ADCu)00.511.522.533.5310·counts (arb)300-250-200-150-100-50-050100150 (ns)start - t0x17t10210310410counts (arb)300-250-200-150-100-50-050100150 (ns)start - t0x19t10210310410counts (arb)Δ𝑇 < −140 ns, and the "long lifetime" window between −130 ns < Δ𝑇 < 60 ns. The resulting

distributions for both time windows is shown in Figure 8.17. This data clearly showed the continuous

energy distribution in the data that resembles the simulations from Chapter 7. There was also a

dramatic decrease in the 511-1275 keV coincidences. This demonstrates that the planned detector

configurations will be able to separate the continuous energy distribution of 3-𝛾 decay of ortho-

positronium from the spurious coincidences from the source, and from the internal radioactivity

(which is entirely absent in these plots). Even with the DAQ un-optimized, and the start detector

in particularly performing below our planned specifications, we were still able to resolve this

distribution.

(a) −156 ns < dT < −140 ns

(b) −156 ns < dT < −140 ns

(c) −130 ns < dT < 60 ns

(d) −130 ns < dT < 60 ns

Figure 8.17 The 2-D Energy distribution between two LYSO crystals with a 3 hit coincidence
condition where the third hit is in the start detector. Cuts refer to the time spectra in Figure 8.16. (a)
and (b) cut on the prompt peak. (c) and (d) cut on the long lifetime component of ortho-positronium.

135

00.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)05010015020025030000.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)05010015020025030000.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)051015202500.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)0510152025The spectra shown in Figure 8.17 look much cleaner than the energy spectra shown in the Tokyo

experiment (Ref. [39]). This is a misleading comparison as the setup presented here had the source

about 26.7 mm below the center of the detector ring. This geometry favors seeing 3-𝛾 events and

largely removes 2-𝛾 events, and does not represent the planned placement of the 𝛽 source and

powder.

8.8 22Na vs. 68Ga

We also performed these studies with a 68Ga source for comparison with the 22Na data. This

source has a halflife of 270 days compared to 22Na with 2.6 years. It also had a much higher activity

at nearly 50 𝜇Ci compared to the sodium source at 7.5 𝜇Ci. This created substantially more pileup

in the start detector. The time spectrum is shown in Figure 8.18, and it was indeed less clean than

those for the sodium source. The structures correspond to improperly digitized start signals that

lead to a misconstructed time spectrum, in particular due to retriggering.

(a)

(b)

Figure 8.18 Time distributions between LYSO-plastic obtained with for the 68Ga source. This uses
the same coincidence trigger as Figure 8.16.

The start detector and powder corresponds to Design (A) from Chapter 6, and as shown there

is immense spreading of the positronium throughout the powder for this source. The spreading

of the positronium leads to the possibility of seeing back-to-back photons. The illustration shown

in Figure 7.15 actually corresponds to the exact geometry of this current setup. The expected

spreading of the positronium in this powder is given in Figure 6.12, and it clearly extends all the

136

300-250-200-150-100-50-050100150 (ns)start - t0x17t10210310410counts (arb)300-250-200-150-100-50-050100150 (ns)start - t0x19t10210310410counts (arb)way out to beyond the 2 cm radius. As mentioned, this source and powder are below the plane of

the detector ring. The observed energy distribution for both time cuts is shown in Figure 8.19, and

can be compared with those in Figure 8.17. This data displayed many more back-to-back events

with the gallium source than the data for the sodium source.

(a) −156 ns < dT < −140 ns

(b) −156 ns < dT < −140 ns

(c) −130 ns < dT < 60 ns

(d) −130 ns < dT < 60 ns

Figure 8.19 2-D energy distribution obtained with a 68Ga source. These can be directly compared
to the plots in Figure 8.17.

Changing of the source required removing the start detector and the powder, and the demon-

strator had no mechanism to ensure a careful alignment of the source and powder in the center of

the ring. This was evident in the gallium results where the (0x00,0x19) pair saw a much larger

contribution from back-to-back photons. This is in agreement with the picture of the setup in Figure

8.8, where the start detector and powder (and most likely the source as well) are offset from the

center of the ring towards the right of the box. This means it was closer to the 0x19 detector than

137

00.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)020040060080010001200140000.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)020040060080010001200140000.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)02040608010012014016018000.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)020406080100120140160180the 0x17 detector.

8.9 Simulation

We are able to specify this setup in the Geant4 simulation, and run the expected 3-𝛾 and 2-𝛾

events the 𝛽 source using Design (A), offset below the ring to appropriately match the setup. Since

this powder had a lifetime of 70 ns there is a 50/50 weighting of the two spectra. The 2-𝛾 and 3-𝛾

distributions for both sources are shown in Figure 8.20.

(a) 22Na 2-𝛾

(b) 22Na 3-𝛾

(c) 68Ga 2-𝛾

(d) 68Ga 3-𝛾

Figure 8.20 Monte-Carlo simulation of the 2-D histograms obtained with a 22Na and a 68Ga source
under the condition of the test, for 2-𝛾 and 3-𝛾 events. The combinations are shown in Figures 8.21
and 8.22.

This simulation takes the stopping positions throughout the powder and generates either a 2-𝛾

event at that position or a 3-𝛾 event. The powder and start detector are not currently implemented

in the 𝛾-simulation, they are included in the 𝛽-stopping simulation, but then when importing that

138

00.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)0102030405060708000.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)0102030405060708000.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)02040608010012014016018020022000.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)020406080100120140160180200220into the second simulation we currently treat the positronium as if it is decaying in air. Indeed there

are no inactive elements in the current simulation, and therefore the coincidences where one photon

scattered off of material in the demonstrator will not be present. The combined spectra along with

both measured distribution are shown for 22Na and 68Ga in Figures 8.21 and 8.22 respectively.

(a) Data for (0x10,0x17)

(b) Data for (0x10,0x19)

Figure 8.21 (a), (b) Measured energy distribution for events in the long lifetime component. (c)
Simulation of the combined results for 2-𝛾 events and 3-𝛾 events for 22Na from Figure 8.20.

(c) Simulation

We are able to reproduce the combined peak and continuum structure. The source was not

carefully centered which created an asymmetry in the coincidences between the two detector pairs.

The gallium data showed extra structure that matches up with backscatter peaks, where one 511

keV photon hit one detector, and the other scattered off of material in the setup and then hits the

other detector.

139

00.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)051015202500.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)051015202500.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)01020304050607080(a) Data for (0x10,0x17)

(b) Data for (0x10,0x19)

Figure 8.22 (a), (b) Measured energy distribution for events in the long lifetime component. (c)
Simulation of the combined results for 2-𝛾 events and 3-𝛾 events for 68Ga from Figure 8.20.

(c) Simulation

140

00.511.522.533.544.555.56310·Energy Detector 0x17 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)02040608010012014016018000.511.522.533.544.555.56310·Energy Detector 0x19 (ADCu)00.511.522.533.544.555.56310·Energy Detector 0x00 (ADCu)02040608010012014016018000.10.20.30.40.50.6Energy Detector 0x17 or 0x19 (MeV)00.10.20.30.40.50.6 Energy Detector 0x10 (MeV)020406080100120140160180200220240CHAPTER 9

NEW PHYSICS IN 3-𝛾 DECAY OF POSITRONIUM

We have now demonstrated that we can observe the continuous 2-D energy distribution from 3-𝛾

decay. It is necessary to investigate how symmetry violations would affect this distribution. The

full theoretical description of symmetry violations in angular correlations of 3-𝛾 decay of ortho-

positronium (when photon polarizations are not measured) was worked out by Bernreuther, Löw,

Ma, and Natchmann in Ref. [19]. This work was reviewed in Chapter 2. In this chapter we connect

the measurement of an asymmetry with the extraction the CP-violating form factors proposed in

Ref. [19]. This requires rethinking some aspects of the analysis. Much of this chapter is also

directly applicable to searches for the form factor 𝑏3 (referred to as "the correlation ˆs · ˆn" that has

been searched for in Refs. [37, 41, 42, 43, 44] and the planned upcoming searches in Refs. [78, 79].

However, to date, no model has been proposed for this form factor. For the purposes of this work,

we will focus on the CP-violating signature the present project will search for.

We first provide a brief review of the form factors describing 3-𝛾 spin-1 positronium decay,

and which form factors correspond to the signal. The rest of this chapter relates the measurement

to extracting these form factors. This requires including detector acceptances into the calculation

of asymmetries. This is pursued as far as possible without assuming a model. Finally we end the

chapter by interpreting the experiment in the context of CP-violating state mixing in positronium.

9.1 Form factors in ortho-positronium decay

All measurements of angular distributions of the photon momenta in the 3-𝛾 decay of spin-1

positronium to 3 photons can be fully described by 9 irreducible tensors each multiplied by a form

factor (that is itself a function of the energies of the photons). The structure of the irreducible

tensors follows from the kinematics and are determined by the rotational properties of the system.

The form factors are determined by the dynamics, either by the Standard Model or new beyond

Standard Model physics. Below the form factors are tabulated, including the positronium trait that

drives the distortion, and the kinematic terms that generate nonzero expectation values from these

141

Form factor Tensor structure Ps dependence Expectation values
Tr(𝜌)
𝑎(𝜔1, 𝜔2)
𝑏1(𝜔1, 𝜔2)
s𝑖
𝑏2(𝜔1, 𝜔2)
s𝑖
𝑏3(𝜔1, 𝜔2)
s𝑖
𝑠𝑖 𝑗
𝑐1(𝜔1, 𝜔2)
𝑠𝑖 𝑗
𝑐2(𝜔1, 𝜔2)
𝑠𝑖 𝑗
𝑐3(𝜔1, 𝜔2)
𝑠𝑖 𝑗
𝑐4(𝜔1, 𝜔2)
𝑠𝑖 𝑗
𝑐5(𝜔1, 𝜔2)

𝑁3𝛾
⟨𝜅𝑎𝑖⟩
⟨𝜅𝑎𝑖⟩
⟨𝑛𝑖⟩
⟨𝜅𝑎𝑖𝜅𝑏 𝑗 ⟩, ⟨𝑛𝑖𝑛 𝑗 ⟩
⟨𝜅𝑎𝑖𝜅𝑏 𝑗 ⟩, ⟨𝑛𝑖𝑛 𝑗 ⟩
⟨𝜅𝑎𝑖𝜅𝑏 𝑗 ⟩, ⟨𝑛𝑖𝑛 𝑗 ⟩
⟨𝜅𝑎𝑖𝑛 𝑗 ⟩
⟨𝜅𝑎𝑖𝑛 𝑗 ⟩

13𝑥3
𝜅1𝑖
𝜅2𝑖
𝑛𝑖
𝜅1𝑖𝜅1 𝑗
𝜅2𝑖𝜅2 𝑗
𝜅1𝑖𝜅2 𝑗
𝜅1𝑖𝑛 𝑗
𝜅2𝑖𝑛 𝑗

Table 9.1 The nine form factors fully describing angular distributions of spin-1 positronium decay
to 3-𝛾. 𝑎 and 𝑏 can stand for any of the 3 photons. Any object with two indices should be interpreted
as a traceless symmetric tensor.

terms. Following Chapter 2, we denote the components of the unit vectors,

ˆk𝑎 = 𝜅𝑎𝑥 ˆx + 𝜅𝑎𝑦 ˆy + 𝜅𝑎𝑧 ˆz

n = ˆn = 𝑛𝑥 ˆx + 𝑛𝑦 ˆy + 𝑛𝑧 ˆz

(9.1)

(9.2)

the normal to the decay plane n is always taken to be normalized. For brevity, the notation for

traceless symmetric tensors is shortened, for instance we write ⟨𝜅𝑎𝑖𝜅𝑏 𝑗 + 𝜅𝑎 𝑗 𝜅𝑏𝑖 − 2
3

ˆk𝑎 · ˆk𝑏𝛿𝑖 𝑗 ⟩ →

⟨𝜅𝑎𝑖𝜅𝑏 𝑗 ⟩. These terms are given in Table 9.1

The term 𝑎(𝜔1, 𝜔2) is the Ore-Powell distribution for isotropic ortho-positronium decay. 𝑏1

and 𝑏2 are CP-violating vector terms. 𝑏3 is indicative of new physics as it is very small in the

Standard Model, being induced by final state photon-photon scattering.

It is not indicative of

CPT -violation as argued in Refs. [37, 42, 41, 44, 80]. 𝑐𝑖 for 1 ≤ 𝑖 ≤ 3 are distortions to the

angular distribution induced by QED. Finally 𝑐4 and 𝑐5 correspond to the CP-violating form factors

we intend to search for. Note that the definition of the tensor polarization in Ref. [19] and used

here differs from the alignment 𝑃2 as defined in Refs. [38, 39] by 𝑃2 = 3𝑠𝑧𝑧. The form factors

𝑐4 and 𝑐5 are "clean signatures" of CP-violation, in that they are not T -odd signatures that then

invoke CPT -symmetry to equate with a violation with CP-violation. As such they cannot be

mimicked by radiative corrections and final state interactions, unless those interactions violate CP

themselves [18, 19].

142

The expected rate for an event with photons having momenta (k1, k2, k3) is given as,

𝑁 (k1, k2, k3) = 𝑎(𝜔1, 𝜔2) + s · B(k1, k2, k3) + 𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3)

(9.3)

In order to turn this into an asymmetry in counts, we need to incorporate a description of the

detector placement and energy cuts into the theoretical description.

9.2 Detector acceptance and phase space cuts

All real experiments have finite detector acceptances and efficiencies. These manifest as

restrictions to the phase space integration. This is included in the description by following Appendix

D in Ref. [19]. We define the "characteristic function," for a pair of detectors with a highest energy

detector with placement Ω1 and a second highest energy detector with placement Ω2,

𝜒𝑎 (k1, k2, k3) =

1,

0,





ˆk1 ∈ Ω1, ˆk2 ∈ Ω2

o/w

(9.4)

this is an estimation of the apparatus if the detectors have perfect detection efficiency, and are only

affected by solid angle and geometry. Specifically it returns 1 when both ˆk1 hits the highest energy

detector and ˆk2 hits the second highest energy detector, or it returns 0 otherwise. In essence, this

means that, by specifying the detector placement and energy cuts we are selecting a region of phase

space. For this reason the authors of Refs. [18, 19, 20] refer to these as "phase space cuts." The

number of coincidence counts for a given pair 𝑎 is,

∫

∫

∫

𝑁𝑎 =

=

=

d 𝑓 3𝛾 𝜒𝑎 (k1, k2, k3)𝑅𝑖 𝑗 (k1, k2, k3) 𝜌 𝑗𝑖

d 𝑓 3𝛾

Ω1Ω2

d 𝑓 3𝛾

Ω1Ω2

𝑅𝑖 𝑗 (k1, k2, k3) 𝜌 𝑗𝑖

(cid:0)𝑎(𝜔1, 𝜔2) + s · B(k1, k2, k3) + 𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3)(cid:1)

(9.5)

(9.6)

(9.7)

where ∫ d 𝑓 3𝛾

Ω1Ω2

is the phase space restricted to the regions where the two photons hit the detectors

(and pass the energy cuts). This is the coincidence counts for a pair of detectors after integrating

over the energy cuts. We consider a second pair of detectors that is the CP-image of pair 𝑎. This is

referred to in ref. [18, 81] as "CP-invariant phase space cuts." Holding the highest energy detector

143

Figure 9.1 Sketch of the detector rings showing the highest energy detector to lie in the x-z plane
and the two pairs are related by flipping the y-component of ˆk2. This is true for Symmetric and
Asymmetric configurations, and all 16 sets can be related by an azimuthal rotation.

the same at Ω1, and choosing a second detector to form a pair, Ω′

2 (Figure 9.1), gives,

∫

∫

𝑁𝑏 =

=

d 𝑓 3𝛾
Ω1Ω′
2

d 𝑓 3𝛾
Ω1Ω′
2

𝑅𝑖 𝑗 (k1, k2, k3) 𝜌 𝑗𝑖

(cid:0)𝑎(𝜔1, 𝜔′

2) + s · B(k1, k2, k3) + 𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3)(cid:1)

(9.8)

(9.9)

The detector placement is chosen such that between two pairs the terms we are not interested in

are equal, and the terms we are interested in are equal but opposite in sign. This will be expanded

upon in the context of the experiment below. To lighten the notation we shorten Ω1Ω2 to simply Ω.

9.3 Extraction of count asymmetry terms

We now consider a positronium state with no vector polarization, and no off-diagonal terms in

the tensor polarization. This gives, 𝑠𝑖 = 0, 𝑠𝑥𝑦 = 𝑠𝑦𝑧 = 𝑠𝑧𝑥 = 0. There is a net tensor polarization,

𝑠𝑧𝑧, and compensating diagonal terms to ensure traceless-ness 𝑠𝑥𝑥 = 𝑠𝑦𝑦 = − 1
2

𝑠𝑧𝑧. We follow the

convention for tensor polarization used in Ref.

[19], −2/3 < 𝑠𝑧𝑧 < 1/3. This is addressed in

Appendix A.

Working out the term that depends on tensor polarization gives,

𝑠𝑖 𝑗𝐶𝑖 𝑗 (k1, k2, k3) = 𝑠𝑥𝑥𝐶𝑥𝑥 + 𝑠𝑦𝑦𝐶𝑦𝑦 + 𝑠𝑧𝑧𝐶𝑧𝑧
(cid:19)
(cid:18)

= 𝑠𝑧𝑧

𝐶𝑧𝑧 −

(𝐶𝑥𝑥 + 𝐶𝑦𝑦)

1
2

(9.10)

(9.11)

144

the same simplification applies for 𝐶𝑥𝑥 + 𝐶𝑦𝑦 = −𝐶𝑧𝑧 (due to the tracelessness of the tensor). This

gives the tensor distribution,

𝑠𝑖 𝑗𝐶𝑖 𝑗 =

(cid:26) (cid:18)

𝑠𝑧𝑧

3
2

1
3

)𝑐1(𝜔1, 𝜔2) + (𝜅2

2𝑧 −

1
3

)𝑐2(𝜔1, 𝜔2) + (2𝜅1𝑧𝜅2𝑧 −

ˆk1 · ˆk2)𝑐3(𝜔1, 𝜔2)

(cid:19)

2
3

+

2𝜅1𝑧𝑛𝑧𝑐4(𝜔1, 𝜔2) + 2𝜅2𝑧𝑛𝑧𝑐5(𝜔1, 𝜔2)

(cid:19)(cid:27)

(9.12)

(𝜅2

1𝑧 −
(cid:18)

In the context of this experiment it is beneficial to break this object up into two separate terms.

Consider the first three terms as the QED induced anisotropy, and the second two terms as the

CP-violating signal,

𝑠𝑖 𝑗𝐶𝑖 𝑗 = 𝑠𝑧𝑧 𝐷 (k1, k2, k3) + 𝑠𝑧𝑧𝐶 (k1, k2, k3)

(9.13)

where 𝐷 corresponds to the terms that are CP-even (and non-zero in QED), and 𝐶 is the terms

that are CP-odd (and zero in QED). Taking the detector response to be the same as above gives an

updated 3-𝛾 distribution,

∫

𝑁Ω =

d 𝑓 3𝛾
Ω

𝑎(𝜔1, 𝜔2) + 𝑠𝑧𝑧

∫

d 𝑓 3𝛾
Ω

𝐷 (k1, k2, k3) + 𝑠𝑧𝑧

∫

d 𝑓 3𝛾
Ω

𝐶 (k1, k2, k3)

= AΩ + 𝑠𝑧𝑧DΩ + 𝑠𝑧𝑧CΩ

(9.14)

where the last term is a definition, writing a shorthand for the function of energies integrated over

the detector placement and energy cuts. This gives AΩ as the isotropic Ore-Powell contribution,

the term DΩ is a P-even anisotropy in the distribution of ˆk𝑖 over the detector solid angles, finally

CΩ is the part of the distribution sensitive to CP-violation. CΩ is the combination of the two

form factors that produce the asymmetry. This is the quantity that most closely corresponds to the

"coefficient multiplying the angular correlation," written as 𝐶𝐶𝑃𝜅1𝑧 ( ˆk1 × ˆk2)𝑧 in Refs. [38, 39],

but has a much more complicated form,

CΩ = 3

∫

(cid:18)

d 𝑓 3𝛾
Ω

𝜅1𝑧𝑛𝑧𝑐4(𝜔1, 𝜔2) + 𝜅2𝑧𝑛𝑧𝑐5(𝜔1, 𝜔2)

(cid:19)

(9.15)

This cannot be separated into a purely geometric factor and a purely energy dependent factor,

because the tensor factors are different. Further we cannot make the simplifying assumption that

only one term exists and then recover a form more closely related to the coefficient searched for in

145

the past. This is because 𝑐4 and 𝑐5 are related by photon indistinguishability, as demonstrated in

Appendix B. We conclude that all experiments to date were sensitive to a combination of "𝜅1𝑧𝑛𝑧"

and "𝜅2𝑧𝑛𝑧."

Considering any of the configurations shown in Figure 7.2, we take ˆk1 to lie in the 𝑥 − 𝑧 plane,

then the difference between any two pairs of detectors in a configuration is flipping the sign of 𝜅2𝑦

(which therefore flips the sign 𝑛𝑧). This is illustrated in Figure 9.1. Measuring with these two pairs

of detectors only changes the sign of the CP-violating term that arises from a tensor polarized

source,

𝑁Ω = AΩ + 𝑠𝑧𝑧DΩ + 𝑠𝑧𝑧CΩ

𝑁Ω′ = AΩ + 𝑠𝑧𝑧DΩ − 𝑠𝑧𝑧CΩ

Measuring a count asymmetry between these detector pairs gets a signal that goes as,

𝐴 =

𝑁Ω − 𝑁Ω′
𝑁Ω + 𝑁Ω′

= 𝑠𝑧𝑧

CΩ
AΩ + 𝑠𝑧𝑧DΩ

(9.16)

(9.17)

(9.18)

This isolates the CP-violating part of the decay matrix. The term in the numerator is the term of

interest. The term in the denominator is the non-cancelling backgrounds.

Now we are in a position to study, for a given detector placement and energy cuts, what size of

signal we expect and finally arrive at a method to derive a sensitivity to CP-violating physics. We

will do this in the context of a specific model. Before proceeding to do this, we first comment on a

different method that could be used to perform this measurement, as the two methods are distinct

but have gotten mixed up in the literature.

9.4 Count asymmetry versus expectation value

Another way to do this experiment would be to measure the expectation value of an observable

that is sensitive to the symmetry violation. For example the expectation value ⟨𝜅1𝑧𝑛𝑧⟩ could be

measured. This would require recording 𝜅1𝑧𝑛𝑧 for each event and calculating the average of the

recorded values. If good phase space cuts are chosen such that there is equal acceptances for events

with negative value and positive value then a non-zero expectation value would be indicative of

146

CP-violation. However,the expectation value cannot be directly compared to a count asymmetry

for the same phase space cuts.

We demonstrate this by considering a simpler analogy: searching for a P-violating interaction

in 1-D quantum mechanics. We start with a state, 𝜓(𝑥) with a well defined parity that evolves into

a new state that may or may not have definite parity. We could measure an operator that is odd

under parity, like 𝑥, meaning measure many equivalent systems and record the value of 𝑥 for each
measurement and calculate ⟨𝑥⟩ = (1/𝑁) (cid:205)𝑁

𝑖 𝑥𝑖. Another method would be to simply count how

many times the particle has a positive value for 𝑥, 𝑁+, or a negative value for 𝑥, 𝑁−. The difference

in counts, 𝐴 = (𝑁+ − 𝑁−)/(𝑁+ + 𝑁−) would also indicate parity violation. Crucially these two

quantities are not equal, 𝐴 ≠ ⟨𝑥⟩. The two are roughly related as ⟨𝑥⟩ ≈ ⟨|𝑥|⟩ 𝐴 if the variation in

⟨|𝑥|⟩ is small.

Now we consider measuring a count asymmetry between how many ortho-positronium decays

have the decay plane along +ˆz versus −ˆz. Compare that to measuring ⟨ˆz · ˆn⟩, the latter quantity

will always be smaller because each event is weighted by a number less than one. This makes

comparison of the searches for 𝑏3 complicated. Three groups to date measured a simple count

asymmetry [37, 41, 43]. One group weighted each event by three different functions of energies

[42], but were careful to explain what their analysis involved and which weightings can be compared

to previous measurements. This weighting is distinct from the functions 𝑏𝑖, 𝑐𝑖 that appear in the

decay matrix. The most recent search measured an expectation value and equate it with previous

count asymmetry measurements [44]. A direct comparison of the expectation value versus the

count asymmetry artificially inflates the sensitivity of the search measuring an expectation value,

since ⟨|ˆs · ˆn|⟩ = ⟨|cos(𝜃)|⟩ ≤ 1.

In short, it is important to distinguish between "measuring a term in the decay distribution

that goes as s · ˆn" and "measuring the expectation value of the observable ⟨s · ˆn⟩." This is another

reason we find that describing these quantities as "form factors" is more clear than calling them

"coefficients of angular correlations" as has been the norm in these measurements. The latter is the

standard nomenclature in nuclear 𝛽-decay, but in such cases the form factors are simply coefficients,

147

not energy dependent functions, at least at tree level [82].

9.5 Application to the analysis

Now we adapt the theory description to the detector array. We hld 𝜅1 to always be in the same

detector and lie in the 𝑥 − 𝑧 plane. There are two detectors for ˆk2 that only change the sign of 𝜅2𝑦

but not the other components. This is illustrated in Figure 9.1. This also changes 𝑛𝑥 → −𝑛𝑥 and

𝑛𝑧 → −𝑛𝑧. The angular distribution we aim to measure does change sign, 𝜅𝑎𝑧𝑛𝑧 → −𝜅𝑎𝑧𝑛𝑧 where

𝑎 = 1, 2. Here we demonstrate the structure of these energy dependent functions (𝑎, 𝑏𝑖, 𝑐𝑖...).

This is shown for the Symmetric 157.5◦ pairs of detectors (0x00,0x27) and (0x00,0x29) and the

Asymmetric 157.5◦ pairs of detectors (0x00,0x17) and (0x00,0x19). This includes the multiplicity

cuts described in Chapter 7.

9.5.1 Phase space volume

All of these form factors are defined on the decay phase space, described by the true kinematic

variables. However cuts are applied on detector level variables. The two are not in direct corre-

spondence, they are instead related by some form of a convolution matrix 𝜒(𝐸𝑎, 𝐸𝑏, 𝜔1, 𝜔2). This

maps the kinematic phase space into the detector level variables. In Figure 9.2a the phase space

defined on the kinematic variables is plotted, and in Figure 9.2b the "volume of phase space" that

the 157.5◦ Symmetric detector pair selects is plotted (for each two energies).

This is purely kinematics: any positronium physics would manifest in structure on top of this

distribution. Any "phase space cuts" are applied to detector level variables, such as coincidence

conditions and energy cuts. Figure 9.3 shows the phase space volume selected by the Symmetric

and Asymmetric 157.5◦ pairs. They indicate that going from the Symmetric to the Asymmetric

confugration decreases the opening angle, and the energy distribution selected moves closer to the

diagonal 𝐸1 + 𝐸2 = 𝑚𝑒. This is simply the energy-angle relationship shown in Figure 3.4. Any

positronium physics would manifest as distortions on top of this distribution by the matrix element

for positronium decay. The form factors are evaluated event by event using the distribution of events

shown above.

148

(a)

(b)

Figure 9.2 (a) The phase space defined in terms of the photons energies 𝜔1 and 𝜔2. (b) That phase
space mapped into detector level variables, namely the energy deposited in both detectors.

9.5.2 QED terms

The Ore-Powell distribution for isotropic decay is shown in Figure 9.4a on the phase space, and

in Figure 9.4b shows this mapped into structure on the coincidence distribution for the detector

pairs. The difference from phase space (Figure 9.3) is not very large for the Ore-Powell distribution.

This is shown for the Symmetric and Asymmetric 157.5◦ configurations in Figure 9.5. This gives

the expected distribution for isotropic ortho-positronium decay and corresponds to the plots shown

in Chapter 7. The induced structure over the phase space is small.

A net tensor polarization induces a CP-symmetric contribution to the counts. This is the DΩ

term as defined above. These are shown in Figure 9.6 for a tensor polarization of 𝑠𝑧𝑧 = +1/3

(maximal positive alignment). This is a combination of the form factors 𝑐1, 𝑐2, and 𝑐3 each

multiplied by their relevant kinematic tensor.

It induces a P-even anisotropy to the individual

photons distributions, and the decay plane distribution. This is exactly the angular anisotropy

measured in Refs.

[35, 36]. This distortion is symmetric between the detector pairs within a

configuration, and therefore should cancel in the numerator of an asymmetry. The size of the

distortion has a nontrivial energy dependence that differs from the isotropic energy dependence.

At this point we can extract a form of sensitivity for the experiment. If we measure an asymmetry,

149

00.10.20.30.40.50.6 (MeV)2w00.10.20.30.40.50.6 (MeV)1w00.10.20.30.40.50.60.70.80.91(a) Symmetric 157.5◦ configuration

(b) Asymmetric 157.5◦ configuration

Figure 9.3 Phase space selected by the detector pairs for both the Symmetric and Asymmetric
157.5◦ configurations.

𝐴𝑎, for configuration 𝑎, the induced asymmetry is,

𝐴Ω = 𝑠𝑧𝑧

CΩ
AΩ + 𝑠𝑧𝑧DΩ

(9.19)

The tensor polarization 𝑠𝑧𝑧 is known (taken as a given for now, discussed in Chapter 10), and we

can estimate AΩ + 𝑠𝑧𝑧DΩ from these simulations as they are determined by the Standard Model.

This enables the extraction the purely CP-violating contribution to the asymmetry CΩ.

CΩ =

1
𝑠𝑧𝑧

(AΩ + 𝑠𝑧𝑧DΩ) 𝐴Ω

(9.20)

For the experiment as outlined, and a source with tensor polarization perfectly aligned along the ˆ𝑧-

axis, this term can only be induced by CP-violating physics. Formally this term is given in Equation

150

(a)

(b)

Figure 9.4 (a) The QED distribution for isotropic ortho-positronium decay plotted on the phase
space. This distribution is sharply peaked at high energies. (b) The same distribution mapped into
detector level variables.

9.15, however without a specific model we do not have the form of 𝑐4(𝜔1, 𝜔2) and 𝑐5(𝜔1, 𝜔2) and

this cannot be simplified further. Since the term CΩ still contains effects of detector placement

and energy cuts it cannot be directly compared between experiments that used different cuts. This

makes the comparison more complicated than when considering "𝐶𝐶𝑃" as in Refs. [38, 39]. In fact,

each detector configuration within the current experiment selects different areas of phase space,

and therefore different CΩ and CΩ′

. Two models will predict different magnitudes of these terms,

and therefore the measurements from the different configurations cannot be combined into a net

result in a model independent way.

We now proceed to consider a specific model, given in Ref. [19] for mixing of positronium

states. We identify the phase space dependence, and what signal it would induce in the planned

experiment.

9.5.3 CP-violating mixing of 13𝑆1 and 21𝑃1 positronium states

In Ref. [19] the authors propose searching for indirect CP-violation in ortho-positronium decay.

Direct violation in 3-𝛾 decay would be dominated by a permanent electric dipole moment (eEDM)

which has been excluded to a high precision [16]. Instead there could be possible combinations

151

00.10.20.30.40.50.6 (MeV)2w00.10.20.30.40.50.6 (MeV)1w020004000600080001000012000(a) Symmetric 157.5◦ configuration

(b) Asymmetric 157.5◦ configuration

Figure 9.5 For each event in Figure 9.3 (meaning a 3 photon event that survives the phase space
selection) the function 𝑎(𝜔1, 𝜔2) is evaluated and the value recorded to the bin for the corresponding
detector variables (𝐸𝑎, 𝐸𝑏). This gives the QED distribution of events for unpolarized positronium.

of terms such that the production of an eEDM is suppressed, but CP-violating state mixing is not

suppressed. They considered the CP-violating mixing of the 13𝑆1 and 21𝑃1 states. Note that the

decay of 21𝑃1 positronium is suppressed due to the atomic structure of positronium, this supresses

the effects of state mixing by a factor of 𝛼/(2

√

8) ≈ 1/775. The phenomenology of direct versus

indirect CP-violation is explored in Appendix B, and the source of this suppression is illustrated.

This large factor dominates any estimated sensitivity, effectively reducing it by a factor of 103.

With a specific model, the count asymmetry can be translated into a sensitivity to a new physics

parameter. Here the parameter is the real part of the mixing term between the two Ps states, Re(𝛿1)

152

(a) Symmetric 157.5◦ configuration

(b) Asymmetric 157.5◦ configuration

Figure 9.6 For each event in Figure 9.3 (meaning a 3 photon event that survives the phase space
selection) the appropriately summed form factors 𝑐𝑖 (𝜔1, 𝜔2) for 0 ≤ 𝑖 ≤ 3, weighted by their
respective kinematic tensors, is evaluated and the value recorded to the bin for the corresponding
detector variables (𝐸𝑎, 𝐸𝑏). This gives the QED distortion for maximally aligned positronium.

[19]. We can pull this term out of the model dependent function of energies,

𝑚𝑖𝑥 = Re(𝛿1) ¯CΩ
CΩ
𝑚𝑖𝑥

(9.21)

Now we evaluate ¯CΩ

𝑚𝑖𝑥 over the detector placement and energy cuts, which produces a signal with

size 𝑠𝑧𝑧Re(𝛿1) ¯CΩ

𝑚𝑖𝑥. This is plotted in Figure 9.7. It is immediately evident that, in contrast to

Figures 9.5 and 9.6, the CP-violating distribution does indeed change sign between the two detector

pairs within a configuration and will therefore create an asymmetry in coincidence counts.

Now an "analyzing power" can be defined for this specific model. A measured asymmetry 𝐴𝑎

153

(a) Symmetric 157.5◦ configuration

(b) Asymmetric 157.5◦ configuration

Figure 9.7 Taking the events from Figure 9.3 and evaluating the CP-violating functions that arise
in state mixing for ortho-positronium given in [19]. Unlike the QED contributions in Figures 9.5
and 9.6, this does create an asymmetry between the detector pairs.

can be related to the new physics parameter as,

𝐴𝑎 = 𝑠𝑧𝑧Re(𝛿1)

(cid:33)

(cid:32)

¯CΩ
𝑚𝑖𝑥
AΩ + 𝑠𝑧𝑧DΩ

(9.22)

where the term in parenthesis is (the inverse of) the analyzing power. This means we want to

maximize ¯CΩ

𝑚𝑖𝑥 and minimize AΩ + 𝑠𝑧𝑧DΩ. This cannot be called a geometric analyzing power as

there is no clean way to separate the energy dependence and the geometric tensor objects (since

ˆk1 · ˆk2 ≠ 0). If ¯CΩ

𝑚𝑖𝑥 only got contributions from one term, this would be possible (for example this

separation can be made in the searches for the form factor ˆs · ˆn𝑏3(𝜔1, 𝜔2)).

Comparing Figure 9.7 to Figure 7.13, we see that the analysis in this chapter produces a

much smoother and continuous distribution over phase space. Whereas in Chapter 7 we saw

154

discontinuities in the size of the signal at the 𝜔1 ≈ 𝜔2 diagonal line. The current plots have

corrected an inconsistency in the previous analysis. This chapter has included the two CP-

violating form factors 𝑐4 and 𝑐5, whereas the analysis of the previous chapter only considered a

signal of the form 𝜅1𝑧𝑛𝑧. Inclusion of the two terms 𝜅1𝑧𝑛𝑧 and 𝜅2𝑧𝑛𝑧 has produced a continuous

distribution in the two energies. This follows from the fact that crossing the 𝜔1 = 𝜔2 diagonal

line, meaning moving from the red triangle to the purple triangle in Figure 9.7, corresponds to

swapping ˆk1 and ˆk2. The current analysis includes both form factors and therefore restores the

needed symmetry properties of the system. Ultimately the discontinuities in the analysis of Chapter

7 were unphysical, and were induced by the analysis considering an unphysical model for the CP-

violating physics. The distribution must be continuous and follow certain symmetry properties due

to photon indistinguishablity, as argued in Appendix B.

configuration
Sym 157.5◦
Sym 135◦
Sym 112.5◦
Asym 157.5◦
Asym 135◦
Asym 112.5◦
Asym 157.5◦
Asym 135◦
Asym 112.5◦

hit order
(0,2)
(0,2)
(0,2)
(0,1)
(0,1)
(0,1)
(1,0)
(1,0)
(1,0)

∫
Ω d 𝑓3𝛾 (%)
2.91 × 10−2
1.35 × 10−2
2.35 × 10−2
1.67 × 10−2
4.90 × 10−3
5.3 × 10−7
1.67 × 10−2
4.40 × 10−4
5.1 × 10−7

DΩ

AΩ

¯CΩ
𝑚𝑖𝑥
2.70 × 10−3 −9.31 × 10−4 −8.19 × 10−7
1.24 × 10−3 −4.94 × 10−4 −2.50 × 10−7
−7.8 × 10−5 −1.81 × 10−8
2.11 × 10−4
2.54 × 10−8
1.53 × 10−3
6.75 × 10−5
7.88 × 10−9
4.45 × 10−4 −1.23 × 10−4
–
–
1.48 × 10−3 −1.34 × 10−4 −2.22 × 10−7
4.40 × 10−3 −1.38 × 10−4 −5.37 × 10−8
–

–

–

–

Table 9.2 Each function from Figures 9.3 through 9.7 integrated over the "phase space" triangle
that imposes the ordering of the hits by energy.

It is clear looking at Figure 9.7 that strictly cutting along the "phase space" triangle as in Chapter

7 does not necessarily line up with the contributions of the CP-violation. For the Asymmetric pair

there is almost a full cancellation within one of the considered regions. It is also worth considering

extending to the regions where we detect ( ˆk2, ˆk3) and ( ˆk3, ˆk1), for the Asymmetric events these get

a large contribution. Those regions get contributions from 2-𝛾 decays where one Compton scatters

off of inactive material which could obfuscate the signal contribution.

The absolute values of the analyzing power and Figure of Merit for all configurations is given

155

(a) Sensitivities

(b) Figure of Merit

Figure 9.8 (a) Absolute value of the analyzing power for each configuration showing which config-
urations will have the greatest effect from new physics. (b) Absolute value of the Figure of Merit
for each configuration showing which configurations have the greatest statistical sensitivity to new
𝑁 where 𝑁 is the counts in one pair if there were no
physics. Figure of Merit is calculated as 𝑆𝑎𝑛
new physics. S157 refers to "Symmetric 157.5◦," A157+ refers to "Asymmetric 157.5◦ when ˆk1
is in the outer ring," and A157- refers to ˆk1 in the middle ring. All plotted configurations have a
partner configuration with the same analyzing power using the lower ring instead of the upper ring,
doubling the statistics.

√

Figure 9.8. The analyzing power is calculated using the last factor in Equation 9.22. This is using

the "phase space triangle" cuts, which are not optimized. The contributions for the Asymmetric

112.5◦ are artificially set to zero for this plot because the value is so low that this simulation did

not have high enough statistics to get an accurate estimate. This Figure of Merit is calculated as

if we perform a simple asymmetry measurement. It scales the analyzing power by

√

2𝑁 where 𝑁

is the counts in one pair if there were no new physics. These numbers are for all 16 azimuthally

related configurations summed for 109 ortho-positronium decays. Equivalently this corresponds to

the counts for one configuration for 1.6 × 1010 ortho-positronium decays.

9.5.4 Statistical sensitivity estimate

We intend to use a source with an activity of 1.85 MBq, estimating that at least 50% of these

𝛽’s will reach the powder, we can expect around 40% of those will form positronium. As proposed

in this chapter, we allow the pseudo-triplet to decay so we are only recording 1/2 of the positronium

156

S157S135S112A157+A135+A112+A157-A135-A112-00.020.040.060.080.13-10·Analyzing PowerS157S135S112A157+A135+A112+A157-A135-A112-00.10.20.30.40.5Figure of Meritpopulations. For a lifetime of 135 ns 95% of the ortho-positronium decay to 3-𝛾, and if the initial

time cut will be at 20 ns (to allow all pseudo-triplet to decay with a high B-field value), then 86%

of the 𝑚 = ±1 positronium will decay in the time window. This gets a rate of good 𝑚 = ±1

positronium decays of 1.5 × 105 per second.

The expected counts in each configuration is 𝑁𝑎 = 𝑅𝑇 (AΩ +𝑠𝑧𝑧DΩ), where 𝑅 = 150×103(1/𝑠)

and 𝑇 is the time (35 days). We can then estimate the uncertainty on the asymmetry as 𝛿 𝐴𝑎 =

√

1/

2𝑁𝑎. The expected sensitivities for each configuration is quoted in Table 9.3, with the sensitivity

including the 32 sets of configurations with identical sensitivities.

hit order AΩ + 𝑠𝑧𝑧DΩ

configuration
Sym 157.5◦
Sym 135◦
Sym 112.5◦
Asym 157.5◦
Asym 135◦
Asym 157.5◦
Asym 135◦
Combined

(0,2)
(0,2)
(0,2)
(0,1)
(0,1)
(1,0)
(1,0)
–

𝑆𝑎𝑛
2.39 × 10−3 −1.14 × 10−4
1.08 × 10−3 −7.72 × 10−5
1.85 × 10−4 −3.26 × 10−5
5.61 × 10−6
1.51 × 10−3
4.04 × 10−4
6.50 × 10−6
1.44 × 10−3 −5.14 × 10−5
3.94 × 10−4 −4.54 × 10−5
–

–

𝑁𝑎 𝜎Re(𝛿1)
0.19
0.41
2.37
4.82
8.03
0.54
1.34
0.10

1.08 × 109
4.89 × 108
8.39 × 107
6.85 × 108
1.83 × 108
6.51 × 108
1.79 × 106
–

Table 9.3 Estimated analyzing power and number of events for each detector configuration. Finally
translated into an expected optimal statistical sensitivity for the model of CP-violating state mixing.

All combined configurations can reach a sensitivity of 0.1 for the CP-violating mixing between

the positronium states. In the end the sensitivity for this model is dominated by the factor of 1/775

that arises from the atomic physics of positronium. This is discussed in Appendix B.

At this point it is worth emphasizing that all of the values in Table 9.2 are purely determined

by the geometry, energy cuts, stopping positions, and the model used for the new physics. The

effect of positronium state populations (and therefore alignment) and the branching ratios of 2-𝛾

and 3-𝛾 decays enter in as weightings when these quantities are summed together to get the final

coincidence counts for a given configuration. The relative weights due to state populations and 2-𝛾

branching ratios are intrinsically related, and determined by the B-field and the time cuts. All time

dependence is carried by the positronium state populations and therefore only the relative weighting

of these terms will change in time.

157

CHAPTER 10

TIME DEPENDENCE IN A MAGNETIC FIELD

The time dynamics of the symmetry violating signal was treated inconsistently between the previous

two experiments [38, 39]. This signal requires tensor polarization for the positronium being studied.

However, the time dependence of the signal is not proportional to time dependence of the tensor

polarization. In Ref. [19], the authors showed that the tensor anisotropies in 3-𝛾 decay of ortho-

positronium have the same dependence on the state population, for both the QED anisotropies, and

possible CP-violating anisotropies. The early precision tests of QED in Refs. [35, 36] measured

the QED induced anisotropy and observed a net anisotropy in the distribution of final state photons.

The searches for the QED anisotropy and the CP-violating distribution started with unpolarized

positronium in a magnetic field. If the angular anisotropy is proportional to the instantaneous tensor

polarization then both experiments would see a net isotropic distribution, since the positronium

starts evenly populated and eventually all the states decay. This conflict arises from invoking

rotational invariance to calculate the time dependence of the angular anisotropies. The addition of

an external magnetic field has broken rotational invariance in these experiments.

In this chapter we extend the theoretical analysis to include the non-trivial time dynamics

induced by the magnetic field. This gives the time dependence of the signal and background

contributions for an initially unpolarized positronium atom in a static magnetic field along the

𝑧-axis in terms of the magnitude of the field, and the initial and final time of the integration window.

This introduces changing contributions from the signal and increasing 2-𝛾 dilution. As the

analysis of Chapter 9 is directly applicable to a system with the triplet decays isolated from the

pseudo-triplet, it is first worth identifying shortcomings in performing the measurement this way,

and how including the pseudo-triplet events gives further handles on systematics. This chapter

concludes with a discussion of a few systematics related to the magnetic field, namely the 2-𝛾

dilution, and the effect of a field misalignment.

158

10.1 Detector efficiencies inducing false asymmetries

The planned analysis will require measuring coincident events with two pairs of detectors.

Calculating the sensitivity to a specific model required including the acceptances and efficiencies

of the detectors. This was done by introducting the "characteristic function" for the setup as

𝜒(k1, k2, k3). This function carries the geometric acceptances and detection efficiencies. We

assumed perfect detectors, but in reality each detector 𝛼 will have some energy dependent intrinsic

efficiency 𝜖𝛼 (𝜔). Worse yet there can be correlated efficiencies for the pair of detectors 𝜖𝛼𝛽 (𝜔𝛼, 𝜔𝛽).

We update the definition of the characteristic function to,

𝜒𝑎 (k1, k2, k3) =

𝜖𝛼𝛽 (𝜔1, 𝜔2),

ˆk1 ∈ Ω1, ˆk2 ∈ Ω2

0,

o/w





This gives an updated estimate of coincidences,

∫

𝑁Ω =

d 𝑓 3𝛾
Ω

𝜖𝛼𝛽 (𝜔1, 𝜔2)𝑎(𝜔1, 𝜔2) + 𝑠𝑧𝑧

∫

d 𝑓 3𝛾
Ω

𝜖𝛼𝛽 (𝜔1, 𝜔2)𝐷 (k1, k2, k3)

∫

+ 𝑠𝑧𝑧

d 𝑓 3𝛾
Ω

𝜖𝛼𝛽 (𝜔1, 𝜔2)𝐶 (k1, k2, k3)

(10.1)

(10.2)

We assume the detector efficiencies are smooth functions of energy multiplied by some intrinsic

efficiency, further take the energy dependence to be flat 𝜖𝛼𝛽 ≈ ¯𝜖𝛼𝛽. For two configurations (with

the same high energy detector) this gives the following coincidence counts,

𝑁𝛼𝛽 = ¯𝜖𝛼𝛽 (AΩ + 𝑠𝑧𝑧DΩ + 𝑠𝑧𝑧CΩ)

𝑁𝛼𝛽′ = ¯𝜖𝛼𝛽′ (AΩ + 𝑠𝑧𝑧DΩ − 𝑠𝑧𝑧CΩ)

(10.3)

(10.4)

The isotropic counts no longer cancel in an asymmetry and therefore the CP-violating signal is

not cleanly isolated. Changing the sign of 𝑠𝑧𝑧 would allow the measurement of two asymmetries,

one between detector pairs, and one between the differing values of 𝑠𝑧𝑧. This would facilitate

canceling the leading effect of the multiplicative efficiencies.

We will demonstrate in this chapter that the contribution from the CP-violating form factor

does not change sign between two time windows. It does have a time dependence that is induced by

159

the B-field. This will give a handle on the signal to change its magnitude (by varying the magnetic

field or changing time windows) to separate it from the isotropic distribution while still using the

same detectors with the same intrinsic efficiencies.

We treat this problem in an overly formal manner for the purposes of this chapter. In Appendix A

we extend the calculation to derive the full time dependence of all angular distributions (including

possible new physics form factors) for a positronium atom formed with a polarized 𝛽+ hitting

unpolarized electrons. Such dynamics have been studied in Refs.

[83, 84, 85, 86], but to our

knowledge this is the first calculation to include possible symmetry violating beyond Standard

Model factors.

10.2 Decaying systems

Introducing a few formal terms and ideas that will make the discussion easier. For this entire

chapter, there is no implied summation on repeated indices. The dynamics of a decaying system

can be approximated using a non-Hermitian Hamiltonian,

𝐻 = 𝑀 +

𝑖

2

Γ

(10.5)

where 𝑀 is the "mass matrix", and Γ as the absorptive part of the Hamiltonian (reserving the term

"decay matrix" to refer to the quantity 𝑅𝑖 𝑗 (k1, k2, k3) defined in Ref. [19]). The eigenvectors have

definite energies and lifetimes,

|𝜓𝛼, 𝑡⟩ = 𝑒−𝑖𝑡𝜔 𝛼− 1

2

𝑡Γ𝛼 |𝜓𝛼, 𝑡 = 0⟩

(10.6)

where 𝜔𝛼 + 𝑖

2 Γ𝛼 is the 𝛼th eigenvalue of 𝐻.

The elements of the absorptive part of the Hamiltonian are Γ𝑖 𝑗 = (cid:205) 𝑓 ⟨ 𝑓 | 𝒯 |𝑖⟩∗ ⟨ 𝑓 | 𝒯 | 𝑗⟩ =

(cid:205) 𝑓 Γ

𝑓
𝑖 𝑗 , where Γ

𝑓
𝑖 𝑗 corresponds to the partial decay to the final state 𝑓 . This gives a density matrix

with time evolution,

𝜌(𝑡) =

∑︁

𝛼𝛽

𝑒−𝑖𝑡 (𝜔 𝛼−𝜔𝛽)− 1

2

𝑡 (Γ𝛼+Γ𝛽) 𝜌𝛼𝛽 (𝑡 = 0) |𝜓𝛼⟩ (cid:10)𝜓𝛽

(cid:12)
(cid:12)

(10.7)

for the states 𝜓𝛼 that are eigenvectors of the Hamiltonian. This gives the instantaneous rate of

population of a given final state,

(cid:164)𝑁 𝑓 (𝑡) = −Tr(Γ 𝑓 𝜌(𝑡))

(10.8)

160

10.3 Combined spin-0 and spin-1 Hilbert Space

The magnetic field induces a mixing between spin-0 state and the 𝑚 = 0 substate of spin-1

positronium. Therefore the time dynamics should be treated in the combined Hilbert space. Indeed

this is the usual starting point when considering the individual spins {|↑↑⟩ , |↓↑⟩ , |↑↓⟩ |↓↓⟩}, then the

combinations with definite 𝐽2 are identified and considerations are restricted to those subspaces.

In the absence of a magnetic field, the Hamiltonian takes the simple form,

𝐻𝐵=0 =

(cid:18)

𝜔 𝑝−𝑃𝑠 +

(cid:19)

Γ𝑝−𝑃𝑠

𝑖

2

14𝑥4 +

(cid:18)

𝛿𝜔 +

(cid:19)

𝛿Γ

𝐽2

𝑖

2

(10.9)

This differs from the standard case of a magnetic dipole interaction with an external field. Two

components of angular momentum (those that are not parallel to the field) no longer commute with

the Hamiltonian and are therefore no longer conserved. The standard magnetic dipole interaction

goes as ˆJ · B, and still conserves the magnitude of angular momentum 𝐽2, since [𝐽2, ˆ𝐽𝑖] = 0. This

is not true for the positronium interaction with an external field. This interaction takes the form of

𝜇( ˆJ𝑒 − ˆJ ¯𝑒) · B. It is not dependent on a component of angular momentum, J = ˆJ𝑒 + ˆJ ¯𝑒, but instead

on the difference of the two particles angular momenta. This difference does not commute with

𝐽2, and as such the dynamics of the system is not confined to a subspace of the Hilbert space with

definite 𝐽2. This is equivalent to saying that the para-positronium and ortho-positronium states mix

despite having different angular momenta, and that the mixed states can decay to 2-𝛾 or 3-𝛾 despite

those having 𝐽 = 0 and 𝐽 = 1 respectively.

The addition of an external magnetic field induces off-diagonal terms in 𝑀. Representing

the states in the pseudo-singlet and pseudo-triplet basis diagonalizes the mass matrix specifically.

Writing this out explicitly in the combined spin-0 and spin-1 Hilbert space (using the Cartesian

basis for the spin-1 states) gives,

𝑀 =

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝜔 𝑝−𝑃𝑠

0

0

0

−𝜇𝑧 𝐵𝑧

𝜔𝑜−𝑃𝑠

0

0

161

0

0

𝜔𝑜−𝑃𝑠

−𝜇𝑧 𝐵𝑧

0

0

0

𝜔𝑜−𝑃𝑠

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(10.10)

Γ =

Γ𝑝−𝑃𝑠

0

0

0

0

Γ𝑜−𝑃𝑠

0

0

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

0

0

Γ𝑜−𝑃𝑠

0

0

0

0

Γ𝑜−𝑃𝑠

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(10.11)

For S-wave positronium we take 𝜇𝑧 ≈ 𝜇𝑒𝑠𝑒

𝑧 − 𝜇 ¯𝑒𝑠𝑒

𝑧 = 𝜇𝑒 (𝑠𝑒

𝑧 − 𝑠 ¯𝑒

𝑧 ), where 𝜇𝑒 is the magnetic moment

of the electron. This ignores any structure effects that renormalize the electron magnetic moment

in positronium. These two matrices do not commute. This means representing the time evolution

in the pseudo-singlet and pseudo-triplet basis gives off-diagonal elements to the absorptive part of

the Hamiltonian.

Following the discussion in Chapter 2, the absorptive part of the Hamiltonian Γ𝑖 𝑗 is the sum of

the partial decay rates. The partial decay rate for ortho-positronium to a given 3-𝛾 state (summing

over photon polarizations) is given as,

Γ𝑖 𝑗 (k1, k2, k3) = Γ𝑜−𝑃𝑠 𝑅𝑖 𝑗 (k1, k2, k3)

(10.12)

where 𝑅𝑖 𝑗 is the decay matrix as defined in Ref.

[19]. This matrix carries the form factors 𝑎,

𝑐1, 𝑐2, etc. Representing this matrix acting on the combined spin-0/spin-1 Hilbert space in the

pseudo-singlet/pseudo-triplet basis makes the decay matrix have many complicated entries. It is

easier for this purpose to stay in the spin-0 and spin-1 basis in which Γ𝑖 𝑗 (k1, k2, k3) is simple, and

calculate the time dependence of 𝜌(𝑡) in this basis.

The non-Hermitian Hamiltonian method (the Wigner-Weisskopf method) is an approximation

but it is well founded in this case as the binding energy released in the decay is small compared to the

mass [87]. The use of pseudo-singlet and pseudo-triplet states is also an approximation as they do

not diagonalize 𝐻 = 𝑀 + (𝑖/2)Γ, but only 𝑀. This means all effects of level broadening are ignored

[87, 88]. This effect is unimportant for the current calculation as it mostly results in energy level

shifts. There is a very small correction to the mixing coefficients that turns them slightly imaginary

and would result in a small correction to the density matrix terms. One further approximation is

that the magnetic field only affects the states but does not affect the matrix elements for the decay.

162

This is a safe estimation for this current calculation purposes, but may not be a small effect for

para-positronium to 3-𝛾. This is unimportant for this discussion but is discussed in Ref. [87].

Generically the density matrix will have time dependent entries,

𝜌(𝑡) =

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝜌00(𝑡) 𝜌0𝑥 (𝑡) 𝜌0𝑦 (𝑡) 𝜌0𝑧 (𝑡)
0𝑥 (𝑡) 𝜌𝑥𝑥 (𝑡) 𝜌𝑥𝑦 (𝑡) 𝜌𝑥𝑧 (𝑡)
𝜌∗
0𝑦 (𝑡) 𝜌∗
𝑥𝑦 (𝑡) 𝜌𝑦𝑦 (𝑡) 𝜌𝑦𝑧 (𝑡)
𝜌∗
0𝑧 (𝑡) 𝜌∗
𝜌∗

𝑦𝑧 (𝑡) 𝜌𝑧𝑧 (𝑡)

𝑥𝑧 (𝑡) 𝜌∗

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

however, the partial decay matrices to 2-𝛾 and 3-𝛾 final states remain simple,

Γ2𝛾 = Γ𝑝−𝑃𝑠

Γ3𝛾 (k1, k2, k3; 𝑡) = Γ𝑜−𝑃𝑠

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

0

0

0

0

0 𝑅𝑥𝑥 𝑅𝑥𝑦 𝑅𝑥𝑧

0 𝑅∗

𝑥𝑦 𝑅𝑦𝑦 𝑅𝑦𝑧

0 𝑅∗

𝑥𝑧 𝑅∗

𝑦𝑧 𝑅𝑧𝑧

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(10.13)

(10.14)

(10.15)

giving the time dependence of the 2-𝛾 events, and the 3-𝛾 events including the full angular

distribution as,

(cid:164)𝑁2𝛾 (𝑡) = Γ𝑝−𝑃𝑠 𝜌00(𝑡)

(10.16)

(cid:164)𝑁3𝛾 (k1, k2, k3; 𝑡) = Γ𝑜−𝑃𝑠

(cid:26)

𝑅𝑥𝑥 𝜌𝑥𝑥 (𝑡) + 2Re(𝑅𝑥𝑦 𝜌∗

𝑥𝑦 (𝑡)) + 2Re(𝑅𝑥𝑧 𝜌∗

𝑥𝑧 (𝑡))

+ 𝑅𝑦𝑦 𝜌𝑦𝑦 (𝑡) + 2Re(𝑅𝑦𝑧 𝜌∗

𝑦𝑧 (𝑡)) + 𝑅𝑧𝑧 𝜌𝑧𝑧 (𝑡)

(cid:27)

(10.17)

This has simplified the problem down to calculating 𝜌00(𝑡) and 𝜌𝑖 𝑗 (𝑡) in terms of the states with

simple time evolution (pseudo-singlet and pseudo-triplet). These two bases are related by the

163

matrix,

where 𝑐 = cos(𝜁) = 1√
2

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)
√︃

𝜓 𝑝𝑆,𝑚=0

𝜓𝑆=1,𝑥

𝜓𝑆=1,𝑦

𝜓 𝑝𝑇,𝑚=0

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

=

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

𝑐 0 0 −𝑠

0 1 0

0 0 1

𝑠 0 0

0

0

𝑐

1 + 1√

1+𝑥2 and 𝑠 = sin(𝜁) =

𝜓𝑆=0,𝑚=0

𝜓𝑆=1,𝑥

𝜓𝑆=1,𝑦

(cid:170)
(cid:169)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:174)
(cid:173)
(cid:171)
(cid:172)
√
1 − 𝑐2 and 𝑥 = |𝐵|/(3.63 Tesla) [45].

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

𝜓𝑆=1,𝑧

(10.18)

This calculation is extended in Appendix A to include an arbitrary net positron polarization and

identify the full time dependence for comparison to related experiments on the "pulsing" angular

distribution in positronium decay measured in Refs. [84, 85, 86]. The final experiment will have

two powder containers, each of which will have a net vector polarization due to parity violation

in nuclear 𝛽-decay. For this work the polarizations are treated to be exactly opposite in sign and

therefore cancel.

10.4 Time evolution for unpolarized positronium

Starting with an even state population the density matrix in the pseudo-singlet/pseudo-triplet at

time 𝑡 is very simple,

𝜌 𝑝𝑆,𝑝𝑇 (𝑡) =

𝑒−𝑡Γ𝑝𝑆

1
4

0

0

0

0

𝑒−𝑡Γ𝑜− 𝑃𝑠

1
4

0

0

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

0

0

𝑒−𝑡Γ𝑜− 𝑃𝑠

1
4

0

0

0

0

𝑒−𝑡Γ𝑝𝑇

1
4

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(10.19)

where Γ𝑝𝑆 is the pseudo-singlet lifetime, Γ𝑝𝑇 is the pseudo-triplet lifetime, and Γ𝑜−𝑃𝑠 is the triplet

lifetime, here taken to be in vacuum. The state represented in the definite angular momentum basis

has a more complicated structure. The full time dependence is quoted below,

𝜌𝑆=0,𝑆=1(𝑡) =

𝜌00(𝑡) =

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)
1
4

𝜌00(𝑡)

0

𝜌𝑥𝑥 (𝑡)

0

0

0

0

0

𝜌𝑦𝑦 (𝑡)

0𝑧 (𝑡)
𝜌∗
0
0
𝑐2𝑒−𝑡Γ𝑝𝑆 + 𝑠2𝑒−𝑡Γ𝑝𝑇 (cid:17)

(cid:16)

164

𝜌0𝑧 (𝑡)

0

0

𝜌𝑧𝑧 (𝑡)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(10.20)

(10.21)

𝑒−𝑡Γ𝑝𝑆 + 𝑒−𝑡Γ𝑝𝑇 (cid:17)

(cid:16)

𝑐𝑠

𝜌0𝑧 (𝑡) =

1
4
𝜌𝑥𝑥 (𝑡) = 𝜌𝑦𝑦 (𝑡) =

𝑒−𝑡Γ𝑝𝑇

1
4

𝜌𝑧𝑧 (𝑡) =

1
4

(cid:16)

𝑠2𝑒−𝑡Γ𝑝𝑆 + 𝑐2𝑒−𝑡Γ𝑝𝑇 (cid:17)

(10.22)

(10.23)

(10.24)

Restricting to times greater than 1 ns means all pseudo-singlet contributions can be neglected.

This gives an instantaneous rate of 2-𝛾 events, and also 3-𝛾 events with angular distribution

(k1, k2, k3),

(10.25)

(10.26)

(cid:164)𝑁2𝛾 (𝑡) =

(cid:164)𝑁3𝛾 (k1, k2, k3; 𝑡) =

1
4
1
4

Γ𝑝−𝑃𝑠 (cid:0)𝑠2𝑁 𝑝𝑇 𝑒−𝑡Γ𝑝𝑇 (cid:1)

Γ𝑝−𝑃𝑠 (cid:0)𝑅𝑥𝑥 (k1, k2, k3) + 𝑅𝑦𝑦 (k1, k2, k3)(cid:1)𝑒−𝑡Γ𝑜− 𝑃𝑠

1
4
The branching ratio and quenched lifetimes are directly related by 𝑐2Γ𝑜−𝑃𝑠 = 𝐵𝑅 𝑝𝑇

3𝛾 Γ𝑝𝑇 , and
following Chapter 9, 𝑅𝑖𝑖 = 𝑎(𝜔1, 𝜔2) + 𝐶𝑖𝑖 and 𝐶𝑥𝑥 + 𝐶𝑦𝑦 + 𝐶𝑧𝑧 = 0. This simplifies Equation 10.26

Γ𝑜−𝑃𝑠 𝑅𝑧𝑧 (k1, k2, k3) (𝑐2𝑒−𝑡Γ𝑝𝑇 )

+

to,

(cid:164)𝑁3𝛾 (k1, k2, k3; 𝑡) =

(cid:26)

𝑎(𝜔1, 𝜔2) (cid:0)2Γ𝑜−𝑃𝑠𝑒−𝑡Γ𝑜−𝑃𝑠 + 𝐵𝑅 𝑝𝑇

3𝛾 Γ𝑝𝑇 𝑒−𝑡Γ𝑝𝑇 (cid:1)

+𝐶𝑧𝑧 (k1, k2, k3) (cid:0)2Γ𝑜−𝑃𝑠𝑒−𝑡Γ𝑜−𝑃𝑠 − 2𝐵𝑅 𝑝𝑇

3𝛾 Γ𝑝𝑇 𝑒−𝑡Γ𝑝𝑇 (cid:1)

(cid:27)

(10.27)

Integrating this in a finite time window reproduces the quantity in Equation 2.39, but now

correctly including the effects of mixing/different branching ratios. This means that the contribution

from the pseudo-triplet to the angular distortion is reduced by the relevant branching ratio. Both

the QED anisotropy and the CP-violating tensor form factors are contained in 𝐶𝑧𝑧 (k1, k2, k3). The

negative contribution from the pseudo-triplet is smaller than the positive contribution from the

triplet decays at all times. There is not just a net anisotropy (as measured in Ref. [36]), but the

distortion term never changes sign between time windows.

10.5

Including pseudo-triplet events

Returning to the analysis from Chapter 9, we consider the detector configuration with solid

angles (Ω1, Ω2) = Ω. The coincidence events are recorded within a time window Δ𝑇 = [𝑡𝑖, 𝑡 𝑓 ].

165

This gives the coincident counts for this detector pair in this time window,

𝑁 Δ𝑇

Ω =

∫

d𝑡

∫ 𝑡 𝑓

𝑡𝑖

d 𝑓 3𝛾
Ω

(cid:164)𝑁3𝛾 (k1, k2, k3; 𝑡)

(10.28)

Following the analysis in Chapter 9, now we consider a second detector configuration Ω′ such

that the symmetry violating term changes sign. The symmetry violating term can be extracted by

forming an asymmetry in counts,

𝐴Δ𝑇
Ω =

𝐴Δ𝑇
Ω =

Ω − 𝑁 Δ𝑇
𝑁 Δ𝑇
Ω′
𝑁 Δ𝑇
Ω + 𝑁 Δ𝑇
Ω′
𝜂CΩ
AΩ + 𝜂DΩ

(10.29)

(10.30)

This matches Equation 9.19 with the replacement of 𝑠𝑧𝑧 with the term 𝜂 which depends on the field

strength and the time window. This term goes as,

𝜂𝐵,𝑡𝑖,𝑡 𝑓 =

2𝑒−𝑡Γ𝑜−𝑃𝑠 |𝑡𝑖
2𝑒−𝑡Γ𝑜−𝑃𝑠 |𝑡𝑖

𝑡 𝑓 − 2𝐵𝑅 𝑝𝑇
𝑡 𝑓 + 𝐵𝑅 𝑝𝑇

3𝛾 𝑒−𝑡Γ𝑝𝑇 |𝑡𝑖
3𝛾 𝑒−𝑡Γ𝑝𝑇 |𝑡𝑖

𝑡 𝑓

𝑡 𝑓

(10.31)

This factor is reminiscent of the calculation of the "averaged tensor polarization" in Equation 2.39,

but now it includes the effect of the 2-𝛾 branching ratio. The time dependence of the asymmetry

is shown in Figure 10.1, for fields ranging from 0 T to 0.5 T. This is plotted for the Symmetric

157.5◦ configuration assuming that CΩ is proportional to the phase space selected. Any model

would result in a rescaling of this plot vertically.

This is in disagreement with the claim from the Michigan experiment that the signal reverses

between time windows [38]. The quantity 𝜂 that linearly scales the signal matches the method that

the Tokyo group used to calculate the tensor polarization when extracting their sensitivity in Ref.

[39]. We do not agree with calling this quantity a "tensor polarization" or an "alignment", as those

are intrinsic quantities to the positronium state that quantify its state populations, and are defined

irrespective of any decay channel.

10.6 Energy cuts and 2-𝛾 dilution

The 2-𝛾 decays that survive the energy cuts have no signal sensitivity and therefore reduce the

final sensitivity. The 3-𝛾 decays and 2-𝛾 decays have distinct energy distributions, but, due to

166

Figure 10.1 The time dependence of an asymmetry in counts in the presence of new physics.
This plots Equation 10.30 for the Symmetric 157.5◦ configuration assuming the new physics is
proportional to the volume of phase space selected. The magnetic field strength varies from 0 T
(red) to 0.5 T (green) and the width of the integration window is taken as 1 ns.

finite energy resolution, some of the 2-𝛾 events will leak the energy cuts. Estimating the reduction

to sensitivity requires estimating the spreading of the positronium, the acceptance of 2-𝛾 events,

the finite energy resolution, and the relative weighting of 2-𝛾 to 3-𝛾 events. The 2-𝛾 decays

occur because quenching of the lifetime from the magnet. This induces a time dependence for the

weighting of 2-𝛾 to 3-𝛾 events. Equation 10.30 can be extended to include the 2-𝛾 dilution,

𝐴𝑎 =

𝜂CΩ
AΩ + 𝜂DΩ + 𝜉F Ω

(10.32)

where F Ω is the proportion of 2-𝛾 events that survive the phase space cuts, and 𝜉 is the relative

weighting of 2-𝛾 decays to 3-𝛾 decays. This gives,

𝜉𝐵,𝑡𝑖,𝑡 𝑓 =

𝐵𝑅 𝑝𝑇
2𝑒−Γ𝑜− 𝑃𝑠𝑡 |𝑡𝑖

2𝛾 𝑒−Γ𝑝𝑇 𝑡 |𝑡𝑖
𝑡 𝑓 + 𝐵𝑅 𝑝𝑇

𝑡 𝑓

3𝛾 𝑒−Γ𝑝𝑇 𝑡 |𝑡𝑖

𝑡 𝑓

(10.33)

This quantity never becomes large due to a magnetic field. At most if all pseudo-triplet decays to

2-𝛾 then 𝜂 → 1

2. This definition does not include the 2-𝛾 decays induced by interaction with the

powder.

The 2-𝛾 dilution will be largest for the Symmetric 157.5◦ configuration. In principle the other

configurations will have a small amount as well, but the 3-𝛾 distribution for those detector pairs

does not extend up towards the 511-511 keV region and can be more cleanly separated from the

2-𝛾 events. Given that this model of CP-violation is peaked going to high energies, the Symmetric

167

157.5◦ configuration warrants special attention. A cut on the summed energies can be added to

separate the region that is primarily 2-𝛾 events from the region that is primarily 3-𝛾 events. The

values of AΩ, DΩ, CΩ, and F Ω will all shift as the energy cut is varied. Figure 10.2 shows the

isotropic distribution, the QED anisotropy, the CP-violating signal, and the 2-𝛾 events for the

Symmetric 157.5◦ configuration using events sampled from the stopping distribution for Design

(C) with a 22Na source. The value of these functions for each summed energy cut is given in Table

10.1 for a 22Na source, and Table 10.2 for a 68Ga source.

𝐸𝑠𝑢𝑚 cut (MeV)
1.022
1.000
0.975
0.950
0.925
0.900

¯CΩ
𝑚𝑖𝑥

AΩ

DΩ
2.14 ∗ 10−3 −7.27 ∗ 10−4 −6.77 ∗ 10−7
2.14 ∗ 10−3 −7.26 ∗ 10−4 −6.76 ∗ 10−7
2.12 ∗ 10−3 −7.22 ∗ 10−4 −6.69 ∗ 10−7
2.07 ∗ 10−3 −7.05 ∗ 10−4 −6.44 ∗ 10−7
1.97 ∗ 10−3 −6.71 ∗ 10−4 −5.96 ∗ 10−7
1.78 ∗ 10−3 −6.08 ∗ 10−4 −5.17 ∗ 10−7

F Ω
3.74 ∗ 10−7
3.34 ∗ 10−7
2.33 ∗ 10−7
1.33 ∗ 10−7
8.75 ∗ 10−8
7.13 ∗ 10−8

Table 10.1 The variation of each function as the summed energy cut is changed for the symmetric
157.5◦ configuration. Increasing the cut dramatically increases the amount of 2-𝛾 events. These
are for the 22Na source using the Design (C) start detector and powder.

𝐸𝑠𝑢𝑚 cut (MeV)
1.022
1.000
0.975
0.950
0.925
0.900

¯CΩ
𝑚𝑖𝑥

AΩ

DΩ
2.09 ∗ 10−3 −5.82 ∗ 10−4 −5.96 ∗ 10−7
2.09 ∗ 10−3 −5.82 ∗ 10−4 −5.95 ∗ 10−7
2.07 ∗ 10−3 −5.79 ∗ 10−4 −5.89 ∗ 10−7
2.03 ∗ 10−3 −5.67 ∗ 10−4 −5.68 ∗ 10−7
1.93 ∗ 10−3 −5.43 ∗ 10−4 −5.27 ∗ 10−7
1.76 ∗ 10−3 −5.00 ∗ 10−4 −4.61 ∗ 10−7

F Ω
1.31 ∗ 10−6
1.16 ∗ 10−6
7.61 ∗ 10−7
4.39 ∗ 10−7
2.95 ∗ 10−7
2.40 ∗ 10−7

Table 10.2 The variation of each function as the summed energy cut is changed for the symmetric
157.5◦ configuration. Increasing the cut dramatically increases the amount of 2-𝛾 events. These
are for the 68Ga source using the Design (C) start detector and powder.

These energy cuts are not tuned, for instance there is large overlap with the Compton continuum

along the 511-keV line. This simulation does not include any inactive material, but the final

experiment will have a supporting structure and an inner module that 511 keV photons will scatter

off of. This will generally make F Ω larger, and impede efforts to lower the energy cuts to the

regions that select ( ˆk2, ˆk3). Tables 10.1 and 10.2 show that F Ω is very small compared to AΩ.

168

(a) Isotropic distribution

(b) QED anisotropy

(c) CP-violating signal

(d) 2-𝛾 events

Figure 10.2 The effect of varying the summed energy cut for the symmetric 157.5◦ configuration.
All four functions have different energy dependence’s, and must be considered when we decide on
the optimal energy cuts.

Since 𝜉 never becomes large the 2-𝛾 distortion is a small shift to the backgrounds, substantially

smaller than the QED anisotropy distortion to counts. Proceeding from here we ignore the 2-𝛾

dilution as it is a small effect for all time cuts (so long as they are sufficiently far from the peak).

169

10.7

Intrinsic detection efficiency

Returning to the discussion of finite detection efficiency, the asymmetry takes the following

form (to first order in CΩ),

𝐴𝑎 =

𝜂CΩ
AΩ + 𝜂DΩ

+

𝜖𝛼𝛽 − 𝜖𝛼𝛽′
𝜖𝛼𝛽 + 𝜖𝛼𝛽′

(10.34)

The quantity 𝜂 is dependent on the value of the B-field and our time cuts. This means it is a tuneable

parameter that can separate out the signal from systematics. In this case the efficiencies giving

an asymmetry value when the field is not applied (and therefore when there is no CP-violating

signal). This is a large benefit of using an electromagnet. Compare with the previous experiments

in Refs. [38, 39], we will be able to vary the magnetic field value without physically moving any

components of the experiment, which could change the scattering characteristics for the setup.

Figure 10.3 shows the asymmetry as a function of time, assuming a signal on the scale estimate

in Chapter 7, and including a 1% difference in efficiency between the two detectors. The new

physics manifests as a time dependent shift to the asymmetry that is distinct from the constant offset

of mismatched detector efficiencies.

Figure 10.3 Estimation of the time dependent asymmetry induced in the 157.5◦ configuration
assuming a 1% difference in detection efficiency and new physics at the scale of sensitivity estimated
in Chapter 9. This uses 1 ns time windows.

10.7.1 All form factors that contribute to an asymmetry

The final systematic we address in this work is the count asymmetry induced by a misaligned

detector array. For the purposes of this discussion, it is easier to fix the coordinates to the detector

170

array, and consider a misalignment of the field axis. We consider the detector pairs shown in Figure

9.1. The highest energy photon is fixed to lie in the x-z plane, and there are two detectors for the

second highest energy photon that differ by changing the sign of 𝜅𝑦. This also changes the sign of

𝑛𝑥 and 𝑛𝑧. The following expectation values change sign between our detector pairs (terms with

two indices should be interpreted as components of symmetric traceless tensors):

1. Vector terms – ⟨𝜅2𝑦⟩, ⟨𝑛𝑥⟩, ⟨𝑛𝑧⟩

2. CP conserving tensor terms – ⟨𝜅1𝑥 𝜅2𝑦⟩, ⟨𝜅1𝑧𝜅2𝑦⟩, ⟨𝜅2𝑥 𝜅2𝑦⟩, ⟨𝜅2𝑧𝜅2𝑦⟩, ⟨𝑛𝑥𝑛𝑦⟩, ⟨𝑛𝑦𝑛𝑧⟩

3. CP violating tensor terms – ⟨𝜅1𝑧𝑛𝑧⟩, ⟨𝜅2𝑧𝑛𝑧⟩, ⟨𝜅1𝑥𝑛𝑥⟩, ⟨𝜅1𝑥𝑛𝑧⟩, ⟨𝜅1𝑧𝑛𝑥⟩, ⟨𝜅2𝑥𝑛𝑥⟩, ⟨𝜅2𝑥𝑛𝑧⟩,

⟨𝜅2𝑧𝑛𝑥⟩, ⟨𝜅2𝑦𝑛𝑦⟩

Any term with one index could be induced by a vector polarization in that direction, and any

term with two indices could be induced by a magnetic field with components along those axes.

This relationship is summarized in Table 9.1. Of the 18 correlations that could contribute to an

asymmetry in counts between the detector pairs, 10 of them correspond to new physics. The first

vector term corresponds to CP-violation, the second two to new physics (though not necessarily

new physics that violates any discrete symmetries). The last 9 tensor correlations that involve

combinations of one photon momentum and the normal to the decay plane are the signal we will

search for. We have optimized the geometry to be most sensitive to the 𝑧𝑧 terms.

The primary concern is the 6 tensor terms in the second line. These are all induced in QED,

and appear if the positronium has an off-diagonal term in the tensor polarization. Off-diagonal

terms can be induced by the B-field not being aligned perfectly along the ˆ𝑧-axis. Just as in Ref.

[19], the signal is proportional to a tensor quantity from the positronium state, however now we

argue that it is no longer the net tensor polarization for a time window, but instead this quantity 𝜂

from Equation 10.31. Just as is the case with the alignment compared to the tensor polarization,

this is really a component of a second rank tensor, 𝜂 → 𝜂𝑧𝑧. We can now calculate the other

components of this tensor given an arbitrary field direction. Taking the magnetic field axis in the
direction ˆB = (sin(𝜃)cos(𝜙), sin(𝜃)sin(𝜙), cos(𝜃))𝑇

would induce the off-diagonal components

171

with the following magnitude,

𝜂𝑥𝑦/𝜂 =

𝜂𝑦𝑧/𝜂 =

𝜂𝑧𝑥/𝜂 =

3
4
3
4
3
4

sin2(𝜃)sin(2𝜙)

sin(2𝜃)sin(𝜙)

sin(2𝜃)cos(𝜙)

(10.35)

(10.36)

(10.37)

where 𝜂 is the quantity from Equation 10.31, and 𝜂𝑖 𝑗 is the 𝑖 𝑗-component if there is a misalignment.

We expect the field will be very well aligned so it is safe to assume 𝜃 is small, however the value

of 𝜙 can fall anywhere between 0 and 2𝜋. This makes the 𝑥𝑦 term is very small. The 𝑦𝑧 and

𝑧𝑥 terms, while not large, are not suppressed either. This analysis is considering a single set of

detectors. Each detector configuration has 16 equivalent configurations related by an azimuthal

rotation. This means that a misalignment of the field relative to the array induces an asymmetry, but

that asymmetry would have a sinusoidal modulation for two detector configurations related by an

axial rotation around the ring. The CP-violating signal (along the 𝑧𝑧-direction) is unchanged by an

axial rotation. This means that the small count asymmetries that a misalignment of the field could

induce will be distinguishable from the signal due to the axial symmetry of our detector array.

172

CHAPTER 11

SUMMARY AND FUTURE WORK

We have presented the initial design, prototyping, and testing of a dedicated apparatus to search for

CP-violation in the 3-𝛾 decay of ortho-positronium. We reviewed some of the history of angular

correlations in o-Ps decay, and identified some inconsistencies in the literature. This motivated an

in-depth reanalysis of the planned experiment.

The array will consist of 48 𝛾-detectors arranged in three rings, the two outer rings will be tilted

towards a central source. The entire array will fit in the FRIB Positron Polarimeter magnet. The

central source will contain a 𝛽+ emitting source, plastic scintillators for triggering, and containers

of powder for positronium formation. We achieved a 50% formation fraction for low density SiO2

powder, and under a roughing pump we achieved a lifetime at 95% the vacuum lifetime. Similar

formation fractions have been achieved at much higher density of MgO powder, if that powder is

baked in a vacuum oven.

The 𝛾-detectors are LYSO crystals in a partially tapered design to maximize detection efficiency

while still leaving room for mounting. These crystals are read out by Silicon photomultipliers that

will allow the full array to be placed directly in the warm bore of the magnet. The final detector

modules tested achieved a 12% FWHM energy resolution at 511 keV. The full set of detectors is

currently in production at Wittenberg University and will soon be ready to install in the magnet.

These detectors can be reliably digitized and read out in coincidence with the plastic start detector

using NSCLDAQ and 250 MSPS PIXIE-16 boards. Reading the 𝛾-detectors in coincidence with

the plastic detector entirely removed the backgrounds from the intrinsic radioactivity of the crystals.

The continuous 2-D energy distribution was separated from the 511 keV annihilation events by

cutting on the long lifetime component in the lifetime spectrum.

The design of the inner module and the 𝛾-detector array was studied in Monte-Carlo simulations

using Geant4. This facilitated the study of both the spreading out of the position of decay, and the

geometric acceptance and detection efficiency for 2-𝛾 and 3-𝛾 decays. This included finite energy

resolution, and the various coincidence conditions.

173

The model dependence of this search was highlighted, in particular how the physics manifests in

"form factor", as opposed to similar searches for "coefficients" in nuclear 𝛽-decay. This was followed

by a discussion of some systematic effects for the measurement, including misalignment of the field,

differing detector resolutions, and dilution from 2-𝛾 events. These will leave distinctive signatures

in the count asymmetries, and incorporating the time dependence induced by the magnetic field

will facilitate separating the signal from some systematic offsets.

This work has outlined how to account for various systematics, and how to perform the analysis

accounting for model dependence. The Monte Carlo framework as constructed is flexible enough

to run an arbitrarily polarized source sampling an arbitrary position for the decay. This will allow

quantitative studies of systematic effects such as offsets of the source and misalignment of the

detector array. The primary bulk of the work to be performed before data collection are completion

of construction for the 𝛾-detectors and testing and construction of a higher rate start detector read

out by photomultiplier tubes.

With the design we will be able to achieve an over 10-fold improvement on current limits for

this signal, and due to the highly symmetric design of the detector array we will have a strong

ability to separate signal from systematics.

174

BIBLIOGRAPHY

[1] M. E. Peskin and D. V. Schroeder. An Introduction to quantum field theory. Addison-Wesley,

Reading, USA, 1995.

[2] B. G. Chen, D. Derbes, D. Griffiths, B. Hill, R. Sohn, and Y.S. Ting. Lectures of Sidney

Coleman on Quantum Field Theory. WORLD SCIENTIFIC, 2018.

[3] Stib. World line, 2007. https://commons.wikimedia.org/wiki/File:World_line.svg.

[4] S. Weinberg. The Quantum Theory of Fields. Cambridge University Press, 1995.

[5] R. P. Feynman and S. Weinberg. Elementary Particles and the Laws of Physics: The 1986

Dirac Memorial Lectures. Cambridge University Press, 1987.

[6] T. D. Lee and C. N. Yang. Question of parity conservation in weak interactions. Phys. Rev.,

104:254–258, Oct 1956.

[7] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson. Experimental test of

parity conservation in beta decay. Phys. Rev., 105:1413–1415, Feb 1957.

[8] E. D. Commins and P. H. Bucksbaum. Weak Interactions of Leptons and Quarks. Cambridge

University Press, 1983.

[9]

J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay. Evidence for the 2𝜋 decay of the
𝐾 0

2 meson. Phys. Rev. Lett., 13:138–140, Jul 1964.

[10] R. L. Workman et al. Review of Particle Physics. PTEP, 2022:083C01, 2022.

[11] K. Abe, R. Akutsu, A. Ali, C. Alt, C. Andreopoulos, L. Anthony, M. Antonova, S. Aoki,
A. Ariga, T. Arihara, Y. Asada, Y. Ashida, E. T. Atkin, Y. Awataguchi, S. Ban, M. Barbi,
G. J. Barker, G. Barr, D. Barrow, C. Barry, M. Batkiewicz-Kwasniak, A. Beloshapkin,
F. Bench, V. Berardi, S. Berkman, L. Berns, S. Bhadra, S. Bienstock, A. Blondel, S. Bolog-
nesi, B. Bourguille, S. B. Boyd, D. Brailsford, A. Bravar, D. Bravo Berguño, C. Bronner,
A. Bubak, M. Buizza Avanzini, J. Calcutt, T. Campbell, S. Cao, S. L. Cartwright, M. G.
Catanesi, A. Cervera, A. Chappell, C. Checchia, D. Cherdack, N. Chikuma, M. Cicerchia,
G. Christodoulou, J. Coleman, G. Collazuol, L. Cook, D. Coplowe, A. Cudd, A. Dabrowska,
G. De Rosa, T. Dealtry, P. F. Denner, S. R. Dennis, C. Densham, F. Di Lodovico, N. Dokania,
S. Dolan, T. A. Doyle, O. Drapier, J. Dumarchez, P. Dunne, A. Eguchi, L. Eklund, S. Emery-
Schrenk, A. Ereditato, P. Fernandez, T. Feusels, A. J. Finch, G. A. Fiorentini, G. Fiorillo,
C. Francois, M. Friend, Y. Fujii, R. Fujita, D. Fukuda, R. Fukuda, Y. Fukuda, K. Fusshoeller,
K. Gameil, C. Giganti, T. Golan, M. Gonin, A. Gorin, M. Guigue, D. R. Hadley, J. T. Haigh,
P. Hamacher-Baumann, M. Hartz, T. Hasegawa, S. Hassani, N. C. Hastings, T. Hayashino,
Y. Hayato, A. Hiramoto, M. Hogan, J. Holeczek, N. T. Hong Van, F. Iacob, A. K. Ichikawa,
M. Ikeda, T. Ishida, T. Ishii, M. Ishitsuka, K. Iwamoto, A. Izmaylov, M. Jakkapu, B. Jamieson,

175

S. J. Jenkins, C. Jesús-Valls, M. Jiang, S. Johnson, P. Jonsson, C. K. Jung, X. Junjie, P. B. Jurj,
M. Kabirnezhad, A. C. Kaboth, T. Kajita, H. Kakuno, J. Kameda, D. Karlen, S. P. Kasetti,
Y. Kataoka, T. Katori, Y. Kato, E. Kearns, M. Khabibullin, A. Khotjantsev, T. Kikawa,
H. Kikutani, H. Kim, J. Kim, S. King, J. Kisiel, A. Knight, A. Knox, T. Kobayashi, L. Koch,
T. Koga, A. Konaka, L. L. Kormos, Y. Koshio, A. Kostin, K. Kowalik, H. Kubo, Y. Kudenko,
N. Kukita, S. Kuribayashi, R. Kurjata, T. Kutter, M. Kuze, L. Labarga, J. Lagoda, M. Lam-
oureux, M. Laveder, M. Lawe, M. Licciardi, T. Lindner, R. P. Litchfield, S. L. Liu, X. Li,
A. Longhin, L. Ludovici, X. Lu, T. Lux, L. N. Machado, L. Magaletti, K. Mahn, M. Malek,
S. Manly, L. Maret, A. D. Marino, L. Marti-Magro, J. F. Martin, T. Maruyama, T. Matsubara,
K. Matsushita, V. Matveev, K. Mavrokoridis, E. Mazzucato, M. McCarthy, N. McCauley,
J. McElwee, K. S. McFarland, C. McGrew, A. Mefodiev, C. Metelko, M. Mezzetto, A. Mi-
namino, O. Mineev, S. Mine, M. Miura, L. Molina Bueno, S. Moriyama, J. Morrison,
Th. A. Mueller, L. Munteanu, S. Murphy, Y. Nagai, T. Nakadaira, M. Nakahata, Y. Nakajima,
A. Nakamura, K. G. Nakamura, K. Nakamura, S. Nakayama, T. Nakaya, K. Nakayoshi, C. Nan-
tais, C. E. R. Naseby, T. V. Ngoc, K. Niewczas, K. Nishikawa, Y. Nishimura, E. Noah, T. S.
Nonnenmacher, F. Nova, P. Novella, J. Nowak, J. C. Nugent, H. M. O’Keeffe, L. O’Sullivan,
T. Odagawa, K. Okumura, T. Okusawa, S. M. Oser, R. A. Owen, Y. Oyama, V. Palladino, J. L.
Palomino, V. Paolone, M. Pari, W. C. Parker, S. Parsa, J. Pasternak, P. Paudyal, M. Pavin,
D. Payne, G. C. Penn, L. Pickering, C. Pidcott, G. Pintaudi, E. S. Pinzon Guerra, C. Pis-
tillo, B. Popov, K. Porwit, M. Posiadala-Zezula, A. Pritchard, B. Quilain, T. Radermacher,
E. Radicioni, B. Radics, P. N. Ratoff, E. Reinherz-Aronis, C. Riccio, E. Rondio, S. Roth,
A. Rubbia, A. C. Ruggeri, C. A. Ruggles, A. Rychter, K. Sakashita, F. Sánchez, G. Santucci,
C. M. Schloesser, K. Scholberg, J. Schwehr, M. Scott, Y. Seiya, T. Sekiguchi, H. Sekiya,
D. Sgalaberna, R. Shah, A. Shaikhiev, F. Shaker, A. Shaykina, M. Shiozawa, W. Shorrock,
A. Shvartsman, A. Smirnov, M. Smy, J. T. Sobczyk, H. Sobel, F. J. P. Soler, Y. Sonoda,
J. Steinmann, S. Suvorov, A. Suzuki, S. Y. Suzuki, Y. Suzuki, A. A. Sztuc, M. Tada, and
The T2K Collaboration. Constraint on the matter–antimatter symmetry-violating phase in
neutrino oscillations. Nature, 580(7803):339–344, Apr 2020.

[12] W. Bernreuther. CP violation and baryogenesis, 2002. arXiv, hep-ph/0205279.

[13] A. D. Sakharov. Violation of CP Invariance, C asymmetry, and baryon asymmetry of the

universe. Pisma Zh. Eksp. Teor. Fiz., 5:32–35, 1967.

[14] R. Gannouji. Introduction to Electroweak Baryogenesis. Galaxies, 10(6), 2022.

[15] M. Kobayashi and T. Maskawa. CP-Violation in the Renormalizable Theory of Weak Inter-

action. Prog. Theor. Phys., 49(2):652–657, 02 1973.

[16] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. Boon Ng, N. Schloss-
berger, S.n Yool Park, Anzhou Wang, Jun Ye, and Eric A. Cornell. An improved bound on
the electron’s electric dipole moment. Science, 381(6653):46–50, 2023.

[17] A. De Rújula, J.M. Kaplan, and E. de Rafael. Elastic scattering of electrons from polarized

protons and inelastic electron scattering experiments. Nucl. Phys. B, 35(2):365–389, 1971.

176

[18] W. Bernreuther and O. Nachtmann. A note on CP-odd and T-odd observables at 𝑒+𝑒− colliders.

Phys. Lett. B, 268(3):424–428, 1991.

[19] W. Bernreuther, U. Löw, J. P. Ma, and O. Nachtmann. How to test CP, T, and CPT invariance
in the three photon decay of polarized 3S1 positronium. Z. Phys.C, 41(1):143–158, Mar 1988.

[20] W. Bernreuther, U. Löw, J. P. Ma, and O. Nachtmann. CP violation and Z boson decays. Z.

Phys.C, 43(1):117–132, Mar 1989.

[21] S. S. Schweber. QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga,

volume 104. Princeton University Press, 1994.

[22] G. F. Knoll. Radiation Detection and Measurement. Wiley, 4th edition, 2010.

[23] C. D. Anderson. The positive electron. Phys. Rev., 43:491–494, Mar 1933.

[24] J. A. Wheeler. Polyelectrons. Ann. N. Y. Acad. Sci., 48(3):219–238, 1946.

[25] A. E. Ruark. Positronium. Phys. Rev., 68:278–278, Dec 1945.

[26] A. Ore and J. L. Powell. Three-photon annihilation of an electron-positron pair. Phys. Rev.,

75:1696–1699, Jun 1949.

[27] D. J Griffiths. Introduction to elementary particles; 2nd rev. version. Physics textbook. Wiley,

New York, NY, 2008.

[28] L. D. Landau. On the angular momentum of a system of two photons. Dokl. Akad. Nauk

SSSR, 60(2):207–209, 1948.

[29] C. N. Yang. Selection rules for the dematerialization of a particle into two photons. Phys.

Rev., 77:242–245, Jan 1950.

[30] M. Deutsch. Evidence for the formation of positronium in gases. Phys. Rev., 82:455–456,

May 1951.

[31] M. Deutsch. Three-quantum decay of positronium. Phys. Rev., 83:866–867, Aug 1951.

[32] A. Zichichi. The Superworld II. Springer New York, NY, Jan 1990.

[33] M. Deutsch and E. Dulit. Short range interaction of electrons and fine structure of positronium.

Phys. Rev., 84:601–602, Nov 1951.

[34] R. M. Drisko. Spin and polarization effects in the annihilation of triplet positronium. Phys.

Rev., 102:1542–1544, Jun 1956.

177

[35] J. Wheatley and D. Halliday. The quenching of ortho-positronium decay by a magnetic field.

Phys. Rev., 88:424–424, Oct 1952.

[36] V. W. Hughes, S. Marder, and C. S. Wu. Static magnetic field quenching of the orthopositro-

nium decay: Angular distribution effect. Phys. Rev., 98:1840–1848, Jun 1955.

[37] B. K. Arbic, S. Hatamian, M. Skalsey, J. Van House, and W. Zheng. Angular-correlation test

of CPT in polarized positronium. Phys. Rev. A, 37:3189–3194, May 1988.

[38] M. Skalsey and J. Van House. First test of CP invariance in the decay of positronium. Phys.

Rev. Lett., 67:1993–1996, Oct 1991.

[39] T. Yamazaki, T. Namba, S. Asai, and T. Kobayashi. Search for CP violation in positronium

decay. Phys. Rev. Lett., 104:083401, Feb 2010.

[40] H. Schieck and Paetz gen. Schieck H. Nuclear Physics with Polarized Particles, Lecture Notes

in Physics v. 842, volume 842. Springer, 01 2012.

[41] S.K. Andrukhovich, N. Antovich, and A.V. Berestov. A spectrometer for the study of angular
correlations in polarized-orthopositronium decay. Instrum. Exp. Tech., page 453–459, 2000.

[42] P. A. Vetter and S. J. Freedman. Search for 𝐶𝑃𝑇-odd decays of positronium. Phys. Rev. Lett.,

91:263401, Dec 2003.

[43] N. M. Antović and S. K. Andrukhovich. Asymmetry in experiments testing CPT in ortho-Ps

decay. Rad. & App., 1:177–182, Feb 2016.

[44] P. Moskal, A. Gajos, M. Mohammed, J. Chhokar, N. Chug, C. Curceanu, E. Czerwiński,
M. Dadgar, K. Dulski, M. Gorgol, J. Goworek, B. C. Hiesmayr, B. Jasińska, K. Kacprzak,
Ł Kapłon, H. Karimi, D. Kisielewska, K. Klimaszewski, G. Korcyl, P. Kowalski, N. Krawczyk,
W. Krzemień, T. Kozik, E. Kubicz, S. Niedźwiecki, S. Parzych, M. Pawlik-Niedźwiecka,
L. Raczyński, J. Raj, S. Sharma, S. Choudhary, R. Y. Shopa, A. Sienkiewicz, M. Silarski,
M. Skurzok, E. Ł Stępień, F. Tayefi, and W. Wiślicki. Testing CPT symmetry in ortho-
positronium decays with positronium annihilation tomography. Nat. Commun., 12(1):5658,
Sep 2021.

[45] J. Govaerts. Positronium spectroscopy in a magnetic field. 1993. arXiv, hep-ph/9308247.

[46] R. Mao, L. Zhang, and R.Y. Zhu. Optical and scintillation properties of inorganic scintillators

in High Energy Physics. IEEE Trans. Nucl. Sci., 55(4):2425–2431, 2008.

[47] A. Zatcepin and S. I. Ziegler. Detectors in positron emission tomography. Z Med Phys,

33(1):4–12, October 2022.

[48] J. Du, Y. Wang, L. Zhang, Z. Zhou, Z. Xu, and X. Wang. Physical properties of LYSO

178

scintillator for NN-PET detectors. 2009 2nd Int. Conf. on Bio. Eng. and Info., pages 1–5,
2009.

[49] LYSO:ce Crystal - LYSO(ce) scintillator - Epic Crystal. https://www.epic-crystal.com/oxide-

scintillators/lyso-ce-scintillator.html.

[50] R. Mao, L. Zhang, and R.Y. Zhu. LSO/LYSO crystals for future HEP experiments. JPCS,

293(1):012004, Apr 2011.

[51] J. Chen, L. Zhang, and R.Y. Zhu. Large size LSO and LYSO crystal scintillators for future
high-energy physics and nuclear physics experiments. Nucl. Instrum. Meth. A, 572(1):218–
224, 2007. Frontier Detectors for Frontier Physics.

[52] H. Alva-Sánchez, A. Zepeda-Barrios, V.D. Díaz-Martínez, and et al. Understanding the
intrinsic radioactivity energy spectrum from 176Lu in LYSO/LSO scintillation crystals. Sci.
rep., 8, 2018.

[53] I. Mouhti, A. Elanique, M.Y. Messous, A. Benahmed, J.E. McFee, Y. Elgoub, and P. Griffith.
Characterization of CsI(Tl) and LYSO(Ce) scintillator detectors by measurements and monte
carlo simulations. Appl. Radiat. Isot., 154:108878, 2019.

[54] F. Acerbi and S. Gundacker. Understanding and simulating SiPMs. Nucl. Instrum. Meth. A,
926:16–35, 2019. Silicon Photomultipliers: Technology, Characterisation and Applications.

[55] S. Gundacker and A. Heering. The silicon photomultiplier: fundamentals and applications of

a modern solid-state photon detector. Phys. Med. Biol., 65(17):17TR01, 2020.

[56] A. Douraghy, D. L. Prout, R. W. Silverman, and A. F. Chatziioannou. Evaluation of scintillator
afterglow for use in a combined optical and PET imaging tomograph. Nucl. Instrum. Meth.
A, 569(2):557–562, 2006. Proc. of the 3rd Int. Conf. on Imag. Tech. in Bio. Sci.

[57] https://www.ketek.net/wp-content/uploads/2018/12/KETEK-PM3325-WB-D0-

Datasheet.pdf.

[58] Fast Acquisition System for nuclEar Research. http://faster.in2p3.fr/.

[59] S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce, M. Asai, D. Axen,
S. Banerjee, G. Barrand, F. Behner, L. Bellagamba, J. Boudreau, L. Broglia, A. Brunengo,
H. Burkhardt, S. Chauvie, J. Chuma, R. Chytracek, G. Cooperman, G. Cosmo, P. Degtyarenko,
A. Dell’Acqua, G. Depaola, D. Dietrich, R. Enami, A. Feliciello, C. Ferguson, H. Fesefeldt,
G. Folger, F. Foppiano, A. Forti, S. Garelli, S. Giani, R. Giannitrapani, D. Gibin, J.J. Gómez
Cadenas, I. González, G. Gracia Abril, G. Greeniaus, W. Greiner, V. Grichine, A. Grossheim,
S. Guatelli, P. Gumplinger, R. Hamatsu, K. Hashimoto, H. Hasui, A. Heikkinen, A. Howard,
V. Ivanchenko, A. Johnson, F.W. Jones, J. Kallenbach, N. Kanaya, M. Kawabata, Y. Kawa-
bata, M. Kawaguti, S. Kelner, P. Kent, A. Kimura, T. Kodama, R. Kokoulin, M. Kossov,

179

H. Kurashige, E. Lamanna, T. Lampén, V. Lara, V. Lefebure, F. Lei, M. Liendl, W. Lock-
man, F. Longo, S. Magni, M. Maire, E. Medernach, K. Minamimoto, P. Mora de Freitas,
Y. Morita, K. Murakami, M. Nagamatu, R. Nartallo, P. Nieminen, T. Nishimura, K. Ohtsubo,
M. Okamura, S. O’Neale, Y. Oohata, K. Paech, J. Perl, A. Pfeiffer, M.G. Pia, F. Ranjard,
A. Rybin, S. Sadilov, E. Di Salvo, G. Santin, T. Sasaki, N. Savvas, Y. Sawada, S. Scherer,
S. Sei, V. Sirotenko, D. Smith, N. Starkov, H. Stoecker, J. Sulkimo, M. Takahata, S. Tanaka,
E. Tcherniaev, E. Safai Tehrani, M. Tropeano, P. Truscott, H. Uno, L. Urban, P. Urban,
M. Verderi, A. Walkden, W. Wander, H. Weber, J.P. Wellisch, T. Wenaus, D.C. Williams,
D. Wright, T. Yamada, H. Yoshida, and D. Zschiesche. Geant4–a simulation toolkit. Nucl.
Instrum. Meth. A, 506(3):250–303, 2003.

[60] J. Allison, K. Amako, J. Apostolakis, P. Arce, M. Asai, T. Aso, E. Bagli, A. Bagulya,
S. Banerjee, G. Barrand, B.R. Beck, A.G. Bogdanov, D. Brandt, J.M.C. Brown, H. Burkhardt,
Ph. Canal, D. Cano-Ott, S. Chauvie, K. Cho, G.A.P. Cirrone, G. Cooperman, M.A. Cortés-
Giraldo, G. Cosmo, G. Cuttone, G. Depaola, L. Desorgher, X. Dong, A. Dotti, V.D. Elvira,
G. Folger, Z. Francis, A. Galoyan, L. Garnier, M. Gayer, K.L. Genser, V.M. Grichine,
S. Guatelli, P. Guèye, P. Gumplinger, A.S. Howard, I. Hrivnácová, S. Hwang, S. Incerti,
A. Ivanchenko, V.N. Ivanchenko, F.W. Jones, S.Y. Jun, P. Kaitaniemi, N. Karakatsanis,
M. Karamitros, M. Kelsey, A. Kimura, T. Koi, H. Kurashige, A. Lechner, S.B. Lee, F. Longo,
M. Maire, D. Mancusi, A. Mantero, E. Mendoza, B. Morgan, K. Murakami, T. Nikitina,
L. Pandola, P. Paprocki, J. Perl, I. Petrović, M.G. Pia, W. Pokorski, J.M. Quesada, M. Raine,
M.A. Reis, A. Ribon, A. Ristić Fira, F. Romano, G. Russo, G. Santin, T. Sasaki, D. Sawkey,
J.I. Shin, I.I. Strakovsky, A. Taborda, S. Tanaka, B. Tomé, T. Toshito, H.N. Tran, P.R.
Truscott, L. Urban, V. Uzhinsky, J.M. Verbeke, M. Verderi, B.L. Wendt, H. Wenzel, D.H.
Wright, D.M. Wright, T. Yamashita, J. Yarba, and H. Yoshida. Recent developments in geant4.
Nucl. Instrum. Meth. A, 835:186–225, 2016.

[61] I. Kawrakow, D.W.O. Rogers, E. Mainegra-Hing, F. Tessier, R.W. Townson, and B.R.B.
Walters. EGSnrc toolkit for Monte Carlo simulation of ionizing radiation transport, 2000.

[62] E. Auffray, F. Cavallari, M. Lebeau, P. Lecoq, M. Schneegans, and P. Sempere-Roldan. Crystal
conditioning for high-energy physics detectors. Nucl. Instrum. Meth. A, 486:22–34, 2002.

[63] A. S. Carnoy, J. Deutsch, T. A. Girard, and R. Prieels. Limits on nonstandard weak currents
from the positron decays of 14O and 10C. Phys. Rev. Lett., 65:3249–3252, Dec 1990.

[64] N. Severijns, J. Deutsch, F. Gimeno-Nogues, B. H. King, I. Pepe, R. Prieels, P. A. Quin,
J. Camps, P. De Moor, P. Schuurmans, W. Vanderpoorten, L. Vanneste, J. Wouters, M. Allet,
O. Navilliat-Cuncic, and B. R. Holstein. Limits on right-handed charged weak currents from
a polarization-asymmetry correlation experiment with 107In. Phys. Rev. Lett., 70:4047–4050,
Jun 1993.

[65] G. Consolati, R. Ferragut, A. Galarneau, F. Di Renzo, and F. Quasso. Mesoporous materials

for antihydrogen production. Chem Soc Rev, 42(9):3821–3832, December 2012.

180

[66] D. Trezzi. Study of Positronium Converters in the Aegis Antimatter Experiment. PhD thesis,

Milan U., 2010.

[67] Y C Jean, P E Mallon, and D M Schrader. Principles and Applications of Positron and

Positronium Chemistry. WORLD SCIENTIFIC, 2003.

[68] D. W. Gidley, H.G. Peng, and R. S. Vallery. Positron annihilation as a method to characterize

porous materials. Annu. Rev. Mater. Res., 36(Volume 36, 2006):49–79, 2006.

[69] K. Shibuya, H. Saito, F. Nishikido, M. Takahashi, and T. Yamaya. Oxygen sensing ability of

positronium atom for tumor hypoxia imaging. Commun. Phys., 3(1):173, Oct 2020.

[70] S. Lipschutz, R.G.T. Zegers, J. Hill, S.N. Liddick, S. Noji, C.J. Prokop, M. Scott, M. Solt,
C. Sullivan, and J. Tompkins. Digital data acquisition for the low energy neutron detector
array (LENDA). Nucl. Instrum. Meth. A, 815:1–6, 2016.

[71] C.J. Prokop, S.N. Liddick, B.L. Abromeit, A.T. Chemey, N.R. Larson, S. Suchyta, and J.R.
Tompkins. Digital data acquisition system implementation at the National Superconducting
Cyclotron Laboratory. Nucl. Instrum. Meth. A, 741:163–168, 2014.

[72] C.J. Prokop, S.N. Liddick, N.R. Larson, S. Suchyta, and J.R. Tompkins. Optimization of the
National Superconducting Cyclotron Laboratory Digital Data Acquisition System for use with
fast scintillator detectors. Nucl. Instrum. Meth. A, 792:81–88, 2015.

[73] L. Dick, L. Feuvrais, L. Madansky, and V.L. Telegdi. A novel efficient method for measuring

the polarisation of positrons. Phys. Lett., 3(7):326–329, 1963.

[74] D. Freytag and K. Ziock. Der einfluß eines magnetfeldes auf die lebensdauer von positronen

in fester materie. Z. Phys., 153(1):124–128, Feb 1958.

[75] https://eljentechnology.com/products/plastic-scintillators/ej-200-ej-204-ej-208-ej-212.

[76] X. Mougeot. Towards high-precision calculation of electron capture decays. Appl. Radiat.

Isot., 154:108884, 2019.

[77] X. Mougeot. Atomic exchange correction in forbidden unique beta transitions. Appl. Radiat.

Isot., 201:111018, 2023.

[78] H. W. Park, D. W. Jeong, J. Jegal, A. Khan, and H. J. Kim. A novel hermetic detection system
of KAPAE for measuring multiphoton decays of positronium. JINST, 18(03):P03011, Mar
2023.

[79] C. Bartram and R. Henning. Caliope:

a search for CPT-violation in o-ps.

JPCS,

1342(1):012106, Jan 2020.

181

[80] S. Bass. QED and fundamental symmetries in positronium decays. Acta. Phys. Pol. B, 50,

2019.

[81] M. Diehl and O. Nachtmann. Optimal observables for the measurement of three gauge boson

couplings in e+e−→𝑊 +𝑊 −. Z. Phys.C, 62(3):397–411, Sep 1994.

[82] M. González-Alonso, O. Naviliat-Cuncic, and N. Severijns. New physics searches in nuclear

and neutron β decay. Prog. Part. Nucl. Phys,, 104:165–223, 2019.

[83] V G Baryshevsky, O N Metelitsa, and V V Tikhomirov. Oscillations of the positronium decay
γ-quantum angular distribution in a magnetic field. J. Phys.B, 22(17):2835, Sep 1989.

[84] V.G. Baryshevsky, O.N. Metelitsa, V.V. Tikhomirov, S.K. Andrukhovich, A.V. Berestov,
B.A. Martsinkevich, and E.A. Rudak. Observation of time oscillation in 3γ-annihilation of
positronium in a magnetic field. Phys. Lett. A, 136(7):428–432, 1989.

[85] E. Ivanov, I. Vata, D. Dudu, I. Rusen, and N. Stefan. Quantum beats in the 3γ annihilation decay
of positronium observed by perturbed angular distribution. Appl. Surf. Sci., 255(1):179–182,
2008.

[86] Y. Sasaki, A. Miyazaki, A. Ishida, T. Namba, S. Asai, T. Kobayashi, H. Saito, K. Tanaka,
and A. Yamamoto. Measurement of positronium hyperfine splitting with quantum oscillation.
Phys. Lett. B, 697(2):121–126, 2011.

[87] A. P. Mills. Line-shape effects in the measurement of the positronium hyperfine interval.

Phys. Rev. A, 27:262–267, Jan 1983.

[88] A. Rich. Corrections to the measured 𝑛 = 1 hyperfine interval of positronium due to annihi-

lation effects. Phys. Rev. A, 23:2747–2750, May 1981.

[89] J. J. Sakurai and J. Napolitano. Modern Quantum Mechanics. Cambridge University Press, 3

edition, 2020.

[90] H. Wahl. First observation and precision measurement of direct CP violation: the experiments

NA31 and NA48. Phys. Rep., 403-404:19–25, 2004.

[91] C. Itzykson and J. B. Zuber. Quantum Field Theory. McGraw-Hill International Book Co,

1980.

[92] J. Pestieau and C. Smith. Soft photon spectrum in orthopositronium and vector quarkonium

decays. Phys. Lett. B, 524(3–4):395–399, January 2002.

[93] P. D. Ruiz-Femenia. Orthopositronium decay spectrum using NRQED, 2004. arXiv, hep-

ph/0410211.

182

APPENDIX A

POSITRONIUM SPIN STATE FOR SPIN-0 AND SPIN-1

A.1 Density matrix

The density matrix is the most general description of a quantum state [40, 89], and is needed

for a full description of an arbitrary positronium state [19]. A "pure" state |𝑎⟩ can be represented

as,

𝜌 = |𝑎⟩ ⟨𝑎|

(A.1)

The expectation value of an operator is then given as ⟨𝐴⟩𝜌𝑎 = ⟨𝑎| 𝐴 |𝑎⟩ = Tr( 𝐴𝜌𝑎). A mixed state
is a statistical mixture of pure states with weights 𝑝𝑖 such that (cid:205)𝑖 𝑝𝑖 = 1. This warrants defining

the density matrix,

∑︁

𝜌 =

𝑝𝑖 |𝑖⟩ ⟨𝑖|

𝑖

(A.2)

where |𝑖⟩ does not need to be orthogonal to | 𝑗⟩. This formal description of a state is minimal and

complete. It allows a closer contact between a state and it’s expectation value for observbales. A

quantum state can be fully and uniquely described by all of its expectation values.

We now consider a finite dimensional system, for example spin-𝑠 with 2𝑠 + 1 basis states in

the Hilbert space. A general observable on this space is a (2𝑠 + 1) × (2𝑠 + 1) matrix. We shorten

this to an 𝑛-dimensional Hilbert space, and 𝑛 × 𝑛 matrix. There is a complete set of operators,

where completeness is defined with resepect to orthogonality in a trace, Tr( 𝐴𝑖 𝐴 𝑗 ) = 𝛿𝑖 𝑗 . There are

𝑛 × 𝑛 such independent matrices including the unit element 1𝑛×𝑛. This means all other observables

must be traceless Tr( 𝐴𝑖1𝑛×𝑛) = Tr( 𝐴𝑖) = 0. Any observable (or matrix on this 𝑛 × 𝑛 space) can be

represented as a linear sum of these observables,

𝐵 = 𝑏01𝑛×𝑛 +

∑︁

𝑏𝑖 𝐴𝑖

𝑖

(A.3)

The density matrix itself is a Hermitian 𝑛 × 𝑛 matrix defined on this Hilbert space, and can be

represented as,

𝜌 = 𝜌01𝑛×𝑛 +

∑︁

𝜌𝑖 𝐴𝑖

𝑖

183

(A.4)

A normalized state has Tr(𝜌) = 1 meaning 𝜌0 = 1/𝑛. Since Tr( 𝐴𝑖 𝐴 𝑗 ) = 𝛿𝑖 𝑗 the weights in the linear

sum satsify 𝜌𝑖 = ⟨𝐴𝑖⟩𝜌. This shows that any state can be uniquely represented by its expectation

values on a complete set of observables

𝜌 =

1
𝑛 1𝑛×𝑛 +

∑︁

𝑖

⟨𝐴𝑖⟩𝜌 𝐴𝑖

(A.5)

A.1.1 Cartesian basis for spin-1 basis states

The calculations are presented in the "Cartesian basis" for spin-1. A spin-1 system has 3

degrees of freedom, meaning that the rotation matrices acting on the Hilbert space can take the

same form as rotations acting on normal vectors. The relationship is the same as that between unit

vectors and spherical harmonics. Starting with the space of states that diagonalize the ˆ𝑆𝑧 operator,

{|𝑆 = 1, 𝑚 = +1⟩ , |𝑆 = 1, 𝑚 = 0⟩ , |𝑆 = 1, 𝑚 = −1⟩}, we define the Cartesian basis kets as,

|𝑆 = 1, 𝑥⟩ =

(|𝑆 = 1, 𝑚 = +1⟩ + |𝑆 = 1, 𝑚 = −1⟩)

1
√

2
|𝑆 = 1, 𝑦⟩ = −𝑖 1
√
2

(|𝑆 = 1, 𝑚 = +1⟩ − |𝑆 = 1, 𝑚 = −1⟩)

|𝑆 = 1, 𝑧⟩ = |𝑆 = 1, 𝑚 = 0⟩

In this basis finite rotations act on the quantum state in the following m,anner,

ˆ𝑈 (𝑅𝑥 (𝜃)) =

ˆ𝑈 (𝑅𝑦 (𝜃)) =

ˆ𝑈 (𝑅𝑧 (𝜃)) =

1

0

0

0 cos(𝜃) −sin(𝜃)

0 sin(𝜃)

cos(𝜃)

cos(𝜃)

0 sin(𝜃)

0

1

0

−sin(𝜃) 0 cos(𝜃)

cos(𝜃) −sin(𝜃) 0

sin(𝜃)

cos(𝜃)

0

0

0

1

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

184

ˆ𝑆𝑥 =

ˆ𝑆𝑦 =

ˆ𝑆𝑧 =

0 0

0

0 0 −𝑖

0 𝑖

0

0

0

0 𝑖

0 0

−𝑖 0 0

0 −𝑖 0

𝑖

0

0

0

0

0

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

these have the same form as finite rotations acting on Cartesian spatial vectors. Further this gives

the ( 𝑗, 𝑘) component of the matrix representation of the angular momentum along the 𝑖th axis as,

( ˆ𝑆𝑖) 𝑗 𝑘 = 𝑖𝜖𝑖 𝑗 𝑘 . This follows from spin-1 being the adjoint representation of SO(3).

A.1.2 A general observable defined on spin-1 basis states

A general observable acting on this space can be represented as a 3×3 Hermitian operator.

There are exactly 9 independent numbers needed to specify a general observable.

We define the tensor polarization as the symmetric combination,

ˆ𝑆𝑖 𝑗 =

1
2

(cid:0) ˆ𝑆𝑖 ˆ𝑆 𝑗 + ˆ𝑆 𝑗 ˆ𝑆𝑖) −

𝑆2𝛿𝑖 𝑗

1
3

(A.12)

the trace of this tensor is zero in that ˆ𝑆𝑥𝑥 + ˆ𝑆𝑦𝑦 + ˆ𝑆𝑧𝑧 = 0 on every state. Each element represented

as a matrix acting on a Hilbert space is also a traceless matrix, but strictly speaking these are two

different traces that should not be confused.

The angular momentum operators satisfy Tr(𝑆𝑖) = 0, and Tr(𝑆𝑖𝑆 𝑗 𝑘 ) = 0. The elements of the

tensor polarization are more complicated, there are 6 elements and one constraint ( ˆ𝑆𝑥𝑥 + ˆ𝑆𝑦𝑦+ ˆ𝑆𝑧𝑧 = 0)

giving 5 independent elements. We take these as ˆ𝑆𝑖 𝑗 where 𝑖 ≠ 𝑗, ˆ𝑆𝑧𝑧, and finally ˆ𝑆Δ = ˆ𝑆𝑥𝑥 − ˆ𝑆𝑦𝑦.

These are five independent components with mutually vanishing traces.

A.1.3 Description of Ps state

The density matrix is a Hermitian operator defined on the Hilbert space and as such all the

above machinery directly carries over. Taking the (normalized) density matrix as a combination of

the independent operators gives,

𝜌 =

1
3

13×3 +

∑︁

𝑖

𝑠𝑖 ˆ𝑆𝑖 +

𝑠𝑖 𝑗 ˆ𝑆𝑖 𝑗

∑︁

𝑖 𝑗

(A.13)

where 𝑠𝑖 = ⟨ ˆ𝑆𝑖⟩𝜌 and 𝑠𝑖 𝑗 = ⟨ ˆ𝑆𝑖 𝑗 ⟩𝜌. If we represent this as a matrix acting on the Hilbert space,

choosing the Cartesian basis it takes the particularly simple form,

𝜌𝑖 𝑗 =

𝛿𝑖 𝑗 +

1
3

1
2𝑖

𝜖𝑖 𝑗 𝑘 𝑠𝑘 − 𝑠𝑖 𝑗

(A.14)

where 𝑖, 𝑗 ∈ (𝑥, 𝑦, 𝑧), and 𝑠𝑥𝑥 + 𝑠𝑦𝑦 + 𝑠𝑧𝑧 = 0.

185

A.2 Ps density matrix for combined spin-0 and spin-1

In the Cartesian basis the f,ull density matrix is a 4×4 matrix and therefore needs 16 − 1

components. The two block diagonal subspaces are the spin-0 and spin-1 spaces. These must

correspond to operators with well defined angular momentum, J and 𝐽2. There are still 6 parameters

as yet unidentified. Now consider that J = s𝑒 + s ¯𝑒, there are exactly 6 combinations of s𝑒 and s ¯𝑒 that

do not correspond to J, those being s𝑒 − s ¯𝑒 and s𝑒 × s ¯𝑒.

The density matrix has the form,

𝜌 = (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝜌00 u†

u

ˆ𝜌

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(A.15)

where ˆ𝜌 corresponds to the 3×3 density matrix of spin-1 discussed above, with components given as

𝜌𝑖 𝑗 . Considering the combinations of the electron and positron spins gives exactly 15 independent

combinations. These are largely determined by the rotational properties in the Cartesian basis. 𝜌00

is unchanged under rotations, 𝜌𝑖 𝑗 rotates like a second rank tensor, and u has three components

that rotate between each other (but do not correspond to definite angular momentum). Direct

computation gives the following,

𝜌00 =

u𝑖 =

𝜌𝑖 𝑗 =

(cid:19)

− s𝑒 · s ¯𝑒

(cid:18) 1
4
1
(cid:0)s𝑒
𝑖 − s ¯𝑒
𝑖
2
(cid:26) (cid:18) 1
4

+ s𝑒 · s ¯𝑒

(cid:1) − 𝑖 (cid:0)s𝑒 × s ¯𝑒(cid:1)

𝑖

(cid:19)

𝛿𝑖 𝑗 − (cid:0)s𝑒

𝑖 s ¯𝑒

𝑗 + s𝑒

𝑗 s ¯𝑒
𝑖

(A.16)

(A.17)

(A.18)

(cid:1) −

𝑖

2

(cid:27)

𝜖𝑖 𝑗 𝑘 (cid:0)s𝑒

𝑘 + s ¯𝑒
𝑘

(cid:1)

This encapsulates the earlier description in Equation A.13 by restoring the dependence on

J = s𝑒 + s ¯𝑒. The two spin-1/2 operators individually satisfy s𝑒

𝑗 = 1
4
𝛿𝑖 𝑗 . The terms corresponding to the spin-0 and spin-1 subspaces

𝑘 . This means that

𝛿𝑖 𝑗 + 𝑖 1
2

𝜖𝑖 𝑗 𝑘 s𝑒

𝑖 s𝑒

(s𝑒

𝑖 s𝑒

𝑗 + s𝑒

𝑗 s𝑒
𝑖 ) = 1

2 (J𝑖J 𝑗 + J 𝑗 J𝑖) − 1

2

can be simplified to,

𝜌00 = 1 −

𝐽2

1
2

𝜌𝑖 𝑗 =

1
6

𝐽2𝛿𝑖 𝑗 −

(cid:18) 1
2

(cid:0)J𝑖J 𝑗 + J 𝑗 J𝑖(cid:1) −

(cid:19)

−

𝐽2𝛿𝑖 𝑗

1
3

𝑖

2

𝜖𝑖 𝑗 𝑘 J𝑘

(A.19)

(A.20)

186

which reproduces the expected behavior for 𝐽2 = 2 and 𝐽2 = 0. This shows that in vacuum a tensor

polarized positronium source cannot be made without both polarized positrons and electrons. This

is why all such experiments to date must use an external field. The off-block-diagonal terms

depending on s𝑒 − s ¯𝑒 also agrees with the calculations of the Hamiltonian including a B-field, which

is proportional to this term and generates an off-diagonal term between the 𝑆 = 0 and 𝑆 = 1 spaces.

A.3 Polarized positronium in a B-field

Now consider positrons that are partially polarized in the x-z plane, and unpolarized electrons.

This is described by the density matrix,

𝑥

1
4
2s ¯𝑒
− 1
0
2s ¯𝑒
− 1

𝑧

𝜌 =

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

2s ¯𝑒
− 1

𝑥

0

1
4
𝑖s ¯𝑒
𝑧

1
2

0

− 1
2

𝑖s ¯𝑒
𝑧

1
4
𝑖s ¯𝑒
𝑥

1
2

𝑧

2 s ¯𝑒
− 1
0

− 1
2

𝑖s ¯𝑒
𝑥

1
4

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(A.21)

For a B-field in the ˆ𝑧 direction, each component has time dependence. For the sake of studying

the decay products, we only need to consider the 𝜌00(𝑡) and 𝜌𝑖 𝑗 (𝑡), as the 𝜌0𝑖 (𝑡) do not survive the

trace with the decay matrix. We take the energy differences as 𝜔 𝑝𝑆 − 𝜔𝑇 = 𝜔1, 𝜔 𝑝𝑆 − 𝜔 𝑝𝑇 = 𝜔2,

and 𝜔 𝑝𝑇 − 𝜔𝑇 = 𝜔3, and similarly for the lifetimes, Γ𝑝𝑆 + Γ𝑇 = 2Γ1, Γ𝑝𝑆 + Γ𝑝𝑇 = 2Γ2, and

Γ𝑝𝑇 + Γ𝑇 = 2Γ3. We are only interested in times > 1 ns, which means the terms that decay as Γ𝑝𝑆,

Γ1, and Γ2 are negligible. Only keeping the longer lifetime components gets,

𝜌00(𝑡) = 𝑠2

(cid:19)

𝑒−𝑡Γ𝑝𝑇

− 𝑠𝑐s𝑧

(cid:18) 1
4

𝜌𝑥𝑥 (𝑡) = 𝜌𝑦𝑦 (𝑡) =

𝑒−𝑡Γ𝑇

1
4
s𝑧𝑒−𝑡Γ𝑇

𝑖

2
1
2
𝑖

𝜌𝑥𝑦 (𝑡) = −

𝜌𝑥𝑧 (𝑡) = −

𝜌𝑦𝑧 (𝑡) = −

𝜌𝑧𝑧 (𝑡) = 𝑐2

𝑠𝑐s𝑥𝑒+𝑖𝜔3𝑒−𝑡Γ3

𝑐2s𝑥𝑒+𝑖𝜔3𝑒−𝑡Γ3
2
(cid:18) 1
4

− 𝑠𝑐s𝑧

(cid:19)

𝑒−𝑡Γ𝑝𝑇

187

(A.22)

(A.23)

(A.24)

(A.25)

(A.26)

(A.27)

These are a scalar and the components of a tensor, and therefore any arbitrary direction for ˆ𝐵 and

ˆs can be described by the corresponding rotation of the above components. Meaning if the new

coordinate system is related by a rotation of angle 𝜃 around the axis (cid:174)𝑎, 𝑅( (cid:174)𝑎, 𝜃), the components of

the density matrix would be given by, 𝜌′

00 = 𝜌00 and ˆ𝜌′ = 𝑅( (cid:174)𝑎, 𝜃) ˆ𝜌𝑅−1( (cid:174)𝑎, 𝜃).

This distribution is mapped into the final state photon distribution by,

(cid:164)𝑁 (k1, k2, k3) =

∑︁

𝑖 𝑗

𝑅𝑖 𝑗 (k1, k2, k3) 𝜌𝑖 𝑗 (𝑡)

(A.28)

simplify the notation for the energy and angle dependent terms, 𝑎(𝜔1, 𝜔2) = 𝑎, 𝐵𝑖 (k1, k2, k3) = 𝐵𝑖,

and 𝐶𝑖 𝑗 (k1, k2, k3) = 𝐶𝑖 𝑗 .

(cid:164)𝑁3𝛾 (k1, k2, k3) = Γ3𝛾

(cid:19)

𝑒−𝑡Γ𝑝𝑇

(cid:18) 1
4
( 𝐴 − 𝐶𝑥𝑥)𝑒−𝑡Γ𝑇 + 𝐵𝑧s𝑧𝑒−𝑡Γ𝑇 + 𝐶𝑥𝑧𝑠𝑐s𝑥cos(𝜔3𝑡)𝑒−𝑡Γ3

− 𝑠𝑐s𝑧

(cid:164)𝑁2𝛾 = Γ2𝛾 𝑠2
(cid:26) 1
4
+ 𝐵𝑦𝑠𝑐s𝑥sin(𝜔3𝑡)𝑒−𝑡Γ3 +

( 𝐴 − 𝐶𝑦𝑦)𝑒−𝑡Γ

1
4

+ 𝐶𝑦𝑧𝑐2s𝑥sin(𝜔3𝑡)𝑒−𝑡Γ3 + 𝐵𝑥𝑐2s𝑥cos(𝜔3𝑡)𝑒−𝑡Γ3
(cid:19)

(cid:27)

+ ( 𝐴 − 𝐶𝑧𝑧)𝑐2

− 𝑠𝑐s𝑧

𝑒−𝑡Γ𝑝𝑇

(cid:18) 1
4

(A.29)

(A.30)

Since 𝐶𝑖 𝑗 is a traceless tensor, the components must satisfy 𝐶𝑥𝑥 + 𝐶𝑦𝑦 + 𝐶𝑧𝑧 = 0. This simplifies

the "diagonal term,"

(cid:164)𝑁3𝛾 (k1, k2, k3) = Γ3𝛾

(cid:26) 1
4
1
4

+

𝐴(2𝑒−𝑡Γ𝑇 + 𝑐2(1 − 4𝑠𝑐s𝑧)𝑒−𝑡Γ𝑝𝑇 )

𝐶𝑧𝑧 (𝑒−𝑡Γ𝑇 + 𝑐2(1 − 4𝑠𝑐s𝑧)𝑒−𝑡Γ𝑝𝑇 )

+ 𝐵𝑥𝑐2s𝑥cos(𝜔3𝑡)𝑒−𝑡Γ3 + 𝐵𝑦𝑠𝑐s𝑥sin(𝜔3𝑡)𝑒−𝑡Γ3 + 𝐵𝑧s𝑧𝑒−𝑡Γ𝑇

+ 𝐶𝑥𝑧𝑠𝑐s𝑥cos(𝜔3𝑡)𝑒−𝑡Γ3 + 𝐶𝑦𝑧𝑐2s𝑥sin(𝜔3𝑡)𝑒−𝑡Γ3

(cid:27)

(A.31)

This reproduces the dependence of the time spectrum on the 𝑧-component of the positron polar-

ization as measured in Refs. [63, 64], and this reproduces the measured pulsing of the distribution

from Ref. [84] (the 𝐶𝑦𝑧 term). However this disagrees by a factor of 1

2 on the 𝐶𝑥𝑧 term compared

to the calculation in Ref. [84].

188

Equation A.31 can be quickly interpreted as follows: consider the term 𝐵𝑧s𝑧𝑒−𝑡Γ𝑇 , this says

that the correlations ⟨𝜅1𝑧⟩, ⟨𝜅2𝑧⟩, and ⟨𝑛𝑧⟩ could be present for a source with polarization s𝑧 in the
z-direction, with time dependence 𝑒−𝑡Γ𝑇 . For this term all three correlations correspond to new

physics.

For CP-violation from 13𝑆1 − 21𝑃1 mixing the tensor term gets a contribution from Re(𝛿1),

and the vector term (manifesting as oscillations in ⟨ ˆk1⟩ and ⟨ ˆk2⟩) get contributions from Im(𝛿1).

To date there have been no searches for the vector correlations indicative of CP-violation. Simple

arguments might imply that such a correlation, ⟨ˆs · ˆk1⟩, would indicate CPT -violation. The

argument goes as, the spin and momentum change sign under T but only the momentum changes

sign under P, therefore this correlation picks up a minus sign and would indicate CPT . This

argument is wrong, and the addition of new physics will generally induce both 𝐵𝑖 and 𝐶𝑖 𝑗 correlations

at similar magnitudes. For the example above, both are proportional to the mixing coefficient, the

imaginary and real parts respectively [19].

189

APPENDIX B

MATRIX ELEMENT AND 3-𝛾 PHASE SPACE

Here we provide a more technical discussion of the phase space and the matrix element. We first

discuss the non-trivial implications of photon indistinguishability that are hidden by imposing an

energy ordering. Following this we give a discussion of the structure of direct versus indirect CP-

violation in ortho-positronium decay, ultimately showing where this prefactor of 1/775 originates

from.

B.1 Photon indistinguishability

The final state contains three indistinguishable particles. This manifests in multiple places,

for example the structure of the matrix element as calculated from Feynman diagrams requires

permutation of the interaction vertices as shown in Figure B.1. However it also manifests as

the definition of the phase space, and it is this manifestation that we are primarily addressing

here. Consider for example 2-𝛾 decay, there are two photons each with 511 keV, restricted to be

back-to-back. Label the two photons 𝑎 and 𝑏, the two photon phase space goes as,

d 𝑓 2𝛾 = 𝜁

d𝑘 3
𝑎
(2𝜋)32𝜔𝑎

d𝑘 3
𝑏
(2𝜋)32𝜔𝑏

𝛿(3) (k𝑎 − k𝑏)

(B.1)

This can be integrated over k𝑏, and we are left with integrating k𝑎 over all possible directions

(with k𝑏 = −k𝑎). The appropriate scaling for 𝜁 is 1/2 due to photon indistinguishability. The

contribution from ( ˆk𝑎 = + ˆ𝑧, ˆk𝑏 = − ˆ𝑧) is double counted by considering ( ˆk𝑎 = − ˆ𝑧, ˆk𝑏 = + ˆ𝑧) as a

distinct final state.

This is not as straightforward when the photons can have different energies. A photon with

energy 𝜔1 can be distinguished from one with 𝜔2. In the definition of the phase space, as shown in

𝑃𝑠

𝑒

𝑒

k𝑎, 𝜖𝜖𝜖 𝑎

k𝑏, 𝜖𝜖𝜖 𝑏

k𝑐, 𝜖𝜖𝜖 𝑐

𝑃𝑠

𝑒

𝑒

k𝑎, 𝜖𝜖𝜖 𝑎

k𝑏, 𝜖𝜖𝜖 𝑏

k𝑐, 𝜖𝜖𝜖 𝑐

Figure B.1 Two of the six Feynman diagram for 3-𝛾 decay of ortho-positronium in QED. The other
4 are generated by cyclic permutation of which photon couples to which vertex.

190

Figure B.2 The six-fold redundant full kinematically allowed phase space. We can restrict ourselves
to considering any individual cell (in effect choosing an ordering for the photons). In principle we
could consider the dynamics on the full space so long as we account for the redundancy.

Figure B.2, when we impose the ordering on the energies 𝜔𝑎 > 𝜔𝑏 > 2𝑚𝑒 − 𝜔𝑎 − 𝜔𝑏, every event

occurs in the red triangle. We only consider the matrix element defined on this triangle. This is

however, a choice of coordinates for phase space and not something fundamental. We could instead

just consider 𝜔𝑎 and 𝜔𝑏 as the energies of two of the photons, with no ordering. Then we would

consider the full region of 𝜔𝑎 < 𝑚𝑒, 𝜔𝑏 < 𝑚𝑒, and 2𝑚𝑒 − 𝜔𝑎 − 𝜔𝑏 < 𝑚𝑒 as the kinematically

allowed phase space. In this case, the matrix element is defined over the entire region, but since

there is no physical meaning to the labels 𝑎 and 𝑏 the matrix element must be symmetric under

their interchange, 𝑅𝑖 𝑗 (k𝑎, k𝑏) = 𝑅𝑖 𝑗 (k𝑏, k𝑎). This is why there are six redundant regions in the

phase space that correspond to the six ordering of the photons. We could restrict our consideration

to any one of the six, or consider the whole space and divide the final result by 1/6.

The decay matrix and the irreducible tensors are unchanged except we replace the momentum

and energy labels 1 and 2 with 𝑎 and 𝑏,

𝑅𝑖 𝑗 (k𝑎, k𝑏) = 𝑎(𝜔𝑎, 𝜔𝑏)𝛿𝑖 𝑗 + 𝜖𝑖 𝑗 𝑘 𝐵𝑘 (k𝑎, k𝑏) + 𝐶𝑖 𝑗 (k𝑎, k𝑏)

(B.2)

To recover the form factors as defined in Ref.

[19] we must impose a partial ordering. They

describe the decay plane as an oriented vector defined as ˆn = ˆk𝑎 × ˆk𝑏/|sin(𝜓𝑎𝑏)|. This has defined

an orientation to the 3-𝛾 phase space. This orientation is flipped under the interchange of any

191

two labels, meaning ˆn → − ˆn under 𝑎 ↔ 𝑏. If the labels are arbitrary then the dynamics must be

unchanged under the switch of labels. Switching the labels for the scalar term,

𝑎(𝜔𝑎, 𝜔𝑏) → 𝑎(𝜔𝑏, 𝜔𝑎)

(B.3)

requires 𝑎(𝜔𝑎, 𝜔𝑏) = 𝑎(𝜔𝑏, 𝜔𝑎). For the vector term,

ˆk𝑎𝑏1(𝜔𝑎, 𝜔𝑏) + ˆk𝑏𝑏2(𝜔𝑎, 𝜔𝑏) + ˆn𝑏3(𝜔𝑎, 𝜔𝑏)

→ ˆk𝑏𝑏1(𝜔𝑏, 𝜔𝑎) + ˆk𝑎𝑏2(𝜔𝑏, 𝜔𝑎) + (− ˆn)𝑏3(𝜔𝑏, 𝜔𝑎)

(B.4)

this requires that 𝑏1(𝜔𝑎, 𝜔𝑏) = 𝑏2(𝜔𝑏, 𝜔𝑎) and 𝑏3(𝜔𝑎, 𝜔𝑏) = −𝑏3(𝜔𝑏, 𝜔𝑎). The CP-odd tensor

terms behave as,

( ˆk𝑎𝑖 ˆn 𝑗 + ˆn𝑖 ˆk𝑎 𝑗 )𝑐4(𝜔𝑎, 𝜔𝑏) + ( ˆk𝑎𝑖 ˆn 𝑗 + ˆn𝑖 ˆk𝑎 𝑗 )𝑐5(𝜔𝑎, 𝜔𝑏)

→ ( ˆk𝑏𝑖 (− ˆn 𝑗 ) + (− ˆn𝑖) ˆk𝑏 𝑗 )𝑐4(𝜔𝑏, 𝜔𝑎)+( ˆk𝑏𝑖 (− ˆn 𝑗 ) + (− ˆn𝑖) ˆk𝑏 𝑗 )𝑐5(𝜔𝑏, 𝜔𝑎)

(B.5)

which requires 𝑐4(𝜔𝑎, 𝜔𝑏) = −𝑐5(𝜔𝑏, 𝜔𝑎). This reproduces the properties quoted in Chapter 2.

We can now choose the ordering of the photons and restrict the phase space to a single cell, but that

does not change the properties of the matrix element, only the domain on which it is evaluated.

The reason for explicitly working this out is to derive that 𝑐4(𝜔1, 𝜔2) = −𝑐5(𝜔2, 𝜔1) quite

generally, not just for the model provided in Ref.

[19]. This means that it is inconsistent to

consider a model where only one of these terms is zero, to do so is to assume that the photons

are distinguishable.

It is for this reason that we cannot consider separate contributions for the

term 𝜅1𝑧𝑛𝑧 from the term 𝜅2𝑧𝑛𝑧, but instead we always get contributions from both. This further

leads to why the analyzing powder cannot be factored into a purely energy dependent part factor

multiplying a purely geometric factor, as the weighting of the two terms is energy dependent and

model dependent. For a signal like 𝑏3(𝜔1, 𝜔2) such a split is possible as it gets contributions from

a single term.

B.2 Direct versus indirect CP-violation in ortho-positronium decay

In exact analogy with neutral Kaon physics, CP-violation can manifest through two different

mechanisms in 3-𝛾 decay of spin-1 positronium. The CP eigenvalues are determined by the

192

spin and orbital angular momentum of the positronium, and by the number of photons and their

polarization/angular distribution of the final state photons. Direct CP-violation means that an

initial state with a definite CP eigenvalue transitions to a final state that does not have a matching

CP eigenvalue, for example the spin-1 ground state positronium decaying to 3-𝛾 with a parity odd

angular distribution. There can also be indirect CP-violation where the positronium states are

mixed by CP-violating physics, and therefore do not have a well defined CP eigenvalue before

decaying.

Most likely if direct violation occurs, then indirect violation occurs as well. However, the

magnitude of the two are not easily related, and is always dependent on the specific model and

system being studied. CP-violation was discovered in 1964 in neutral Kaons [9], but the relative

magnitudes of the direct versus indirect CP-violation was not determined until 1991 [90]. The

difference is where the new physics manifests.

If it is direct, then it manifests in the Feynman

diagram mapping ortho-positronium→ 3𝛾. If it is indirect, then it manifests in the positronium

Hamiltonian governing the time evolution of the positronium energy eigenstates before decay.

As argued in Ref.

[19], direct CP-violation would be dominated by a permanent electric

dipole moment of the electron, which has been excluded to a high level of sensitivity [16]. They

therefore considered exclusively indirect CP-violation in the positronium Hamiltonian that induces

mixing between the 𝑛3𝑆1 and 21𝑃1 states, quantified by a mixing parameter 𝛿𝑛. We are studying

ground state positronium so we are sensitive to 𝛿1. They calculated the form factors 𝑐4(𝜔1, 𝜔2) and

𝑐5(𝜔1, 𝜔2) for this state mixing. For the purposes of this work, that is all that is needed. We can

now include those form factors in our simulation and extract the sensitivity to this specific model.

It is worth commenting on the phenomenology of direct versus indirect CP-violation in positro-

nium. Firstly the decay of positronium is generally calculated at tree level by calculating the

amplitude for free electron-positron annihilation, ⟨𝑛𝛾| 𝒯 |𝑒−, p, 𝑠; 𝑒+, −p, 𝑠′⟩ = M (p). The initial

momentum distribution is taken as the (Fourier transform of the) Hydrogen wavefunction with the

reduced mass for positronium, ˜𝜓𝑛𝐽 𝐿𝑆𝑚 𝐽 (p) = ˜Ψ𝐼 (p), where 𝐼 = {𝑛, 𝐽, 𝐿, 𝑆, 𝑚𝐽 } [91]. The kinetic

energy is highly non-relativistic and it is very safe to take a low momentum approximation. Most

193

derivations of the S-wave annihilation only keep the first term, but for this discussion it is vital to

consider the first two terms [1],

⟨𝑛𝛾| 𝒯 |𝑃𝑠, 𝐼⟩ =

=

=

∫

∫

∫

d3 𝑝ℳ(p) ˜Ψ𝐼 (p)

d3 𝑝 (cid:0)M|p=0 + ∇𝑝M |p=0 · p + ...(cid:1) ˜Ψ𝐼 (p)

d3 𝑝𝑒−𝑖p·0M|p=0 ˜Ψ𝐼 (p) +

∫

d3 𝑝𝑒−𝑖p·0∇𝑝M |p=0 · p ˜Ψ𝐼 (p) + ...

= M|p=0Ψ𝐼 (x = 0) + (∇𝑝M |p=0) · (∇𝑥Ψ𝐼 (x = 0)) + ...

(B.6)

(B.7)

(B.8)

(B.9)

The lowest order goes as the value of the wavefunction at the origin. This is reasonable as this

is where the electron and positron could be considered "in contact". The second order term goes as

the derivative of the wavefunction at the origin. Now inspecting the wavefunctions, the S-wave is

the only wavefunction that does not vanish at the origin, but its derivative is zero, and the P-wave

has vanishing value at the origin, but non-vanishing derivative. Generally the structure of the radial

wavefunctions go as Ψ(𝑟) ∝ 𝑅(𝑟/𝑎𝑃𝑠)𝑒−𝑟/2𝑎 𝑃𝑠 where 𝑅(𝑥) is some polynomial in 𝑥, and 𝑎𝑃𝑠 is the

Bohr radius for positronium. Each derivative carries a factor of 1/𝑎𝑃𝑠, and the Bohr radius carries

a fine structure constant, 𝑎𝑃𝑠 ∝ 1/𝛼. Each higher order term in the low momentum expansion

carries an extra suppression by the fine-structure constant. Therefore 21𝑃1 decay is suppressed by

a factor of 𝛼 relative to 13𝑆1 purely from the atomic structure of positronium. This suppression is

unavoidable and will suppress any sensitivity that angular correlations in 3-𝛾 decay have to state

mixing.

If the asymmetry gets a numerator with contributions from new physics and a denominator from

QED then indirect CP-violation would give a contribution of the form 𝐴 ∝ −Re(𝛿1)𝛼/(2

√

8) ∗

𝑓 (𝜔1, 𝜔2) where 𝛿1 is the mixing parameter, 𝛼/(2

√

8) is the combination of prefactors arising

from the atomic physics, and 𝑓 (𝜔1, 𝜔2) is the quotient of two polynomials of energies defined on

the phase space (that also depends on the normal vectors for the momenta). All-in-all this gives a

contribution that is suppressed by a factor of 𝛼/(2

√

8) ≈ 1/775.

In indirect CP-violation, the 3-𝛾 decay of ortho-positronium has 3 QED vertices and therefore

carries 𝑒6 = 𝛼3 from the diagram squared, then includes the wavefunction effects (which brings

194

in more factors of 𝛼). Direct CP-violation (at the tree level) means replacing one or more QED

vertices by the "new physics" vertices. However, as we are not considering state mixing the atomic

structure effects are the same. This means that an asymmetry where the new physics contribution

is in the numerator and QED contribution is in the denominator would go as, ( 𝜒𝑒6−𝑛)/(𝑒6), where

𝜒 is the "new physics parameter", and 0 ≤ 𝑛 ≤ 3 is the number of QED vertices replaced by the

new vertex. This is just to illustrate that the system in general has enhanced sensitivity for direct

CP-violation and suppressed sensitivity for indirect. Nevertheless, the model provided in Ref.

[19] is for indirect violation through state mixing, on the grounds that there could be cancellations

or enhancements in the positronium physics that allow CP-violation to manifest as state mixing

without inducing a permanent electron electric dipole moment. They carefully worked through

general parameter space, and a few specific models and do find that a high level of fine tuning is

needed for this to occur.

B.3 Translating matrix element into probability distribution function

One difficulty is that the calculated amplitude for annihilation of P-wave positronium generically

diverges as the energies approach 511 keV [19]. This is due to neglecting the binding energy of the

positronium. For the purpose of Monte Carlo simulation, we need to turn this into a probability

distribution function to sample events from. To handle the divergence, we do not consider any 3-𝛾

events where any photon has energy greater than 𝑚𝑒 - 6.8 eV. Even in QED, none of the predictions

can be trusted in this energy range, and it is an experimentally inaccessible range (since we cannot

distinguish it from back-to-back 2-𝛾 decay). It is well known that the 3-𝛾 distribution calculated

in the method outlined above does not correctly reproduce the low energy soft-photon physics

described by Low’s Theorem [2, 92], and that more involved methods utilizing nonrelativistic QED

are required [93]. For our purposes this region of phase space is completely negligible.

B.4 Form factors

Here we recite the explicit energy dependence of the form factors as given in Ref. [19]. We

follow their notation by using,

𝑥 = 𝜔1/𝑚𝑒

195

𝑦 = 𝜔2/𝑚𝑒

𝑧 = 𝜔3/𝑚𝑒 = 2 − 𝑥 − 𝑦

The angle between ˆk1 and ˆk2 is determined by energy and momentum conservation. It is given

as,

ˆk1 · ˆk2 = cos(cid:0)𝜓12

(cid:1) = 1 − 2

𝑥 + 𝑦 − 1
𝑥𝑦

(B.10)

this is denoted as 𝑔(𝑥, 𝑦) in Ref. [19].

The QED form factors are scaled to be unitless, ˆ𝑓 = Γ𝑜−𝑃𝑠 𝑓 /𝒩2

1𝑆, where 𝒩1𝑆 = −𝑒3𝜓1𝑆 (0)/(

and 𝜓1𝑆 (0) = (𝜋𝑟 3
0

)−1/2, and 𝑟0 = 2(𝑚𝑒𝛼)−1/2 is the Bohr radius for positronium:

√

2𝑚2

𝑒),

ˆ𝑎(𝑥, 𝑦) =

(cid:0)𝑥4 + 2𝑥3𝑦 − 4𝑥3 + 3𝑥2𝑦2 − 9𝑥2𝑦 + 7𝑥2 + 2𝑥𝑦3 − 9𝑥𝑦2

1
𝑥2𝑦2𝑧2

64
3
+ 13𝑥𝑦 − 6𝑥 + 𝑦4 − 4𝑦3 + 7𝑦2 − 6𝑦 + 2(cid:1)

ˆ𝑐1(𝑥, 𝑦) =16

1
𝑦2𝑧2

(cid:0)2𝑥2 + 2𝑥𝑦 − 4𝑥 + 𝑦2 − 2𝑦 + 2(cid:1)

ˆ𝑐2(𝑥, 𝑦) =𝑐2(𝑦, 𝑥)

ˆ𝑐3(𝑥, 𝑦) =16

1
𝑥𝑦𝑧2

(cid:0)𝑥2 + 2𝑥𝑦 − 2𝑥 + 𝑦2 − 2𝑦 + 1(cid:1)

For models of CP-Violation, they scale by ˆ𝑓 = − 𝑓 𝑚𝑒Γ𝑜−𝑃𝑠/(𝒩1𝑆𝒩2𝑃𝜁), where 𝒩2𝑃 = −
𝑅′
2𝑃 (0) = (24𝑟 5

)−1/2, and 𝜁 = Im(𝛿1) for 𝑏1 and 𝑏2, and 𝜁 = Re(𝛿1) for 𝑐4 and 𝑐5.

0

√

ˆ𝑏𝑚𝑖𝑥
1

(𝑥, 𝑦) =

8
𝑥2𝑦2𝑧3

(4 − 18𝑦 − 16𝑥 + 32𝑦2 + 44𝑥𝑦 + 26𝑥2 − 28𝑦3 − 55𝑥𝑦2

− 45𝑥2𝑦 − 22𝑥3 + 12𝑦4 + 32𝑥𝑦3 + 36𝑥2𝑦2 + 24𝑥3𝑦 + 10𝑥4 − 2𝑦5

− 7𝑥𝑦4 − 10𝑥2𝑦3 − 10𝑥3𝑦2 − 5𝑥4𝑦 − 2𝑥5

ˆ𝑏𝑚𝑖𝑥
2

ˆ𝑐𝑚𝑖𝑥
4

(𝑥, 𝑦) = ˆ𝑏𝑚𝑖𝑥

(𝑦, 𝑥)
1
(cid:1)
sin(cid:0)𝜓12
𝑥𝑦𝑧3

(𝑥, 𝑦) =8

(cid:0)2 − 6𝑥 − 7𝑦 + 7𝑥2 + 13𝑥𝑦 + 9𝑦2 − 4𝑥3 − 9𝑥2𝑦

− 9𝑥𝑦2 − 5𝑦3 + 𝑥4 + 2𝑥3𝑦 + 3𝑥2𝑦2 + 2𝑥𝑦3 + 𝑦4(cid:1)

ˆ𝑐𝑚𝑖𝑥
5

(𝑥, 𝑦) = − ˆ𝑐𝑚𝑖𝑥

4

(𝑦, 𝑥)

196

√

4𝜋𝑚2

𝑒),

(B.11)

(B.12)

(B.13)

(B.14)

6𝑒3𝑅′

2𝑃 (0)/(

(B.15)

(B.16)

(B.17)

(B.18)

Note that this gets,

𝑐𝑚𝑖𝑥
(𝑥, 𝑦)
4
𝑎(𝑥, 𝑦)

=

𝒩1𝑆𝒩2𝑃Re(𝛿1)
−𝑚𝑒Γ𝑜−𝑃𝑠

Γ𝑜−𝑃𝑠
𝒩2
1𝑆

ˆ𝑐𝑚𝑖𝑥
(𝑥, 𝑦)
4
ˆ𝑎(𝑥, 𝑦)
√
√︃

𝜋𝑟 3
0

2𝑚2
𝑒
−𝑒3

ˆ𝑐𝑚𝑖𝑥
(𝑥, 𝑦)
4
ˆ𝑎(𝑥, 𝑦)

(B.19)

(B.20)

(B.21)

= −Re(𝛿1)

= −Re(𝛿1)

√

−

6𝑒3
√

√︃

4𝜋𝑚3
24𝑟 5
𝑒
0
ˆ𝑐𝑚𝑖𝑥
(𝑥, 𝑦)
4
ˆ𝑎(𝑥, 𝑦)

8

𝛼
√
2

197