Jagiellonian University
Faculty of Physics, Astronomy and Applied Computer Science
Daria Kisielewska
Studies of CPT symmetry violation in
matter-antimatter systems
Doctoral thesis
Supervised by
prof. dr hab. Pawel Moskal
and
dr Eryk Czerwi ´nski
Cracow 2018
DECLARATION
Wydział Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Jagiello ´
nski
O´swiadczenie
Ja ni zej podpisana Daria Kisielewska (z domu Kami ˙´
nska) (nr indeksu: 1055739) doktorantka Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiello´ze przedło ˙
nskiego o´swiadczam, ˙zona przeze mnie rozprawa doktorska pt. Studies of CPT symmetry violation in matter-antimatter systems jest oryginalna i przedstawia wyniki bada ´
n wykonanych przeze mnie osobi´scie, pod kierunkiem prof. dr hab. Pawła Moskala i dr Eryka Czerwi ´
nskiego. Prac ˛e napisałam samodzielnie. O´swiadczam, ˙a
ze moja rozprawa doktorska została opracowana zgodnie z Ustaw ˛
o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z pó´zniejszymi zmianami). Jestem ´swiadoma, ˙
ze niezgodno´s´c niniejszego o´swiadczenia z prawd ˛a ujawniona w dowolnym czasie, niezale ˙acych z ww. ustawy, mo ˙c
znie od skutków prawnych wynikaj ˛ze spowodowa´uniewa ˙
znienie stopnia nabytego na podstawie tej rozprawy.
Kraków, dnia
Daria Kisielewska
I believe in evidence. I believe in observation, measurement, and reasoning, confirmed by independent observers. I’ll believe anything, no matter how wild and ridiculous, if there is evidence for it. The wilder and more ridiculous something is, however, the firmer and more solid the evidence will have to be.
— Isaac Asimov, The Roving Mind
This thesis is dedicated to my boys.
ABSTRACT
In this thesis systems made of quark-antiquark and lepton-antilepton were studied for CPT symmetry violation effects.
The first study was preformed in neutral K meson pairs by comparing the asymmetries constructed from the decay rates into the two CP conjugated semileptonic final states, π−e+ν and π+e−ν¯. If the CPT symmetry holds, then the asymmetry constructed for short-lived kaon (AS) and long-lived kaon (AL) are expected to be identical. At present, the most precise measurement of AL has been performed by the KTeV collaboration: AL =(3.322 ± 0.058stat ± 0.047syst) × 10−3 . The measurement of its counterpart is experimentally more difficult since it requires a very pure KS beam which can be realised only by exploiting the entangled neutral kaons pairs produced at a φ-factory. The first measurement of AS has been performed by the KLOE collaboration in 2006 using 410 pb−1 of integrated luminosity: AS =(1.5 ± 9.6stat ± 2.9syst) × 10−3, with an accuracy dominated by the statistical uncertainty. The new measurement reported in this thesis is based on a four times larger data sample, corresponding to an integrated luminosity of 1.63 fb−1. The final value AS =(−4.9 ± 5.7stat ± 2.6syst) × 10−3 improves statistical accuracy by a factor of almost two with respect to the previous KLOE result. The combination of these two measurements gives AS =(−3.8 ± 5.0stat ± 2.6syst) × 10−3 and allows to determine the new limits on CPT violating parameters Re(x−)=(−2.0 ± 1.4) × 10−3 , and Re(y)=(1.7 ± 1.4) × 10−3. The obtained results are in agreement with CPT invariance and with statistical uncertainty almost a factor of two smaller with respect to the former measurements.
The second part of this work comprised a demonstration of the feasibility of using the J-PET detector to test the CPT violation in correlations of momenta of photons originating from ortho-positronium annihilation and the spin of ortho-positronium. For this purpose simulations of the o-Ps formation and its annihilation into three photons were performed taking into account distributions of photons’ momenta as predicted by quantum electrodynamics and the response of the J-PET tomograph. The results indicate that the J-PET detector has a realistic chance to improve best present limits established for CPT symmetry violations in decays of positronium by more than an order of magnitude. This can be achieved by the application of plastic scintillators which have superior time resolution and allow to create a setup with high granularity of detection strips and low detection pile-ups, which allows to overcome the limitation on the source activity. In addition, the improved angular resolution combined with the excellent timing, and with the possibility of triggerless registration of all events allow for suppression and monitoring of background events.
STRESZCZENIE
W prezentowanej pracy przedstawione s ˛a wyniki bada n stopnia łamania symetrii
´CPT w systemach kwark-antykwark oraz lepton-antylepton.
Prowadzone pomiary polegaj ˛a na porównaniu asymetrii ładunkowej w rozpadach półleptonowych π−e+ν i π+e−ν¯neutralnych mezonów K. Je´sli symetria CPT jest zachowana w tym procesie, to asymetria ładunkowa wyznaczona dla kaonu krótko-˙s´a asymetrii ładunkowej kaonu
zyciowego (AS) ma warto´c równ ˛długo-˙
zyciowego (AL). Dotychczas najdokładniejszy pomiar AL uzyskany został w eksperymencie KTeV: AL =(3.322 ± 0.058stat ± 0.047syst) × 10−3 . Pomiar AS przeprowadzony został przez zespół KLOE: AS =(1.5 ± 9.6stat ± 2.9syst) × 10−3 przy u ˙danych. Uzyskany wynik zdominowany jest przez nie
zyciu 410 pb−1 pewno´s´c statystyczn ˛a. Niniejsza praca prezentuje analiz ˛e 1.63 fb−1 danych zebranych detektorem KLOE w latach 2004-2005. Otrzymana warto´s´c wynosi: AS =(−4.9 ± 5.7stat ± 2.6syst) × 10−3 i w poł ˛aczeniu z poprzednim pomiarem pozwala na uzyskanie najbardziej precyzyjnej warto´sci tej wielko´sci na swie
´cie: AS =(−3.8 ± 5.0stat ± 2.6syst) × 10−3 . ze parame-
Wyznaczone zostały tak ˙try opisuj ˛ace stopie n łamania symetrii CPT : Re(x−)=
´(−2.0 ± 1.4) × 10−3,i Re(y)=(1.7 ± 1.4) × 10−3 . Uzyskane wyniki nie wskazuj ˛a na łamanie symetrii CPT .
Druga cz˛e´c pracy skupia si˛e na wykazaniu mo ˙sci u ˙
s´zliwo´zycia detektora J-PET do testu symetrii CPT . Badan˛aobserwabl˛ajestkorelacjami˛edzywektoramip˛edu kwantów gamma pochodz ˛acych z anihilacji stanu orto-Pozytonium a wektorem spinu. W tym celu stworzone zostały kompleksowe symulacje komputerowe uwzgl ˛edniaj ˛ace rozkłady p ˛edów kwantów anihilacyjnych wynikaj ˛ace z przewidywa ´sci detektora J-PET.
n elektrodynamiki kwantowej oraz własno´Otrzymane wyniki wskazuj ˛a na mo ˙s´acych
zliwo´c polepszenia aktualnych parametrów opisuj ˛łamanie symetrii CPT o rz ˛ad wielko´zliwe to b ˛eki zastosowaniu w
sci. Mo ˙edzie dzi ˛detektorze J-PET scyntylatorów plastikowych, które pozwol ˛a na u ˙´
zycie zródła o wysokiej aktywno´sci. Dodatkowymi atutami s ˛a: dobra rozdzielczo´s´c k ˛atowa, wysoka dokładno´s´c pomiaru czasu oraz zapisywanie wszystkich zebranych zdarze n,´pozwalaj ˛ace na separacj ˛e fotonów pochodz ˛acych z anihilacji stanu orto-Pozytonium
+
i bezpo´sredniej anihilacji ee− oraz monitorowanie tła.
ACKNOWLEDGMENTS
First of all, I would like to thank Dr. Eryk Czerwi ´
nski and Prof. Paweł Moskal without whose supervision this thesis would not exist. I owe an enormous gratitude to Eryk who has been there as my supervisor for the past years and has been relentless in his support and constructive critique. I am greatly indebted also to Prof. Moskal for giving me the opportunity to work within his research group and for his support and guidance during the preparation of this thesis.
I would also like to extend my appreciation to Prof. Antonio Di Domenico and Dr. Erika de Lucia for their constructive comments and careful supervision of my analysis.
Special thanks to Prof. Giogio Capon, Prof. Wojciech Wi´slicki, Dr. Michał Silarski, Dr. Wojciech Krzemie ´
n, Dr. Antonio De Santis for their suggestions and remarks during our group meetings.
I also thank my Colleagues from Laboratories of Frascati: Dr. Elena Perez del Rio, Dr. Marcin Berłowski and Dr. Gianfranco Morello as well as from the J-PET experiment: Dr. Aleksander Gajos, Ewelina Kubicz, Monika Pawlik-Nied´zwiecka, Szymon Nied´zwiecki, Dominika Alfs, Krzysztof Kacprzak, Dr. Magdalena Skurzok, Dr. Grzegorz Korcyl and Dr. Sushil Sharma. I thank them for their companionship and for providing a pleasurable and friendly working atmosphere.
Finally, I wish to thank my family: Justyna, Andrzej and Konrad for their support and countless Skype connections. To Katarzyna and Michał for their assistance and patience throughout these years. To Kaja and Gapa for cycling trips and boardgame parties. To Danuta and Helena for visits and kindness.
Special thanks to Tomek, my best friend and husband, and the newest family additions: our two little boys, who have been kind enough to let me finish this thesis.
This work was supported by the Polish National Science Centre through the Grants Number 2014/14/E/ST2/00262, 2016/23/N/ST2/01293, 2016/21/B/ST2/01222, and by the Ministry of Science and Higher Education through the grant 7150/E338/M/2017.
CONTENTS
1 discrete symmetries in physics 1
1.1 NeutralKmesonssystem ......................... 3
1.2 Positroniumatom . . .... . .... . . .... . .... . . .... . . 6
i measurement of the charge asymmetry for the KS → πeν
decay and test of cpt symmetry with the kloe detector 2 the kloe experiment 11
2.1 DriftChamber .. . . .... . .... . . .... . .... . . .... . . 13
2.2 ElectromagneticCalorimeter ....................... 13
2.3 Triggersystem.. . . .... . .... . . .... . .... . . .... . . 14
2.4 Offline reconstruction filter and event classification algorithm . . . . 15
3 registration of the φ → KL KS → KL (crash)πeν processes at kloe 17
3.1 KL crashselection ............................. 17
3.2 Correction of momenta distribution in Monte Carlo simulation . . . 20
3.3 Selection of KS → πeν events ....................... 21
3.4 Particle identification with the Time of Flight method . . . . . . . . . 22
3.5 Signalextraction. . . .... . .... . . .... . .... . . .... . . 26
3.6 Selection of the KL → πeν controlsample... . . .... . .... . . 28
4 results 31
4.1 Systematic uncertainties on AS ...................... 34
4.2 Charge asymmetry for short-lived kaon and test of CPT symmetry . 35
ii feasibility study of o-Ps → 3γ measurement with the j-pet
detector 5j-pet detector 39
5.1 Designdetails ................................ 40
5.2 J-PETdetectorproperties ......................... 41
5.3 Softwareanalysis-Framework ...................... 42
6 performance assessment: monte carlo simulation 43
6.1 Program architecture and simulated geometry . . . . . . . . . . . . . 43
6.2 Positroniumformation ... . .... . . .... . .... . . .... . . 43
6.3 Simulation of back-to-back annihilation events . . . . . . . . . . . . . 45
6.4 Simulation of 3γ events .......................... 46
7 feasibility study 49
7.1 Polarizationcontrol. .... . .... . . .... . .... . . .... . . 50
7.2 Backgroundreduction ........................... 51
7.3 J-PET efficiency studies with Monte Carlo simulations . . . . . . . . 54
7.4 Discussionandprospects ......................... 54
8 conclusions 57
bibliography
1
DISCRETE SYMMETRIES IN PHYSICS
The idea of symmetry is widely used in modern physics, since a set of physical phenomena exhibits common symmetries. Symmetry arguments can be guiding principles to understand new phenomena and put limitations on considered theories.
According to the Noether’s theorem, symmetries are associated with a transformation of a system and for every global continuous symmetry there exists an associated time independent quantity [1]. Thus the invariances of laws of physics under translations of time, space or rotation imply conservation of energy, momentum, or angular momentum, respectively. Discrete symmetries of nature such as charge conjugation (C), parity inversion (P) or time reversal (T ) do not lead to new conserved quantities. To the best of our knowledge, few symmetries are really exact in nature. The C, P and CP are violated in the weak interactions. An example studied in this thesis is the CPT symmetry, which is strongly embedded into the theoretical framework of modern physics. This is expressed in the CPT theorem1 that can be stated as follows [5]: Any quantum theory, formulated on flat space time is symmetric under the combined action of CPT transformations, provided the theory respects
(i) Locality, (ii) Unitarity (i.e. conservation of probability) and (iii) Lorentz invariance.
The formulation of this theorem may raise doubts as to why one should consider testing the CPT invariance, particularly taking into account the fact that all our phenomenology to date has been based on quantum theories with built in CPT conservation. Speculations on CPT symmetry violation recently arose in discussions of possible:
•
testing the fundamentals of today’s physics. A theorem presented by Greenberg [6] shows that violation of CPT automatically indicates violation of the Lorentz symmetry. For this reason the studies of Lorentz invariance and CPT tests are strongly linked. Searches in a wide range of different systems are summarized in Reference [7].
•
searching for physics beyond the Standard Model. The spontaneous Lorentz and CPT violation could occur in more extended models. For example the string structure of the universe may cause spontaneous CPT symmetry breaking [8, 9].
•
looking for the sources of matter-antimatter imbalance. In principle this can be explained by the Sakharov conditions [10]. However, the amount of CP violation predicted by the Standard Model is not sufficient to generate the observed overabundance of matter. The CPT violation could create the mechanism in the early Universe that lead to the baryon asymmetry observed nowadays [11].
1 There exist several proofs of CPT theorem and they are based on slightly different initial general assumptions [2–4]. However, in most self-consistent theories the CPT theorem is automatically valid.
1
discrete symmetries in physics
The size of possible CPT violation effects can be discussed only in the frame of an explicit fundamental theory. To date no evidence for CPT violation has been found, so any effects must be miniscule.
In this thesis two systems will be studied for possible CPT violation effects. The first one is the neutral kaon system. The performed test is based on comparison between semileptonic asymmetry in KS decays:
Γ(KS → π−e+ν) − Γ(KS → π+e−ν¯)
AS = , (1.1)
Γ(KS → π−e+ν)+ Γ(KS → π+e−ν¯)
and its counterpart AL [12] constructed for KL. To date, the most precise measurement of AL has been preformed by the KTeV collaboration: AL =(3.322 ± 0.058stat ± 0.047syst) × 10−3 [13]. The precision achieved for AL is two orders of magnitude better compared to the precision of AS =(1.5 ± 9.6stat ± 2.9syst) × 10−3 determination [14]. The present accuracy of AS determination is dominated by the statistical uncertainty. The measurement reported in this thesis is based on a four times larger data sample, corresponding to an integrated luminosity of 1.63 fb−1 , which allows to reach a twice smaller statistical error.
The second system is a purely leptonic electron-positron bound state called positronium. Its structure is analogous to the Bohr atom. The ortho-positronium triplet (o-Ps) and para-positronium singlet (p-Ps) states can be distinguished with different properties, determined by their spin. Due to the charge conjugation conservation the o-Ps can decay only into an odd number of photons, while the p-Ps into an even number, and the mean lifetime of o-Ps state in vacuum is longer (140 ns) than for p-Ps state (120 ps) [15]. Studies of discrete symmetries violation in an ortho-positronium state were proposed by Bernreuther et al. in 1988 [16]. The signals for discrete symmetries violation in a spin-polarized ortho
positronium will be visible in a selected set of angular correlations consisting of the i-th photon momentum ski (photons are ordered by decreasing energy) and the ortho-positronium spin ss. The evidence for discrete symmetry violations could be observed in a non-vanishing value of one of the forbidden correlations,
e.g. ss · kˆ1 × kˆ2 for CPT symmetry. Previous measurement of the CPT violation coefficient was conducted by Vetter and Freedman using the Gammasphere detector and resulted in violation amplitude parameters CCPT = 0.0026 ± 0.0031 consistent with zero [17]. In this thesis we will explore the possibility to study the CPT violation coefficient by the J-PET detector built at the Jagiellonian University.
The thesis consists of seven chapters. In the next chapter a brief introduction on phenomenology of semileptonic decays of neutral kaon and ortho-positronium annihilation is given. The next three chapters, comprising the first part, are dedicated to measurement of the charge asymmetry for the KS → πeν decay and a test of CPT symmetry with the KLOE detector. The second chapter contains a description of the KLOE apparatus where the characteristics of the main subdetectors are reported. The third chapter consists of data analysis description: an identification of KS meson by its long-lived counterpart and criteria used to select KS → πeν decays. The obtained value of AS and determined CPT symmetry violation parameters are presented in the fourth chapter. The second part of the
1.1 neutral k mesons system
thesis consists of three chapters dedicated to the study of angular correlations in the positronium decay at the J-PET detector. Chapter five is devoted to a description of the J-PET detector made from plastics scintillators. A description of the preformed Monte Carlo simulation is given in chapter six, leading to conclusions and further plans summarized in the last chapter.
This thesis incorporates material from the following papers2:
•
Measurement of the charge asymmetry for the KS → πeν decay and test of CPT symmetry with the KLOE detector KLOE-2 Collaboration: A. Anastasi, (...), D. Kisielewska-Kami ´
nska et al. JHEP 9 (2018) 21 [18]
•
KS semileptonic decays and test of CPT symmetry with the KLOE detector
D. Kami ´
nska for the KLOE-2 Collaboration
Acta Phys. Polon. B46 (2015) 19 [19]
•
A feasibility study of ortho-positronium decays measurement with the J-PET scanner based on plastic scintillators J-PET Collaboration: D. Kami ´
nska et al.
Eur. Phys. J. C76 (2016) 445 [20]
•
Searches for discrete symmetries violation in ortho-positronium decay using the J-PET detector J-PET Collaboration: D. Kami ´
nska et al. Nukleonika 60 (2015) 729 [21] and conference reports [22–25].
The first part of the thesis uses materials from the short report of the KS → πeν analysis status [19], and a final measurement of AS value with the KLOE dataset [18]. The second part of the thesis is based on prospects of using the J-PET detector for ortho-positronium decays measurement [20] and discrete symmetries studies [21]. Some material from each of these papers has also been incorporated into this introductory chapter. In all cases the thesis author is the leading author of the papers and the main contributor to the conducted studies.
The developed Monte Carlo simulations, reported in the second part of the thesis, were used for exploring the possibility of three gamma photon imaging based on the ortho-positronium annihilation. These studies result in a patent [26] and additional article [27], not described in this dissertation.
1.1 neutral k mesons system
A neutral kaon system plays a special role in CPT violation searches. Due to its sensitivity to a variety of symmetry violation effects, it is one of the best candidates for such kind of studies. One of the possible tests is based on the comparison between semileptonic asymmetry in short-lived kaon meson (KS) decays (AS) and the analogous asymmetry in long-lived kaon meson (KL) decays AL [12]. To date,
2 Please note that the surname of the dissertation author ’Kami ´nska’ was changed to ’Kisielewska’ due to marriage.
discrete symmetries in physics
the AL [28] was determined with a precision more than two orders of magnitude better than AS [14]:
AL =(3.322 ± 0.058stat ± 0.047syst) × 10−3 , (1.2)
AS =(1.5 ± 9.6stat ± 2.9syst) × 10−3 . (1.3)
The present accuracy of AS determination is dominated by the statistical uncertainty. Therefore, the aim of this work was a determination of AS with two times smaller statistical error due to four times bigger data sample and improved systematical uncertainties.
1.1.1 Charge asymmetry in neutral kaon semileptonic decays
Neutral kaons are the lightest particles which contain a strange quark. Observed ¯
KS and KL are linear combinations of flavour (strange) eigenstates (K0 and K0):
1
¯
|KS) = (1 + ES ) |K0) +(1 − ES ) |K0),
2(1 + |ES|2)
(1.4)
1
¯
|KL) = (1 + EL) |K0)− (1 − EL) |K0).
2(1 + |EL|2)
where the introduced small parameters ES and EL can be rewritten to separate CP and CPT violation parameters EK and δK , respectively:
ES = EK + δK,
(1.5)
EL = EK − δK.
¯
In the Standard Model a decay of K0 (or K0) state is associated with the transition of the s¯quark into u¯quark (or s into u) and emission of the charged boson. Change of strangeness (ΔS) implies the corresponding change of electric charge (ΔQ) (see Figure 1.1). This effect is referred to as the ΔS = ΔQ rule. Therefore, decays of K0 → π−e+ν and K¯0 → π+e−ν¯are possible but K0 → π+e−ν¯and K¯0 → π−e+ν are forbidden.
¯
1.1 neutral k mesons system
¯
Decay amplitudes for semileptonic decays of states |K0) and |K0) can be written as follows [12]:
(π− e +ν| Hweak |K0) = A+,
− ¯¯¯
(π+ eν| Hweak |K0) = A−,
(1.6)
(π+ e −ν¯| Hweak |K0) = A−,
¯
(π− e +ν| Hweak |K0) = A¯+,
where the Hweak is the term of Hamiltonian corresponding to the weak interaction and A+, A¯−, A−, A¯+ parametrize semileptonic decay amplitudes. It is useful to introduce the following notation:
A¯+
x = ,
A+
∗
A−
x¯=,
A¯−
(1.7)
A¯∗ − −A+
y = ,
A¯∗
− + A+
∗
∗
¯
x ± x¯1 A+ A−
x± == ±.
¯
22 A+ A−
For further considerations, rules for applying symmetry operators to amplitudes of two spin zero systems A and B (and corresponding anti-systems A and B) could be summarized as:
(T B|T HweakT −1 |T A) =((B|T HweakT −1 |A)) ∗
(CPB| CPHweakCP−1 |CPA) = (B| CPHweakCP−1 |A) (1.8)
(CPT B| CPT HweakCPT −1 |CPT A) =((B| CPT HweakCPT −1 |A)) ∗ .
One obtains the relation between the semileptonic amplitudes and conservation of a particular symmetry by applying the rules presented above to the states defined in Equation 1.6. These considerations are summarized in Table 1.1.
Table 1.1: Relations between discrete symmetries and semiletponic amplitudes
Conserved quantity Required relation
ΔS = ΔQ rule CPT symmetry CP symmetry T symmetry x = ¯x = 0 x = ¯x ∗ , y = 0 x = ¯x, y = Im(y) y = Re(y)
The measured value of lepton charge asymmetry can be expressed in terms of x−, x+ and y:
AS = 2 [Re (EK )+ Re (δK ) − Re(y)+ Re(x−)] ,
(1.9)
AL = 2 [Re (EK ) − Re (δK ) − Re(y) − Re(x−)] .
discrete symmetries in physics
From the sum and difference of AS and AL one can extract parameters accounting for possible violations of the CPT symmetry, either in the decay amplitudes (y) or in the mass matrix (δK):
(AS − AL)/4 = Re(δK )+ Re(x−), (1.10) (AS + AL)/4 = Re(EK ) − Re(y). (1.11)
The charge asymmetry for KL decays was precisely determined by the KTeV experiment at Fermilab [28]. At present the most accurate measurement of KS charge asymmetry was obtained by the KLOE collaboration [14]. The achieved charge asymmetry for KS decays is consistent within error limits with charge asymmetry for KL decays, which suggest conservation of CPT symmetry. However, this result is dominated by a statistical uncertainty which is three times larger than the systematic one, and it can be improved by analysing the 1.63 fb−1 total luminosity data sample acquired in 2004 and 2005 by the KLOE experiment.
1.2 positronium atom
Interaction between electron-positron pair leads to direct annihilation into photons or creation of a bound state called positronium. This system decays through the annihilation of e+ and e− into photons depending on positronium’s quantum mechanical state Φn,l,m(sr) |S, Sz), where the orbital wave function Φ is the hydrogen atom wave function with the electron mass replaced by the reduced mass of the electron-positron pair and where n, l, and m are the usual principle, orbital and magnetic quantum numbers, respectively. The spin is a linear combination of electron and positron spins, of which there are four possibilities:
|S = 1, Sz = 1) = |↑) |↑) ,
1
|S = 1, Sz = 0) = √ (|↑) |↓) + |↓) |↑)) ,
2
(1.12)
|S = 1, Sz = −1) = |↓) |↓) ,
1
|S = 0, Sz = 0) = √ (|↑) |↓) − |↓) |↑)) ,
2
1
where |↑) and |↓) denote Sz =+ and Sz = −1 for a single electron (positron). The
22
triplet state is called ortho-positronium, while anti-aligned singlet state is called para-positronium. Constrained by conservation laws, the ortho-positronium state can annihilate only to an odd number of photons, while the para-positronium state can decay only to an even number of photons. In practice, final states with larger photon numbers are suppressed by few orders of magnitude and the positronium annihilations are dominated by p-Ps→ 2γ and o-Ps→ 3γ.
Positronium is a purely leptonic state which allows to determine its properties using quantum electrodynamics alone. The decay rates were found to be [29, 30]:
2α5
1 mec−1
Γ (p-Ps → 2γ)= = 8.032 × 10−9 s ,
2¯h
(1.13)
2(π2 − 9) mec2α6
−1
Γ (o-Ps → 3γ)= = 7.211 × 10−6 s ,
9π ¯h
1.2 positronium atom
where me is the mass of electron, c is the speed of light, α is the fine structure constant, and ¯h is the reduced Planck constant. The additional power of α in Γ (o-Ps → 3γ) follows directly from the Feynman rules given the additional photon in o-Ps decay. The inverse of the decay rates give the mean lifetimes of the states in vacuum [15]:
τo−Ps = 142 ns,
(1.14)
τp−Ps = 0.125 ns.
1.2.1 Positronium decay correlation test
The positronium annihilation into photons can be used for tests of discrete symmetries invariance in the leptonic sector. Searches for those effects conducted to date, have taken three forms:
•
looking for forbidden decays e.g. o-Ps→ 2γ or o-Ps→ 4γ. The upper limit for those decays is in an order of 10−6 [31],
•
studies of forbidden transitions in positronium energy spectrum [32],
•
tests involving correlations of photons momenta originating form o-Ps anni
hilation. In this thesis we will focus on the last listed test. The general theoretical framework for those studies has been shown in 1988 by Bernreuther et al [16], and has introduced the simplest correlation between the spin polarization vector s
of o-Ps and a vector normal to the decay plane:
s
s
· k1 × sk2 (1.15)
where sk1 and sk2 are the momenta of the two most energetic annihilation photons (see Figure 1.2). The CPT invariance implies vanishing of the aforementioned angular correlation.
A non-zero observed effect would be an evidence for a new interaction, and one needs to carefully investigate if this interaction violates CPT symmetry. The effects that may mimic the signal from CPT violation are following:
•
the effects in final state interactions, originating from virtual creation of charged particle pairs. This effect is expected to be at the level of 10−9 [33],
•
the weak interaction amplitudes, such as parity mixing and weak decays of positronium state. Those effects turn out to be at the level of 10−12 [16].
discrete symmetries in physics
Figure 1.2: The vectors in an o-Ps→ 3γ event (in o-Ps frame of reference) used to construct operators used in searches for CPT symmetry violation. Momentum vectors are ordered by magnitude (green arrows; dashed, dotted, dashed-dotted). Vector product sk1 × sk2 is a violet arrow, while the red arrow denotes the spin Ssof the decaying ortho-positronium. The top-left panel shows the reference state, while the others present the system after applying discrete symmetry operators: T , P or CPT .
Part I
MEASUREMENT OF THE CHARGE ASYMMETRY FOR
THE K S → πeν DECAY AND TEST OF CPT SYMMETRY
WITH THE KLOE DETECTOR
2
THE KLOE EXPERIMENT
The KLOE detector operated at the DAΦNE electron-positron collider (see Figure 2.1) localized at National Laboratory in Frascati near Rome. The energy of two colliding beams is adjusted to produce the φ meson which decays predominantly into a pair of neutral or charged kaons. Main decay modes of the φ meson and neutral kaons are presented in Table 2.1. Since the beams cross at an angle of 2 × 12.5 mrad the φ-meson is produced with a small momentum of pφ ≈ 13 MeV/c [34].
Figure 2.1: Schematic view of the DAΦNE accelerator complex. The positrons are created in an intermediate stage of the linear accelerator (LINIAC). Both electrons and positrons are accelerated in LINIAC, transfered to the accumulator ring and then injected into DAΦNE storage rings, at an energy of 510 MeV. The positrons and electron beams are circulating in separate rings with two intersection points. Figure adapted from Ref. [35].
The DAΦNE accelerator with average luminosity of 5 × 1032 cm−2s−1 is capable of producing about 1300 kaon pairs per second. During two data taking campaigns,
11
the kloe experiment
Table 2.1: Main branching ratios of the φ meson and neutral kaons [36].
φ KS KL
Channel BR (%) Channel BR (%) Channel BR (%)
K+K− KSKL ρπ ηγ 48.9 ± 0.5 34.2 ± 0.4 15.32 ± 0.32 1.309 ± 0.024 π+π− π0π0 π=e±ν π=µ±ν 69.30 30.69 7.04 × 10−2 4.69 × 10−2 π=e±ν π=µ±ν 3π0 π+π−π0 40.55 27.04 19.52 12.54
in 2001-2002 and 2004-2005, KLOE collected a data sample which corresponds to the integrated luminosity of 2.5 fb−1 .
The KLOE detector consists of two main components: the cylindrical drift chamber and the electromagnetic calorimeter, both surrounding the beam pipe. All elements are immersed in a 0.52 T magnetic field created by a superconducting coil placed along the beam axis. A schematic cross-section side view of the KLOE detector is shown in Figure 2.2.
+
of the detector is an ee− interaction point surrounded by the spherical beam pipe. Along the beam pipe the focusing quadrupoles are mounted instrumented with compact tile calorimeters (QCAL). The main components of the KLOE detector are a large drift chamber (DC), filled with a helium-based gas mixture, and an electromagnetic calorimeter (EMC) surrounding the DC. All detectors are immersed in a solenoidal magnetic field (0.52 T), to allow charged particle’s momenta measurement. Figure adapted from Ref. [37].
2.1 drift chamber
2.1 drift chamber
The KLOE drift chamber (DC) [38] is a 3.3 m long cylinder with internal and external radii of 0.25 m and 2 m, respectively. The mechanical support of the DC consists mainly of two endplates and 12 external panels stretched between them. The gas sealing is provided by a 750 µm thick aluminated carbon fiber cylinder. A gas mixture composed of helium (90%) and isobutane (10%) acts as a quencher. These features maximize transparency to photons and reduce charged particle multiple scattering and KL → KS regeneration. About 40% of produced KL mesons decay inside the DC volume, while most of the surviving KL’s interact and are detected in the electromagnetic calorimeter.
Between the DC endplates around 12500 sense wires are stretched creating cells, which are organized in coaxial layers with two different dimensions of a transverse plane: 2 × 2 cm2 (12 inner layers) and 3 × 3 cm2 (46 outer layers). In order to define the position along the z axis of the detector the neighbouring layers are twisted in the opposites directions by a small stereo angle [38].
The spacial resolution obtained in the r, ϕ plane is better than 200 µm, a resolution along the z axis of ∼ 2 mm and the resolution of the decay vertex determination of ∼ 1 mm. Moreover, the curvature of the reconstructed tracks allows to determine the particle momentum with a relative accuracy of 0.4 % [38].
The drift chamber allows to reconstruct the charged particle tracks and momenta while the calorimeter enables recording of time and energy of both charged and neutral particles.
2.2 electromagnetic calorimeter
The geometrical acceptance of the KLOE electromagnetic calorimeter (EMC) is almost 4π. The EMC consists of two main parts: 24 trapezoidal modules arranged into a barrel and 32 C-shaped modules that create the two endcaps closing the barrel. Each module consists of lead (48%), scintillating fibers (42%) and glue (10%). A schematic view is shown in Figure 2.3. For each module the readout is provided from both ends by photomultipliers connected to the module by the light guides.
The KLOE calorimeter has been designed to have an excellent accuracy of energy determination and time resolution: σ(E)/E = 5.7%/ E[GeV], σt =
the kloe experiment
54 ps/ E[GeV] ⊕ 140 ps [40] in order to register the hits of neutral particles and provide a possibility of the Time of Flight (TOF) measurement (details in Section 3.4).
2.3 trigger system
The start of data acquisition is preceded by a two levels trigger system [41]. The first level trigger (T1) is a fast trigger with a minimal delay which starts the acquisition at the front-end electronics. It requires two local energy deposits above threshold in the EMC (50 MeV on the barrel, 150 MeV on the end-caps) and hit multiplicity information from the drift chamber. The trigger time is determined by the first particle reaching the calorimeter and is synchronized with the DAΦNE radio frequency (RF) signal.
The second level trigger (T2) uses information from both the drift chamber and the electromagnetic calorimeter. Both the triggers’ decision can be vetoed if the events were recognized as Bhabha scattering or cosmic ray event (see Figure 2.4). For control purposes those events are accepted and saved as a dedicated downscaled sample. The background events from the Bhabha scattering, cosmic rays or machine background that survive the trigger requirements are rejected at the beginning of the offline reconstruction by the background filter. Without the background rejection at the trigger level the background rate would be almost 20 times greater than the φ meson production rate.
+
majority of ee− → φ decays, and provide efficient rejection of the two main sources of background: small angle Bhabha scattering and particles lost from DAΦNE beams. Both T1 and T2 triggers are based on the topology of energy deposits in the EMC and on the hit multiplicity in the DC. Figure adapted from Ref. [39].
2.4 offline reconstruction filter and event classification algorithm
2.3.1 Determination of the global time offset
The time interval between bunch crossings (Tbunch = 2.715 ns) is smaller than the time spread of the registered signals originating from KLKS events that can reach 30-40 ns. The offline reconstruction procedure therefore has to determine the true bunch crossing time T0 for each event and correct all times related to that event accordingly. In standard reconstruction algorithms the T0 time is determined by using the information coming from the electromagnetic calorimeter. In the studied channel, since the KS decay time is smaller than KL interaction time in the calorimeter, the T0 time has to be corrected in an offline analysis. Details will be given in Section 3.4.
2.4 offline reconstruction filter and event classification algorithm
Signals gathered by the electronics are translated into quantities connected with the detector using the detector maps. Then information associated with the calorimeter is reconstructed first in order to produce the preliminary estimate of T0 (see Section 2.3.1). The events identified as Bhabha scattering, cosmic rays and machine background are rejected based on the reconstructed information. Next, the CPUintensive procedures are invoked: reconstruction of the charged particles’ tracks and vertices in the drift chamber.
All collected events are classified into dedicated streams:
•
KPM φ → K+K− ,
•
KSL φ → KLKS,
•
RPI φ → ρπ,
•
RAD φ → radiatives (ηγ, η/γ, π0γ...),
•
CLB Bhabha and cosmic events used as calibration samples,
•
UFO UnidentiFied Objects.
The main selection criteria are based on a robust set of cuts in order to allow users to perform more sophisticated analyses [42–44]. To maximize streaming efficiency a single event can be tagged by more than one algorithm. At the same time, the streams overlap is kept below the one per cent level. For the efficiency estimation procedure every 10th reconstructed event is stored, regardless of decisions of the classification algorithm.
The data sample used for this analysis has been processed and filtered with the KLOE standard reconstruction software and the event classification procedure. The simulated data samples are based on the Monte Carlo (MC) GEANFI program [45]. The program included a full description of the KLOE detector simulating the responses of all detectors and it accounts for their efficiency and resolutions. Monte Carlo simulations took into account -run by run -data taking conditions such as the φ momentum, beam spot size and position, background levels, trigger thresholds and dead channels.
3
REGISTRATION OF THE φ → K L K S → K L ( crash ) πeν PROCESSES AT KLOE
The charge asymmetry for the short-lived neutral kaon is given by:
N +/E+ − N−/E−
AS = , (3.1)
N+/E++ N−/E−
− ¯
where N + and N− are the numbers of observed KS → π−e+ν and KS → π+eν decays, respectively, while E+ and E− are the corresponding analysis efficiencies. Negative and positive charged pions interact differently in the detector material,
− ¯
therefore the efficiency is separately estimated for π−e+ν and π+eν final charge states. It should be noted that the AS value depends only on the ratio of π+/π− (e+/e−) efficiencies and not on the absolute values. The ratio E(π+)/E(π−) -for the part which depends on the different nuclear interactions of positive and negative pions -has been determined directly from data using a control sample because the MC simulation was not fully reliable on this point. On the other hand, e+ and
+
e− interactions are charge independent (aside of the negligible contribution of eannihilation in flight) so the MC can safely be used for their estimate.
The measurement of AS requires a very pure KS beam which can only be realised exploiting the entangled neutral kaons pairs produced at a φ-factory. This property is used in the so-called tagging technique -identification of KL(KS) meson on one side allows to select a KS(KL) meson on the other side of the φ meson decay point (see Figure 3.1). The performed analysis is based on an identification of KS through the detection of KL interaction in the calorimeter. In order to select semileptonic decays of KS mesons, an additional kinematic selection is applied. It starts from a requirement of a vertex formation by tracks of two oppositely charged particles near the φ meson decay point. In the next step, the signals from drift chamber and Time of Flight technique are applied to improve signal over background ratio and to attribute the recorded tracks and EMC interactions to particles from the semileptonic decay.
3.1 KL crash selection
About 60% of produced KL mesons reach the calorimeter and deposit energy there, which is referred to as Kcrash. Due to that, the selection of KL candidates takes into account only particle interactions in EMC (commonly referred to as calorimeter clusters) with energy
Eclu(Kcrash) � 100 MeV. (3.2)
It is also required that the cluster is not connected with any track reconstructed in drift chamber. The obtained energy distribution of Kcrash is shown in Figure 3.2. The visible discrepancy between data (dashed histogram) and MC simulations (thin,
registration of the φ → KL KS → KL (crash)πeν processes at kloe
Figure 3.1: Transverse view of an exemplary signal event. The figure was obtained using the Event Display for the KLOE experiment [46]. The neutral kaons identification is simplified by the difference in its mean life times -KS decays close to the interaction point (IP). Therefore in the analysis the position of a KL vertex can be limited to distances larger than a few τKS from the IP. In the conducted analysis the KS is identified using the KL interaction in the calorimeter. In the next step, the semileptonic decays of KS mesons are selected by requiring vertex near the IP. Further sample cleaning is provided by applying kinematical cuts and the Time of Flight technique.
solid histogram) is due to a simplified description of the calorimeter in the simulation. Therefore, the energy distribution was modified by the phenomenological correction [47]:
Enew clu (Kcrash) · (p1 + Int(Eold p2), (3.3)
clu (Kcrash)= Eold clu − 100) ·
where Eold (Kcrash) stands for cluster energy before and after
clu (Kcrash) and Enew
clu
correction, Int returns an integer part of the expression, while values of p1 and p2 were obtained from previous KLOE analysis and are equal to 1.025 and 0.0003, respectively [48].
Since the angular distribution of kaon emission is described by characteristic
dN
p-wave angular distribution: ∝ sin2 θ, where θ is a polar angle, most of the
dΩ
kaons are emitted in the direction perpendicular to the beam axis. Therefore, a part of background is rejected by choosing KL clusters only in the barrel part of the calorimeter.
In the next step, for each cluster candidate the velocity of the contributing particle is calculated as:
Rclu
β = , (3.4)
c · tclu
where Rclu and tclu denote the distance and time of flight of the KL between the interaction point (obtained run by run from Bhabha events) and cluster position respectively, and c is the speed of light. In the φ meson rest frame KL mesons have
3.1 KL crash selection
low velocity β ∼ 0.22. To compare the obtained β value with the expected one, the transformation to the φ rest frame is applied [47, 49]:
β2 + β2 + 2ββφ cos α
φ
β ∗ = , (3.5)
1 − βφβ cos α
where α is the angle between the φ meson momentum vector and the direction connecting the interaction point with the cluster position.
The obtained β∗ distribution (Figure 3.3, left) peaks at two different values due to the procedure used to determine T0, which assumes the first cluster to be generated by a prompt photon which does not occur in case of the semileptonic decays. If the T0-cluster is generated by the fastest pion, which has a time of flight of ∼ 10 ns (instead of 6 − 7 ns for a prompt photon), then the bunch crossing number differs from the correct value by (-1). Example of simulated β∗ distribution for different bunch numbers is shown in Figure 3.3, right. Hence, the applied cut
0.18 <β ∗ < 0.27 (3.6)
is chosen to contain both peaks for further analysis. If more than one cluster fulfilled the above criteria then the one with the smallest time of arrival (tclu) is chosen as corresponding to the interaction of the KL meson.
3.1.1 Kaons momentum determination
After KL identification, its momentum is determined by considering the two body decay in the φ rest frame:
(ECM /γφ)βCM (βCM )2M2
cos θ + K cos2 θ +(ECM /γφ)2 − M2
K KK
pK = , (3.7)
1 − (βCM )2 cos2 θ
registration of the φ → KL KS → KL (crash)πeν processes at kloe
Figure 3.3: Left: distribution of velocity of the tagging KL meson for experimental (blue points) and simulated histograms. Solid black line represents all simulated events, while red dashed corresponds to the semileponic decay. The double peak structure results from the bunch number assignment during time reconstruction. Right: Monte Carlo simulation showing dependency between position of peak structure and assigned bunch crossing number.
where the ECM is the kaon energy in the center of mass and θ is the angle of the
K
kaon with respect to the φ meson boost direction.
The momentum components of both kaons are given by:
spK1 = pK sxK − sxφ |sxK − sxφ|, spK2 = spφ − spK1 , (3.8) (3.9)
where sxK and sxφ are position of KL interaction vertex and φ meson production point, respectively.
In order to do a transformation from the φ meson center of mass frame to the laboratory frame the position and momentum vector of the φ meson and the KL cluster position are needed. The φ meson parameters are obtained run by run from Bhabha scattering events.
3.2 correction of momenta distribution in monte carlo simulation
To improve Monte Carlo simulations with data agreement for the final distribution the reconstructed MC track momentum components pi have been smeared using three Gaussian functions:
3
3
new
p = pi × (1 + αp) × (1 + Δ · fj · G(0, σj )), (i = x, y, z) (3.10)
i j=1
where G(0, σj ) is the Gaussian distribution with zero mean and standard deviation σj, fj is its amplitude, while Δ is the fractional uncertainty on the track curvature.
3.3 selection of KS → πeν events
The momentum shift αp and the Gaussian parameters are tuned using the KL → πeν control sample (see Section 3.6). The fit yields f1 = 96%, σ1 = 0.34, f2 = 3.2%, σ2 = 9.74, f3 = 0.8%, σ3 = 71.2 and αp = 1.37 · 10−4 . The smearing scheme was adapted from [50].
3.3 selection of KS → πeν events
Selection of semileptonic events starts from a requirement of two oppositely charged particles with tracks forming a vertex close to the Interaction Point (IP) (see Figure 3.4):
ρvtx < 15 cm,
(3.11)
|zvtx| < 10 cm,
where ρvtx is a radial distance in the x − y plane between the selected vertex and
+
the ee− interaction point. The main background source for KS → πeν decays is the KS → π+π− process with misidentification of π as e, since BR(KS → π+π−) is
registration of the φ → KL KS → KL (crash)πeν processes at kloe
Figure 3.5: Left: Simulated distribution of the angle between charged secondaries in KS rest frame. Right: Simulated distribution of invariant mass calculated under the assumption that both registered particles were pions. In both figures black solid lines represent all events, red dashed lines show semileptonic decays and blue points are the collected data. Vertical dashed lines represent the cuts described in the text.
around 104 times larger than for the signal events. The first cut applied to reject KS → π+π− events at preselection stage is:
70◦ <α< 175◦ , (3.12)
where α is the angle between momenta of charged secondaries in KS rest frame. The obtained α distribution is shown in the left panel of Figure 3.5. In case of two body decay (such as KS → π+π−) α takes the value of ∼ 180◦ and in case of three body decay (KS → πeν) it is spanned over a large range (dashed histogram). The applied cut allows to reduce the remaining KS → ππ background by 80% while only 3% of signal is rejected. The next cut is on an invariant mass Minv(π, π), calculated using momenta of the particles which track form a vertex, assuming that both particles were pions:
300 1.5 ns (3.16)
is applied for further KS → π+π− events rejection (see Figure 3.6). The remaining
pairs of tracks are tested under pion-electron δt(π, e) and electron-pion δt(e, π) hypothesis (see Figure 3.7):
|δt(e, π)| < 1.3 ns ∧ δt(π, e) < −3.4 ns
or (3.17)
δt(e, π) > 3.4 ns ∧|δt(π, e)| < 1.3 ns
Once particle identification has been performed, the time differences δt(e) and δt(π) are reevaluated using the identified particle masses and subtracting the T0 of the event:
δt(mπ)+ δt(me)
T0 = Nint × Tbunch, (3.18)2 · Tbunch
registration of the φ → KL KS → KL (crash)πeν processes at kloe
where Nint stands for the nearest integer and Tbunch = 2.715 ns is the minimum bunch crossing period. Events are then selected within the circle in the δt(e) − δt(π)
plane:
δt(e) − 0.07 ns 0.6 ns 2 + δt(π) − 0.13 ns 0.6 ns 2 < 1 (3.19)
as shown in Figure 3.8. Position of the center was determined from the data sample.
This cut keeps the number of background events selected for normalization under
control. Details of the normalization procedure are presented in the next Section.
3.4 particle identification with the time of flight method
registration of the φ → KL KS → KL (crash)πeν processes at kloe
3.5 signal extraction
After the presented event selection criteria, the remaining residual background components are:
•
KS → π+π−, where one of the pion tracks is badly reconstructed and classified as an electron by the Time of Flight procedure,
•
KS → π+π− → πµ, where π decay occurred before entering the drift chamber,
•
KS → π+π−γ,
• other, mainly φ → K+K− decays.
The best separation between signal and background components is provided by the variable:
M2(e)= E2(e) − p 2(e)
=(EKS − E(π) − Emiss)2 − p 2(e)
=(EKS − E(π) − pmiss(π, e))2 − p 2(e), (3.20)
3
pmiss =(pKS,i − pe,i − pπ,i)2, (3.21) i=x,y,z
presented in Figure 3.9. Its sensitivity lies in the fact that during the Time of Flight procedure a wrong mass hypothesis is usually assigned to the particle classified as an electron. So, muons (from πµ background category) or badly reconstructed pions (from ππ background category) can be separated based on their mass difference. In case of a proper assignment, Emiss and pmiss correspond to the energy and momentum of a neutrino.
The signal yield is estimated by fitting the M2(e) data distribution with the sum of the corresponding simulated distributions for signal and background channels with free normalization and accounting for a finite size of the Monte Carlo sample [51, 52]. In order to obtain a χ2 distributed variable the log-likelihood is normalized as explained in Ref. [53].
The result of the fit for the signal events is 34579 ± 251 for KS → π−e+ν and 36874 ± 255 for KS → π+e−ν¯, with total χ2/ndof = 118/109, summing on the two final charge states (see Figure 3.9).
3.5 signal extraction
registration of the φ → KL KS → KL (crash)πeν processes at kloe
3.6 selection of the KL → πeν control sample
The efficiencies of the selection of the semileptonic channels will differ due to different registration probabilities of positive and negative pions in the detector. To account for this effect a data sample of KL → πeν decay, which is a dominant decay mode of KL meson (see Table 2.1), is selected and used as a control sample. Moreover, the AL value was estimated in order to rule out any possible bias in the analysis scheme.
The events of the control sample are tagged by the neutral KS decay KS → π0π0. Selection of KS → π0π0 events is provided by the standard KLOE tagging algorithm [42, 43] which looks for clusters in the barrel of the electromagnetic calorimeter not associated to any tracks. The total deposited energy should be greater than 300 MeV and each of the candidates has to deposit energy between 20 MeV and 300 MeV. The last step is the evaluation of the KS invariant mass mK which should satisfy the (390 and 3.4. The normalization requires two scaling factors only (signal and background, see Figure 3.10), because the semileptonic channel is a main decay mode of KL meson.
The purity (S/(S + B)) of the sample reaches 96.62%. The final number of selected events is 522834 ± 1106 and 544183 ± 1122 for KL → π−e+ν and KL → π+e−ν¯decays, respectively. This allows to determine the value of lepton charge asymmetry for the long-lived kaon (analogously as in Equation 3.1) to be:
AL =(1.7 ± 2.7stat) · 10−3 , (3.22)
which is consistent with the best measurement of this value provided by the KTeV [28] within one standard deviation.
3.6.2 Sample selection for efficiency determination
The events of the control sample are used to estimate the efficiencies for positive and negative pions. To this aim a single track selection scheme is developed and applied, after vertex reconstruction and cuts on the opening angle in the KL rest frame and Minv(π, π), as described in Section 3.3.
At this stage we require that at least one track reaches the calorimeter with TCA. For this track the δt(e) and δt(π) variables are constructed (see Equation 3.14). A sample of electrons (positrons) is then selected by requiring (see Figure 3.11):
[(δt(e) − 0.07 ns)/1.2 ns]2 + [(δt(π)+ 4 ns)/3.2 ns]2 < 1. (3.23)
As the aforementioned procedure does not require extrapolation of the pion tracks towards the calorimeter, the selected sample is used for TCA and TOF
3.6 selection of the KL → πeν control sample
− ¯
efficiency evaluation separately for negative and positive pions. Details are given in Section 4.
registration of the φ → KL KS → KL (crash)πeν processes at kloe
Figure 3.11: The δt(π) vs δt(e) distribution for all particles of the control sample with a calorimeter cluster associated to their corresponding DC track. Events within the ellipse contain KL → πeν events and are chosen for efficiency estimation. Distributions were made for data (top left), total Monte Carlo events (top right), MC KL → πeν events (bottom, left) and MC background events (bottom right). It should be noted that plots were made for a single track with two different mass hypotheses.
4
RESULTS
The analysis efficiency is estimated as follows:
E = ET EC · ET AG · EANA, (4.1)
where ET EC stands for trigger and event classification efficiency, while ET AG and EANA denote tagging and analysis efficiencies, respectively. The analysis efficiency EANA can be expressed as a product of four contributions:
•
kinematical cuts (EKC ): cuts on reconstructed vertex fiducial volume, opening angle α, and Minv(π, π) (see Section 3.3);
•
Track to Cluster Association algorithm (ETCA);
•
Time of Flight cuts (ET OF );
•
fit range (EFR) of the M2(e) variable.
The efficiency ET EC is evaluated using downscaled minimum-bias data samples without event classification and background rejection filters applied. The estimation of ET AG, EKC and EFR are based on MC simulation. ET OF are determined using the KL → πeν control sample with the method described in Section 4. ETCA consists of
=
the product of ET CA(π±), evaluated from control sample, and ETCA(e ) determined from MC:
EKS MC
= EKS TCA(e)= EKL DATA TCA (π) × EKS MC
ETCA TCA(π) × EKS TCA (π) × EKL MC (π) T CA (e). TCA
(4.2)
Then, the Time of Flight efficiency is determined using the KL → πeν data control sample in a similar manner:
EKS MC
EKS = EKL DATA TOF
× . (4.3)
TOF TOF EKL MC
T OF
The total efficiency is (7.39 ± 0.03)% and (7.81 ± 0.03)% for KS → π−e+ν and KS → π+e−ν¯, respectively. The evaluated efficiencies for the different analysis steps are presented in Table 4.1.
Using these efficiencies in Eq. 3.1
the result for AS is:
AS =(−4.9 ± 5.7stat) × 10−3 . (4.4)
4.0.1 Stability of the final result in time
In order to study AS variation during the measurement time, the whole analysed sample was divided into ten subsamples equal in luminosity. For each of them
Table 4.1: Efficiencies (%) for the different analysis steps.
Efficiency (%) KS → π−e+ν KS → π+e− ¯ν
trigger and event classification (ET EC ) 99.80 ± 0.02 99.80 ± 0.02
KS tagging (ET AG) 36.54 ± 0.05 36.67 ± 0.05
kinematical cuts (EKC ) 75.60 ± 0.07 75.62 ± 0.07
Track to Cluster Association (ET CA) 42.22 ± 0.08 41.85 ± 0.08
Time of Flight (ET OF ) 64.03 ± 0.19 67.96 ± 0.18
Fit range (EF R) 99.16 ± 0.03 99.17 ± 0.02
the AS (i) was determined separately (see Figure 4.1). The mean value of AS (i) is consistent with the result obtained by analyzing the whole data set. Moreover, no time-dependent effect is visible in analysis efficiencies (see Figure 4.2). Due to different probability of interaction of negative and positive pions in the detector material (see Section 3), the discrepancy between efficiencies for both final charge states is observed.
results 33
4.1 systematic uncertainties on AS
In order to estimate the contributions to the systematic uncertainty of the result, the full analysis chain is repeated varying all the analysis cut values of selection variables by +/− an amount comparable to their experimental resolution. The contributions from the stability of M2(e) distribution fit, momenta smearing, trigger and event classification procedures are also estimated separately.
The systematic uncertainties are classified into the following groups (see Table 4.2):
•
Trigger and event classification:
– Systematic effects originating from the trigger and the event classification procedure are estimated with prescaled data samples (with a prescaling factor of 1/20). The analysis of the prescaled samples follows the standard analysis chain. The systematic contribution (σT EC) is estimated to be 0.28 × 10−3 .
•
Tagging and preselection:
– The KL deposited energy cut (see Eq. 3.2) is changed to the values:
Eclu(crash)= {95, 105, 110, 115, 150, 200} MeV.
The stability of the result is checked within this range. The systematic uncertainty is evaluated by changing the cut by ±5 MeV.
–
The β∗ interval (see Eq. 3.6) is enlarged or shrunk by 0.02 (1σ) on each side:
0.18 = 0.02 <β ∗ < 0.27 ± 0.02. The stability of the result is checked up to a variation of ±5σ.
–
The zvtx and ρvtx cuts for the reconstructed KS → πeν decay vertex position (see Eq. 3.11) are each independently varied by ±0.2 cm (±1σ). The stability of the result is checked against a variation of ±5σ.
–
The range of the opening angle α of the charged secondaries in the KS rest frame (see Eq. 3.12) is enlarged or shrunk by 2◦ (1σ) on each side:
70 = 2◦ <α< 175 ± 2◦ . The stability of the result is checked up to a variation of ±5σ with the constraint of the upper bound not exceeding 180◦ .
–
The Minv(π, π) interval (see Eq. 3.13) is enlarged or shrunk by 1 MeV (1σ) on each side:
300 = 1 MeV 3.4 = 0.1 ns, |δt(π, e)| < 1.3 ± 0.1 ns].
The stability of the result is checked up to variations of ±0.4 ns.
– The circular region for selection of the signal in the {δt(e), δt(π)} plane (see Eq. 3.19) is enlarged or shrunk by varying its radius of ±0.1 ns. The stability of the result is checked for variations ranging from −0.3 ns to +0.4 ns.
•
Momenta smearing:
– The KL → πeν control sample is divided into ten subsamples equal in luminosity. The momenta smearing parameters are tuned separately for each subsample and obtained results are consistent within systematical uncertainty. From the standard deviation of the results the systematic contribution (σMS) is estimated to be 0.58 × 10−3 .
•
Fit procedure:
–
The systematic uncertainty from the histogram bin width σHBW is determined by varying the bin width from 0.8 to 1.6 MeV2/1000 (this variation corresponds to the M2(e) resolution evaluated from MC). σHBW is estimated to be 0.61 × 10−3 . The stability of the result is checked for variations of the bin width from 2σ to 5σ.
–
The systematic uncertainty from the fit range is evaluated by varying it from [−24 : 24] MeV2/1000 to [−28 : 28] MeV2/1000 or [−20 : 20] MeV2/1000. The stability of the fit procedure is checked for histogram ranges from [−36 : 36] MeV2/1000 to [−12 : 12]) MeV2/1000, while keeping the nominal bin size.
The total systematic uncertainty is estimated as the sum in quadrature of the contributions listed above and reported in Table 4.2.
4.2 charge asymmetry for short-lived kaon and test of cpt symmetry
The value of the KS → πeν charge asymmetry has been measured with the KLOE detector based on the data sample of 1.63 fb−1 integrated luminosity. The final value:
AS =(−4.9 ± 5.7stat ± 2.6syst) × 10−3 , (4.5)
is consistent with the previous determination performed by the KLOE group [14] and improves its statistical accuracy by a factor of almost two.
Due to similar analysis schemes, parts of the systematical uncertainty of both KLOE measurements related to the applied cuts are correlated. Taking this information into account, the combined result of the reported measurement and the previous result is:
AS =(−3.8 ± 5.0stat ± 2.6syst) × 10−3 . (4.6)
Table 4.2: Summary of contributions to the systematic uncertainty on AS .
Contribution Systematic uncertainty (10−3)
Trigger and event classification σT EC 0.28
Tagging and preselection Eclu(crash) 0.55
" β∗ 0.67
" zvtx 0.01
" ρvtx 0.05
" α 0.46
" Minv(π, π) 0.20
Time of flight selection δt(π, π) 0.71
" δt(e, π) vs δt(π, e) 0.87
" δt(e) vs δt(π) 1.82
Momenta smearing σMS 0.58
Fit procedure σHBW 0.61
" Fit range 0.49
Total 2.6
The sum and difference of AS and AL allow to search for possible violation of the CPT symmetry, either in the decay amplitudes or in the mass matrix. Using the AL result from KTeV [28] with the KLOE AS value, determined in this work 4.6, the results are:
(AS − AL)/4 = Re(δK )+ Re(x−)=(−1.8 ± 1.4) × 10−3 , (4.7) (AS + AL)/4 = Re(EK ) − Re(y)=(−0.1 ± 1.4) × 10−3 . (4.8)
Using Re(δK )=(2.5 ± 2.3) × 10−4 [36] and Re(EK )=(1.596 ± 0.013) × 10−3 [54] the Re(x−) and Re(y) parameters are extracted:
Re(x−)=(−2.0 ± 1.4) × 10−3 , (4.9) Re(y)=(1.7 ± 1.4) × 10−3 . (4.10)
The obtained Re(x−) and Re(y) values are in agreement with the previous results but the uncertainty of the result from this thesis is by almost a factor of two smaller with respect to the former measurements [14, 36]. The obtained result agrees with the CPT symmetry invariance. Improvements are expected in the future with the analysis of the additional ∼ 5.5 fb−1 of data collected by the KLOE-2 experiment [55–57].
Part II
FEASIBILITY STUDY OF o -P s → 3 γ MEASUREMENT
WITH THE J -PET DETECTOR
5
J-PET DETECTOR
The Jagiellonian Positron Emission Tomograph (J-PET) is a cylindrically shaped
detector built from plastic scintillators (see Figure 5.1). The novel concept of using
Figure 5.1: Left: Photo of the Jagiellonian Positron Emission Tomograph. The J-PET detector is made of three cylindrical layers of EJ-230 plastic scintillator strips (black) and Hamamatsu R9800 vacuum tube photomultipliers (grey). The signals from photomultipliers are probed in the voltage domain at four thresholds with the timing accuracy of about 30 ps [58]. The data acquisition system is working in the trigger-less mode [59]. Right: Arrangement of the scintillator strips with denoted coordinate system. The strips are oriented along the Z axis.
plastic polymers in place of conventional crystal scintillators was announced in 2009 [60]. Replacing the series of expensive scintillator crystals along the z axis by a single plastic scintillator strip allows to reduce the price of the whole device as well as the complexity of the readout system. This solution enables to build a device from larger scintillators resulting in greater field of view (FOV) and usage of the time of flight (TOF) technique, which significantly improves the resolution of the tomographic image [61]. Commercial TOF-PET systems (developed by GE, Siemens, Philips) have resolution along the line of response (LOR) between 300 and 550 ps, which corresponds to a spatial resolution between 4.5 and 8 cm [61]. The single
strip J-PET prototype shows ability to reach 4.9 cm [62], with the possibility of further improvement by more precise determination of gamma quanta interaction along the strip [63] and by application of other reconstruction methods such as
e.g. the compressive sensing theory [64–66]. The J-PET detector was designed as a cost-effective scanner for simultaneous metabolic imaging of the whole human body. However, its superior time resolution, high granularity of detection strips and lower detection pile-ups provide new research opportunities for discrete symmetries violation studies. The detailed J-PET program is described in the Reference [67].
39
j-pet detector
5.1 design details
+
Gamma quanta from ee− annihilation interact with plastic scintillator strips, and cause emission of photons from the visible light spectrum. Usually a few thousand photons are emitted isotropically for 511 keV gamma quanta as a result of the scintillation. The optical signal from the sctintillator is read out at both of its ends by the Hamamatsu R9800 vacuum tube photomultipliers (PMT). In order to decrease photon losses the sides along a scintillator strip are covered with reflective foil. Polymers absorb internally less light emitted by scintillation from radiation than crystals, therefore the usage of longer polymer scintillator strips is possible. The J-PET detector consists of three layers of EJ-230 plastic scintillator strips with dimensions of 7 × 19 × 500 mm3 . The innermost and middle layers consist of 48 strips each, while the outermost layer -96 strips. Another advantage of the plastic scintillators over commonly used crystals is their lower price and shorter duration of signals (about 5 ns compared to 45 ns for LYSO crystal) [68]. This allows to use high activity sources and fast digital electronics readout [58, 59]. The J-PET Data Acquisition System is build out of Trigger Readout Board v3 hardware equipped with Time-to-Digital Converters (TDC) and Field Programmable Gate Array devices for the TDC readout and data transmission. Each analog signal from the PMT is sampled in the voltage domain at four thresholds. This gives 4 points on the leading edge and 4 points at the trailing edge of the signal. A scheme of the registration process is shown in Figure 5.2. Probing signals at four thresholds
Figure 5.2: Left: Incident gamma quantum (red) interacts with detector strip and causes the emission of photons, later registered by a photomultipliers (PMT). Right: Recorded signal is sampled at four voltage threshold levels (blue lines). Each signal crossing a given threshold is registered at both leading and trailing edge (black and grey dots respectively). Recorded times at both PMT’s (tA and tB ) are used for determination the gamma quantum interaction place and time (tγ ) along the scintillator strip (see Eq. 5.2
and 5.1). The value of deposited energy corresponds to the sum of registered times over threshold (TOT) for all thresholds crossed by the signal.
allows to reconstruct the original signal shape [66]. Data is collected continuously (in a trigger-less mode) to ensure that the registered information is preserved and stored for further, high-level processing preformed by dedicated analysis software described in Section 5.3.
5.2j-pet detector properties
5.2j-pet detector properties
The time and position of a gamma quantum interaction (referred to as a hit further in the text) in a scintillator can be calculated from the time of scintillation light registration in photomultipliers located at the ends of a single plastic scintillator strip (tA, tB). The distance (Δz) along the strip between its center and the hit position (see Figure 5.2, left) can be expressed as:
(tA − tB ) · v
Δz = , (5.1)
2 where v is the effective light signal velocity in the plastic scintillator [69]. The time of interaction tγ is obtained from a sum of the light registration times at sides A and B of the scintillator:
tA + tB L
tγ = − , (5.2)
22v where L is the length of scintillator strip.
Gamma quanta interact with polymer scintillators mainly via the Compton effect and the characteristic spectra of deposited energy are described with a differential cross section for scattering in a solid angle dΩ given by the Klein-Nishina formula [70]:
2 E/ 2 E/
dσrE
0
=+ − sin2 φ, (5.3)
dΩ 2 E EE/
where r0 is the classical electron radius, E and E/ denote energies of the primary and scattered photon and φ is the planar angle between their momenta.
The J-PET detector measures the deposited energy using time over threshold (TOT) [71]. The TOT value for a single threshold is determined based on registered times on leading (tL) and trailing (tT ) edges of the signal:
L
T OT = tT − t. (5.4)
The total energy deposited by gamma quanta is correlated with the sum of TOTs at all thresholds crossed by the signals on both A and B sides of a scintillator.
The obtained energy and time resolution of registered gamma quanta were experimentally determined and within the range of deposited energy Edep which corresponds to (200, 340) keV, are equal to [72]:
σ(T 0 (5.5)
hit) ≈ 80 ps,
σ(E) 0.44
= √ . (5.6)
EE [MeV]
For lower energies the time resolution can be expressed as a function of deposited energy (Edep)[20]:
σ(T 0 )[ps]
hit
σ(Thit(Edep)) = 1 . (5.7) Edep[keV]
270
Considering the most challenging time reconstruction for gamma quanta with low energies (around 50 keV), one can see that the J-PET detector provides a precision on the level of 200 ps. In the commercial PET systems the events with an energy deposition lower than about 400 keV are discarded [73, 74].
j-pet detector
5.3 software analysis -framework
The J-PET data reconstruction is a multi-stage process based on dedicated analysis software, the J-PET Analysis Framework [75]. It is developed in a C++ programming language and is based on the Open Source library ROOT [76]. The source code of the project is available on the GitHub service [77] under the Apache Licence.
The analysis of data with the J-PET Framework consists of several modules. Each of them corresponds to a particular computing task e.g. calibration procedure or reconstruction algorithm. This approach allows the user to choose between available reconstruction algorithms or create a dedicated analysis module and easily implement it into the data processing chain. A scheme of the flow of data delivered by the data acquisition system (DAQ) is shown in Figure 5.3.
Figure 5.3: Scheme of data processing by the J-PET Analysis Framework software. The input data (recorded TDC times) are delivered by data acquisition system (DAQ). Pink rectangles represent computational task responsible for e.g. calibration or a reconstruction algorithm. Violet rectangles stand for reconstructed physical quantities e.g. gamma interaction points in the scintillator strip (referred as hits). The TDC signals are used to reconstruct shapes of electric signals from photomultipliers. Next, pairs of signals from a single photon hit in a scintillator strip allow for reconstruction of the hit time and position along the strip. Finally, hits identified to originate from a single annihilation event are merged into event structures.
6
PERFORMANCE ASSESSMENT: MONTE CARLO SIMULATION
Part of the presented work was to develop the Monte Carlo simulation package of the J-PET detector. The program is based on the Geant4 simulation package [78], which controls the tracking of particles through detector geometry and uses well tested routines to simulate interactions.
In the further section, an integration of the output from the developed simulation program with the existing J-PET analysis software will also be presented. The preformed simulation results in an estimation of the accuracy of CPT violation parameters as a function of the number of registered o-Ps→ 3γ decays.
6.1 program architecture and simulated geometry
The core of the Monte Carlo simulation is conducted by the Geant 4 simulation software [78], while the detector description and details on the o-Ps→ 3γ annihilation process (not included in Geant4 physics) are implemented as a part of this dissertation. Code is available under an Apache Licence on the GitHub service [79].
From the user point of view, the files containing MC-generated events have to be processed in the same manner as collected data (see section 5.3). For this purpose a dedicated module is created as a part of the J-PET Analysis Framework. As an input it uses structures created by the Geant4 software with generated information about gamma quanta interactions. In further steps of the analysis chain, those hits are processed in the same manner as experimental data. A scheme of the data flow is shown in Figure 6.1. The Monte Carlo simulations account for: angular and energy
+
distributions of gamma quanta originating from direct ee− or ortho-positronium annihilation, Compton interactions of emitted gamma quanta in the detector built from plastic scintillators, determination of gamma quanta hit-position and hit-time in the detector with resolutions known from experiment, multiple scattering and accidental coincidences, as well as reconstruction of registered gamma quanta four-momenta.
Simulations were performed assuming a detector geometry with three cylindrical detection layers that corresponds to the J-PET detector. In addition, the supporting frame was created based on Computer Aided Design (CAD) technical drawings and loaded directly while executing the simulations. An exemplary view of a simulated geometry is shown in Figure 6.2.
6.2 positronium formation
An isotope undergoing a β+ decay emits a positron that travels through matter, scatters and slows down reaching thermal energies. Then it undergoes free annihilation or forms a positronium [80]. In water at 20◦C the positron has about 64% chance of undergoing free annihilation [81]. The positronium is produced mostly in
43
6.3 simulation of back-to-back annihilation events
the ground state forming para-Positronium (1S0, p-Ps) or ortho-positronium (3S0, o-Ps) with probabilities of 25% and 75%, respectively. The annihilation of these states is leading predominantly to an emission of two or three gamma quanta for p-Ps or o-Ps states, respectively. However, the interactions with matter can lead to inversion of the ortho-positronium spin or to the pick-off processes and, as a result, can affect the relative ratio of 3γ/2γ annihilation. The fraction of atoms annihilating into 3γ quanta is described by following equation [82]:
1 − P 3 τo−Ps
f3γ =+ P , (6.1)
372 4 τvacc
where P denotes o-Ps formation probability, τo−Ps and τvacc is the o-Ps lifetime in a sample and in vacuum, respectively. The effective yield of annihilation into 3γ in most of non-metallic substances is of the order of 1%, although in some cases, as for example fine powders of alkaline oxides, it can reach even 29% as recently shown for the amberlite porous polymer XAD-4 (CAS 37380-42-0)[83].
6.3 simulation of back-to-back annihilation events
+
In the PET tomography the registration of two gamma quanta from ee− annihilation is essential. Due to momentum conservation they are emitted back-to-back. Annihilation quanta interact with the plastic scintillators used in J-PET. However, usage of plastics scintillators, due to their low density, results in a lower registration probability. Linear absorption coefficient of 511 keV gamma quanta for plastic scintillators amounts to 0.098 cm−1 [84], and is more than eight times smaller than for the LSO crystal amounting to 0.821 cm−1 [85].
In polymer scintillators, incident gamma quanta are mainly interacting through Compton scattering described by the Klein-Nishina formula [70].
performance assessment: monte carlo simulation
Figure 6.3: Left: Registration efficiency for 2 γ quanta originating from direct annihilation. The efficiency map was estimated with 0.5 mm × 0.5 mm bin size. Right: Sensitivity map in X-Y plane in the central part of the detector (z = 0). Result obtained as a part of derivation of a statistical model for the J-PET detector necessary for image reconstruction purposes. Right figure adapted from [86]. The spatial configuration of J-PET detector strips and the requirement of two hits registration along the LOR results in characteristic structures visible on the sensitivity map.
Registration of 2γ annihilation events requires reconstruction of signals from two strips. Non-solid detection layers coverage with scintillator strips results in "blind spots" -areas lying inside the detector where registration probability is zero or close to zero and emitted gamma pairs are not detected by any of the scintillator pairs. The J-PET detector configuration of detection strips results in the efficiency map presented in Figure 6.3.
6.4 simulation of 3γ events
Positronium is the lightest purely leptonic system, and it can annihilate only into gamma quanta. In case of the decay into three photons, they are coplanar in the Center of Mass (CM) frame due to momentum conservation. The double differential cross section as a function of photons’ energies E1 and E2 is expressed as [87]:
62 2 2
d2σ 18eme − E3 me − E2 me − E1
= ++ , (6.2)
dE1dE2 6 vm2 E1E2 E1E3 E2E3
e
where e and me are electron charge and mass, respectively, v denotes electronpositron relative velocity, Ei (i=1,2,3) are gamma photon energies, and E1 + E2 + E3 = 2me follows from energy conservation. In above formula, the conservation of four-momentum results in the characteristic energy distribution of gamma quanta (see Figures 6.4). Energy of single gamma quanta from ortho-positronium annihilation is within the [0, 511] keV range and the spectrum of deposited energy is presented in Figure 6.5.
6.4 simulation of 3γ events
Figure 6.4: Left: Energy spectrum of photons originating from three-photon annihilation of an electron and a positron. Right: Dalitz plot for the o-Ps→ 3γ annihilation. In both figures the non-homogeneity of the density distribution is due to the energy dependence of the o-Ps→ 3γ transition amplitude (see Eq. 6.2) which was taken into account in simulations according to the predictions based on quantum electrodynamics [87].
The registration efficiency of the photons from the o-Ps→ 3γ events depends on the energy deposition threshold used in the front-end electronics. The hardware threshold at the order of 10 keV [72] is set to discriminate the experimental noise and further selection threshold based on the measured energy deposition is applied. The probability of registration of 1, 2 or 3 gamma quanta originating from o-Ps→ 3γ annihilation as a function of applied selection threshold is shown in Figure 6.6
(right panel). The left panel shows the efficiency map in the X − Y plane in the center of detector. The o-Ps→ 3γ is a three-body annihilation, therefore the observed efficiency map has a uniform distribution.
performance assessment: monte carlo simulation
Figure 6.5: Distribution of energy deposited in plastic scintillators by gamma quanta originating from o-Ps→ 3γ annihilations for single gamma quanta (left panel) and for two gamma quanta randomly selected (right panel). The shown spectrum is a convolution of the energy distribution of gamma quanta from the o-Ps→ 3γ decay (Figure 6.4, left panel) and the Klein-Nishina distribution of kinetic energy of electrons acquired via Compton scattering.
Figure 6.6: The o-Ps→ 3γ registration efficiency (determined taking into account geometrical acceptance, probability of gamma quanta registration in the plastic scintillator and J-PET detector resolution) for transverse view of a central plane of the detector (left panel) and as a function of the applied threshold (right panel). The shown dotted, dashed and solid lines indicate efficiency assuming that at least one, two or three photons deposited energy above the threshold, respectively. While reconstruction of o-Ps→ 3γ events requires low experimental threshold, its efficiency map is free of the blind spots which results from detector arrangements in case of 2γ back to back reconstruction (compare Figure 6.3)
7
FEASIBILITY STUDY
sk2
Recent studies on CPT odd triple correlations s
· k1 × s, where s
is the spin of the ortho-positronium, and sk1 and sk2 are the momenta of the two most energetic annihilation photons, were suggested in [16] and performed by Vetter and Freedman using the Gammasphere detector [17]. Gammasphere is an array consisting of 110 high purity germanium (HPGe) detectors. Each Ge detector assembly consists of a 7 cm diameter and 8 cm long cylindrical HPGe detector surrounded by six bismuth germanate (BGO) scintillators on the sides and one BGO scintillator in the back [88]. The probability that a detected 511 keV photon deposits its full energy is roughly 15%. In the experiment with Gammasphere, the 0.37 MBq source of 68Ge or 22Na was placed underneath a thin (0.2 mm) plastic scintillator and a hemisphere of silicon dioxide aerogel. Positrons from β decay were identified by around 70 keV energy deposition in the scintillator. The average ortho-positronium polarization (P ) was 21.5% for 22Na and 30.5% for 68Ge. The magnitude of the source polarization is reduced by accepting positrons in a solid angle of 2π. During the 36 day run, the Gammasphere reconstructed around 2.65 × 107 o-Ps→ 3γ annihilation events [17]. The observable measured by the Gammasphere was the asymmetry:
N+ − N−
A = , (7.1)
N++ N−
where N+ and N− denote number of decays with the normal to the decay plane parallel (+) and antiparallel (-) to the o-Ps spin direction, respectively (see Figure 1.2). The angular correlation between spin and decay plane is related to the count asymmetry observed in the experiment and the average polarization of the o-Ps (P ), by [17]:
CCPT = A/ (P ) = 0.0026 ± 0.0031. (7.2)
The obtained result is the most precise measurement to date.
The rate limitation of the previous experiment can be overcome by the J-PET detector due to its much higher granularity and about one to two orders of magnitude shorter duration of signals (plastic scintillators at the J-PET [68, 72] vs. HPGe/BGO at Gammasphere [17]) leading to significant reduction of pile-ups. It is also important to stress that the J-PET detector is characterized by about 3 times higher angular resolution and time resolution (∼ 0.1ns) [68, 72] is improved by about a factor of ten with respect to the Gammasphere detector [17]. An improvement on the precision of CPT symmetry violation test is also expected because of about two orders of magnitude larger statistics, which can be achieved due to the possibility of longer runs and due to the usage of the higher activity of positron source (10 MBq at the J-PET vs. 0.37 MBq at Gammasphere [17]).
49
feasibility study
7.1 polarization control
During past experiments, the ortho-positronium decay point was assumed to lie within the aerogel targets, e.g. hemisphere in Gammasphere, and its exact time and spatial coordinates were not reconstructed. In J-PET, however, due to its relatively high angular acceptance and timing resolution, a reconstruction of the o-Ps→ 3γ process is possible by means of a new trilateration-based reconstruction method. The algorithm was developed and tested, and obtained results are presented in References [89] and [90].
The method based on trilateration allows for a simultaneous reconstruction of both location and time of the annihilation based on time and interaction position of gamma quanta in the J-PET detector. Gamma quanta from the o-Ps→ 3γ annihilation travel on a distance between the annihilation point (which needs to be localized) and the detector where the places and times of their interaction are recorded and serve as reference points. An additional constraint is given by the fact that all three photons are produced in a three-body decay and thus their momenta as well as the o-Ps decay point are contained within a single plane in the frame of reference of the decaying positronium atom.
s
Experimental realization of triple correlation measurements s
· k1 × sk2 requires a vacuum chamber whose walls are coated on the inner side with a porous medium for o-Ps production (see Figure 7.1). In the center of the chamber a β+ source is
located, and emitted positrons from β+ decay are polarized along their momentum s
with P = sv/c, where the sv and c denotes positrons velocity and speed of light, respectively. The average polarization can be estimated from the average velocity of the positrons emitted in the β+ decays. If positrons are emitted in a cone with an opening angle of 2α, then the aforementioned equation can be expressed as:
v
(P ) =(1 + cos α) /2. (7.3)
c
The polarization of the ortho-positronium is (from a statistical argument) simply 2/3 of the average positron polarization [33].
The capability of reconstruction of ortho-positronium annihilation coordinates allows for a more precise polarization estimation, presented schematically in Figure 7.2.
7.2 background reduction
Figure 7.2: Left: Illustration of ortho-positronium spin determination. In the center of the detector a β+ source is located (red dot), and the emitted positron may form an ortho-positronium state in the aerogel (yellow band). Dashed lines represent gamma quanta originating from ortho-positronium annihilation. Emitted gamma quanta are registered by scintillator strips represented by the circularly arranged blue squares. Based on registered hits positions and times, the annihilation vertex is reconstructed on the cylinder, which, in turn, allows to estimate the positron momentum direction and ortho-positronium spin direction. Figure adapted from [89]. Right: Visualization of the detector with an annihilation chamber placed at its center.
With the J-PET detector and the cylindrical positronium target with a radius of 10 cm the uncertainty of determination of positron direction will amount to about 15◦ [89]. In a conducted test measurement a large annihilation chamber and 9 MBq sodium source were used. The source was closed in the Kapton foil inside the cylindrical aluminum chamber. In the first tests, positrons were annihilating in the aluminum wall [90]. In the next experiments, the annihilation chamber will be realized as the positron source with a porous material cylinder around it, where we plan to use porous target materials like polymer XAD-4 (CAS 37380-42-0)[83] on the internal surfaces of the chamber.
7.2 background reduction
Annihilation into 2γ may mimic a registration of 3γ annihilation due to the secondary scatterings in the detector. Such scattering is shown pictorially in Figure 7.3. For the reduction of this background the following complementary methods can be considered, based on information of:
•
relation between position of the individual detectors which recorded hits and the time difference between registered hits,
•
angular correlation of relative angles between the gamma quanta propagation directions,
•
the distance between the origin of the annihilation (position of the annihilation
chamber) and the decay plane. In Figure 7.4
example spectra of θ23 vs θ12 distribution are shown, where θij are the opening angles order that θ12 <θ23 <θ13 (see Figure 7.3) between registered
feasibility study
+
ee− annihilation into 2γ. Circularly arranged squares represent scintillator strips -purple and green colors indicate strips where the gamma quanta were or were not registered, respectively. For clarity, the elements are shown not to scale. Only a single layer with selected number of scintillators and increased X − Y dimensions is shown. The arrows represent the actual gamma quanta occurring in the events, while dotted lines indicate naively reconstructed gamma quanta. Examples of primary and secondary scatterings are depicted.
gamma quanta. For the o-Ps→ 3γ process, due to the momentum conservation, θ23 > 180◦ − θ12 and therefore events corresponding to the o-Ps→ 3γ decay will lie above the diagonal, as shown in green colour in Figure 7.4.
Most of the background events will correspond to points at the diagonal (θ23 = 180◦ − θ12) and below diagonal (θ23 < 180◦ − θ12) as can be inferred from the middle and left panel of Figure 7.3. Therefore, one of the possible selection cuts was applied on ordered opening angles (θ12 <θ23 <θ13) between registered gammas, and resulted in a decrease of background by a factor 104 while rejecting only 3% of signal events (see Figure 7.4). Combining aforementioned criterion with the requirement that registered time difference (Δt) as a function of the difference between sequential numbers of the detectors (ΔID) is small (Δt< 0.3 ns) allows for total reduction of the instrumental background by a factor of 109. Therefore, even though the 3γ events are expected to constitute only about 0.5 % of 2γ events, the background due to the 2γ annihilation associated with a secondary scattering in the detector can be reduced to a negligible level.
However, we have to take into account that the remaining background is caused not only by misidentified 2γ events, but also by true annihilations into 3γ which may originate from the interaction of the positronium with surrounding electrons and hence will constitute a background for studies of discrete symmetries. Interaction of ortho-positronium with matter is classified into: pick-off annihilations and orthopara spin conversion. Contribution from these processes depends on the used target material, e.g. in aerogel IC3100 and amberlite porous polymer XAD-4 about 7% and 36% of ortho-positronium undergo these interactions, respectively [83]. The events originating from the true o-Ps→ 3γ annihilation process (No−Ps) can be misidentified with the events from the following processes: pick-off process with direct annihilation to 3γ (N3γ pick−off ); pick-off process with annihilation to 2γ misidentified as 3γ due to secondary scatterings (N2γ pick−off ); conversion of ortho-positronium to para-positronium with subsequent C symmetry violating decay to 3γ (N3γ conv); conversion of ortho-positronium to para-positronium with
7.2 background reduction
+
ee− → 2γ annihilation is registered in the detector while the other is scattered and cause signals in two detectors, lie on the diagonal of the plot. Events where one gamma is missing detection, and the other undergoes two scatterings are localized below the diagonal line. Example of an analysis cut, rejecting 3% of signal and reducing background by factor 104 is shown as a dashed purple line. The presented distribution includes the angular resolution of the J-PET detector.
subsequent annihilation to 2γ misidentified as 3γ due to the secondary scatterings
(N2γ conv). The conservative upper limit on these background contributions may be estimated as:
N2γ conv N2γ pick−off N3γ conv N3γ pick−off
< << , (7.4)
No−Ps No−Ps No−Ps No−Ps
where:
N3γ pick−off τmatter
< 1 − /370 ≈ 2 · 10−4(IC3100) < 10−3(XAD-4);
No−Ps τvacuum
N2γ pick−off
< 0.07 · 10−9(IC3100) < 0.36 · 10−9(XAD-4);
No−Ps
N3γ conv
< 0.07 × 2.8 · 10−6(IC3100) < 0.36 × 2.8 · 10−6(XAD-4);
No−Ps
N2γ conv
< 0.07 · 10−9(IC3100) < 0.36 · 10−9(XAD-4).
No−Ps
(7.5)
In the above estimations the factor 10−9 denotes the reduction power of the 2γ events and 2.8 · 10−6 stands for the upper limit on the C symmetry violation via the p-Ps → 3γ process [91]. The precise control of these contributions will be provided by the measurement of the true 2γ events with high statistics.
feasibility study
7.3j-pet efficiency studies with monte carlo simulations
The rate of registered o-Ps→ 3γ events in general can be expressed by the formula:
Ro−Ps→3γ = A · fo−Ps→3γ · Edet(th) · Eana, (7.6)
where A is the total annihilation rate (fast timing of applied plastic scintillators allows for usage of a 10 MBq positron source), fo−Ps→3γ is the fraction of annihilations via o-Ps→ 3γ process in the target material, Edet(th) is the detector efficiency as a function of applied detection threshold while Eana denotes selection efficiency used to discriminate between 3γ and 2γ events.
The Edet efficiency of the o-Ps→ 3γ reconstruction will depend on the energy deposition threshold used in the analysis. The probability of registration of 1, 2 or 3 gamma quanta originating from o-Ps→ 3γ annihilation (Edet) as a function of the applied selection threshold is shown in Figure 6.6
(right panel). The efficiency Edet contains contribution from geometrical acceptance, probabilities of gamma quanta interaction in the plastic scintillators used and it was determined taking into account the J-PET detector resolution. In evaluation of Edet, it is assumed that the event selection threshold is set to 50 keV. A fraction of annihilations via o-Ps→ 3γ process is estimated taking into account only the longest lived component in two selected materials IC3100 (fo−Ps→3γ = 16.6%) and XAD-4 (fo−Ps→3γ = 28.6%)[83]. The expected rate of registered signal events per second is 15 for IC3100 and 25 for XAD-4. Using amberlite porous polymer XAD-4 instead of aerogel IC3100 as target material in the experiment, allows to collect the required statistics almost twice faster, however, resulting in higher systematic uncertainties due to the interaction of positronium with the target material, as discussed in Section 7.2.
7.4 discussion and prospects
The CPT violating parameter uncertainty depends on the number of reconstructed o-Ps annihilations and spin measurement accuracy. This dependency for the Gammasphere detector is shown in Figure 7.5.
Based on the preformed Monte Carlo simulations it was concluded that the J-PET multipurpose detector constructed at the Jagiellonian University allows for exclusive registration of the decays of ortho-positronium into three photons (o-Ps→ 3γ). Using XAD-4 as the target material, the 109 o-Ps→ 3γ events will be reconstructed after 15 months of continuous data taking. This time can be reduced to 50 days by adding two additional layers of scintillation strips [20]. The achieved results indicate that the J-PET detector gives a realistic chance to improve the best present limits established for the CPT symmetry violation in the decays of positronium [17] by more than an order of magnitude. This can be achieved by (i) collecting at least two orders of magnitude higher statistics, due to the possibility of using a β+ source with higher rate (10 MBq at J-PET vs 0.37 MBq at Gammasphere [17] experiment), (ii) the enhanced fraction of 3γ events by the use of the amberlite polymer XAD-4, (iii) measurements with a few times improved angular resolution and (iv) about two times higher degree of o-Ps polarization, as shown recently in Reference [89]. The limitation on the source activity can be overcome by the J-PET detector due to the application of plastic scintillators that
7.4 discussion and prospects
Figure 7.5: Dependency between the number of reconstructed o-Ps→ 3γ events and the uncertainty of CPT violating parameter (red line). Plot is made assuming that uncertainty on the asymmetry goes as the inverse of the square root of the total number of events and that detector and analysis parameters follow the Gammashpere. Result obtained by Vetter and Freedman [17] is denoted by black square.
are characterized by about two orders of magnitude shorter duration of signals, thus decreasing significantly the pile-up problems with respect to the crystal based detector systems. In addition, the improved angular resolution combined with the superior timing of the J-PET detector (improved by more than order of magnitude with respect to the crystal detectors) and with the possibility of the triggerless registrations [59] of all kind of events with no hardware coincidence window allow for suppression and monitoring of the background due to misidentification of 2γ events and possible contribution from 3γ pick-off annihilations.
8
CONCLUSIONS
The aim of this thesis was to investigate the CPT symmetry violation effects in two matter-antimatter systems created by different fundamental components: quark-antiquark and lepton-antilepton.
In the first part, a neutral kaon system, formed from combination of ds¯and
¯
ds quarks, was used to determine the lepton charge asymmetry value for the short-lived kaon. The difference between AS and the corresponding asymmetry for its long-lived counterpart is sensitive to the CPT symmetry violation effects. The uncertainty of the test preformed until now was limited by the AS uncertainty originating mainly from the size of the data sample.
Studies presented in this thesis were conducted on data gathered by the KLOE detector in 2004-2005, and the obtained result:
AS =(−4.9 ± 5.7stat ± 2.6syst) × 10−3 , (8.1)
improved on the accuracy of the most precise result by a factor of almost two. The combination of the results obtained by the KLOE collaboration gave:
AS =(−3.8 ± 5.0stat ± 2.6syst) × 10−3 , (8.2)
and allowed to determine new limits on CPT violating parameters:
Re(x−)=(−2.0 ± 1.4) × 10−3 , (8.3) Re(y)=(1.7 ± 1.4) × 10−3 , (8.4)
which are in agreement with CPT invariance within the achieved precision.
It is worth mentioning that the data taking campaign has been restarted in 2014 and until march 2018 the KLOE-2 detector collected an additional dataset with an integrated luminosity of 5.5 fb−1 . KLOE-2 provides not only larger statistics but also improved event reconstruction due to the new Inner Tracker detector. The detector aims at improving the tracking and vertexing resolution close to the KS → πeν decay point [92].
Another part of the KLOE-2 program is a test of CPT symmetry in transitions. The difference between the semileptonic asymmetries for KS and KL is also accessible while measuring the double ratio of CPT -violating ratios [93]. Analysis details are presented in Reference [90].
In the second part of this thesis, a feasibility study of using the J-PET detector to test the CPT violation manifested as a non-vanishing angular correlation of photons’ momenta was carried out. For this purpose a dedicated Monte Carlo simulation was created. Special emphasis was put on describing the ortho-positronium annihilation into three gamma quanta and the response of the J-PET tomograph. The efficiency maps of the central region of the J-PET detector were determined for two and three
57
gamma quanta annihilations. The conducted studies allowed to determine the radius of the cylindrical aluminium chamber which maximizes the 3γ/2γ ratio. Additionally, the expected measurement time to improve presently most precise result [17] was determined for the currently tested experimental setup, which uses the XAD-4 target material in the cylindrical annihilation chamber.
The expected CPT violation effects are highly model dependent and no single theory predicting them is known. Therefore, searches for CPT violations have to be conducted through a broad range of systems. In this thesis more precise limitations were given on parameters Re(x−) and Re(y) in the neutral kaon system, and a possible search in the decay of ortho-positronium atoms was discussed. Further results in this field are anticipated, among others, in neutrino oscillations [94] and atomic physics experiments [95].
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