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Uppsala University Department of and Astronomy Master Thesis

Search for the C-violating φ → ωγ decay and acceptance studies of the rare ω → l+l−π0 decay with the KLOE experiment

Author: Walter Andersson Ikegami Supervisor: Andrzej Kupsc

Abstract

This thesis is a groundwork for a search for the C-violating φ → ωγ decay and the rare ω → l+l−π0 decay with the KLOE detector. A feasibility study of the detection acceptance for the ω → π+π−π0 and ω → l+l−π0 decays produced in + − + − 0 the e e → ωγISR and e e → ωπ production channels. A study of the main + − background to the forbidden φ decay, the e e → ωγISR process, is performed −1 using a data sample with an integrated luminosity√ of L = 1.6 fb collected by the KLOE detector at center of mass energy s = 1019 MeV. Table of contents

1 Popul¨arvetenskaplig sammanfattning 5

2 Summary 7

3 Theoretical Introduction 9 3.1 The ...... 9 3.2 Form Factors ...... 11 3.3 Symmetries ...... 12 3.4 C-violating φ → ωγ decay ...... 12 3.5 Initial State Radiation ...... 13 3.6 Decay kinematics and differential decay rates ...... 14 3.7 The ω → π+π−π0 decay ...... 15 3.8 The ω → l+l−π0 conversion decay ...... 16 3.9 The π0 → γγ decay ...... 18

4 The KLOE detector at DAΦNE and Software Tools 19 4.1 The DAΦNE Collider ...... 19 4.2 The KLOE Detector ...... 20 4.3 KLOE Monte Carlo program GEANFI ...... 21 4.4 Data Sample used in this Thesis ...... 22 4.5 ROOT ...... 23 4.6 Phokhara ...... 23

+ − 5 Feasibility study of the e e → ωγISR reaction 24 5.1 Constructing the ISR event generator ...... 24 5.2 Acceptance ...... 25 5.3 Preselection criteria for a potential new data sample ...... 27

6 Monte Carlo analysis of the ωγ final state in the KLOE detector 29 6.1 Reconstructing of four-momentum ...... 29 + − 6.2 Extraction of e e → ωγISR signal from MC ...... 29 6.3 Background rejection cut: 3 ...... 30 6.4 pairing ...... 31 6.5 Background rejection cut: Monochromatic photon energy ...... 32 6.6 Background rejection cut: γω angle difference ...... 32 6.7 Background rejection cut: Charged identification (CPI) . . 33 6.8 Efficiencies ...... 34

+ − 7 Results on Monte Carlo study of e e → ωγISR process 36

2 8 Feasibility studies of ω conversion decays 39 8.1 The event generators ...... 39 8.2 Acceptance ...... 39 8.3 Preselection criteria for a potential new data sample ...... 41

9 Outlook 42 9.1 New data sample ...... 42 9.2 Kinematic fit ...... 42 9.3 Simulation of a hypothetical φ → ωγ decay ...... 43 9.4 Fitting Monte Carlo sample to the data ...... 43 9.5 Energy scan ...... 44

10 Summary and Conclusion 45

3 Acknowledgement

I would like to express my gratitude to my supervisor, Andrzej Kupsc, for his continued support throughout the making of this thesis. I would also like to thank Li Caldeira Balkest˚ahl,Lena Heijkenskj¨old, Joachim Pettersson for always being available for questions, assistance and discussions re- lating to my work. Finally, I would like to thank Karin Sch¨onning for proof-reading my thesis and providing valuable suggestions.

4 1 Popul¨arvetenskaplig sammanfattning

Standardmodellen ¨ar en teorin som beskriver alla elementarpartiklar och hur de v¨axelverkar. I denna teori kan processer som till exempel partikels¨onderfall ber¨aknas. F¨or att testa Standardmodellen j¨amf¨or man dess f¨oruts¨agelser med experimentel- la resultat. Eftersom s˚am˚anga f¨oruts¨agelser st¨ammer v¨al ¨overens med experi- mentella m¨atningar med h¨og precision s˚aanses Standardmodellen vara den mest framg˚angsrika beskrivningen av universums minsta byggstenar. Experimentella m¨atningar utf¨ors oftast i partikelacceleratorer d¨ar partiklar ac- celereras till h¨oga energier och krockar med varandra. I dessa krockar skapas nya partiklar som m¨ats med toppmoderna detektorer. Detektorerna brukar utformas med n˚agrahuvudsyften men kan anv¨andas f¨or att utf¨ora studier p˚aalla m¨ojliga re- aktioner, s˚avidadessa f¨oljer experimentets grundf¨oruts¨attningar. N¨ar man har en ny id´eom n˚agotman vill unders¨oka brukar man utf¨ora effektivitetsstudier. Dessa visar om en reaktion har tillr¨ackligt goda f¨oruts¨attningar f¨or att kunna m¨atas med detektorn. I den h¨ar rapporten har studier genomf¨orts p˚atv˚aspecifika s¨onderfall hos tv˚a partiklar; omega-mesonen (betecknas med grekiska bokstaven ω) och fi-mesonen (betecknas φ). S¨onderfallet d¨ar en φ- s¨onderfaller till en ω-meson och en foton bryter mot en fundamental symmetrilag inom Standardmodellen. Konsekvensen av detta symmetribrott ¨ar att sannolikheten f¨or att φ meson s¨onderfallet sker blir extremt litet. Sannolikheten f¨or detta s¨onderfall uppm¨attes senast ˚ar1966. Denna m¨atning baserades p˚aett litet stickprov med d˚aliguppl¨osning och d¨arf¨or ¨ar os¨akerheten i m¨atningen mycket stor. D¨arf¨or g˚ardet inte att dra slutsatsen om formuleringen av symmetrin ¨ar korrekt eller om fysik bortom Standardmodellen p˚averkar sanno- likheten f¨or att s¨onderfallet kan ske. φ-mesoner bildas i acceleratorn DAΦNE i Italien. Positroner och elektroner accelereras i DAΦNE till mycket h¨oga energier for att sedan krocka och bilda φ-mesoner. Genom MonteCarlosimuleringar har skillnaden mellan φ-s¨onderfallet och den s˚akallade ISR-processen studerats. ISR ¨ar ett specialfall d˚aelektronen eller positronen skickar ut en foton innan en meson bildas. N¨ar en ω-meson bildas i ISR-processen s˚abildas samma partiklar som φ-meson s¨onderfallet. Experimentellt ¨ar dessa tv˚ah¨andelser mycket sv˚araatt skilja ˚atd˚ade ger liknande signaler i detektorn. Det f¨orv¨antade antalet ISR-h¨andelser i DAΦNE anses vara tillr¨ackligt h¨ogt f¨or att kunna studeras och ge en b¨attre f¨orst˚aelseav processen. Jag har ocks˚agjort f¨orstudier som kommer underl¨atta den f¨orest˚aendem¨atningen av φ-s¨onderfallet, d¨ar en b¨attre m¨atning p˚a φ-s¨onderfallets sannolikhet ska kunna g¨oras. En s˚adan m¨atning skulle kunna styrka Standardmodellens legitimitet om den ¨overensst¨ammer med Standardmodellens f¨oruts¨agelser. Om den inte g¨or det, tyder det p˚afysik bort- om Standardmodellen.

5 Uppskattningar av antalet ω-s¨onderfall har ocks˚agenomf¨orts i denna rapport. I detta arbete s˚avisas att antalet ω meson s¨onderfall som man kan f¨orv¨anta sig i den datam¨angd som insamlades i DAΦNE experimentet ¨ar v¨aldigt f˚a.D¨aremot framkom att fler h¨andelser skulle kunna erh˚allas om man ¨andrar p˚avissa paramet- rar som anv¨ands f¨or att filtrera den reaktion man vill ˚at.Det antal ISR-processer som f¨orv¨antas i datam¨angden kan anv¨andas av andra forskare f¨or att utf¨ora en f¨orb¨attrad m¨atning av φ mesonens sannolikhet att s¨onderfalla.

6 2 Summary

The Standard Model is the primary theory describing the elementary and their interactions via the electromagnetic, weak and strong forces. The discoveries of new elementary particles have given further credence to the Standard Model. While the Standard Model has been successful in predicting the electromagnetic, weak interactions and the strong interactions at high energies, it does not include the gravitational force. Moreover, there are some phenomena that the Standard Model leaves unexplained, such as dark . The generation of mass of particles held together by the is also not explained in the Standard Model. Searches for suppressed and forbidden processes provide precision tests of the Standard Model predictions. In this thesis I have investigated one such process; namely φ → ωγ. The φ → ωγ decay violates charge conjugation C- and is therefore forbidden in strong and electromagnetic interactions. Consequently, its branching ratio is expected to be very small. Currently, an experimental upper limit for the decay branching ratio is < 5% with a 84% confidence level which, for a theoretically forbidden reaction, is a very weak constraint. It is based on an experiment with a bubble chamber exposed to a K− beam carried out at Brookhaven National Laboratory in 1966 [1]. The DAΦNE - collider which is designed to operate at the cen- ter of mass energy corresponding to the φ meson mass is an excellent facility which could significantly improve this upper limit. The DAΦNE collider is located at La- boratori Nazionale di Frascati (LNF)) . During data taking runs, DAΦNE has reached luminosities up to 1.3 · 1032 cm−2s−1 and between 1999 and 2005 KLOE, the general purpose detector located in DAΦNE, recorded an integrated luminosity of ∼ 2.5 fb−1 [2]. −1 For this study, a data sample√ of L = 1.6 fb integrated luminosity collected at the center of mass energy s = 1019 MeV by the KLOE detector is used. The data sample was originally prepared for φ → ηγ → π+π−π0γ measurements by removing unwanted events. This was done by requiring two charged tracks and at least three neutral clusters where one of the clusters has energy greater than 250 MeV in every event. The most important source of background to the forbidden φ → ωγ decay is the + − initial state radiation process (ISR) e e → ωγISR which has the same final state particles but, in contrast to the φ → ωγ decay, conserves C-parity. Understanding this process in the data sample is the first step towards constraining the branching ratio of the φ → ωγ decay, which can be done in multiple ways. + − Prior to the analysis of the data sample, a feasibility study for the e e → ωγISR process was done as a prestudy to determine whether a sufficient number of events were collected for the improvement of the upper bound for the branching ratio of the φ → ωγ decay. The φ decay could still occur via the weak force. However, such a decay would be heavily suppressed and the expected branching ratio is much

7 smaller than the current upper limit. A larger branching ratio of the φ → ωγ decay than expected could imply new physics. In this thesis, I have also investigated whether the data sample contains suffici- ent number of ω- for studies of the rare ω → l+l−π0 conversion decay where l± stands for a . This decay could play a significant role in the theoretical prediction of the magnetic moment of the which, to this date, does not agree with experimental results [3]. With a sufficient number of ω → l+l−π0 decays a competitive measurement of the transition form factor could be performed. This could lead to an improved prediction of the anomalous magnetic moment of the muon in agreement with experiment or point towards physics beyond the Standard Model. The thesis is structured as follows. In Section 3, the theoretical background and tools are presented. Section 4 introduces the experimental facility DAΦNE, the KLOE detector as well as the software tools used in this thesis. In Section 5, + − the feasibility study of the e e → ωγISR process is presented. In Section 6, the analysis of the ISR process in the KLOE data sample is performed. In Section 7, a summary and a discussion of the results are given. In Section 8 the feasibility study of the ω → l+l−π0 conversion decay produced in the ISR as well as the e+e− → ωπ0 production channel is performed. In Section 9, an outlook is given. A conclusion is given in Section 10.

8 3 Theoretical Introduction

In this section, the theoretical background for the study of ω meson production channels and decays present in the DAΦNE experiment is presented.

3.1 The Standard Model The Standard Model is a framework describing elementary particles and their interactions using quantum field theories. In 1961 Sheldon Glashow discovered a way of combining the electromagnetic and into one unified theory called the electroweak theory [4]. In 1969 Steven Weinberg and Abdus Salam [5] incorporated the Higgs mechanism [6] and later quantum chromodynamics (QCD) for the strong interaction[7] [8] developed in 1974 was included after the existence of was discovered [9] [10]. The Standard Model is a gauge quantum field theory with internal symmetries based on a local gauge group SU(3) ⊗ SU(2) ⊗ U(1) (1) where SU(3) corresponds to the strong interaction, SU(2) corresponds to the weak interaction and U(1) corresponds to the electromagnetic interaction. The Standard Model Lagrangian includes kinetic terms which describe particle motion, and mass terms which describe the particle mass. The particle couplings are represented by interaction terms. In Fig.11 all elementary particles included in the Standard Model are presented. The gauge are mediators of their corresponding forces and are respon- sible for particle interactions. There are twelve vector bosons ( 1) and one scalar (spin 0), shown in red and yellow in Fig.1: • Photons (γ), the which mediates the electromagnetic force • 8 (g), the vector bosons which mediates the strong force • W +, W −, and Z0, the vector bosons which mediates the weak force • , the which arises from the Higgs field that generates masses of the elementary particles In addition to the scalar- and vector bosons the Standard Model includes twelve spin 1/2 ; six quarks and six , shown in purple and green in Fig.1. which constitute the observable matter in our universe. The electron e, muon µ and tauon τ interact via the weak or electromagnetic force. Their counterparts can only interact via the weak force. The quarks carry, in addition to , color charge and can also interact via the strong force.

1Picture taken from Wikipedia - Standard Model of elementary particles. Accessed 29 July, 2014

9 Fig 1: Elementary particles included in the Standard Model.1

The electric charge of a particle can be either positive or negative. However, the color charge comes in six varieties; red, blue and green as well as their respective anticolor antired, antiblue and antigreen. Quarks (antiquarks) carry one distinct color (anticolor) while the gluons consist of a superposition of color-anticolor pairs. Color charged particles exhibit a phenomenom called confinement. Unlike the electromagnetic force whose strength (i.e. its coupling constant) decreases with distance (i.e. increases with energy), the strength of the strong force increases with distance. This is due to the itself having color charge, in contrast to the electrically neutral photon, and thus being able to self couple. The consequence is that as two quarks are being separated, one will eventually reach a point where it is energetically more favorable for new bound states to be created than to separate the quarks further. Thus quarks can never be observed as free par- ticles, but always confined in bound, colorless states called . Color charged particles also exhibit asymptotic freedom. As the distance decreases and energy increases, the strong coupling constant gets smaller and the quarks behave as free particles. Until very recently, all observed hadrons were found to be consistent with either quark- antiquark combinations, or three-quark states. and antibaryons are three-quark and three-antiquark bound states, respectively, while a meson is a quark-antiquark . The hadrons are ”colorless”, which means that the sum of the color charges of the constituent quarks add up to white e.g. red, blue

10 and green for a or blue and antiblue for a meson add up to be color neutral (white), analogous to the properties of the three colors humans can percieve. The hadrons are ordered according to their symmetry group representations into multiplets. In these multiplets, the hadrons are presented according to their quantum numbers. This thesis will treat the lightest pseudoscalar (total spin 0 and odd parity) and vector mesons (total spin 1 and odd parity) involving only u, d and s quarks. The lightest pseudoscalar- and vector mesons can be represented in a multiplet based on their charge and . These representations are called nonets and are shown in Fig.22 and Fig.33.

Fig 2: The nonet.2 Fig 3: The nonet.3

3.2 Form Factors At high energies the strong coupling constant becomes small and perturbative QCD (pQCD) can be applied The predictions of pQCD have been successfully tested by experiment. At low energies, the coupling constant grows larger which means that QCD becomes non-perturbative. Alternative methods have to be ap- plied in order to make theoretical predictions. Some of the more common methods are effective field theories (EFT) such as Chiral Perturbation Theory (χPT), nu- merical models like Lattice QCD (LQCD), and phenomenological models such as Vector Meson Dominance (VMD). Performing an explicit calculation in these frameworks is not possible. Form factors are one way to describe low energy QCD processes and are usually intro- duced, especially in EFT’s. The form factor, usually given as a function of the momentum transfer q2, describes scattering processes without a complete picture of the underlying physics. It parametrizes the matrix element and is written as the difference between a point-like and the experimentally observed cross section dσ dσ exp = point |F (q2)|. (2) dq2 dq2 2Picture taken from Wikipedia - Meson. Accessed 31 October, 2014 3Picture taken from Wikipedia - Meson. Accessed 31 October, 2014

11 Thus, the form factor is a quantity that is measurable in experiments. Usually form factors are classified as either space-like or time-like depending on the momentum transfer in the Feynman diagram they appear in. A Feynman diagram is constructed by writing down the intitial and final state particles and then connecting them with propagators and vertices which are allowed by con- servation laws e.g. charge, , flavor (in the vertex with a connected photon). In Fig.4 space-like and time-like form factors are shown in an s-channel and a t-channel Feynman diagram, respectively.

γ∗(q) f γ∗(q) f

Fig 4: Left: Feynman diagram which give rise to a Space-like form factor. Right: Feynman diagram which give rise to a Time-like form factor.

3.3 Symmetries Symmetries are an important part of the Standard Model. A symmetry can be understood as a quantity of a system that remains unchanged when the system undergoes a transformation. Theoretically, the symmetry of a theory can be chec- ked by applying the corresponding transformation to the lagrangian [11]. If the transformations leave the lagrangian unchanged, the symmetry is preserved in the theory. Symmetries provide a powerful tool to understand whether a certain reaction is expected to occur or not. Predictions using symmetries have introduced a plet- hora of precision experiments that have been or will be carried out to improve our understanding and the viability of the symmetries. Parity is an example of a symmetry which corresponds to the flip of the signs on all spatial coordinates (x → −x, y → −y, z → −z). Charge conjugation is another symmetry where all charges, e.g. electric charge, , strangeness etc. changes sign. In principle, this turns a particle into its own . Its corresponding operator returns the eigenvalue, the C-parity ηC = ±1, when acting on a state.

3.4 C-violating φ → ωγ decay The φ → ωγ is a C-violating decay. This can be seen by looking at the C-parity ηC of each particle. For a system of n (free) particles, the C-parity is given by

C |ψ1...ψni = ηC1 ...ηCn |ψ1...ψni . (3)

12 The φ, ω, and γ all have ηC = −1 [12]. Looking at the decay, the initial state has C-parity −1 but the final state has C-parity +1, since for free particles ηC is multiplied to obtain the C-parity of the system. The reaction should not occur in strong and/or electromagnetic interactions as these conserve charge parity. The reaction could still take place via the weak interaction since it does not conserve charge parity, but the branching ratio would be very small. This can be estimated from one of the leading order Feynman diagrams given in Fig.5.

γ s u u

W −

s u

Fig 5: Feynman diagram of a φ → ωγ decay. The φ and ω mesons are represented by their quark constituents ss and uu, respectively.

To get a rough idea of how suppressed this reaction is one can look at the number of vertices in the Feynman diagram. Each vertex connected to the γ propagator introduces a coupling constant factor e, the electron charge, to the matrix element. Likewise, an electromagnetic coupling constant factor g, which is related to e by the Weinberg angle e = g sin θW , appear in the matrix element from the vertex connected to the W − propagator. The W − propagator introduces a mass factor 2 1/MW which greatly suppresses the matrix element. however, the calculation of this Feynman diagram is beyond the scope of this thesis.

3.5 Initial State Radiation One of the main focus of this thesis is Initial state radiation (ISR). ISR occurs when a photon is emitted from one of the leptons before they annihilate in an e+e− collider experiment such as DAΦNE. The ISR process can produce one or several photons before annihilation. The leading order Feynman diagram (one emitted photon) is shown in Fig.6.

e+ γ∗ ω

e−

e− γISR

+ − Fig 6: Feynman diagram of the leading order e e → ωγISR production channel.

13 For the leading order diagram, the probability of emitting a photon is given by the radiator function [13]:

x2 2 x2 4 α (1 − x + 2 ) sin θ − 2 sin θ w0(θ, x) = [ 2 − πx 2 4me 2 2 (sin θ + s cos θ)

2 2 2 4 4me (1 − 2x) sin θ − x cos θ − 2 ], (4) s 2 4me 2 2 (sin θ + s cos θ) where s is the center of mass energy squared of the colliding beams, α is the fine-structure constant, me is the electron mass, and x is the fraction of energy transferred from the beam to the ISR photon which can be expressed as s − Q2 x = , (5) s where Q2 is the squared of the hadronic system. The probability of an ISR photon to be emitted within a certain polar angle region is obtained by integrating the radiator function

Z π−θ0 W0(θ0, x) = w0(θ, x) sin θdθ. (6) θ0 The normalized probability distribution function is then defined as

W0(θ0, x) P (θ0, x) ≡ (7) W0(0, x)

3.6 Decay kinematics and differential decay rates In quantum field theory particle decays and scattering processes are described by the differential decay rate. Kinematical distributions of particles, such as momen- tum or scattering angle, can be calculated from the differential decay rate. For a particle with mass M decaying into n particles with masses mi,(i = 1, ..., n), three-momenta ~pi and energy Ei, the differential decay rate is given by (2π)4 dΓ = |M |2dΦ (P ; p , ..., p ), (8) 2M 1→n n 1 n where M1→n is the Feynman matrix element and P is the initial particle four- momentum. The dΦn element is the n-body phase space given by

n n 3 X Y d ~pi dΦ (P ; p , ..., p ) = δ4(P − p ) . (9) n 1 n i (2π)32E i=1 i=1 i

14 In particular, for a two-body decay into particle 1 and 2, the differential decay rate is given by [12] 1 |~p | dΓ = |M |2 1 dΩ, (10) 32π2 1→2 M 2 where dΩ = dφ1d(cos θ1) is the solid angle element of particle 1 in the initial particle rest frame. The Feynman matrix element depends on the Lorentz invariant combination of 2 2 the particle four momentum (excluding pi = mi constants). The matrix element 1 can depend on 2 n(n − 1) − 1 variables. For a two-body decay, the matrix element is constant. In the case of a three-body decay the matrix element depends on two indepen- 2 2 2 2 dent variables m12 = (p1 + p2) and m23 = (p2 + p3) . The differential decay rate of a three-body decay can be written as [12]: 1 1 dΓ = |M |2dm2 dm2 , (11) (2π)3 32M 3 1→3 12 23 where the matrix element has been averaged over initial spin states. The limits of 2 2 m23 for a given value of m12 is given by energy and momentum conservation: q q 2 2 ∗2 ∗2 2 ∗2 2 ∗2 2 (m23)max = (E2 + E3 ) − E2 − m2 − E3 − m3 , (12) q q 2 2 ∗2 ∗2 2 ∗2 2 ∗2 2 (m23)min = (E2 + E3 ) − E2 − m2 + E3 − m3 , (13)

∗2 2 2 2 ∗2 2 2 where E2 = (m12 −m1 +m2)/2m12 and E3 = (M −m12 −m3)/2m12 are energies of particle 2 and 3 in the p1 + p2 rest frame.

3.7 The ω → π+π−π0 decay The dominant decay channel of the ω meson is the three- decay ω → π+π−π0 with a branching ratio of BR(ω → π+π−π0) = 89.2±0.7% [12]. When searching for decays involving ω mesons e.g. φ → ωγ it is beneficial to select the most common decays to optimise the statistical precision. Taking particle spin into account, the differential decay rate of the ω → π+π−π0 decay can be written as [14]

2 2 2 2 2 2 2 2 2 dΓ mπ+ mπ− mπ0 + 2p12p23p13 − mπ+ p23 − mπ− p13 − mπ0 p12 2 2 = 3 3 , (14) dm12m23 3 · (2π) · 32mω

15 where the following variables are introduced (p · p )2 − m2 − m2 p = 1 2 π+ π− 12 2 (p · p )2 − m2 − m2 p = 2 3 π− π0 23 2 (p · p )2 − m2 − m2 p = 1 3 π+ π0 . 13 2 When studying a three-body decay, it is convenient to graphically represent the kinematics using a Dalitz plot. A Dalitz plot is a two-dimensional plot where the 2 2 axes of the plot are the Lorentz invariant variables m12 and m23. By using the differential decay rate Eq.(14) the Dalitz plot density distribution is well repro- duced according to experimental results in contrast to what a flat phase space distribution would yield.

3.8 The ω → l+l−π0 conversion decay The ω → l+l−π0 conversion decay can provide information about the hadronic contribution to the anomalous magnetic moment of the muon [3]. The branching ratio of the decay is BR(ω → e+e−π0) = (7.7±0.6)×10−4 and BR(ω → µ+µ−π0) = (1.3 ± 0.4) × 10−4 [12]. The differential decay width of the hadronic light-by-light decay ω → l+l−π0 is given by [15]

dΓA→Bl+l− 1 1 + − 2 2 2 = 3 3 |MA→Bl l | (15) dml+l− dml−B (2π) 32mA where M is the matrix element given by [16] h  2 1 4 2 2 2 2 2 2 4 4 4 |M + − | = e |f (q )| q + 2(m − m − m ) + 2m + m + m A→Bl l 3 AB 23 B A l B A  1 − 2m2(m2 + m2 ) − 2(2m2 + m2 + m2 )m2 + 2m4 l B A l B A 23 23 q2 1 i + 2m2(m2 − m2 )2 , (16) l A B q4 2 2 where e is the elementary charge, fAB(q ) is the transition form factor and q is 2 the dilepton invariant mass ml+l− . A normalized form factor is defined as 2 2 fAB(q ) FAB(q ) ≡ (17) fAB(0) such that FAB(0) = 1. The normalization factor fAB(0) is related to the partial decay width of the A → Bγ decay through the following expression [17] 2 2 3 2 (mA − mB) e 2 ΓA→Bγ = 3 |fAB(0)| . (18) 96πmA

16 At low energies, QCD becomes non-perturbative due to asymptotic freedom and other models have to be used. VMD is a theory developed by J.J. Sakurai in 1960 [18] and has successfully predicted the transition form factor of π0-, η- and the η0-meson [19]. However, in the case of the ω-meson, the situation is different. As the invariant for the lepton pair ml+l− increases, the transition form factor for the ω → l+l−π0 decay predicted by VMD diverges from the experimentally measured form factor obtained in the NA60 experiment as shown in Fig.7.

Fig 7: NA60 measurement of the transition form factor [20]. The older measurement performed 2 by the Lepton G experiment is also shown. M is the dilepton invariant mass ml+l− .

In the NA60 as well as the Lepton G measurement [21], the form factor is obtained from fitting using the pole approximation [22]

2 2 2 2 −2 |FAB(q )| = (1 − q /Λ ) , (19) where Λ is the fitted constant. An alternative expression of the form factor is derived in [17]. In this scheme, a more detailed evaluation of the VMD contribution and systematic corrections to it are introduced. The transition form factor in this scheme is given by

17 2 2 2 2 mρ + q |FAB(q )| = 2 2 , (20) mρ − q where mρ is the ρ-meson mass. This form factor describes the NA60 data well and gives a simple way of evaluating the form factor for all values of q2. Thus, this form factor is used in the feasibility studies in Section 5.

3.9 The π0 → γγ decay The neutral pion is the lightest observed meson. The neutral pion decays electro- magnetically into two photons almost exclusively, BR(π0 → γγ) = 98.823±0.034% [12]. For the same reason as the ω → π+π−π0 decay it is convenient to tag π0- mesons through this decay.

18 4 The KLOE detector at DAΦNE and Software Tools

The DAΦNE (Double Annular Φ factory for Nice Experiments) is an e+e− collider located at the LNF (Laboratori Nazionali di Frascati) in Frascati, Rome. The col- lider was primarily constructed for direct CP violation studies in produced from φ meson decays [2]. The KLOE (K LOng Experiment) detector, named after the Greek novel Daph- nis and Chlo¨e, is a general purpose detector, located at one of DAΦNEs interaction points, designed to study physics and φ meson decays as well as other topics. FINUDA (FIsica NUcleare a DAΦNE) and DEAR (DAΦNE Exotic Re- search) were subsequently placed at the second interaction point of DAΦNE for hypernuclear spectroscopy and K-mesic hydrogen research.

4.1 The DAΦNE Collider DAΦNE is designed to operate with a luminosity up to 5 · 1032cm−2s−1. The and are accelerated in a two-stage linear particle accelerator (LINAC) to an energy of 510 MeV with a repetition rate of 50 Hz [2]. The leptons are then fed into the accumulator ring where they are cooled and stacked into bunches. The DAΦNE facility is shown in Fig.8.

Fig 8: Overview of the DAΦNE facility [23].

DAΦNE has two separate rings for electron and positron beams where the bunches are injected. This minimizes the undesired beam-beam interactions. The rings have a circumference of 98 m and intersect at two points at an angle of (π − 0.025) rad. In the KLOE interaction region, the bunches have a horizontal spread of σx ≈ 2 mm and a vertical spread of σy ≈ 10 µm.

19 4.2 The KLOE Detector The KLOE detector, shown in Fig. 9, was designed to measure CP violation in kaon decays. The large drift chamber (DC) has 12,582 drift cells arranged in 58 stereo layers. The cells in the 12 innermost layers have a cross section of 2×2 cm2, whereas the 46 outer layers are 3 × 3 cm2. The stereo angle vary between 60 and 150 mrad, and increases with the layer radius. In total, the DC consists of 52,140 wires 3.3 m in length and an inner and outer radii of 25 and 200 cm, respectively [24]. The DC has a spatial resolution of σz = 2 mm in the beam direction and

Fig 9: Cross-sectional view of the KLOE detector. The interaction region is surrounded by the drift chamber (DC), the electromagnetic calorimeter (EMC), the superconducting coil followed by the magnet return yoke [25].

σxy = 150 µm in the plane transverse to the beam. The momentum resolution of ◦ the DC for charged particles with large scattering angles (θ > 45 ) is σp/p < 0.4%. Close to the interaction region, two focusing quadrupoles are located. In order to increase the hermeticity of the KLOE detector, the quadrupoles are surrounded by two tile calorimeters (QCAL). The QCAL has the shape of a hollow cylinder connected to a truncated cone surrounding the quadrupole [26]. The electromagnetic calorimeter (EMC) surrounds the DC and has a solid- angle coverage of about 94%. The EMC consists of three parts; the calorimeter

20 barrel and the two calorimeter endcaps. There are 24 and 32 detector modules in the barrel and in the endcaps, respectively. These modules are built from layers of 1 mm scintillating fibres and 0.5 mm thick lead foils in between. The scin- tillating fibres in the modules run parallel to the beam-pipe in the barrel and perpendicular in the endcaps. All signals are read out at both ends of the mo- dules with 4880 photomultiplier tubes. The EMC has an energy resolution of p p σE/E = 5.7%/ E (GeV) and a time resolution of σt = 54 ps/ E (GeV)⊕140 ps. The position resolution transverse to the fibres is σxy (barrel) = σxz (endcap) = 1.3 p cm and σz (barrel) = σy (endcap) = 1.2 cm/ E (GeV) for the coordinate along fibres [27]. To filter out noise events, a two-level trigger is used. The trigger conditions are based on information from both the EMC and the DC. The second level trigger initiates the event readout and is satisfied either i) by two cluster hits in the EMC with energy deposit of > 50 MeV in the barrel and > 150 MeV in the endcap or ii) by at least 120 hits in the DC within a 1.2 µs time window [28]. The event information is recorded and stored in a tape library by the data acquisition system (DAQ). The DAQ is designed to sustain a trigger rate of 104 events per second. Given a luminosity of 1032 cm−2s−1, events are recorded at around 2200 Hz [29, 30].

4.3 KLOE Monte Carlo program GEANFI In particle accelerator experiments, Monte Carlo simulations are an essential tool to investigate particle behaviour in the detector. A simulation of an experiment is started with an event generator. Event generators are used to simulate physical re- actions expected to occur in the experiment and produces four-momentum vectors and vertex positions for all particles that were produced. The event generator does not take into account the detector or any other material that could interfere with the reactions. After the event generation, particle propagation through the detector are simu- lated. This process takes into account particle interaction with the detector e.g. influence of the magnetic field, scattering in non-sensitive and active detector ele- ments, pair production or creation/decay including all known effects. At the end of this step, the simulation output is digitized so that it replicates the signal output from the detector. After the digitization, the reconstruction begins. In this step the hits in the DC from charged particles are used to construct tracks which then gives the track momentum. The hits from particles in the EMC crystals are used to construct clusters which contains the position and deposited energy. In KLOE, the simulation and digitization is done using the KLOE Monte Carlo program GEANFI. It is based on the GEANT 3.21 library [31], a simulation pac- kage for particle transport through matter, which is widely used in

21 experiments.

4.4 Data Sample used in this Thesis The data collected with KLOE corresponds to an integrated luminosity of L = 1.6 fb−1 which was collected between 2004 and 2005. It would be time consuming and inefficient to perform an analysis using directly the collected data files. To circumvent this problem, it is common to select a portion of the data containing the desired physical processes with simple set of criteria called preselection. The preselected data which are used in the analysis were originally intended for φ → ηγ studies where the η-meson was tagged by the η → π+π−π0 decay and the π0-meson by the π0 → γγ decay. The preselection criteria which were applied are: 1. Two charged tracks with opposite curvature 2. At least three neutral clusters in the EMC 3. One neutral cluster with deposited energy in the EMC E > 250 MeV

The first two preselection criteria are used to filter√ unwanted events with different final state topology than the η → π+π−π0. For s = 1019 MeV the monochromatic photon of the φ → ηγ decay has the energy Eγη = 363 MeV. The third preselection condition is intended to suppresses background events which do not have high energy photons. In the reconstruction stage, each event is classified into seven different catego- ries, or streams, [32] based on detected particles, energies, momentum etc. This is done by declaring a boolean variable and set it to true if the event fufills the criteria of the corresponding stream. The categories are defined based the most common processes present in DAΦNE, e.g. a φ → K+K− decay should end up in the φ → K+K− stream, although an event could be categorized into none, one or multiple streams. The seven streams are: • Stream 1 (KPM) φ → K+K−

• Stream 2 (KSL) φ → KLKS • Stream 3 (RPI) φ → ρπ + π+π−π0 • Stream 4 (RAD) φ → ηγ, η0γ, π0γ.f 0γ • Stream 5 (CLB) Bhabha and Cosmic events (for calibration) • Stream 6 (UFO) Unidentified Objects • Stream 7 (BHA) Bhabha Scattering events Because these data sample were preselected with similar cuts as the φ → ηγ, i.e. the RAD stream, only these events are selected in the analysis in Section 6.

22 4.5 ROOT In this thesis, the software package ROOT [33] is used. ROOT is a C++ framework developed at CERN and is commonly used for data analysis in particle physics experiments. It includes packages for histogramming, four vector calculation, phase space simulation among other. The event generators developed in this thesis described in Section 5 and 8 are constructed with ROOT. ROOT is also used in the analysis of the KLOE data sample described in Section 6.

4.6 Phokhara Phokhara [34] is a Monte Carlo event generator which simulates ISR as well as Final State Radiation (FSR) for certain final states for e+e−processes. The generator can simulate the ISR process up to Next-to-Leading Order (NLO). Among the 0 possible final states Phokhara can simulate are K+K−, K0K ,ΛΛ, π+π−π0. In the input file, the following information is specified: • Number of events to generate • LO or NLO simulation • The final state which shall be simulated • Inclusion of FSR • Experimental settings such as center-of-mass energy or angular cut on the ISR photon When FSR is included the interference effect between ISR and FSR is auto- matically taken into account, however the presence of FSR is beyond the scope of this thesis. By default, Phokhara provides the cross section and its error as a function of q2 which, in this case, is the invariant mass of the hadronic system. Additional information can be specified as well.

23 + − 5 Feasibility study of the e e → ωγISR reaction

+ − 0 The ωγISR → π π π γISR production channel has the same final state as the forbidden φ → ωγ decay and is the main background. Because DAΦNE operates at the φ-meson mass, the photons from the ISR process and the forbidden φ decay will have the same energies, making them hard to separate. If the ISR process is well undersood in the KLOE data sample, then the φ → ωγ decay can be separated from the background and branching ratio of the decay can be determined with much higher accuracy with e.g. background subtraction or signal to background fitting. The aim of this thesis is to study the ISR reaction in the KLOE data sample. + − 0 + − 0 The φ → ηγ → π π π γ and ωγISR → π π π γISR decays have the same final + − state. However, for the e e → ωγISR process the monochromatic photon has the energy EγISR = 209 MeV and will therefore not fulfill the preselection condition. The ISR events can however still pass the preselection criteria if any of the two photons originating from the π0 decay has E > 250 MeV or when the ISR photon energy is not correctly reconstructed in the detector. To be sure that it is worthwhile to analyse the ISR process in the KLOE data sample, a feasibility study should first be performed to estimate the number of ISR events. In this section an ISR event generator is constructed and the expected yield of ISR events in the KLOE data sample is presented. Generating a new data sample is beyond the scope of this thesis, however a new set of preselection criteria are suggested in this section.

5.1 Constructing the ISR event generator The decay scheme that is constructed in this section is shown in Fig.10. The ω- and the π0-meson are tagged by their most common decay mode. The charged

γ γISR π0 γ + ω π π−

Fig 10: Feynman diagram of the ISR process and the subsequent decays that is constructed in the generator. can only decay through the weak interaction. As a consequence, the charged pions have a lifetime of ∼ 10−8s and will (most likely) reach the electromagnetic calorimeter before decaying and thus their eventual decay is not taken into account.

24 + − For the e e → ωγISR production channel simulation, the ω and γISR momenta are generated with a combined invariant mass of 1019 MeV/c2. The ISR photon in generated isotropically:

px = Eγ sin(θ) cos(φ)

py = Eγ sin(θ) sin(φ) (21)

pz = Eγ cos(θ) and the ω-meson is set to scatter back-to-back to the ISR photon. The events are then assigned a weight based on the radiator function Eq.(4). Since the radiator function peaks at 0 and 180 degrees, the sampling would require a huge number of events to achieve a sufficent statistics. This problem is reduced by generating only ISR photons within the KLOE detector geometric acceptance 23◦ < θ < 157◦. The simulated polar angle distribution of the ISR photons are shown in Fig.11

Fig 11: Simulated polar angle distribution of ISR photons.

The ω-meson decay is simulated by generating the three pions according to a phase space distribution in the ω-meson rest frame. The generated three pions events are then assigned weights calculated from the differential cross section Eq.(14) where spin is taken into account. The simulated Dalitz plot is shown in Fig.12 The π0 decay is simulated to decay into two photons with a phase space distri- bution. The photon energy distribution is shown in Fig.13

5.2 Acceptance One million events are generated with the ISR generator. The total acceptance is calculated by taking the ratio between number of events that passed the prese-

25 Fig 12: Dalitz plot of the ω → π+π−π0 decay taking spin of the ω into account. The invariant mass squared of π+ and π0 is plotted against the invariant mass squared of π+π− lection criteria and the total number of events. Taking into account the weight of each event the efficiency becomes P pass wi  = P = 0.1108 ± 0.0006. (22) all wj The statistical uncertainty is obtained by running the simulation ten times. The collected  values are assumed to have a Gaussian distribution and from them, the mean value and the standard deviation of the acceptance  is calculated. It is important to note that this simulation, effects from the detectors, i.e. detector resolutions and efficiencies, are not taken into account. This result therefore cor- responds to the hypothetical case with an ideal detector, having 100% efficiency and ideal resolution. The acceptance calculated in Eq.(22) gives the detection probability of an event where the ISR photon is emitted inside the KLOE geometry. Therefore, a correc- tion must be added to the efficiency to compensate for this. The probability of an ISR photon to be emitted within the KLOE detector region (not taking into ac- count the tile calorimeters) is calculated analytically using the probability function Eq.(7) W0(θ0, x) ISR = = 0.21, (23) W0(0, x) √ ◦ 2 2 where s = 1019 MeV, x = 0.411, θ0 = 23 and Q = mω have been used. The total efficiency then becomes

tot = ISR ×  = 0.0231 ± 0.0001. (24) To calculate the number of events in the data sample, the ISR cross section + − is needed. The e e → ωγISR cross section is calculated using Phokara event

26 Fig 13: Simulated energy of photons in the KLOE reference frame originating from the π0 → γγ decay. Small values in the y−axis are due to the weights applied to every events. generator and is found to be equal to σ = 2.36 nb. The number of events expected in the data sample is calculated using the following equation,

+ − + − + − 0 N(e e → γISRω → 3γπ π ) = Lσ × BR(ω → π π π )×

0 ×BR(π → γγ) × tot = 76 000 ± 720 (25) −1 where L = 1.6 fb , and the efficiency tot is from Eq.(24). The uncertainties in the number of events is calculated by error propagation. The error of the PDG branching ratios and the efficiency is taken into account and they are assumed to be uncorrelated. The π0 photon energy spectrum in Fig. 13 reveals that there are a few events passing the energy deposit preselection condition, thus one can expect ISR events in the real data sample when performing an analysis. However, this could potentially lead to high systematic errors. An analysis can still be carried out a new data sample should be created with improved preselection criteria in the future.

5.3 Preselection criteria for a potential new data sample The potential to increase the event yield in a new data sample is also simulated by varying the energy requirement as well as the number of required neutral clusters in the preselection criteria. The resulting efficiencies are shown in Table 1. The efficiencies for the alternated preselection cirteria are calculated by simulating 10 000 000 events and the uncertainty was obtained from the mean efficiency of five simulations. Here, the ISR photon is generated for all scattering angles. From Table 1 it is obvious that the preselection criteria applied to the KLOE + − data sample heavily suppresses the e e → ωγISR process. Preselecting a new data sample where the number of required photons or the energy of the highest

27 energetic photon is lowered could increase the event yield with a factor of up to 30.

+ − EγMax (GeV) N(γ) (γISRω → 3γπ π ) 0.25 3 0.0231 ± 0.0001 0.25 2 0.105 ± 0.007 0.20 3 0.149 ± 0.003 0.20 2 0.385 ± 0.013 0.15 2 0.579 ± 0.010 0.10 2 0.698 ± 0.007

+ − + − 0 Table 1: Efficiencies for e e → ωγISR with ω → π π π where the energy requirement and the number of neutral cluster are varied.

28 6 Monte Carlo analysis of the ωγ final state in the KLOE detector

In the last section it was shown that although the photon energy preselection + − criterion heavily suppresses the signal, a significant number of e e → ωγISR events are expected in the KLOE data sample. This feasibility study served as a check and a motivation to continue the analysis in the KLOE data sample. In this + − section, the e e → ωγISR process is studied as a background process in KLOE data sample for the search for the C-violating φ → ωγ decay. + − In an e e collider operating at the mφ peak the main background comes from + − e e → ωγISR. This process has the same final state particles and is allowed as it does not violate C-parity. The first step in the search for the φ → ωγ is to understand the background. Therefore, cuts are applied to maximise the yield of the ISR process and minimise the yield of other reactions. The criteria are developed from Monte Carlo simula- tions and then applied to the real data. The purpose of these cuts is simplify the + − analysis of the channel of interest, in this case, the e e → ωγISR.

6.1 Reconstructing of four-momentum The first step in the analysis is to reconstruct the four-momenta of all particles in every event using the track momentum and energy/position information from the clusters. All charged tracks are assumed to be charged pions. The four-momentum vectors are then constructed using the measured three-momentum of the track and assign m = mπ± to get the complete kinematic information of the charged particle. Photons are reconstructed using the energy deposit and the position of the cluster in the electromagnetic calorimeter. Only the clusters that are not associated to a charged track are used.

+ − 6.2 Extraction of e e → ωγISR signal from MC By Monte Carlo simulations that includes detector response, we learn about the reaction of interest and how to distinguish it from other, less interesting, reactions. If, for example, an invariant mass distribution from data which has a peak in it is described by two different Monte Carlo samples; one with a certain reaction which give rise to a peak and another sample that describes the continuous background, the one can confirm that the peak in the data originates from the reaction in the first MC sample. In the Monte Carlo files, there is no way to tell if an event was generated with or without ISR, which means that the clean sample of Monte Carlo simula- + − ted e e → ωγISR process cannot be obtained. Instead, these events have to be manually extracted.

29 All events have a boolean variable specifying which physics generator that was + − used [35]. A clean e e → ωγISR Monte Carlo sample is produced by first only selecting events generated with the φ → ρπ generator, the generator which is used to produce all ISR events with π+π−π0 in the final state. The sample is further refined by selecting events with exactly three pions (π+π−π0) and one ISR photon.

Fig 14: True invariant mass of π+π−π0. Only events classified as φ → ρπ with exactly three pions and one photon are selected.

+ − To confirm that only e e → ωγISR are left in the sample, the kinematic va- riables from the Monte Carlo simulation before the detector propagation, known as true variables, are studied. The true invariant mass of the three pions shown + − in Fig. 14 reveal that there are still non e e → ωγISR events in the data. These + − 0 events are removed by imposing a mass cut |M(π π π ) − mω| < 10 MeV. After the mass cut 70302 events are left.

6.3 Background rejection cut: 3 photons The first cut introduced in the analysis is to require each event to have exactly th- ree photons. This cut mainly suppresses background originating from e+e− → ωπ0, + − φ → K K and φ → KSKL. In addition, background channels which normally have three photons in the final states are excluded if the production was accompa- nied by additional ISR photon. The resulting number of events after the rejection cut are presented in Table 2 and 3.

30 6.4 Photon pairing The next step in the analysis is assigning the three photons in the final state as either the ISR photon or a decay product of the π0 decay. A pseudo-χ2 variable is defined 2 2 (mγγ − mπ0 ) χγγ = 2 (26) σmγγ 0 where mγγ is the calculated invariant mass of two photons, mπ0 is the π mass and 2 σmγγ is the measurement error squared. The measurement error is defined as σ 1 σ σ  mγγ = Eγ1 ⊕ Eγ2 (27) mγγ 2 Eγ1 Eγ2 where σEγi is the KLOE calorimeter energy resolution for photons and Ei is the photon energy.

2 Fig 15: Invariant mass of two photons with the smallest χγγ value for all data events.

For each event, the pseudo-χ2 variable is calculated for all possible photon pair combinations and the pair with the smallest χ2 value is assumed to originate from the π0 decay. Fig. 15 shows the invariant mass for these photons for every event. After the photon pairing, the third photon is assumed to be the monochromatic photon originating from the ISR process. To improve the correct combination selection, especially if the analysis would be expanded to encompass events with more than three photons, the χ2 variable would be modified by adding a term (E − E )2 γ3 γISR . (28) σ2 EγISR

31 This term would be used to find the photon with energy closest to the expected energy of an ISR photon.

6.5 Background rejection cut: Monochromatic photon energy The most dominating background channel is the φ → ηγ decay. Roughly half of the Monte Carlo sample consists of this decay. The monochromatic photon for this decay has the energy Eγη = 363 MeV, much larger than for the ωγISR process which has EγISR = 209 MeV. The difference in the energies of the monochromatic photons makes it easy to suppress the φ → ηγ channel by introducing a cut on the photon energies. The monochromatic photon in each event is required to have the energy |Eγ −209| < 40 MeV. The cuts are visualised as two vertical lines in Fig. 16 and Fig. 17. The resulting number of events after the rejection cut are presented in Table 2 and 3.

Fig 16: Monochromatic photon energy from Fig 17: Energy of the ISR photon from the + − the φ → ηγ decay channel. The vertical li- e e → ωγISR channel. The vertical lines nes denote the cut applied in the analysis. denote the cut applied in the analysis.

In Fig. 17, the effect of the preselection requirement is visible. The sudden steep peak at Eγ = 250 MeV is due to the requirement of the highest energetic photon to have Eγ,max > 250 MeV.

6.6 Background rejection cut: γω angle difference The next rejection cut introduced in the present analysis is to require the recon- structed ω-meson and the monochromatic photon to scatter back-to-back. Even after requiring exactly three photons in the final state does the e+e− → ωπ0 reac- tion still contribute significantly to the background when one of the four photons in the final state is not detected. + − The e e → ωγISR production channel is a two-body process with its center of momentum system equal to the lab system and thus the ω meson and the ISR photon will scatter back-to-back. Meanwhile, the third photon in e+e− → ωπ0 will have a broader range of angles between the outgoing ω- and π0-mesons. The

32 opening angle between the ω-meson and the supposed ISR photon candidate is ◦ required to be 6 γω > 170 . In Fig. 18 and Fig. 19 the cuts are visualized. The resulting number of events after the rejection cut are presented in Table 2 and 3.

Fig 18: Angle between the reconstructed Fig 19: Angle between the reconstructed ω- ω-meson and the third photon from the meson and the monochromatic photon from e+e− → ωπ0 process in the lab frame. The the ISR process in the lab frame. The verti- vertical line denotes the cut applied in the cal line denotes the cut applied in the ana- analysis. lysis.

Even though the photons from a π0 → γγ decay are distributed isotropically and emitted back-to-back in the π0 rest frame, a photon originating from the π0-meson either from the e+e− → ωπ0 reaction or from the subsequent decay ω → π+π−π0 will have an additional boost and will therefore not necessarily be emitted back-to-back with the reconstructed ω-meson. The resulting number of events after the rejection cut are presented in Table 2 and 3.

6.7 Background rejection cut: Charged particle identification (CPI) Until this point in the analysis, all charged tracks have been assigned the charged pion mass. However, a large amount of background comes from Bhabha scattering with additional radiated photons e.g. e+e− → e+e−γ. These events are removed by performing a charged particle identification. For each event two hypotheses are considered, one where the charged tracks are pions and the other when the charged tracks are electrons. Two time differences are then calculated,

∆tπ = (tπ − r/c) − tcluster (29)

∆te = (te − r/c) − tcluster (30)

where tπ,e are the calculated time of flight assuming a pion and an electron respectively, r is the track length, tcluster the measured travel time and c the . These two values are plotted against each other in Fig.20 and Fig.21. A cut is introduced where if ∆te > −0.5 for both tracks which excludes Bhabha scattering events.

33 Fig 20: The time differences ∆tπ and ∆te Fig 21: The time differences ∆tπ and + − plotted against eachother for e e → ∆te plotted against eachother for bhabha ωγISR events. The horizontal line denotes events. The horizontal line denotes the cut the cut applied in the analysis. applied in the analysis.

6.8 Efficiencies The number of events for all Monte Carlo channels after each analysis cut are summarised in Tables 2 and 3. Monte Carlo channels with less than 3000 events before the cuts are not displayed.

0 + − Signal ωπ K K KSKL ρπ preselection 70302 1141832 199917 1165465 691547 3 photons 67492 327910 116249 329289 611960 ∆Eγ < 40 MeV 34310 34276 9924 37643 51125 ◦ 6 γω > 170 25676 10308 949 3929 18180 CPI 25595 10217 929 3627 18092

Table 2: Number of events for each Monte Carlo channel after each applied cut. Monte Carlo channels with few events (< 3000 events before cuts) are not displayed.

0 + − a0γ η γ ηγ e e γ preselection 19995 22986 6373726 714130 3 photons 2315 20493 6103432 676534 ∆Eγ < 40 MeV 193 3834 141149 51237 ◦ 6 γω > 170 46 243 102157 41487 CPI 45 242 101548 5268

Table 3: Number of events for each Monte Carlo channel after each applied cut. Monte Carlo channels with few events (< 3000 events before cuts) are not displayed.

The efficiencies for each channel after every applied cut is shown in Table 4 and 5. Most background channels have been suppressed enough to be neglected when performing a comparison between Monte Carlo and the experimental data e.g. a0γ. Other channels still contribute a lot to the total sample after the cuts have been

34 applied and, in particular, the φ → ηγ channel is about four times as abundant as + − the e e → ωγISR.

0 + − Signal ωπ K K KSKL ρπ preselection 100 100 100 100 100 3 photons 96.0 28.7 58.1 28.3 88.5 ∆Eγ < 40 MeV 48.8 3.0 5.0 3.2 7.4 ◦ 6 γω > 170 36.5 0.9 0.5 0.3 2.6 CPI 36.4 0.8 0.5 0.3 2.6

Table 4: Efficiencies in % for each Monte Carlo channel after each applied cut. Monte Carlo channels with few events (< 3000 events before cuts) are not displayed.

0 + − a0γ η γ ηγ e e γ preselection 100 100 100 100 3 photons 11.6 89.2 95.8 94.7 ∆Eγ < 40 MeV 9.7 16.7 2.2 7.2 ◦ 6 γω > 170 0.2 1.1 1.6 5.8 CPI 0.2 1.1 1.6 0.7

Table 5: Efficiencies in % for each Monte Carlo channel after each applied cut. Monte Carlo channels with few events (< 3000 events before cuts) are not displayed.

35 + − 7 Results on Monte Carlo study of e e → ωγISR process

In the original Monte Carlo sample, there were 9.7·106 events, where 70302 events + − originated from the e e → ωγISR process. Most of the background comes from the φ → ηγ decay which can be seen in the large peak around the eta meson mass 2 mη = 548 MeV/c in the invariant mass of the three reconstructed pions in Fig. 22. Table 6 summarizes the efficiencies from the cuts introduced in Section 6. after preselection after cuts ISR 70302 25595 Background 9.6 · 106 134794 ISR/Background 0.0073 0.19

Table 6: Number of ISR events and background events as well as the ISR to background ratio.

After applying the selection criteria, 160389 events remain, which is 1.7% of + − the events of the full Monte Carlo sample, where 25595 e e → ωγISR events remain. This corresponds to ∼ 36% of the ISR events before any cuts. The signal to background ratio has thus been improved from 0.0073 before the analysis to 0.20 after the analysis. In Fig. 23 the invariant mass of the three pions after the event selection is shown. In both figures, the background (red histogram) is added + − on top of the e e → ωγISR process (green histogram).

Fig 22: Invariant mass of the three pions Fig 23: Invariant mass of the three pi- π+π−π0 before analysis. The spectrum is ons π+π−π0 after analysis. The number of + − dominated by background, most originating e e → ωγISR processes are comparable to from φ → ωη and e+e− → ωπ0 processes. the number of background events.

In Fig. 24, the scattering angle of the monochromatic photon is shown for the Monte Carlo sample. The background (red histogram) is added on top of the + − e e → ωγISR process (green histogram). On both spectra, there is a ”dip” in number of events around θ = 45◦, 135◦. This is probably due to the transition between the barrel- and the endcap calorimeters but it has yet to be confirmed.

36 Fig 25: Monochromatic photon scattering Fig 24: Monochromatic photon scattering angle for data after applying selection crite- angle for all Monte Carlo channels after ap- ria. The peak at the left end of the spectrum plying selection criteria. is yet to be understood.

In the data shown in Fig. 25 there is a sharp peak on the left side of the spectrum which is not described by the Monte Carlo sample. This needs to be looked into although due to time constraint this was not done in this thesis.

Fig 26: π+π− invariant mass for all Monte Fig 27: π+π− invariant mass for data after Carlo channels after analysis. analysis for data.

In Fig. 26 and Fig. 27 the invariant mass of the charged pions is shown for the Monte Carlo sample and the data, respectively. The background (red histogram) + − is added on top of the e e → ωγISR process (green histogram). Here, the Monte Carlo sample agrees with the data in shape. To get a qualitative picture of whether the Monte Carlo sufficiently describes the data the invariant mass of the ω meson in the Monte Carlo is normalized to the ω meson in the data sample with a weight to the Monte Carlo sample. This is shown in Fig. 28.

37 Fig 28: π+π−π0 invariant mass for data (black dots) and all Monte Carlo channels (colored histograms).

The invariant mass spectrum from the Monte Carlo and the data has a similar shape although the Monte Carlo distribution seem to be slightly shifted to the left. A detailed analysis of the cross sections used in the Monte Carlo sample has yet to be done.

38 8 Feasibility studies of ω conversion decays

+ − + − 0 The ISR channel e e → ωγISR and the e e → ωπ process are two of the most important production channels present at DAΦNE when it comes to production of ω-mesons. These two channels are therefore useful in studies of the ω-meson transition form factor in the ω → l+l−π0 decay. In this section, a feasibility study for ω conversion decays and, in general, the expected yield of omega mesons is presented. The motivation is to estimate the number of rare ω → l+l−π0 decays in the KLOE data sample. A high number of events could also be used for a competitive measurement of the ω-meson transition form factor. As a source for the ω-mesons, Initial State Radiation and the e+e− → ωπ0 production channels are considered. The expected number of ω → l+l−π0 decays in the data sample is studied by constructing similar event generators as in Section 5.

8.1 The event generators Event generators are constructed for each of the following reaction chains similarly + − + − 0 to the e e → ωγISR → π π π γISR generator described in Section 5: e+e− → [π0 → γγ][ω → e+e−[π0 → γγ]] → 4γe+e− (31) e+e− → [π0 → γγ][ω → µ+µ−[π0 → γγ]] → 4γµ+µ− (32) e+e− → [π0 → γγ][ω → π+π−[π0 → γγ]] → 4γπ+π− (33) + − + − 0 + − e e → γISR[ω → e e [π → γγ]] → 3γe e (34) + − + − 0 + − e e → γISR[ω → µ µ [π → γγ]] → 3γµ µ (35) When studying the ω → µ+µ−π0 decay, which has a branching ratio of BR(ω → µ+µ−π0) = (1.3 ± 0.4) × 10−2%, the abundant ω → π+π−π0 decay which has a branching ratio of BR(ω → π+π−π0) = 89.2 ± 0.7% would dominate the back- ground. Since pions and have masses close to each other, they are hard to distinguish. Thus this three pion decay channel is also studied to find out how this contribution compares to the ω → µ+µ−π0 signal decay. The e+e− → ωπ0 production reaction is simulated according to a phase space distribution. The ω → l+l−π0 decay configurations are first simulated with a phase space distribution and then weighted using the matrix element Eq.(16) and the form factor Eq.(20). The simulated dilepton invariant masses are shown in Fig. 29 and Fig. 30. The ω → π+π−π0 and π0 → γγ decays are simulated as in Section 5.

8.2 Acceptance

+ − + − 0 In Table 6 all total acceptances calculated for e e → ωγISR and e e → ωπ with the following decay channels ω → π+π−π0, ω → π0e+e− and ω → π0µ+µ− are

39 Fig 29: Invariant mass of e+e− from the ω → e+e−π0 decay.

Fig 30: Invariant mass of µ+µ− from the ω → µ+µ−π0 decay. presented. The error in the number of events are calculated in the same way as in Eq.(25) i.e. the errors in the PDG branching ratios and the statistical uncertainty of the efficiency are taken into account. From Table 6, it is noted that the number of expected ω → l+l−π0 events is probably not sufficient to enable an analysis on the ω transition form factor. In particular, at the DAΦNE energy ranges, charged pions and muons behave similarly due to similar masses and both particles usually being completely stopped in the calorimeter. Thus it is very hard to distinguish charged pions from muons and an analysis on ω → µ+µ−π0 will have a large background from the ω → π+π−π0 decay. It could be more prominent to study the ω-meson transition form factor in an experiment with dedicated muon detectors such as BES III. Then, the distinction between charged pions and muons would not be an issue.

40 Channel Events Efficiency e+e− → π0ω → 4γe+e− 3 000 ± 240 0.5711 ± 0.0018 e+e− → π0ω → 4γµ+µ− 240 ± 74 0.2684 ± 0.0004 e+e− → π0ω → 4γπ+π− 660 000 ± 5 700 0.1074 ± 0.0004 + − + − e e → γISRω → 3γe e 130 ± 10 0.0555 ± 0.0004 + − + − e e → γISRω → 3γµ µ 13 ± 4 0.0266 ± 0.0001

Table 7: Expected number of events and detection efficiencies for all channels. The uncertainties of the PDG branching ratios as well as the error in the efficiency is taken into account in the error of number of events.

8.3 Preselection criteria for a potential new data sample In the same way as in Section 5.3 the prospects of obtaining more ω → l+l−π0 decays by altering the preselection criteria is simulated. 10 000 000 events are ge- nerated and the error of the efficiency is calculated from five repeated simulations. In Table 7, the same behavior of the efficiency is observed. The event yield could be

+ − + − EγMax (GeV) N(γ) (γISRω → 3γe e ) (γISRω → 3γµ µ ) 0.25 3 0.0555 ± 0.0004 0.0266 ± 0.0001 0.25 2 0.247 ± 0.019 0.120 ± 0.002 0.20 3 0.160 ± 0.007 0.154 ± 0.002 0.20 2 0.445 ± 0.019 0.347 ± 0.008 0.15 2 0.517 ± 0.021 0.444 ± 0.007 0.10 2 0.600 ± 0.028 0.544 ± 0.012

+ − + − 0 + − 0 Table 8: Efficiencies for e e → ωγISR with ω → π π π and ω → l l π where the energy requirement and the number of neutral cluster are varied. increased by lowering the energy requirement and the number of neutral clusters in the preselection criteria.

41 9 Outlook

In this section, suggestions for further studies are presented.

9.1 New data sample The preselection criteria present in the data sample used in this thesis results in + − a very low efficiency for the signal process e e → ωγISR. Ideally, a new data sample should be prepared with modified preselection criteria to encompass a + − greater number of e e → ωγISR events. As seen in Table 1, the most obvious change to the preselection is the requi- rement on the highest energetic photon. By lowering the requirement threshold from 250 MeV to below 209 MeV (taking into account the energy resolution of the detector) or reducing the number of neutral clusters required, the signal can be increased while still keeping some background suppressing power. Additionally, a data sample recorded below the mφ threshold with the same preselection should be generated if one wants to perform an energy scan.

9.2 Kinematic fit One particular technique which is widely used in analysis of particle physics data is kinematic fitting. It was not applied to this analysis due to time constraints, but the concept is described here. Kinematic fitting is used to discriminate unwanted events and provide improved measured quantities within the measurement errors based on a priori knowledge e.g. energy-momentum conservation, particles connected to a vertex [36, 37]. The χ2 variable is defined as

N meas 2 M X (Vi − V ) X χ2 = i + λ H (V , ..., V ), (36) σ2 j j 1 N i=1 i j=1

meas where Vi is the variable to be fitted, Vi is the measured value, Hj are the imposed constraints, and λj are the lagrange multipliers. Because the charged particle momentum is measured with higher precision than the photons, only the photon variables have to be fitted. In KLOE, photons are reconstructed from the energy deposit and position of the corresponding cluster, and the time of flight. + − For the three photons in the final state of the e e → ωγISR process there are 15 variables:

1. the energy Ei

2. the interaction to cluster time, ti

42 3. the cluster position, xi, yi zi The constraints for energy-momentum conservation as well as the speed of light of photons are written as

1. ti − ri/c = 0 for all photons √ P 2. s = 3γ Eγ + Eπ+ + Eπ− P 3. 3γ ~pγ + ~pπ+ + ~pπ− = ~pb After the selection cuts have been applied, the most important background channel is the φ → ηγ decay. In many of these events, the energy of the monochromatic photon was measured with a much lower energy (< 250 MeV) and will thereby survive the energy cut applied in the analysis. The χ2 variable is by definition increasing the further a measured quantity is from the hypothetical value. Since the φ → ηγ decay should have a monochromatic photon with energy Eγ = 363 MeV, these events would yield a large χ2 value. Thus to exclude these events, an upper limit on the χ2 variable can be required for every event. The kinematic fit also provides improved values for all fitted quantities which can be used to improve the resolution of the final results at the end of the analysis.

9.3 Simulation of a hypothetical φ → ωγ decay It is important to have an understanding of the φ → ωγ decay in the KLOE data sample e.g. detector and selection efficiencies. This can be done by constructing an event generator for the φ → ωγ decay and then propagate these events through the KLOE detector. In BES3 similar searches for the C-violating decays J/Ψ → γγ and γφ have been performed [38]. In these searches, the decays are simulated according to phase space. The same assumption of the φ → ωγ distribution can be done to get a rough understanding of the behaviour of the process in the KLOE data sample. When performing more precise searches, a more realistic model of the φ → ωγ decay distribution might be required.

9.4 Fitting Monte Carlo sample to the data Once a satisfying Monte Carlo sample has been obtained, the number of e+e− → ωγISR processes in the data can be determined by fitting the Monte Carlo distri- butions to the data. This is done by comparing the spectrum of a quantity, e.g. invariant mass, of the Monte Carlo and data and assigning a weight and see if a good match between data and Monte Carlo can be achieved. A more powerful way to make sure that the Monte Carlo sample gives a suffici- ent description of the data, a two dimensional spectrum can be used, e.g. invariant mass of the charged pions plotted against the scattering angle of the monochroma- tic photon, to cross check that the Monte Carlo accurately describes all physics.

43 9.5 Energy scan

The same analysis can be performed on a data sample recorded below the mφ + − threshold. Since the e e → ωγISR process is well understood in this energy region, the presence of a φ → ωγ can be seen in a larger than expected ωγISR cross section at the mφ peak or a significant change in for example the scattering angle distribution of the monochromatic photon.

44 10 Summary and Conclusion

The ISR process is the main background in the search for the C-violating φ → ωγ decay. In this thesis, the ISR process was studied with the KLOE detector. The study confirms the presence of the e+e− → ωγ process in the DAΦNE experiment. The used data sample in this thesis is not ideal as the energy requi- rement in the preselection criteria heavily suppresses the process. By lowering the energy requirement and/or the required number of neutral clusters, the event yi- eld can be increased by a factor of up to 30. The same analysis performed on the KLOE data sample in this thesis can be used as a foundation for the search of the forbidden φ → ωγ decay. A prospects of measuring the transition form factor of the ω-meson with the KLOE detector was also studied. The number of expected events were found to be insufficient to perform a competitive measurement.

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