### Uppsala University

### Department of Physics and Astronomy

### Master Thesis

## Study of Discrete Symmetries in η ^{0} Meson Decays with BESIII

### Author:

### Viktor Thorén

### Supervisor:

### Andrzej Kupsc, Division of Nuclear Physics Subject Reader:

### Magnus Wolke, Division of Nuclear Physics

### September 20, 2017

Abstract

This thesis studies the rare decay η^{0} → π^{+}π^{−}e^{+}e^{−} using Monte
Carlo simulations and data from the BES-III detector in Beijing, China.

The branching ratio of the reaction was measured at BES-III in 2013
using a data set of225× 10^{6} J/Ψ events. This work lays the ground-
work for an updated branching ratio measurement using the full data
set of1.3× 10^{9} J/Ψ and determines a potential CP-violating asymme-
try in the angle between the decay planes of the π^{+}π^{−}- and e^{+}e^{−}-pairs.

A total of 2558 signal events are observed after cuts, and the asymme-
try parameter is determined to be Aϕ = (1.96± 1.97stat.± 0.4syst.)×
10^{−2}. The result is consistent with zero within the uncertainty.

### Populärvetenskaplig Sammanfattning

### Partikelfysikens Standardmodell (SM) beskriver hur materiens allra minsta beståndsdelar beter sig och interagerar med varandra. En uppsjö av experi- ment har visat att modellens beräkningar stämmer överens med verkligheten med hög precision. I ett flertal fall har SM dessutom förutsagt existensen av partiklar som senare upptäckts. Ett samtida exempel är Higgspartikeln som observerades för första gången vid CERN år 2014.

### Men vår kunskap om naturen är ännu inte komplett. Flera obesvarade frågor kvarstår, och delar av teorin behöver vidare studier. Till exempel beskriver SM bara tre av de fyra fundamentala krafterna: den starka, den svaga och den elektromagnetiska. Gravitationen är inte inkluderad. Inte heller har SM lyckats förklara vad mörk materia består av, eller varför det finns mer materia än antimateria i universum.

### Det finns också problem av mindre filosofisk natur. Vid vissa energier kan man inte göra beräkningar utifrån SM, utan andra metoder behövs. Olika förslag finns; så kallade effektiva fältteorier, och det är nödvändigt att jämföra dem med experimentella resultat för att avgöra om de beskriver verkligheten väl eller inte.

### Ett annat intressant område att studera är symmetrier. I fysiken används dessa generellt för att göra förutsägelser om olika storheters bevarande, eller om hur olika system beter sig om man ändrar någon eller några förutsät- tningar. En viktig symmetri i SM är den postulerade Charge-Parity sym- metrin (CP) som säger att fysikens lagar för ett system är desamma om man spegelvänder systemet och byter ut partiklar mot antipartiklar. Detta är sant i nästan alla fall, men experiment har visat att det finns undantag.

### Man säger att symmetrin är bruten.

### Det sällsynta sönderfallet η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### , kan möjligen uppvisa ett brott av CP-symmetrin. Detta tar sig i uttryck i form av en asymmetri i vinkeln mellan de plan som π

^{+}

### π

^{−}

### - och e

^{+}

### e

^{−}

### -paren färdas i efter att de har bil- dats. Genom att undersöka om CP-brottet verkligen sker i den här processen och hur stor effekten i sådant fall är, kan vi lära oss mer om hur CP-brott fungerar. Är standardmodellens beskrivning tillräcklig, eller tyder resultatet på ny fysik, bortom standardmodellen? En precis mätning av sannolikheten att sönderfallet inträffar skulle dessutom utgöra ett intressant test av de ef- fektiva fältteorier som beskriver det.

### Partikelfysikens främsta verktyg för att studera dessa fenomen är accel-

### eratorer. Där accelereras laddade partiklar till nära ljusets hastighet och

### kollideras med varandra. Den höga energin i kollisionen gör att andra, tyn- gre partiklar kan bildas. Dessa kan i sin tur sönderfalla, och slutresultatet blir en mängd olika intressanta konfigurationer som studeras med hjälp av en detektor.

### I det här examensarbetet används datorsimuleringar och data från en sådan anläggning: detektorn BES-III (Beijing Spectrometer) vid acceleratorn BEPC II (Beijing Electron Positron Collider) för att studera sönderfallet η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### . Med hjälp av simuleringarna undersöks hur reaktionen av intresse kan isoleras från alla andra reaktioner som registreras av detektorn.

### På så sätt kan man vid analys av riktiga data försäkra sig om att felaktigt identifierade reaktioner inte påverkar resultatet. I en tidigare studie vid BES- III mättes sannolikheten att just detta sönderfall inträffar, men bara en del av den totala datamängden användes. Om all tillgänglig data analyseras kan en mer precis mätning åstadkommas.

### Det här arbetet undersöker CP-symmetrin i sönderfallet, och uppmäter

### storleken av CP-brottet, A

^{ϕ}

### , till (1.96 ±1.97

^{stat.}

### ±0.4

^{syst.}

### ) ×10

^{−2}

### , vilket är lika

### med noll inom mätosäkerheten. Resultatet tyder således inte på att något

### brott av CP-symmetrin förekommer i detta sönderfall. Studien utgör också

### ett förarbete till en uppdaterad mätning av sannolikheten att sönderfallet

### inträffar.

### Acknowledgements

### I would like to express my gratitude to my supervisor, Andrzej Kupsc, for his advice throughout the course of this work, and for giving me the opportunity to visit IHEP in Beijing and be part of the BESIII collaboration.

### I would also like to thank Joachim Pettersson and Walter Ikegami An- dersson for sharing the office with me. Thanks to Magnus Wolke for being my subject reader and proof reading this report.

### Furthermore, I would like to extend my gratitude to Prof. Fang Shuang- shi, for being very helpful both before and during my stay at IHEP and for providing valuable advice on my work. Also, thanks to Peter Weidenkaff, and everyone else with whom I shared an office at IHEP, for showing me around and making my stay very enjoyable.

### Finally, my sincere thanks to my family and friends, for their continued

### support in everything I choose to do in life.

### Contents

1 Introduction

### 1

2 Theoretical Background

### 2

2.1 Standard Model

### . . . . 2

2.2 Low Energy QCD

### . . . . 5

2.3 Form Factors

### . . . . 6

2.4 Symmetries

### . . . . 6

2.5 CP-violation

### . . . . 6

3 The Decay η^{0} → π^{+}π^{−}e^{+}e^{−}

### 7

3.1 Amplitude### . . . . 7

### 3.1.1 Form Factors of the Decay η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### . . . . 8

3.2 Predictions

### . . . . 9

3.3 Previous Measurements

### . . . . 9

3.4 CP-violating Asymmetry

### . . . . 10

3.5 Background

### . . . . 11

4 The Decay η^{0} → π^{+}π^{−}γ

### 11

4.1 Amplitude### . . . . 12

5 Experiment

### 13

5.1 BEPC II### . . . . 13

5.2 The BES-III detector

### . . . . 13

### 5.2.1 Main Drift Chamber . . . . 14

### 5.2.2 Time of Flight System . . . . 15

### 5.2.3 Electromagnetic Calorimeter . . . . 15

### 5.2.4 Muon Detector . . . . 15

### 5.2.5 Trigger . . . . 15

5.3 η^{0} Production

### . . . . 16

5.4 ROOT

### . . . . 16

5.5 BESIII Offline Software System

### . . . . 16

### 5.5.1 KKMC . . . . 17

### 5.5.2 BesEvtGen . . . . 17

6 Asymmetry Measurement

### 17

6.1 Statistical Uncertainty### . . . . 19

6.2 Uncertainty in the Efficiencies

### . . . . 19

7 Event Generator for η^{0} → π^{+}π^{−}e^{+}e^{−}

### 20

7.1 Generating Invariant Mass of e^{+}e^{−}

### . . . . 20

7.2 Generated Distributions

### . . . . 21

7.3 Generator For η^{0} → π^{+}π^{−}γ

### . . . . 24

8 Background Discrimination

### 26

8.1 Invariant Mass at Beam Pipe### . . . . 28

8.2 Opening Angle

### . . . . 28

8.3 Cuts

### . . . . 29

8.4 Efficiencies

### . . . . 30

8.5 Signal to Background Ratio

### . . . . 32

9 Systematic Checks

### 33

10 Results### 34

10.1 Total Number of Signal Events### . . . . 34

10.2 Comparison between 2009 and 2012 Data

### . . . . 35

10.3 Asymmetry Parameter

### . . . . 37

11 Discussion

### 38

11.1 Asymmetry Parameter### . . . . 38

11.2 Difference Between 2009 and 2012 Data

### . . . . 38

12 Outlook

### 38

12.1 Branching Ratio Measurement### . . . . 38

12.2 Further Systematic Checks

### . . . . 39

12.3 Form Factor

### . . . . 40

12.4 Event Generator

### . . . . 40

13 Conclusion

### 40

A Manual for the Event Generator### 43

A.1 Code Structure### . . . . 43

A.2 Requirements

### . . . . 43

A.3 Usage

### . . . . 43

A.4 Template Input File

### . . . . 44

B Helix Tracks

### 44

C Maximum log-Likelihood Estimation of the Uncertainty

### 47

### 1 Introduction

### The success of the standard model (SM) of particle physics can hardly be overstated. Not only does it provide a good descriptions for a wealth of experimental results, it has also resulted in a large number of predictions that have later been confirmed by experiment including, but not limited to, the existence of the tau lepton, the top quark, and the Higgs boson.

### However, the SM is not without its problems. To date, it only describes three out of four fundamental forces. Gravity still defies inclusion into its framework. Furthermore, there is a number of questions left unanswered by the SM, such as: what is dark matter, why is the weak force so weak, or why is there more matter than anti-matter? There are also discrepancies between theoretical predictions and experimental results that need to be accounted for. If found to be statistically significant, such results could point to new physics, beyond the SM.

### A fundamental part of any physical theory is symmetries, the property that the physics of a system stay the same under a given transformation. In the SM, one identifies three such symmetries: charge conjugation (C), parity (P), and time reversal (T). It is both intuitive and aesthetically pleasing to have a theory that is invariant under each of these changes on its own, and initially this was believed to be the case. However, nature does not always oblige. First, a violation of the P symmetry was discovered in the 1950’s, and in the 1960’s it was shown that the combination of C and P (CP) was likewise broken. At the present stage, only the combination of all three (CPT) is considered to be a fundamental symmetry of nature.

### CP-violation is particularly interesting, as it might hint at an answer to the disparity between matter and anti-matter in the observable universe.

### Since its discovery, CP-violation has been accomodated into the SM, but attempts at further observations of CP-violation remain an interesting avenue of research, both in terms of increasing our understanding of the mechanisms of CP violation as it occurs within the SM, but also as a possible signal of physics beyond the SM.

### Furthermore, there are limits in which the conventional descriptions of the

### SM are no longer useful. One example is the low energy region of Quantum

### Chromodynamics (QCD), the theory of the strong force. The strong coupling

### constant runs from large to small values as energy increases, and one can

### not use it as an expansion variable in the low energy region. Perfoming

### calculations and making predictions for this region therefore requires other

### methods. Either one can resort to numerical solutions, as in lattice QCD, or one can formulate an effective field theory which approximates physics in this particular energy region, making no attempt to replicate physics at higher energies. Examples of effective field theories include Chiral Perturbation Theory (χPT) and Vector Meson Dominance (VMD) . These theories give predictions for different processes, including for example rare decays, that need to be studied precisely in order to determine to what extent they are realised in nature.

### The neutral pseudoscalar mesons (π

^{0}

### , η, η

^{0}

### ) make excellent laboratories for studies of effective field theories. They can be produced in large numbers in conventional accelator experiments, and their masses are low, meaning that low energy effects will have a comparatively large impact on their decays.

### The BES-III detector at the Beijing Electron-Positron Collider (BEPCII) has been collecting data in the charmonium physics region since 2009. Two runs in 2009 and 2012 were made at the mass of the J/Ψ-resonance, recording a total of 1.3 × 10

^{9}

### J/Ψ events. η

^{0}

### mesons are abundantly produced in the decays of the J/Ψ, allowing for measurements of rare η

^{0}

### decays that have hitherto been impossible.

### This thesis investigates the rare decay η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### using Monte Carlo simulations and data from the BES-III detector. A number of different ef- fective field theories predict the branching ratio for this decay, and a precise measurement can be used to test their accuracy. Furthermore, this decay may exhibit a CP-violating asymmetry between the π

^{+}

### π

^{−}

### and e

^{+}

### e

^{−}

### decay planes. The asymmetry parameter is measured and the groundwork for an updated branching ratio measurement is laid.

### 2 Theoretical Background

### 2.1 Standard Model

### The elementary particle content of the SM, shown in fig. 1, can be divided into

### two general groups: fermions, obeying Fermi-Dirac statistics, and bosons,

### obeying Bose-Einstein statistics.

Figure 1: Particle content of the Standard Model. The blue lines indicate interac-
tions.^{1}

### The fermions all have spin

^{1}

_{2}

### and can be further divided into two cat- egories: leptons that only interact via the electroweak forces, and quarks that carry colour charge and therefore also interact via the strong force. The bosons include the spin 1 gauge bosons responsible for mediating the fun- damental interactions. The photon, γ, carries the electromagnetic force, the W

^{±}

### and Z bosons the weak force and 8 gluons with different colours mediate the stong force.

### The second type of boson is spin 0 scalar bosons, the only confirmed case of which is the Higgs boson responsible for electroweak symmetry breaking and fermion masses via the interactions with the Higgs field.

### Due to the properties of the strong force, only colour neutral (white) objects can be observed. This rule, known as colour confinement, implies

1Image source: Elementary Particle Interactions in the Standard Model, Wikimedia Commons. Accessed 2017-06-01.

### that quarks do not exist as free particles. Instead they form colour neutral bound states known as hadrons. Two types of hadrons make up the vast majority of observed particles:

### • (anti-)baryons, consisting of three valence quarks with colour charges red, blue and green, respectively.

### • mesons,consisting of a quark with any given colour, and an antiquark with the corresponding anti-colour.

### The quantum numbers of hadrons are directly derived from their quark con- tents. Mesons in particular can be categorized according to their spin and parity J

^{P}

### . Relevant for this project are the most common types: pseu- doscalar mesons, with J

^{P}

### = 0

^{−}

### and vector mesons with J

^{P}

### = 1

^{−}

### .

### The three lightest quarks: up, down and strange follow an approximate U(3) symmetry, and mesons therefore combine into a singlet and an octet state:

### 3 × ¯3 = 8 + 1 (1)

### The resulting nine pseudoscalars and nine vector mesons are shown in fig. 2.

### The observable mesons η and η

^{0}

### are however not the exact states derived from the symmetries. Rather, they are mixes of the singlet state η

0### and the octet state η

8### according to:

### η η

^{0}

### = − sin θ

^{mix}

### cos θ

mix### cos θ

mix### sin θ

mix### η

0### η

8### (2) with θ

mix### ≈ −20

^{◦}

### [1].

(a) Nonet of pseudoscalar mesons. ^{2} (b) Nonet of vector mesons.^{3}
Figure 2: Mesons compsed of the u, d and s quarks.

### 2.2 Low Energy QCD

### The coupling constant of the strong force, α

S### , changes with the energy scale, going from (relatively) large to small values as the energy increases, see fig. 3.

### At high energies where the coupling is small, calculations can be performed through the standard, perturbative approach of quantum field theory. At low energies on the other hand, the size of the coupling does not permit perturbative solutions. Instead, one resorts to effective field theories. These describe physics exclusively in this region, and make no attempt at reproduc- ing QCD in the high energy regime. Examples include Chiral Perturbation Theory (χPT) or Vector Meson Dominance (VMD).

Figure 3: The QCD coupling α_{S} as a function of the energy scale [2].

### VMD describes the interactions of hadrons and photons as occuring via the exchange of a vector meson. It provides accurate predictions for the decays of lights mesons and of their transition form factors. There are a number of different VMD models, and experimental results are needed to test to what extent they are realised in nature [3].

3Image source: Meson Nonet - spin 0, Wikimedia Commons. Accessed 2017-06-01.

3Image source: Meson Nonet - spin 1, Wikimedia Commons. Accessed 2017-06-01.

### 2.3 Form Factors

### In effective field theories the goal is to make quantitative predictions for in- teresting interactions without giving a complete picture of the underlying physics. To this end, one encapsulates all pertinent information in form fac- tors specific to the given reaction. This can also be seen as a parametrization of the structure of the interacting particles. VMD has been successful in pre- dicting the form factors for transitions of the light pseudoscalars π

^{0}

### , η, and η

^{0}

### .

### 2.4 Symmetries

### In the context of physical theories, a symmetry is the property of being in- variant under a certain transformation. This allows one to find conservation laws and provides clear conditions to determine whether different processes are physically allowed. In the SM, one identifies three such symmetries: Par- ity (P), Charge conjugation (C), and time reversal (T). P is the mirroring of spatial coordinate, C is the interchange of particles with their anti-particles, and T is the reversal of the ”direction” of time. Initially, it was believed that these were all universal and would be preserved for any process. While this is certainly true for many processes, there is now clear experimental evidence to the contrary. P, C, the combination of C and P (CP), and T are all violated in certain processes, and only the combination of all three (CPT) is believed to be a fundamental symmetry of the SM [4].

### 2.5 CP-violation

### CP is the product of the C and P symmetries, and was proposed to be a fundamental symmtery of nature after experiments in the 1950’s showed the P symmetry to be broken. However not long thereafter, in 1964, Cronin and Fitch [5] discovered an indirect violation of the CP symmetry in the decays of neutral K-mesons to 2π. The physical states K

S### and K

L### are mixes of the CP-eigenstates K

1### and K

2### . It was shown that both K

S### and K

L### could decay into 2π, but not with the same probability. This violates CP indirectly. On the other hand, a decay of the CP-odd K

2### to the CP-even 2π would constitute a direct CP-violation.

### Since then, CP-violation has been observed by several experiments in

### flavor-changing weak decays of both K- and B-mesons. So far, no CP-

### violation has been seen in electromagnetic or strong processes.

### CP-violation can be accommodated into the Standard Model through a complex phase in the Cabbibo-Kobayashi-Maskawa (CKM) matrix, which governs the quark mixing in weak decays. This reproduces the experimental results for K- and B-mesons. However, the origin of CP-violation is still unexplained, and the amount of CP violation observed is too small to account for the asymmetry between matter and antimatter in the universe. Searching for CP-violation in other processes therefore remains an interesting avenue of research for increasing our understanding of CP violation as it occurs within the SM, but also as a possible signal of physics beyond the SM [6].

### 3 The Decay η ^{0} → π ^{+} π ^{−} e ^{+} e ^{−}

### In the low energy limit of QCD the coupling constant becomes large, and one can treat the quark masses as vanishing. The resulting Lagrangian of an effective field theory has an anomalous U(1) axial symmetry, i.e. a classical symmetry that is broken in quantum mechanics. The non-conservation of the axial current leads to interaction terms labeled the triangle and box anomalies on account of the shape of the corresponding Feynman diagrams.

### In the chiral limit, the decay of a neutral pseudoscalar P into π

^{+}

### π

^{−}

### γ

^{(∗)}

### is governed by the latter, see fig. 4.

P

π^{+}

π^{−}
γ^{(}^{∗)}

Figure 4: Feynman diagram for the box anomaly [7].

### The decay η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### occurs when the photon of the box anomaly decay is virtual, and in turn decays into an electron-positron pair : γ

^{(∗)}

### → e

^{+}

### e

^{−}

### [7].

### 3.1 Amplitude

### The spin-summed squared matrix element for the decay P → π

^{+}

### π

^{−}

### l

^{+}

### l

^{−}

### is [7]:

### |A

P →π^{+}π

^{−}l

^{+}l

^{−}

### |

^{2}

### (s

ππ### , s

ll### , θ

π### , θ

l### , ϕ)

### = e

^{2}

### 8 (k

^{2}

### )

^{2}

### (

### |M(s

^{ππ}

### , s

ll### ) |

^{2}

### λ(m

^{2}

_{P}

### , s

ππ### , s

ll### ) 1 − β

l^{2}

### sin

^{2}

### θ

l### sin

^{2}

### ϕ s

ππ### β

_{π}

^{2}

### sin

^{2}

### θ

π### + 4ReM(s

ππ### , s

ll### )(E

_{+}

^{∗}

### − E

−^{∗}

### ) λ

^{1/2}

### (m

^{2}

_{P}

### , s

ππ### , s

ll### )β

_{l}

^{2}

### β

_{π}

^{2}

### √ s

ππ### √

### s

ll### × −

^{1}

_{2}

### (m

^{2}

_{P}

### −s

^{ππ}

### −s

^{ll}

### ) sin θ

π### cos θ

π### cos θ

l### sin θ

l### sin ϕ + √ s

ππ### s

ll### sin

^{2}

### θ

π### sin

^{2}

### θ

l### sin ϕ cos ϕ + 2Re {M(s

^{ππ}

### , s

ll### )(E

_{+}

^{∗}

### + E

_{−}

^{∗}

### ) } √

### s

ππ### s

ll### β

_{l}

^{2}

### β

π### λ(m

^{2}

_{P}

### , s

ππ### , s

ll### ) sin θ

π### sin θ

l### cos θ

l### sin ϕ

### + X

±

### |E

^{±}

### |

^{2}

### λ

^{1/2}

### (s

ππ### , m

^{2}

_{P}

### , k

^{2}

### ) ∓ m

^{2}P

### − s

^{ππ}

### − k

^{2}

### β

π### cos θ

π 2### · 1 − β

l^{2}

### cos

^{2}

### θ

l### + 4s

ππ### s

ll### β

_{π}

^{2}

### sin

^{2}

### θ

π### 1 − β

l^{2}

### sin

^{2}

### θ

l### cos

^{2}

### ϕ

### ∓ 4 λ

^{1/2}

### (s

ππ### , m

P### √

### s

ππ### β

π### √

### s

ll### β

_{l}

^{2}

### sin θ

π### sin θ

l### cos θ

l### cos ϕ

### + 2Re[E

_{+}

^{∗}

### E

−### ]

### λ(s

ππ### , m

^{2}

_{P}

### , k

^{2}

### ) 1 − β

π^{2}

### cos

^{2}

### θ

π### 1 − β

l^{2}

### cos

^{2}

### θ

l### − 4s

ππ### s

ll### β

_{π}

^{2}

### 1 − β

l^{2}

### sin

^{2}

### θ

π### sin

^{2}

### θ

l### cos

^{2}

### ϕ − β

l^{2}

### cos

^{2}

### θ

π### cos

^{2}

### θ

l### − 4 √ s

ππ### √

### s

ll### β

_{π}

^{2}

### β

_{l}

^{2}

### m

^{2}

_{P}

### − s

^{ππ}

### − k

^{2}

### sin θ

π### cos θ

π### sin θ

l### cos θ

l### cos ϕ

### ) .

### (3) where s

ππ### and s

ll### are the invariant masses of the pion and lepton pair respec-

### tively, m

P### is the mass of the decaying pseudoscalar, ϕ is the angle between the π

^{+}

### π

^{−}

### and l

^{+}

### l

^{−}

### planes in the P rest frame, θ

π### is the angle between the π

^{+}

### three momentum and the P three-momentum in the π

^{+}

### π

^{−}

### rest frame, and θ

l### is the angle between the l

^{−}

### three-momentum and the P three-momentum in the l

^{+}

### l

^{−}

### rest frame. λ(x, y, z) is the Källén kinematic function. M and E

±### are magnetic and electric form factors.

### 3.1.1 Form Factors of the Decay η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### The magnetic form factor M can be expressed as [7]:

### M (s

ππ### , s

ll### ) = M × V MD(s

^{ππ}

### , s

ll### ) (4)

### V M D is a model dependent vector meson dominance factor. For the case P = η

^{0}

### , M is given by:

### M = e

### 8π

^{2}

### f

_{π}

^{3}

### √ 1 3

### f

_{π}

### f

8### sin θ

mix### + 2 √ 2 f

π### f

0### cos θ

mix### (5) where f

π### , f

0### , f

8### are the pion, singlet and octet decay constants respectively.

### θ

mix### ≈ −20

^{◦}

### is the η-η

^{0}

### mixing angle. The vector meson dominance factor is:

### V M D(s

ππ### , s

ll### ) = 1 − 3

### 4 (c

1### − c

^{2}

### + c

3### ) + 3

### 4 (c

1### − c

^{2}

### − c

^{3}

### ) m

^{2}

_{V}

### m

^{2}

_{V}

### − s

^{ll}

### − im

^{V}

### Γ(s

ll### ) + 3

### 2 c

3### m

^{2}

_{V}

### m

^{2}

_{V}

### − s

^{ll}

### − im

^{V}

### Γ(s

ll### )

### m

^{2}

_{V}

### m

^{2}

_{V}

### − s

^{ππ}

### − im

^{V}

### Γ(s

ππ### ) (6) where m

V### is the mass of the vector meson and the parameters c

1### − c

^{2}

### , and c

3### are chosen by comparison to data. The values of these parameters for two different models and two fits performed in [8] are given in table 1.

Table 1: The different VMD models.

### Model c

1### − c

2### c

3### Full VMD 1/3 1

### Hidden gauge model 1 1

### Modified VMD (Fit 1) 1.168 ± 0.069 0.927 ± 0.010 Modified VMD (Fit 2) 1.20 ± 0.043 0.930 ± 0.011

### 3.2 Predictions

### Theoretical predictions of the branching ratio are given in table 2.

Table 2: Predictions for the η^{0} → π^{+}π^{−}e^{+}e^{−} branching ratio.

### Unitary χPT[9] Hidden gauge [7] Modified VMD [7]

### BR(η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### ) (10

^{−3}

### ) 2.13

^{+0.17}

_{−0.31}

### 2.17 ± 0.21 2.27 ± 0.13

### 3.3 Previous Measurements

### The reaction was first observed at CLEO in 2009. The branching ratio was

### measured to be (2.5

^{+1.2}

_{−0.9}

### ± 0.5) × 10

^{−4}

### from 7.9

^{+3.9}

_{−2.7}

### signal events. The total

### sample size corresponded to around 4 × 10

^{4}

### η

^{0}

### [10].

### In 2013, another measurement was performed at BESIII using a sample of (225.3 ± 2.8) × 10

^{6}

### J/ψ events. The branching ratio was determined to be (2.11 ± 0.12(stat.) ± 0.15(syst.)) × 10

^{−3}

### from 429 ± 24 signal events [11].

### 3.4 CP-violating Asymmetry

### In a number of previous studies a large CP violating asymmetry in the angle between the π

^{+}

### π

^{−}

### and e

^{+}

### e

^{−}

### planes has been predicted and experimentally confirmed in the decay K

L### → π

^{+}

### π

^{−}

### γ

^{∗}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### (eg. [12]). In [13] this analysis is extended to derive a corresponding CP-violating observable in the decay η

^{(}

^{0}

^{)}

### → π

^{+}

### π

^{−}

### γ

^{∗}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### .

### This CP-violation stems from the mixing of parity-conserving magnetic form factors and parity-violating electric form factors in the decay amplitude, i.e the terms proportional to ReM(s

ππ### , s

ll### )(E

_{+}

^{∗}

### ± E

−^{∗}

### ) in eq. 3. When com- puting the decay rate, two of these terms vanish after integration over θ

π### due to the cos θ

π### sin θ

π### dependence. The remaining term depends on sin ϕ cos ϕ and can only be non-zero after integration if there is an asymmetry in the distribution of the angle ϕ between the π

^{+}

### π

^{−}

### and e

^{+}

### e

^{−}

### decay planes, see fig. 5.

e^{+}

e^{-}
π^{+}
π^{-}

**φ**

Figure 5: Definition of the asymmetry angle ϕ [14].

### One finds an observable by studying the ϕ distribution of the total decay width [13], [7]:

### A

^{CP}

### = < sign(sinϕcosϕ) > (7)

### = R

2π0

dΓ(η^{0}→π^{+}π^{−}e^{+}e^{−})

dϕ

### dϕsign(sinϕcosϕ) R

2π0

dΓ(η^{0}→π^{+}π^{−}e^{+}e^{−})
dϕ

### (8)

### The asymmetry A

^{ϕ}

### has been measured for the η decay by the WASA-at- COSY [14] and KLOE [15] experiments, but no measurement has of yet been performed for η

^{0}

### .

### 3.5 Background

### Table 3 shows all sources of hadronic background as identified via analysis of an inclusive MC sample of 225 × 10

^{6}

### J/Ψ events [11]. Due to the low event counts of all other backgrounds, only the first decay is taken into account in this work.

Table 3: Hadronic background events remaining after selection criteria are applied to inclusive MC sample.

### Decay Chain Number of Events

### η

^{0}

### → γρ

^{0}

### , ρ

^{0}

### → π

^{+}

### π

^{−}

### 498

### η

^{0}

### → γρ

^{0}

### , ρ

^{0}

### → π

^{+}

### π

^{−}

### γ

F SR### 5

### J/ψ → π

^{+}

### π

^{−}

### π

^{0}

### γ

F SR### 2

### η

^{0}

### → γω , ω → π

^{+}

### π

^{−}

### 1

### J/ψ → h

1### (1170)π

^{0}

### , h

1### (1170) → ρ

^{0}

### π

^{0}

### , ρ

^{0}

### → π

^{+}

### 1

### J/ψ → π

^{+}

### π

^{−}

### π

^{0}

### 1

### 4 The Decay η ^{0} → π ^{+} π ^{−} γ

### The main source of background is the decay η

^{0}

### → π

^{+}

### π

^{−}

### γ. A certain number

### of the photons from this decay will convert to an electron-positron pair via

### pair production in the beam pipe material and inner wall of the MDC. Such

### events will have the same signature as the signal. The PDG value for the

### branching ratio is 0.291 ± 0.005 [16], and the combined materials a photon

### needs to pass through before reaching the detector correspond to 1.04 % of

### a radiation length [17]. The percentage of photons that will convert can be

### found as [18]:

### I

conv.### = I

0### 1 − exp

### − 7 9 · l

### x

0### = I

0### exp

### 1 −

### − 7

### 9 · 1.04 · 10

^{−2}

### x

0### x

0### = 0.008056 · I

^{0}

### (9)

### I.e. 0.8056 % of the photons will convert. In total, the branching ratio for a conversion event will be of the same order of magnitude as for the signal.

### 4.1 Amplitude

### This decay can be described by the spin-summed amplitude squared [7]:

2

### X

pol=1

### |A

^{P →π}

^{+}

^{π}

^{−}

^{γ}

### |

^{2}

### ( ˜ E

γ### , θ

π### ) = E ˜

_{γ}

^{2}

### 1 − 2 ˜ E

γ### /m

P### β

_{π}

^{2}

### sin

^{2}

### (θ

π### )

### 4m

^{2}

_{P}

### ( |M

^{G}

### |

^{2}

### + |E

^{G}

### |

^{2}

### ) (10) To first order, the electric form factor can be put to zero. The magnetic form factor takes the form:

### M

G### (s

ππ### ) = m

^{3}

_{P}

### M (s

ππ### , k

^{2}

### = 0) (11) where

### M (s

ππ### , k

^{2}

### = 0) = M × V MD(s

ππ### ), a (12) with

### M = e

### 8π

^{2}

### f

_{π}

^{3}

### √ 1 3

### f

_{π}

### f

8### sin θ

mix### + 2 √ 2 f

π### f

0### cosθ

mix### for P = η

^{0}

### (13) and,

### V M D(s

ππ### ) = 1 − 3 2 c

3### + 3

### 2 c

3### m

^{2}

_{V}

### m

^{2}

_{V}

### − s

^{ππ}

### − im

^{V}

### Γ(s

ππ### ) (14)

### 5 Experiment

### 5.1 BEPC II

### The Beijing Electron Positron Collider (BEPC II) is a circular collider capa- ble of operating at center-of-mass energies between 2 and 5 GeV. Particles are accelerated to up to 1.89 GeV by a linear accelerator before injection into the storage rings. In order to minimize beam-beam interactions, electrons and positrons circulate in two separate rings. This allows for storage of more than one bunch of each. At BEPCII, each ring holds 93 bunches with a spac- ing of 8 ns. The beams collide at an angle of 11 mrad at the interaction point (IP) where events are detected by the Beijing Spectrometer (BES-III, [17]).

### The design luminosity is 1 × 10

^{33}

### cm

^{−2}

### s

^{−1}

### at 1.89 GeV beam energy.

Figure 6: Layout of BEPC [19].

### 5.2 The BES-III detector

### BES-III is a 4π detector built for physics in the tau-charm region, covering

### a wide range of topics from charmonium, and τ -leptons mass to R measure-

### ments and the fairly recently discovered XYZ states [20]. The components

### of BES-III are described below.

Figure 7: The BESIII detector [21].

### 5.2.1 Main Drift Chamber

### Immediately surrounding the interaction point is first the beam pipe, con-

### sisting of 1.4 mm beryllium, 14.6 µm gold, and 0.8 mm of synthetic mineral

### oil [17], followed by the helium-gas based Main Drift Chamber (MDC) cov-

### ering all azimuthal angles and polar angles in the range |cos(θ)|< 0.93. It

### consists of a total of 6796 signal wires arranged in 43 layers, 8 in the inner

### chamber, and 35 in the outer chamber. The inner layers have a cell size of

### about12 × 12 mm

^{2}

### and the outer layers about 16.2 × 16.2 mm

^{2}

### . The inner

### and outer radii are 60 mm and 800 mm, respectively [21]. The inner wall

### consists of carbon fiber and is 1.32 cm thick [17]. The MDC is enclosed in 1 T magnetic field provided by a superconducting solenoid magnet. The spatial resolution is 2 mm in the z-direction, and <120µm in the transverse direction. dE/dx resolution σ

dE/dx### < 6 % for momenta up to 700 MeV/c and the transverse momentum resolution is 0.5 % at 1 GeV/c [21].

### 5.2.2 Time of Flight System

### The Time Of Flight (TOF) system is used for particle identification (PID) of charged particles by comparing the measured time for a particle to pass with the expected time given the momentum of the particle measured in the MDC. This is useful e.g. for distinguishing between pions and kaons.

### The system uses plastic scintillators connected to photomultiplier tubes and consists of two parts: barrel and endcap. The barrel TOF covers

### |cos(θ)|< 0.83 and the endcap 0.85 < |cos(θ)|< 0.95. The total time res- olution is about 100 ps [21].

### 5.2.3 Electromagnetic Calorimeter

### The ElectroMagnetic Calorimeter (EMC) is used for measurements of the positions and energies of electrons and photons. Consisting of 6272 CsI(Tl) crystals distributed over one barrel and two endcap sections, it covers polar angles |cos(θ)|< 0.83 or 93 % of 4π. The range of energies that can be measured are 20 MeV to 2 GeV, and the resolution is 2.5 % . The resolution for electromagnetic shower position is σ

xy### ≤ 6 mm / pE[GeV ] [21].

### 5.2.4 Muon Detector

### The purpose of the muon detector is to discriminate between muons and hadrons. This is accomplished by using of resistive plate chambers [21].

### 5.2.5 Trigger

### The Data Acquisition (DAQ) system can handle a maximum event rate of

### 4 kHz. Therefore one needs to separate the type of events one wants to

### study from the background. This is done by the trigger. BES-III uses a

### two-level trigger scheme composed of a software event filter and a hardware

### trigger analysing signals from the subdetectors to determine if a given event

### is interesting. The necessary conditions for accepting an event have been set

### using Monte Carlo simulations. The trigger efficiency for all J/Ψ decays is 97.7 % [21].

### 5.3 η

^{0}

### Production

### Of particular interest for this project are data from the runs at the mass of the J/ψ resonance, 3.097 GeV, from which η

^{0}

### can be abundantly produced [21].

### The dominating source of η

^{0}

### is the radiative decay J/ψ → γη

^{0}

### , which occurs with a branching ratio of 5.15 ×10

^{−3}

### [16].

### The latest available data sample at BESIII consists of 1.3 × 10

^{9}

### J/ψ collected with BESIII in 2009 (0.225 × 10

^{9}

### ) and 2012. This corresponds to a total of about 6.7 × 10

^{6}

### η

^{0}

### produced in the radiative decay.

### 5.4 ROOT

### ROOT is an object-oriented analysis framework for high energy physics based on C++ developed at CERN [22] (see also http://root.cern.ch). It includes a variety of useful features such as phase space event generation, four-vector algebra, histogram fitting, and handling of data in the form of trees. Trees are data structures for storing large numbers of objects of the same class, such as events from an experiment. This along with I/O and compression features makes it a useful tool for particle physics data analysis.

### In this project, ROOT is used for analysis and to construct a Monte Carlo generator for P → π

^{+}

### π

^{−}

### l

^{+}

### l

^{−}

### and P → π

^{+}

### π

^{−}

### γ.

### 5.5 BESIII Offline Software System

### The BESIII Offline Software System (BOSS) covers all aspects of data pro- cessing and analysis within the BESIII collaboration. It incorporates a num- ber of external libraries, including ROOT, CERNLIB, CLHEP and Geant4 and runs on the operating system Scientific Linux Cern (SLC). The features of BOSS are:

### • Event generation: A variety of Monte Carlo generators are available, covering the whole range of physics at BESIII.

### • Detector simulation: The detector response to generated events is

### simulated using software based on Geant4.

### • Reconstruction: Data or Monte Carlo events are reconstructed from detector response.

### • Calibration: Compute and retrieve calibration constants for each sub- detector.

### • Analysis: Generates event objects for physics analysis and provides tools such as PID and kinematic fitting.

### In this project, the generator KKMC is used to generate J/ψ that are in turn decayed by BesEvtGen.

### 5.5.1 KKMC

### KKMC [23] is used to generate charmonium states such as J/ψ from e

^{+}

### e

^{−}

### collisions using the Electroweak SM.

### 5.5.2 BesEvtGen

### BesEvtGen [24] combines generators from BESIII with classes from Evt- Gen to generate charmonium decays. Kinematics are generated from phase space, and events are accept-rejected one decay sequence at a time based on the amplitude squared. Many decays are implemented from the start, while rare decays need to be implemented manually by the user through the Do It Yourself (DIY) model. To use DIY, the amplitude for the decay in question must be defined in a class that can thereafter be called when running BesEvt- Gen. This functionality is used to generate η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### and η

^{0}

### → π

^{+}

### π

^{−}

### γ decays.

### 6 Asymmetry Measurement

### From identified η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### events, the symmetry can be extracted as:

### A

^{ϕ}

### = N (sinϕcosϕ > 0) − N(sinϕcosϕ < 0)

### N (sinϕcosϕ > 0) + N (sinϕcosϕ < 0) (15) where ϕ is the asymmetry angle in the η

^{0}

### rest frame as defined in figure 5, and N is the number of events that fulfill the given condition.

### From data, the asymmetry is determined by calculating the quantity

### sin ϕ cos ϕ for each event using a method described by [15]. Firstly, the

### four-momenta are boosted to the η

^{0}

### rest frame using the reconstructed η

^{0}

### momentum vector. Subsequently, the unit normal vectors of the two decay planes are determined as:

### ˆ n

e### = p

e^{+}

### × p

e^{−}

### |p

^{e}

^{+}

### × p

e^{−}

### | (16)

### n ˆ

π### = p

π^{+}

### × p

π^{−}

### |p

π^{+}

### × p

_{π}

^{−}

### | (17)

### where p

π^{±}

### , p

e^{±}

### are the momentum three vectors of the pions and electrons respectively. It follows from the properties of the dot product that:

### ˆ n

e### · ˆn

^{π}

### = kˆn

^{e}

### kkˆn

^{π}

### kcos ϕ = cos ϕ (18) where ϕ is the angle between the normal vectors, i.e. the asymmetry angle.

### For the cross product it holds that:

### kˆn

e### × ˆ n

π### k = kˆn

e### kkˆn

π### ksin ϕ (19)

### = ⇒ sin ϕ = kˆn

^{e}

### × ˆ n

π### k (20) In order to take into account that sin ϕ will take positive or negative values depending on which quadrant ϕ lies in, a reference direction is defined as a unit vector along the intersection of the two planes:

### z = ˆ p

e^{+}

### + p

e^{−}

### |p

e^{+}

### + p

e^{−}

### | (21)

### sin ϕ can then be determined with the correct sign as:

### sin ϕ = (ˆ n

e### × ˆ n

π### ) · ˆz (22) The final expression for sin ϕ cos ϕ is:

### sin ϕ cos ϕ = (ˆ n

e### × ˆ n

π### ) · ˆz(ˆn

e### · ˆn

π### ) (23)

### 6.1 Statistical Uncertainty

### The standard error propagation formula gives the uncertainty in A

^{ϕ}

### as:

### σ

A_{ϕ}

### = s

### d A

^{ϕ}

### dN

+ 2### σ

_{N}

^{2}

_{+}

### + d A

^{ϕ}

### dN

− 2### σ

_{N}

^{2}

_{−}

### = s

### 1 N

2### σ

_{N}

^{2}

_{+}

### +

### −1 N

2### σ

^{2}

_{N}

_{−}

### = s

### σ

_{N}

^{2}

_{+}

### + σ

_{N}

^{2}

_{−}

### N

^{2}

### (24)

### If the error in the number of counts N

+### , N

−### are √

### N

+### and √

### N

−### respectively, one finds:

### σ

Aϕ### = r N

+### + N

−### N

^{2}

### = 1

### √ N

### (25)

### 6.2 Uncertainty in the Efficiencies

### Applying of a cut to an event has two possible outcomes: either the event will pass the cut or not, and therefore it can be considered a binomial process.

### The probability of passing the cut is the efficiency . For a sample size N, one finds the uncertainty in the number of surviving events k to be [25]:

### σ

k### = p(1 − )N (26)

### Here is the true efficiency, which is not known. Replacing it with the efficiency estimated from MC,

^{0}

### , one finds:

### σ

^{0}

### = 1 N

gen.### s k

### 1 − k N

gen.### (27)

### Here N

gen.### is the total number of generated events

### 7 Event Generator for η ^{0} → π ^{+} π ^{−} e ^{+} e ^{−}

### A Monte Carlo event generator for the reaction η

^{0}

### → π

^{+}

### π

^{−}

### e

^{+}

### e

^{−}

### was written.

### The program generates kinematics for the four final state particles in two stages using the TGenPhaseSpace class of ROOT. In the first step, the decay η

^{0}

### → π

^{+}

### π

^{−}

### γ

^{∗}

### is calculated in the η

^{0}

### rest frame. Subsequently, the virtual photon is decayed to the e

^{+}

### e

^{−}

### -pair. Pion masses and the invariant mass of the virtual photon are given as input. The latter is generated from a 1/q

^{2}

### distribution for each event using the inverse transform sampling method (see section 7.1).

### Events are accept-rejected based on the full theoretical amplitude, see eq. 3. For each generated event, the amplitude is computed and weighed according to the maximal amplitude which is pre-calculated from a total of 10

^{6}

### events so that an event is accepted if:

### u ≤ |A

^{2}comp

### |

### |A

^{2}max

### | (28)

### where u ∼ Unif(0, 1) is a random number from a uniform distribution be- tween 0 and 1. For a description of the program and its usage, see appendix A.

### 7.1 Generating Invariant Mass of e

^{+}

### e

^{−}

### Knowing that the momentum transfer q

^{2}

### = s

e^{+}e

^{−}

### of the virtual photon will be very small, it would be inefficient to generate events with all kinematically al- lowed values of q

^{2}

### as most events would be rejected. Therefore, q

^{2}

### is assumed to approximately follow a probability distribution function (PDF) 1/q

^{2}

### peak- ing at the threshold for e

^{+}

### e

^{−}

### production (2 × m

^{e}

### ). No random generator for this distribution is available by default in ROOT, and consequently, a new function is written for this purpose, using the inverse transform sampling method.

### If F is a Cumulative Distribution Function (CDF), F

^{−1}

### is the inverse

### CDF of F, and U is a random number from a uniform distribution on [0, 1],

### then X := F

^{−1}

### (U ) is distributed according to F [26].

### The CDF of f (q) = 1/q

^{2}

### is given by:

### F (q) = Z

q−∞

### f (q) = Z

q−∞

### 1

### q

^{2}

### (29)

### = − 1

### q (30)

### Since we are only interested in generating positive numbers, we take the absolute value of the above function.With u ∼ Unif(0, 1), one finds the inverse:

### F F

^{−1}

### (u) = u (31)

### (32) In the case F (q) = 1/q:

### F (F

^{−1}

### (u)) = 1

### F

^{−1}

### (u) = u (33)

### = ⇒ F

^{−1}

### (u) = 1

### u (34)

### In order for the distribution to peak at (2m

e### )

^{2}

### , one can simply multiply by that value.

### 7.2 Generated Distributions

### The generated θ

π### , θ

e### , ϕ, s

ππ### , and s

ll### distributions are presented below:

hThetaP Entries 100000 Mean 0.001142 Std Dev 0.448

θπ

cos

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Events/0.02

0 200 400 600 800 1000 1200 1400 1600

hThetaP Entries 100000 Mean 0.001142 Std Dev 0.448

θπ

Figure 8: Generated distribution of θ_{π}.

hThetaE Entries 100000 Mean 0.001001 Std Dev 0.6258

θe

cos

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Events/0.02

800 900 1000 1100 1200 1300 1400

hThetaE Entries 100000 Mean 0.001001 Std Dev 0.6258

θe

Figure 9: Generated distribution of θ_{e}.

hPhi Entries 100000 Mean 1.565 Std Dev 1.003

[rad]

0 0.5 1 1.5 2 2.5 φ3

Events/0.03

600 700 800 900 1000 1100 1200 1300 1400

hPhi Entries 100000 Mean 1.565 Std Dev 1.003

φ

Figure 10: Generated distribution of ϕ.

π

### s

π0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ππ

### s ∂ / Γ ∂

0 0.5 1 1.5

2 2.5

−6

×10

Theor. decay rate.

Generated Distribution

Figure 11: Comparison of theoretical decay rate and generated s_{ππ}-distribution
from η^{0} → π^{+}π^{−}e^{+}e^{−}. Arbitrary normalisation.

### s

ee 0 0.0020.0040.0060.008 0.010.0120.0140.0160.0180.02ee

### s ∂ / Γ ∂

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

−3

×10

Theor. diff. decay rate.

Generated Distribution

Figure 12: Comparison of theoretical decay rate and generated s_{ee}-distribution
from η^{0} → π^{+}π^{−}e^{+}e^{−}. Arbitrary normalisation.

### 7.3 Generator For η

^{0}

### → π

^{+}

### π

^{−}

### γ

### The corresponding decay where the photon is real is implemented by gener-

### ating the π

^{+}

### π

^{−}

### γ final state directly from phase space using the TGenPhas-

### eSpace class of ROOT, and accept-rejecting based on the amplitude squared

### (see eq. 10).

π

### s

π0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ππ

### s ∂ )/ γ

-### π

+### π → ' η ( Γ ∂

0 0.05 0.1 0.15

0.2 0.25 0.3 0.35 0.4

−3

×10

Theor. decay rate.

Generated Distribution

Figure 13: Comparison of theoretical decay rate and generated s_{ππ}-distribution
from η^{0} → π^{+}π^{−}γ. Arbitrary normalisation.

### E ~

γ0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

γ

### E

### ~

+-### ∂ )/ γ π π → ' η ( Γ ∂

0 0.1 0.2 0.3 0.4 0.5

−3

×10

Theor. decay rate.

Generated Distribution

Figure 14: Comparison of theoretical decay rate and generated E_{γ}-distribution
η^{0} → π^{+}π^{−}γ. Arbitrary normalisation.

### 8 Background Discrimination

### In order to separate conversion events from the signal, one can study the position of the e

^{+}

### e

^{−}

### -vertices. A e

^{+}

### e

^{−}

### -pair from an η

^{0}