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ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology

135

Damian Pszczel

Search for a new light boson

in meson decays

Dissertation presented at Uppsala University to be publicly examined in Room 80101,Ångstrom Laboratory, Lägerhyddsvägen 1, Uppsala, Thursday, 25 October 2018 at 09:00 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: PhD Michel Garcon.

AbstractPszczel, D. 2018. Search for a new light boson in meson decays. Uppsala Dissertations fromthe Faculty of Science and Technology 135. 146 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-0436-6.

The subject of the presented work lies in the field of experimental particle physics. The maintopic is the study of e+ e− pairs from η meson decays. The data sample used in this work wascollected by the WASA-at-COSY collaboration in proton-proton collisions at 1.4 GeV kineticbeam energy. The experiment took place in 2012 at Forschungzentrum Jülich in Germany at theCOSY storage ring. An internal proton beam interacted with a pellet target of frozen hydrogen.

We implemented a set of selection criteria in order to extract the η → e+ e− γ event candidates.This is a rare electromagnetic decay of the η meson with branching ratio equal to 6.9·10−3. Theresulting set of events served as the basis for three analyses.

First, we extracted the η transition form factor that is a function depending on the inner quarkand gluon structure of the meson. We implemented a specific method to reduce the contributionof background channels from direct pion production.

The second analysis was the search for a narrow structure on the e+ e− invariant mass in theselected sample of η → e+ e− γ candidates. Many theoretical models and some astrophysicaland particle physics measurements suggest the existence of a new boson, also called the darkphoton, that couples to both dark and to Standard Model particles. This particle would decayto e+ e− pairs of well-defined mass and therefore could be detected by looking for narrow peaksin the e+ e− invariant mass spectra. Since no statistically significant signal was observed, we setan upper limit on the coupling parameter ε2.

The third objective of this work was to select a sample of η → e+ e− candidates. This is avery rare decay and therefore sensitive to physics beyond the Standard Model. No signal fromη → e+ e− was observed, therefore we were able to set an upper limit on the branching ratiofor this decay.

Keywords: dark matter, dark vector boson, dark photon, U boson, meson decays, form factor,Dalitz decay, eta meson

Damian Pszczel, Department of Physics and Astronomy, Nuclear Physics, Box 516, UppsalaUniversity, SE-751 20 Uppsala, Sweden.

© Damian Pszczel 2018

ISSN 1104-2516ISBN 978-91-513-0436-6urn:nbn:se:uu:diva-359777 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-359777)

To my parents, my wife and my brother. . .

Contents

1 Theory and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 The η meson and its decay channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Dark matter and search for a new light boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Dalitz decays of pseudoscalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.2 Search for a massive dark photon using e+e− invariant

mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4.3 Search for dark photon in π0 → e+e−γ channel . . . . . . . . . . . . . . . . . . 271.4.4 η → e+e−γ channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Rare leptonic decays of pseudoscalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.5.1 π0 → e+e− channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5.2 η → e+e− channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Analysis outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 The COSY storage ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 The WASA detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.1 Pellet Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Central Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.3 Forward Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 The Data Acquisition and Trigger System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4 Description of the trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 Forward Range Hodoscope energy calibration . . . . . . . . . . . . . . . . . . . . 503.1.2 The energy calibration of the Central Detector . . . . . . . . . . . . . . . . . . . . 53

3.2 Track reconstruction in MDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Pluto simulation framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 WASA Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Production of the η meson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4 The decays of η meson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.1 The η → γγ decay channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 The η → e+e−γ decay channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 The η → π+π−π0 decay channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.4 The η → π0π0π0 decay channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.5 The η → π+π−γ decay channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Direct production of π mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5.1 The pp→ ppπ+π− reaction channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5.2 The pp→ ppπ0π0 reaction channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5.3 The pp→ ppπ+π−π0 reaction channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.1 Initial data sample reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Normalization using η → γγ channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 Analysis of the main trigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.1 Trigger efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.2 Trigger stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Analysis of the η → e+e−γ channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1 Multiplicity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Time conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Particle identification method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.1 ΔE-p method - basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3.2 ΔE-p method - development of a systematic procedure . . . . . . 84

6.4 Rejection of the background from external pair production . . . . . . . . . . . . 866.5 Selection based on total missing energy versus total missing mass

plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.6 Angular distributions for the η → e+e−γ channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.6.1 Angle between η and the beam direction in laboratoryframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.6.2 Angle between γ∗ and γ in η meson rest frame . . . . . . . . . . . . . . . . . . . 906.7 Summary of the η → e+e−γ selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.8 Radius of closest approach after selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.9 Method for non-η background suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.10 Combinatorial background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.10.1 Estimate of the number of random coincidences . . . . . . . . . . . . . . . . . 986.10.2 Invariant mass distribution of random coincidences . . . . . . . . . . . 99

7 Analysis of η → e+e− channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.1 Summary of η → e+e− selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.2 Selection of η → e+e− event candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2.1 pp→ ppπ+π− background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2.2 η → e+e−γ background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2.3 Baryonic resonance background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2.4 Combinatorial background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2.5 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3 Summary of η → e+e− selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.1 Determination of the transition form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.1.1 Background subtraction using η → e+e−γ selection cuts 1058.1.2 Background subtraction using the fit to two proton

missing mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.2 U− γ coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3 Search for η → e+e− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3.1 Initial η → e+e− analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.3.2 Optimization of the η → e+e− analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.4 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9 Summary in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.1 The missing mass puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.2 Strategies of dark matter detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.3 The dark photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.4 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10 Summary in Swedish - Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.1 Mysteriet med den saknade massan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.2 Strategier för att detektera mörk materia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.3 Den mörka fotonen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.4 Resultat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

11 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

List of Tables

Table 1.1: Non-exhaustive list of pseudoscalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Table 1.2: Decay modes of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Table 1.3: Theoretical predictions of η → e+e− branching ratio . . . . . . . . . . . . . . . . 33

Table 2.1: Basic characteristics of the PT system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Table 2.2: Stopping power of the FRH for different particles . . . . . . . . . . . . . . . . . . . . . . 46

Table 5.1: Effect of the selection conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Table 5.2: Events in the final sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Table 5.3: Number of η mesons (/106) extracted from the fits with secondorder polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Table 5.4: Number of η mesons (/106) extracted from the fits with thirdorder polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Table 5.5: Number of η mesons (/106) extracted from the fits withoutadditional polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Table 5.6: Number of η mesons (/106) extracted from the fits with thirdorder polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Table 5.7: Neutral selection trigger combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Table 5.8: Charged selection trigger combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Table 6.1: η → e+e−γ and background reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Table 7.1: η → e+e− and background reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Table 8.1: Results of the form factor fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Table 8.2: Results of the form factor fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Table 8.3: FF fits from various experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Table 8.4: Results of the BRlimit fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Table 8.5: Results of the BRlimit fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

List of Figures

Figure 1.1: Elementary particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 1.2: Pseudoscalar meson nonet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 1.3: Positron fraction in cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.4: U boson production mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 1.5: Feynman diagram of the η → e+e−γ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 1.6: Feynman diagram of the η → e+e−γ decay in the VMDmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 1.7: Feynman diagram of the π0 → e+e−γ decay with a couplingbetween SM photon and U boson (dark photon) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 1.8: Invariant mass of e+e− from π0 → e+e−γ decay . . . . . . . . . . . . . . . . . . . . . 27Figure 1.9: Upper limits of (ε2,mA′) parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 1.10: Experimental determination of η transition form factor fromη → μ+μ−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 1.11: Experimental determination of η transition form factor . . . . . . . . . . 30Figure 1.12: P→ l+l− decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 1.13: Theoretical predictions of BR(π0 → e+e−) and KTeVexperimental value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.1: The COSY facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.2: The WASA set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.3: The definition of the coordinate system used in WASA . . . . . . . . . . 37Figure 2.4: The WASA Pellet Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.5: MDC inside Al-Be cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.6: MDC detector tilted layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.7: A schematic view of the forward, central and backward partsof the PS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.8: A longitudinal detection module of PS and the light guides(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.9: The 3D model of the central part of the PS surrounding theMDC detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.10: Calculated distribution of the magnetic flux density in theSCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.11: The electromagnetic calorimeter of WASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 2.12: SEC module, light guide and the photomultiplier tube . . . . . . . . . . . . 43Figure 2.13: Polar angle coverage of the calorimeter layers . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.14: The FWC detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.15: The FWC exploded view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.16: The Forward Proportional Chamber detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 2.17: The Forward Trigger Hodoscope detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 2.18: Schematic view of the Forward Range Hodoscope detector . . . 47Figure 2.19: The data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.20: The trigger system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.1: Illustration of non-uniformity in FRH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.2: Energy deposits in FRH2 vs FRH1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.3: Non-linearity fit graphical interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.4: Energy deposit in calorimeter versus q/e× momentum forη → e+e−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.5: Energy deposit in calorimeter versus q/e× momentum forη → π+π−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.6: Calorimeter calibration plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.1: The cross section for pp→ ppη and pp→ ppη ′ . . . . . . . . . . . . . . . . . . . 57Figure 4.2: The mechanism of pp→ ppη production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.3: Monte Carlo simulation (Pluto) for 106 events: proton kineticenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.4: Monte Carlo simulation (Pluto) for 106 events: proton θangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.5: Monte Carlo simulation (Pluto) for 106 events: θ angle of theη meson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.6: Monte Carlo simulation (Pluto) for 106 events: angle betweendilepton and photon in the laboratory frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.7: Geometrical variables involved in the description of a Dalitzprocess in PLUTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.8: Effect of the transition form factor on the invariant mass ofe+e− spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.9: Missing mass of two protons for pp→ ppπ+π−π0

. . . . . . . . . . . . . . . . 63Figure 5.1: Missing mass of two proton-like tracks in FD . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 5.2: Time difference between charged tracks in FD . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 5.3: Missing mass of two protons - selection with charged decays 67Figure 5.4: Missing mass of two protons - selection with neutral decays . 67Figure 5.5: Missing mass of two protons vs invariant mass of twophotons - selection with neutral decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 5.6: The invariant mass of γγ: data and η → γγ Monte Carlosimulation (WMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 5.7: The invariant mass of γγ: data and the fit result withindividual contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.8: The invariant mass of γγ: fit with different binning . . . . . . . . . . . . . . . . . 71Figure 5.9: The invariant mass of γγ: fit with different fit range . . . . . . . . . . . . . . . . 72Figure 5.10: List of triggers for the experiment with the prescaling factor(fifth column) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.11: The trigger stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 6.1: Invariant mass of two charged particles in CD (with theassumption of electron masses): 5 ·106 η → e+e−γ events . . . . . . . . . . . . . . . . . . . . . . . . . 79

Figure 6.2: Invariant mass of two charged particles in CD (with theassumption of electron masses): 8 ·105 η → π+π−γ events . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 6.3: Invariant mass of two charged particles in CD (with theassumption of electron masses): 1 ·107 η → γγ events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 6.4: Invariant mass of two charged particles in CD (with theassumption of electron masses): 14 ·106 pp→ ppπ+π− events . . . . . . . . . . . . . . . . . 80Figure 6.5: Invariant mass of two charged particles in CD (with theassumption of electron masses): 1 ·106 pp→ ppπ+π−π0 events . . . . . . . . . . . . . . . 81Figure 6.6: Invariant mass of two charged particles in CD (with theassumption of electron masses): 1 ·107 pp→ ppπ0 (→ γγ)π0 (→ e+e−γ)events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 6.7: Invariant mass of two charged particles in CD (with theassumption of electron masses): 106 pp→ pΔ+→ ppe+e− events . . . . . . . . . . . 82Figure 6.8: Time difference between neutral tracks and protons . . . . . . . . . . . . . . . . 83Figure 6.9: Time difference between charged tracks and protons . . . . . . . . . . . . . . 83Figure 6.10: Time difference between charged and neutral tracks in CDbefore time cuts relative to protons in FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 6.11: Time difference between charged and neutral tracks in CDafter time cuts relative to protons in FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 6.12: Energy deposit in SEC versus charge × momentum plot -data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 6.13: Energy deposit in CD versus charge × momentum plot -graphical cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 6.14: Particle identification fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 6.15: Normalized distances from the e+ fitted line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 6.16: Photon conversion into e+e− in the beam pipe material . . . . . . . . . . 87Figure 6.17: WASA MC η → e+e−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 6.18: WASA Monte Carlo η → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 6.19: Data with conversion graphical cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 6.20: Total missing energy versus total missing mass forη → π+π−γ simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 6.21: Total missing energy versus total missing mass forη → e+e−γ simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 6.22: Total missing energy versus total missing mass forpp→ ppπ+π− simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 6.23: Total missing energy versus total missing mass forpp→ ppπ+π−π0 simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 6.24: Theta angle of η meson in laboratory reference frame -Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 6.25: Theta angle of η meson in laboratory reference frame - data . 91Figure 6.26: Angle between γ and γ∗ in η meson rest frame - simulationof η → e+e−γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 6.27: Angle between γ and γ∗ in η meson rest frame - simulationof different channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 6.28: Radius of closest approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 6.29: Missing mass of two protons versus invariant mass of e+e−(data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 6.30: Global fit to the missing mass of two protons . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 6.31: Fit to the missing mass of two protons for IMe+e− in (0,50)MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.32: Fit to the missing mass of two protons for IMe+e− in (50,100)MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.33: Fit to the missing mass of two protons for IMe+e− in(100,150) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.34: Fit to the missing mass of two protons for IMe+e− in(150,200) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.35: Fit to the missing mass of two protons for IMe+e− in(200,250) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.36: Fit to the missing mass of two protons for IMe+e− in(250,300) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.37: Fit to the missing mass of two protons for IMe+e− in(300,350) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.38: Fit to the missing mass of two protons for IMe+e− in(350,400) MeV/c2 range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 6.39: Invariant mass of e+e− - extracted η content and VMDsimulation with form factor corresponding to the ρ meson mass . . . . . . . . . . . . . . . . 97Figure 6.40: Charged tracks multiplicity after particle identification . . . . . . . . . . . 98Figure 6.41: Invariant mass of e+e− distribution for the combinatorialbackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 7.1: Angle between e+e− pair (laboratory frame): η → e+e−simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 7.2: Energy of e+ and e−: η → e+e− simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 7.3: Angle between electrons on OXY plane and applied cut:data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 7.4: Angle between electrons on OXY plane and applied cut:η → e+e− simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 7.5: Sum of energy deposits of the two electrons and applied cut:data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 7.6: Sum of energy deposits of the two electrons and applied cut:η → e+e− simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 7.7: Invariant mass of ee and applied cut: data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 7.8: Invariant mass of ee and applied cut: η → e+e− simulation . 104Figure 8.1: Transition form factor fit (20 MeV/c2 bin width): Sample A 106Figure 8.2: Transition form factor fit (20 MeV/c2 bin width): Sample B 106Figure 8.3: Transition form factor fit (20 MeV/c2 bin width): Sample C 106Figure 8.4: Transition form factor fit (40 MeV/c2 bin width): Sample A 107Figure 8.5: Transition form factor fit (40 MeV/c2 bin width): Sample B 107Figure 8.6: Transition form factor fit (40 MeV/c2 bin width): Sample C 107

Figure 8.7: Form factor resulting from fit based on backgroundsubtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 8.8: Form factor fit based on background subtraction (bin widthmodified for 0−100 MeV/c2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 8.9: Form factor fit based on background subtraction (bin widthmodified for 100−400 MeV/c2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 8.10: Form factor comparison between this work and othermeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 8.11: Invariant mass of e+e− - data and the backgroundsimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Figure 8.12: Invariant mass of e+e− - difference between data and sumof background Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 8.13: The smearing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 8.14: Acceptance of η →U(→ e+e−)γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 8.15: Reconstructed e+e− invariant mass distribution for MU = 50MeV/c2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 8.16: Reconstructed e+e− invariant mass distribution forMU = 100 MeV/c2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 8.17: Reconstructed e+e− invariant mass distribution forMU = 150 MeV/c2






. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 8.23: Upper limit on β as a function of U boson mass MU . . . . . . . . . . . . . 117Figure 8.24: Upper limit on ε2 as a function of e+e− invariant mass(systematics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 8.25: Upper limit on ε2 as a function e+e− invariant mass . . . . . . . . . . . . . 118Figure 8.26: Fit to the η → e+e− two proton missing mass distribution . . . 119Figure 8.27: Fit to two proton missing mass (530−560 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 8.28: Fit to two proton missing mass (530−560 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 8.29: Fit to two proton missing mass (530−560 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 8.30: Fit to two proton missing mass (530−580 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Figure 8.31: Fit to two proton missing mass (530−580 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 8.32: Fit to two proton missing mass (530−580 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 8.33: Fit to two proton missing mass (520−590 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 8.34: Fit to two proton missing mass (520−590 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 8.35: Fit to two proton missing mass (520−590 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 8.36: Energy deposits of charged particles in SEC (RS1 beforecut) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Figure 8.37: Energy deposits of charged particles in SEC (RS2 beforecut) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Figure 8.38: Energy deposits of charged particles in SEC (RS1 after cut) 124Figure 8.39: Energy deposits of charged particles in SEC (RS2 after cut) 124Figure 8.40: Particle identification plot - RS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 8.41: Particle identification plot - RS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 8.42: Fit to two proton missing mass (530−560 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 8.43: Fit to two proton missing mass (530−560 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 8.44: Fit to two proton missing mass (530−560 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 8.45: Fit to two proton missing mass (530−580 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 8.46: Fit to two proton missing mass (530−580 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 8.47: Fit to two proton missing mass (530−580 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 8.48: Fit to two proton missing mass (520−590 MeV/c2): 2MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Figure 8.49: Fit to two proton missing mass (520−590 MeV/c2): 5MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Figure 8.50: Fit to two proton missing mass (520−590 MeV/c2): 10MeV/c2 bin width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Figure 9.1: Invariant mass of e+e− - data and the backgroundsimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 9.2: Upper limit on ε2 as a function e+e− invariant mass . . . . . . . . . . . . . 134Figure 9.3: Form factor comparison between this work and othermeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 10.1: Den invarianta massfördelningen för e+e− - par frånexperimentet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 10.2: Uppmätt övre gräns för den mörka fotonens koppling tillgamma-kvanta i detta arbete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Figure 10.3: η mesonens formfaktor bestämd i detta arbete tillsammansmed andra data och teoretiska beräkningar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

1. Theory and motivation

This chapter is an attempt to contextualize the main topics of this thesis and tosummarize some previous results.

1.1 The Standard ModelThe Standard Model (SM) is a physical theory developed in the second halfof the XXth century. It describes the reality as emerging from different combi-nations of 12 elementary particles, quarks and leptons (left side of figure 1.1),that interact trough the exchange of 5 so called gauge bosons (right side offigure 1.1).

Figure 1.1. Elementary particles.

The Higgs boson occupies a special place in this picture for it is responsiblefor particle masses. Its recent (2012) discovery at CERN [1, 30] reaffirmed theSM is one of the most successful physical model of all time.

1.2 The η meson and its decay channelsThe etymology of the word meson comes from the Greek mesos which meansmiddle. Hideki Yukawa, in the 1930’s, first used this term to characterize

19

Table 1.1. Non-exhaustive list of pseudoscalar mesons.

Particle Quark content Mass (MeV/c2) IG JPC Mean lifetime (ns)

π+/π− ud/ud 139.57 1− 0− 2.6×10−8

π0 uu−dd√2

134.97 1− 0−+ 8.52×10−17

η uu+dd−ss√3

547.86 0+ 0−+ 5.02×10−19

K+/K− us/us 493.68 12 0− 1.24×10−8

K0S

ds−sd√2

497.61 12 0− 8.95×10−11

K0L

ds+sd√2

497.61 12 0− 5.12×10−8

particles with masses between those of electrons and protons. Those parti-cles are known today as pions (π mesons). Mesons interact strongly so theyare hadrons and they are composed of quark-anti-quark pair thus their baryonnumber is zero.

The η particle is a pseudoscalar meson. The list of pseudoscalar mesons,along with their basic characteristics such as the quark content, mass, meanlifetime, isospin (I), G-parity (G), total angular momentum (J), parity (P) andC-parity (C) is presented in table 1.1. The total spin of pseudoscalars is zeroand they have odd parity JP = 0−. The pseudoscalar mesons consisting of up,down, and strange quarks only - pions, kaons , η and η ′ mesons - form a nonet(see figure 1.2). In terms of the SU(3) flavour symmetry group, introducing amixing angle θ , one can represent the η meson as a superposition of a singletand an octet states η1 and η8:

|η1〉= 1√3(uu+dd + ss) (1.1)

|η8〉= 1√6(uu+dd−2ss) (1.2)

and (see [57])

|η〉= cosθ |η8〉− sinθ |η1〉 ≈ 1√3

uu+dd− ss, (1.3)

where the θ value estimations vary between −15.4◦ [40] and −18.4◦ [47]while combined BABAR and CLEO data provide θ ∼−16.84◦.

The approximation that the three lightest quarks (u, d and s) have zeromasses leads to an exact SU(3) flavour symmetry, also known as chiral sym-metry of the quantum chromodynamical (QCD) Lagrangian. In this picture,the pseudoscalar mesons would be the massless Goldstone bosons1. Since we

1Only η ′ meson would have a non zero mass.

20

Figure 1.2. Pseudoscalar meson nonet.

Table 1.2. Decay modes of η .

Decay mode Branching ratio

η → γγ (39.41±0.20)%

η → π0π0π0 (32.68±0.23)%

η → π+π−π0 (22.92±0.28)%

η → π+π−γ (4.22±0.08)%

η → e+e−γ (6.9±0.4)×10−3

know that quark masses are non zero, this symmetry is broken and the pseu-doscalar mesons acquire their masses.The lifetime of the η (see table 1.1) meson is relatively long since all its strong,electromagnetic and weak decays are forbidden in the first-order (C, CP, Gparity conservation). Its main decay channels are shown in table 1.2.

1.3 Dark matter and search for a new light bosonThe SM of particle physics provides a very satisfactory description of the in-teractions between fields and matter that fill the space we live in. However,the unexplained nature of Dark Matter motivates searches for an extension ofthis model to a more fundamental theory. One of many possible approaches tosolve this issue consists in introducing the concept of the Dark Sector.

Dark Sector particles interact weakly with the usual matter through oneor more mediators that are coupled to the SM via a portal. There are dif-ferent types of portals depending on the mediator spin and parity - scalar,pseudoscalar, fermion or vector. Some astrophysics observations such as thepositron and/or electron excesses observed by PAMELA [8] (see spectra in

21

figure 1.3 from [9]), ATIC [29], H.E.S.S. [11] and AMS [70] as well as thenarrow 0.511 MeV γ ray emission from the galactic bulge observed by IN-TEGRAL [52] may indicate the presence of new undiscovered particles thatdecay to e+e− pairs. Those measurements are our principal motivation to fo-cus on the hypothetical vector portal and its associated vector mediator.

Figure 1.3. Positron fraction in cosmic rays (see [9]).

The hypothesis of a new boson solving the problem of the dark matter con-tent in the Universe is an idea that emerged in the 80’s. Multiple authorspostulated an extra U(1)dark abelian gauge field and the associated light vec-tor boson, also called the dark photon, in the O(MeV-GeV) mass range, as apossible extension of the SM [38, 31, 23].

The dark photon is a vector field A′μ with Lagrangian:

LA′ =−14

F ′μνF ′μν +ε

2cosθWBμνF ′μν −

12

m2A′A

′μA′μ , (1.4)

where F ′μν ≡ ∂μA′ν−∂νA′μ is the dark photon field strength and Bμν ≡ ∂μBν−∂νBμ is the SM hypercharge field strength and θW is the Weinberg weak mix-ing angle. This model of the minimal kinetically mixed dark photon [12] isparametrized by the dark photon mass mA′ and the kinetic mixing parameter ε .For mA′ in the range of MeV-GeV, the dominant effect of this kinetic mixing(after electroweak symmetry breaking) is a coupling to the SM electromag-netic field strength Fμν expressed by ε

2 F ′μνFμν . The result of this mixing isthat the dark photon acquires a coupling of strength eε to the electromagneticcurrent. To put it simply, the existence of a dark photon should be taken into

22

account in every phenomenon where photons are involved and, given our ex-perimental tools are sufficiently sensitive (precise), we could detect its effectsin every electromagnetic measurement.

A possible mechanism for dark photon productionDark photons could be produced in a reaction similar to the one presented infigure 1.4. A dark particle χ annihilates with its antiparticle χ producing anintermediate U vector boson. The rest of this work is based upon the hypoth-esis that the dark bosons couple with SM photons that in turn produce e+e−pairs. The e+e− invariant mass corresponds to the mass of the dark boson.

Figure 1.4. U boson production mechanism.

1.4 Dalitz decays of pseudoscalar mesonsA (single) Dalitz decay of a pseudoscalar meson, also called a conversiondecay, is shown in figure 1.5 for η . The meson decays into one real and onevirtual photon. The latter converts into a lepton-antilepton pair (electrons ormuons). The squared four-momentum of the virtual photon, q2 is equal to theinvariant mass of the resulting l+l− system:

q2 = M2l+l− = (El+ +El−)

2− (−→pl+ +−→pl−)

2, (1.5)

where E denotes the particle energy and �p its momentum vector. In the point-like QED approximation, the differential cross section2 of the Dalitz decay isgiven by:

2Normalized with respect to the double photon channel.

23

dΓP→l+l−γ

dq2ΓP→γγ=

2α3πq2

√1− 4m2

lq2

(1+2m2

lq2

)(1− q2

M2P)3/2 (1.6)

The equation 1.6 holds as long as we neglect the inner structure of the pseu-doscalar meson, i.e. its quark and gluon content. In section 1.4.1 we willdescribe the formalism used to take into account this effect.

Figure 1.5. Feynman diagram of the η → e+e−γ decay.

1.4.1 Form factorsElectromagnetic form factor

When we consider the scattering on charged composite particles, we need totake into account the inner electromagnetic structure of those objects. Quan-tum electrodynamics (QED) allows us to calculate the interaction propertiesof point-like particles. The inner electromagnetic structure of a composite par-ticle can be represented by a function called the form factor. The differentialcross-section for the scattering of an electron (elementary point-like particle)with a composite particle (such as protons, mesons, etc) can then be written inthe form:

dσdq2

=

∣∣∣∣ dσdq2

∣∣∣∣QED

∣∣∣F(q2)∣∣∣2 , (1.7)

where∣∣∣ dσ

dq2

∣∣∣QED

is the differential cross-section for the scattering of an elec-

tron on another point-like charged particle calculated in the QED framework,and F(q2) is the electromagnetic form factor function depending on the trans-ferred four-momentum q. In the non-relativistic limit, the electromagneticform factor is the Fourier transform of the charge distribution function.

Transition form factor

In subsection 1.4.1, we have seen how to determine the electromagnetic formfactor of a non-elementary charged particle - we basically just need to bombard

24

it with a structure-less particle such as electrons and calculate the deviation ofthe observed differential cross-section with respect to theoretical predictions.For a pseudoscalar meson such as η this is more complicated because it isshort-lived and electrically neutral. Moreover, the conservation of C-parityforbids the processes involving neutral mesons and a single-photon exchange.In order to investigate the inner (quark and gluon) structure of a pseudoscalarmeson we have to adopt a more subtle approach. We study the decay of theform:

η → γ + γ∗ → γ + e++ e− (1.8)

This is the Dalitz decay of the η meson. As already seen in 1.4, it involves twophotons - one real, massless, and one virtual with non-zero mass that convertsinto an electron-positron pair3. The term transition is generally used in pro-cesses such as A→ Bγ where A and B are neutral mesons. The transition formfactor describes the effects of the electromagnetic dynamic structure arising atthe transition vertex of this process. The Dalitz decay is a special case sinceit only involves one neutral meson. Therefore, the corresponding transitionform factor depends only on the electromagnetic structure of this meson. Thevirtual photon carries a time-like four-momentum q2 > 0, which is definedby the equation 1.5. The probability of formation of a lepton pair with somemass me+e− is proportional to the probability of emission of a virtual photonwith four-momentum q2. In case of a Dalitz decay q2 = m2

e+e− . Analogicallyto the approach presented in section 1.4.1, we can express the invariant massspectrum of e+e− system as a product of two contributions: a QED part corre-sponding to the point-like approximation (see equation 1.6) and the transitionform factor that contains the effect of the inner structure of the η meson.

Vector Meson Dominance model

In the 60’s, J.J. Sakurai predicted the existence of vector mesons coupled tothe hadronic isospin and hypercharge currents [64]. The VMD model was in-troduced in order to explain the fact that the interaction between (energetic)photons and hadrons is much more intense than expected by the sole interac-tion of photons with the hadron’s electric charge4. The model tries to solvethis issue by assuming that the photon (JP = 1−) is a superposition of the pureelectromagnetic photon which interacts only with electric charges and neutralvector mesons (also JP = 1−). It follows that the interaction between photonsand hadrons occurs by the exchange of vector mesons and it is these mesonsthat give rise to the enhancement5. This effect is especially well pronouncedin the case of time-like photons (q2 > 0) when the squared 4-momentum ap-

3More generally, lepton-antilepton pair.4Moreover, the interaction between photons and protons is comparable to the interaction ofphotons with neutrons in spite of the difference of their electric charge structure.5Through their resonant, i.e. possessing a complex pole, propagators.

25

proaches the squared mass of one of the vector meson (e.g. q2 ≈ m2ρ ). In the

framework of the VMD model the form factor takes the form of:

F(q2) = ∑V

M2V

M2V −q2− iMV ΓV (q2)

∼= 1

1− q2

M2V

, (1.9)

where V = ρ, ω, φ 6, ΓV (q2) is the total width and MV the mass of the vector

meson, q2 was already described by 1.5.The illustration of the VMD model in case of η → e+e−γ decay is shown

in figure 1.6. One can compare it to the diagram from figure 1.5.

Figure 1.6. Feynman diagram of the η → e+e−γ decay in the VMD model.

The equation 1.9 is often used as a fit function. The MV vector mesonmass is replaced by a free parameter Λ and, for convenience, the value of Λ−2

(sometimes called bη ) is provided as result of form factor calculations.

1.4.2 Search for a massive dark photon using e+e− invariantmass distribution

The search for a new particle in the Dalitz decays of light mesons is based onthe assumption that if there is a coupling between the new dark boson and theStandard Model photon. Therefore, one could expect to see its signature bylooking at the distribution of the invariant masses of e+e− pairs. Figure 1.7presents the same Feynmann diagram for a Dalitz decay7 that was shown insection 1.4.1 with the only difference that the intermediate vector meson wasreplaced by the hypothetical new particle (with mass lower than the decayingmeson mass).

If this new particle has a well-defined mass, one would expect to find anexcess of event corresponding to its value. We use this fact in our analysisof the η → e+e−γ decay and try to find any more or less narrow structurein the e+e− invariant mass spectra that is not compatible with any knownbackground.

6For the energy range considered in our experiment the most important contribution originatesfrom the ρ meson.7Figure 1.7 shows the π0 → e+e−γ reaction but the discussion that follows is equally true forthe η → e+e−γ decay that we use in our work.

26

Figure 1.7. Feynman diagram of the π0 → e+e−γ decay with a coupling between SMphoton and U boson (dark photon).

1.4.3 Search for dark photon in π0 → e+e−γ channelThe search for a dark photon was already performed by WASA-at-COSY col-laboration in the π0 → e+e−γ channel. The results were published in refer-ence [5]. The idea of the analysis was to search for a narrow structure in theinvariant mass spectrum of e+e− pair (see section 1.4.2) for dark photon massin 20-100 MeV/c2 range (see figure 1.8). For ε2 > 10−6 the average path trav-eled by a dark photon emitted in a low energy π0 decay should be less thana millimeter, therefore we are, in principle, able to detect its decay inside ourdetector. Neglecting higher-order electromagnetic and tiny weak contributionsand assuming that the dark photon doesn’t decay to light dark scalars and/orfermions, the total width of the dark photon can be expressed as:

ΓA′ = ΓA′→e+e− =13

αε2mA′

√1− 4m2

e

m2A′(1+

2m2e

m2A′). (1.10)

A sample of 5 · 105 π0 → e+e−γ decays were selected and since no signal

Figure 1.8. Invariant mass of e+e− from π0 → e+e−γ decay from [5].

from dark photon decay was observed an upper limit for the model parameters(ε2,mA′) was established. This is shown in figure 1.9.

27

Figure 1.9. Upper limits of (ε2,mA′) parameter space from [5].

1.4.4 η → e+e−γ channelThe first observations of η → e+e−γ (and π0 → e+e−γ) channel were per-formed in the 70’s (see references [55] and [51]). They were based on limiteddata samples with high background contribution (mostly from photon con-version in the target and the detector material). A very insightful treatment ofelectromagnetic decays of light mesons, including η → e+e−γ , is presented byLandsberg in [58]. The main reason for present day focus on the η → e+e−γdecay is related to the determination of η transition form factor. This is donethrough two decay modes η → e+e−γ and η → μ+μ−γ . The latter chan-nel was studied in IHEP8 with Lepton-G spectrometer [36] and by NA60 ex-periment at CERN in reference [14] and, more recently [15]. The results ofLepton-G and NA60 experiments are based on 600 and 9000 event candidatesrespectively. They are shown, along with the VMD prediction, in figure 1.10.One can observe there is no information about the form factor below two muonmasses since it corresponds to the lower kinematic limit of the squared four-momentum transfer. The η → e+e−γ channel was analyzed in SND detectorat VEPP-2M collider (Novosibirsk) in 1996 and 1998 [3]. Here, 109 candi-dates were found. Another experiment was performed at MAMI-C acceleratorwith Crystal Ball (CB) and TABS detectors [20]. It collected 1345 η → e+e−γcandidates. Both results, along with NA60 data points and a theoretical cal-culation from reference [68], are shown on figure 1.11. The most precise

8Institute for High Energy Physics located in Protvino (Russia).

28

Figure 1.10. Results of the Lepton-G (open circles) and the NA60 (triangles) mea-surements of the η → μ+μ−γ decay. The solid and dashed-dotted lines are fits to theNA60 data while the dotted line is the VMD model prediction. This figure is takenfrom [36].

29

Figure 1.11. The red circles are the data of [20] (the black curve is the fit to thedata). The green (open) circles show the result of the SND experiment [3]. The blue(inverted) triangles represent the result obtained by NA60 [36]. The green (dashed)curve is a calculation performed by [68]. Picture is taken from [20].

30

measurements of the Λ−2 parameter up to date were performed by the NA60experiment [15]: Λ−2 =

(1.934±0.067stat±0.050sys

)GeV−2. The most re-

cent result is from MAMI (CB/TABS detectors) [7]: Λ−2 = (1.97±0.11tot)GeV−2.The analysis of this channel will be detailed in chapter 6. It will be a doubletrack approach. On the one hand, we will extract the η transition form factorand on the other hand we will search for the dark photon. The latter will pro-ceed similarly to the π0 → e+e−γ analysis described in 1.4.3 as we look for apeak in the invariant mass of e+e− from η → e+e−γ decay.

1.5 Rare leptonic decays of pseudoscalar mesonsThe rare leptonic decays of pseudoscalar mesons, P → l+l−, provide a verysensitive probe of physics beyond the SM. Historically, the first decay of thistype to be discovered experimentally was η → μ+μ− [48]. In those fourth-order electromagnetic processes, the hadron vertex is connected to the lep-tonic pair by two virtual photons (see figure 1.12). This decay cannot proceedthrough one virtual photon (tree level) since a photon cannot couple to spin 0.It is suppressed with respect to P → γγ reaction by two orders of α ∼ 10−2

for it has two more vertices. In the decay into e+e−, highly relativistic, theapproximate conservation of helicity reduces this branching ratio even more,by a factor of 2(me/mP)

2. Those effects are responsible for the extremely lowSM value of those processes. Following reference [58], the branching ratiocan be written in the form:

BR(P→ l+l−) = Γ(P→ l+l−)/Γ(P→ all channels)

= BR(P→ γγ)2α2ξ 2β [|X |2 + |Y |2], (1.11)

where α is the fine structure constant, ξ = ml/mP, β = (1− 4ξ 2)1/2, X andY are the real (dispersive) and imaginary (absorptive) components of the nor-malized dimensionless amplitude in the P→ l+l− decay, respectively. If we

neglect the dispersive part, the imaginary component |Y |2 = 14 β−2

(ln 1+β 2

1−β 2

)2

provides a way, using the so-called unitarity bound9, to set a theoretical lowerlimit on the branching ratio. The dispersive part diverges logarithmically fora pointlike vertex. Therefore we must introduce a cutoff related to the vertexstructure, the form factor FP(q

2γ1

;q2γ2

;m2P) (see figure 1.12). A more detailed

description of the form factor is presented in section 1.4.1.

9Based on the fact that |Y |2 ≥ (ImY )2

31

Figure 1.12. P→ l+l− decay.

1.5.1 π0 → e+e− channelThe unitary bound for this channel constrains its branching ratio: BR(π0 →e+e−) ≥ 4.69× 10−8. Taking into account the data from CELLO and CLEOexperiments [18] and [43] to determine the form factor parameters, a valueof BRth(π0 → e+e−) = (6.23± 0.09)× 10−8 is obtained in [33]. This re-sult is 3.3σ below the experimental value from KTeV experiment at FermilabBRexp(π0 → e+e−) = (7.48±0.29stat ±0.25sys)×10−8 [2]. KTeV measured794 π0 → e+e− events candidates where π0’s were produced and tagged byKL → 3π0 reaction. This enhancement is hard to explain by SM contributions(radiative and mass corrections). On the graph 1.13 from [32] we see manytheoretical calculations of this branching ratio (using different models), theunitary bound and the CLEO bound (form factor data) along with the experi-mental KTeV value. Although some of the theoretical predictions [65] (very

Figure 1.13. Theoretical predictions of BR(π0→ e+e−) and KTeV experimental value(taken from [32]).

large uncertainties) and [42] reported in figure 1.13 are consistent with theexperimental value, the result calculated by [34] in 2007 is the most precise

32

Table 1.3. Theoretical predictions of η → e+e− branching ratio.

Model Ref BR(η → e+e−)×109

LMD large-Nc [54] 4.5±0.02CLOE+OPE [34] 4.6±0.06Modified VMD [60] 4.65±0.01Hidden gauge [60] 4.68±0.01ChPT [65] 5±1Mass corrections [35] 5.24ChPT large-Nc [42] 5.8±0.2

and therefore relevant estimation to searches related to this channel. We canneglect any weak SM contribution since the Z boson is much more massivethan the pion mZ/mπ0 ∼ 103. The discrepancy between theoretical predictionsand experimental results described in this section leads to speculations aboutthe existence of yet undiscovered light particles.

1.5.2 η → e+e− channelThe most recent theoretical calculations of this decay branching ratio providevalues around 10−9. They use different models, among other ChPT [65], mod-ified VMD, hidden gauge [60] and take into account various corrections (suchas mass correction [35]). Those results are listed in table 1.3.

In 2008, a branching ratio upper limit of 2.7 · 10−5 at 90% C.L. was es-tablished for this channel in WASA-at-CELSIUS experiment and publishedin [22]. WASA-at-CELSIUS set-up consisted of the same WASA detector weuse in this work while it operated at the CELSIUS storage ring located in Up-psala (Sweden) at The Svedberg Laboratory. This limit was extracted from asample of 2.41 · 105 η mesons produced in pd→ 3Heη reaction at 893 MeVincident proton energy (close to η production threshold). The best actual up-per limit is < 4.9 · 10−6 (at 90% confidence level) and it was established byHADES experiment in 2012 [10]. HADES operates at GSI research center inDarmstadt (Germany). This result was based on data collected in pp collisionsat 3.5 GeV.Up till now, there are only upper limits for η → e+e− decay but if one wereto determine a BR(η → e+e−) value exceeding ∼ 6 ·10−9 it might be seen asa signature of physics beyond the SM. The analysis of 2012 WASA-at-COSYdata with respect to η → e+e− channel is described in chapter 7.

1.6 Analysis outlookThe structure of this work is the following. In chapter 2, we present the ex-perimental set-up, the technical parameters of the COSY storage ring and a

33

detailed description of the WASA detector together with its data acquisitionsystem. Chapter 3 is dedicated to event reconstruction i.e. the process usedin WASA-at-COSY in order to extract particle tracks from raw detector re-sponses. In chapter 4, we describe the methods of event simulation that areused in this work. We illustrate some of reconstruction effects on those sim-ulations and we briefly discuss the rest gas influence. Chapter 5 presents thedata sample used in this work and describes the initial data reduction. Chapter6 and chapter 7 report the analysis of η → e+e−γ and η → e+e− channelsbased on this data sample, respectively. Results of those analysis and the re-lated discussions are presented in chapter 8. At the end of this chapter, thereis a short summary and the outlook for future activity is sketched.

34

2. Experimental set-up

2.1 The COSY storage ringCOSY (Cooler Synchrotron) is a particle accelerator and a storage ring (184m of circumference) operated by the Institute of Nuclear Physics (IKP) atForschungzentrum Jülich in Germany. It provides polarized and unpolarizedbeams of protons or deuterons with momentum range between 0.3 and 3.7GeV/c.

The schematic view of the COSY facility is shown in figure 2.1. It has theform of a race track with two 40 m long straight sections. COSY is able toprovide beams to both internal and external targets.

The isochronous cyclotron JULIC starts the acceleration process impartingup to 296 MeV/c momenta to ions. The beam is then injected into the COSYring where it is accelerated until the desired momentum is reached. In order toreduce the phase-space volume, down to Δp/p∼ 10−4, electron and stochasticcooling are implemented (see ref [61]).

The COSY ring can store around 1011 particles per accelerated bunch. Thisallows for typical luminosities around 1031cm−2s−1 in case of experimentswith an internal target. The beam intensity is reduced due to collisions withthe internal target (or beam extraction). In case of our experiment, the durationof a typical acceleration cycle is of the order of a few minutes. The averagebeam momentum also decreases with time - this effect is compensated by theuse of “barrier bucket”, a nonlinear radio frequency cavity (see reference [67]for details).

2.2 The WASA detectorWASA (Wide Angle Shower Apparatus) is a large-acceptance detector (almost4π) for charged and neutral particles. It operated at the CELSIUS storagering at The Svedberg Laboratory in Uppsala (Sweden) until June 2005. Afterthe shutdown of CELSIUS, the detector was transported to COSY. Installedin summer 2006, it collected data until middle of 2014. In 2015 the centraldetector and the pellet target were removed while the forward detector is usedas azimuthally symmetric polarimeter for the Electric Dipole Moment (EDM)experiment.

The WASA detector is equipped with an internal target and situated at oneof the straight sections of the COSY storage ring. It was designed to study

35

Figure 2.1. The COSY facility.

36

rare decays of mesons (π0, η , η ′ or ω) and to investigate the structure ofhadrons and symmetry breaking mechanisms. The cross section of the detectortogether with the acronyms of its sub-parts is shown in figure 2.2.

Figure 2.2. The WASA set-up.

In the following sections, we divide the WASA detector into three compo-nents: the pellet target (PT), the central detector (CD) and the forward detector(FD) and we describe their sub-parts.

The WASA coordinate system (used in this analysis) is sketched in fig-ure 2.3.

x

y

z

COSY beam

Pellet beam

Particle trajectory

φθ

Figure 2.3. The definition of the coordinate system used in WASA.

37

2.2.1 Pellet TargetThe WASA target is based on a unique design principle that provides dropletsof frozen hydrogen or deuterium. To form those pellets from an initial jet,a piezoelectric vibrating nozzle is used. The pellets are then collimated andinjected into the scattering chamber where they freeze by evaporation1 andinteract with the accelerator beam. The pellet guiding tube is connected to theCOSY beam pipe tube - to its top - in order to allow the pellet stream injectionfrom above and to its bottom - to collect the droplets in the pellet beam dump.

Figure 2.4. The WASA Pellet Target.

This particular setup was developed and tested at the CELSIUS storage ring(see [69] and [37]). It allows for high luminosities (up to 1032 cm−2 s−1) andsignificantly limits the internal photon conversion inside the interaction region(IR). The system design provides the necessary space to insert the 4π detectoraround the IR. A schematic description of the PT is shown in figure 2.4 andsome of its basic characteristics are listed in table 2.1

2.2.2 Central DetectorThe main objective of this part of the WASA detector is the detection and iden-tification of the particles produced in the collisions of beam and pellet targetstreams. Those particles are directly produced in the interaction process or in-

1The temperature and pressure conditions in the scattering chamber are kept below the hydrogentriple point.

38

Table 2.1. Basic characteristics of the PT system.Pellet parameter Value units

Diameter 25−35 μmFrequency 5−7 kHzVelocity 70−80 m/sStream divergence 0.04 ◦Effective thickness ∼ 1015 atoms/cm2

Figure 2.5. MDC inside Al-Be cylinder. Figure 2.6. MDC detector tilted layers.

directly as a decay product of other particles. The WASA CD, surrounding theinteraction region, consists of three sub-detectors and a solenoid. It is able tomeasure energies and momenta of charged and neutral particles in an almost4π solid angle range.

Mini Drift Chamber

The so-called Mini Drift Chamber (MDC) consists of 1738 aluminized Mylardrift tubes arranged in 17 cylindrical layers with layer radius between 41 mmand 204 mm (see figure 2.5). Each straw (drift tube) is filled with a mixture ofargon and methane (80%-20%) and contains a central anode made of a 20 μmdiameter gold plated tungsten wire. The diameter of those straws varies be-tween 2 mm (five inner layers), 3 mm (six central layers) and 4 mm (six outerlayers). In order to allow for z coordinate determination, the tubes in eightof the layers are slightly skewed (tilted) with respect to the beam direction(from 6◦ to 9◦) forming a hyperboloidal shape (see figure 2.6). The tubes inthe remaining nine layers are aligned with the COSY beam axis. Each layerof straws is held in place by semi-ring plates made of Al-Be (50%-50%) alloyand the whole structure is placed inside an Al-Be cylinder.

The MDC detector surrounds the interaction region. It is used to reconstructvertex positions and charged particle momenta, charges and energies based on

39

the characteristic of their tracks in the magnetic field provided by the solenoid(see section 2.2.2). It covers a polar angle (θ on figure 2.3) range from 24◦to 159◦. The precision of vertex reconstruction for the scattered protons isσx,y ∼ 1 mm (transverse directions) and of σz ∼ 3 mm (beam axis). A detaileddescription of the MDC detector can be found in [49].

Plastic Scintillator Barrel

Figure 2.7. A schematic view of the forward, central and backward parts of the PSdetector.

The PSB is a collection of thin, 8 mm thick, BC-408 plastic scintillators.It has cylindrical shape and is placed directly outside the MDC detector (seefigure 2.9). The central part (PSC), made of 50 rectangular bars2, forms twopartially overlapping layers and surrounds the drift chamber (MDC). Its ex-tremities are closed with forward and backward end caps, consisting of 48trapezoidal shaped elements each. The forward end cap (PSF) is placed at90◦ with respect to the beam axis and the backward cap (PSB) is inclined at30◦ forming a conical shape. A longitudinal PS detection module is shown infigure 2.8 while the figure 2.7 represents the arrangement of all modules.

Superconducting solenoid

To determine sign and momenta of charged particles one needs a magneticfield. Therefore a superconducting solenoid is inserted between the SEC andthe PSB detectors. It is cooled by a liquid Helium cryostat that keeps theoperating temperature at 4.5 K and it provides an axial (parallel to the beam)magnetic field. The flux density in the interaction region can reach 1.3 T (seefigure 2.10). The experiment that provided data for the analysis described hereused a field of 1 T.

In order to reduce the negative effect on the accuracy of the energy mea-surements in SEC, the walls of the SCS are made of 16 mm thick aluminiumwhich is equivalent to 0.18 radiation length. A 5 tons iron yoke, enclosing thewhole CD, provide the return path for the magnetic flux. It also shields thephotomultipliers from the magnetic field.

2Two of the initial 48 elements are divided in the middle in order to make space for the pelletinjection tube.

40

Figure 2.8. A longitudinal detection module formed by the forward (B), central (A),backward (C) elements of PS and the light guides (D).

Figure 2.9. The 3D model of the central part of the PS (blue) surrounding the MDCdetector (light brown).

41

Figure 2.10. Calculated distribution of the magnetic flux density for a coil carryingcurrent of 667 A (see [63]). Contour maxima are indicated by lines marked A-H,where: A=0.10 T, B=0.25 T, C=0.050 T, D=0.75 T, E=1.00 T, F=1.20 T, G=1.30 Tand H=1.50 T.

Scintillating Electromagnetic Calorimeter

The electromagnetic calorimeter (see figure 2.11) is the outermost active com-ponent of the CD placed between the solenoid and the yoke. It is formed by1012 sodium-doped CsI scintillating crystals which have the shape of a trun-cated pyramid (see figure 2.12). They are arranged in 24 ring-layers along thebeam axis and cover an angular range from 20◦ to 169◦. Those layers formthree groups:• the central part (SEC) consists of 17 layers, each with 48 crystals (with

lenght of 30 cm each)• the forward part (SEF) consists of 4 layers, each with 36 crystals (with

lenght of 25 cm each)• the backward part (SEB) consist of 3 layers, two with 24 crystals and

one with 12 crystals (with length of 20 cm each)The crystal length varies for SEF, SEC and SEB - the length is equivalent to

∼ 16 radiation lengths and ∼ 0.8 of hadronic interaction length. The stoppingpower of the crystals is around 190 MeV for pions, 400 MeV for protons and450 MeV for deuterons (see reference [16]). The schematic structure of thecalorimeter layers and their angular coverage is shown in figure 2.13

The electromagnetic calorimeter provides information about the energy de-posited along neutral or charged particle tracks and the angular parameters ofthose tracks (emission angles). It covers 96% of the 4π solid angle with holesfor PS light guides and for pellet injection. The energy resolution for photonsis given by ΔE

E = 5%√E/GeV

while for charged stopped particles it is ∼ 3%.

The angular resolution is limited by the crystal size and thus we have ∼ 5◦resolution in polar angle and ∼ 7.5◦ resolution in azimuthal angle.

42

Figure 2.11. The electromagnetic calorimeter of WASA.

Figure 2.12. A fully equipped SEC module consisting of a CsI crystal, light guide andthe photomultiplier tube.

43

Figure 2.13. Polar angle coverage of the calorimeter layers with the number of con-stituting crystals indicated above.

2.2.3 Forward DetectorThe Forward Detector (FD) is situated down-stream the interaction region andconsists of plastic scintillators (FWC, FTH, FRH and FVH) and a straw tubetracker (FPC). The FD provides information about the energy deposits andangular parameters of charged tracks, mostly scattered protons, deuterons andhelium ions. The angular coverage (polar angle) of the FD detectors is 3◦−18◦and the angular resolution is about 0.2◦.

Forward Window Counter

The FWC is closest to the interaction region. It is formed by two layers of24 pie-shaped elements (see figures 2.15 and 2.14). Each element is a 3 mmBC408 plastic scintillator. The first layer is inclined by 80◦ with respect to thebeam axis (in order to be as close to the interaction region as possible). Thesecond layer is perpendicular to the beam axis. The layers are shifted by halfan element with respect to each other, resulting in an effective granularity of48 elements.

This detector is an important part of the trigger system and for experimentswith 3He production it also provides information for particle identification.

Forward Proportional Chamber

The FPC, located right after the FWC, is a straw tube tracker made of fourmodules of four layers. Layers consist of 122 drift tubes each made of 26 μmaluminized mylar and having 8 mm diameter and a 20 μm stainless steel sens-ing wire. The drift gas is the same a in the MDC (see section 2.2.2), a80%−20% mixture of argon and ethane.

In order to improve the efficiency of track reconstruction the orientation ofthe modules is rotated in the XY plane as seen on figure 2.16 at −45◦ (module1), 0◦ (module 4), 45◦ (module 2) and 90◦ (module 3) with respect to they-axis.

44

Figure 2.14. The FWC detector where aquarter of the second layer was removedto show the structure. Figure 2.15. The FWC exploded view.

Figure 2.16. The Forward Proportional Chamber detector.

45

Table 2.2. Stopping power of the FRH for different particles.Particle Stopping power

π 200 MeVp 360 MeVd 450 MeV

3He 1000 MeV3He 1100 MeV

This detector provides the most precise angular information about the for-ward scattered particles.

Forward Trigger Hodoscope

The Forward Trigger Hodoscope consists of three layers of 5 mm BC408 plas-tic scintillators. There are 48 pie-schaped elements in the first layer and 24elements in the second and third layer. The last two layers have Archimedianspiral shape elements and are oriented clockwise in the second layer and anti-clockwise in the third layer. This geometry is shown in figure 2.17. It createsa special pixel structure where each pixel corresponds to a given combinationof elements and layers. It allows for a fast extraction of angular information,both polar and azimuthal angles, and hit multiplicities needed by the triggersystem.

Figure 2.17. The Forward Trigger Hodoscope detector.

Forward Range Hodoscope

The FRH detector consists of five pie-shaped layers each made of 24 BC400plastic scintillator elements each (see figure 2.18). The first three layers havea thickness of 110 mm and the last two 150 mm. The energy resolution of thestopped particles is about 3%. The stopping power of the FRH for differentparticles is shown in table 2.2.

The main function of the FRH detector is the reconstruction of kinetic en-ergies (from energy deposits) and track parameters of the forward scattered

46

Figure 2.18. Schematic view of the Forward Range Hodoscope detector.

particles. Using the ΔE −E method one can use the signals from differentlayers in order to identify those particles (e.g. disentangle between protonsand deuterons). The coincidence of signals from matching azimuthal (φ ) an-gle segments of FRH, FWC and FTH is used to identify the trajectories ofparticles.

2.3 The Data Acquisition and Trigger SystemThe data flow corresponding to signals from different detector elements ishuge. At the design luminosity3 of 1032 cm−2s−1 the event rate was estimatedto be of the order of 5 · 106 per second (50MHz). The size of one event isof the order of a few kilobytes. The data acquisition system represented onfigure 2.19 is able to handle between 10 and 20MHz. We cannot save allsignals, therefore we need a trigger system that enables the disk writing foronly those events that have a desired signature.

The figure 2.20 illustrates two-level organization of the trigger system. Thefirst level4 consists of fast detectors such as plastic scintillators in FD and thePSB detector in CD. It is based on a set of multiplicity coincidence and trackalignment conditions. It provides the time scale for the event. It is used, forexample, to fix the integration gates for the QDCs. The second, slower5, stageof the trigger system takes into account the cluster multiplicity and the energydeposits in the electromagnetic calorimeter (SEC).

After passing through the Delay Matching unit that corrects for the timedifferences between the first and second trigger level, the information from the

3This is an upper limit never reached in the actual experiment. Luminosity values a few timeslower were used.4The first level processing time is around 200ns5The second level processing time is around 500ns

47

Figure 2.19. The data acquisition system.

Figure 2.20. The trigger system.

48

primary triggers (input triggers) is combined in order to form a more complextriggering pattern. A prescaling factor is applied to all high rate triggers inorder to balance the event composition.

2.4 Description of the triggerThe trigger (TR10) that was selecting the data set used in this work is definedas follows: fhdwr2&frhb2&seh2. Those acronyms stand for:• fhdwr2 - two matching tracks in FD (geometrical overlapping and tem-

poral coincidence between different detectors)• frhb2 - two deposits above a defined threshold in the second layer of the

FRH detector• seh2 - two deposits above a defined (high) threshold (around 50MeV) in

the electromagnetic calorimeterThis trigger doesn’t use any condition on the charge of the tracks in the CD.Therefore, in principle, it should be sensitive for charged and neutral decaysof the η meson such as η → e+e−γ or η → γγ . The analysis of trigger effectsand efficiency is of the uttermost importance since we need to compare ourdata to Monte Carlo simulations. During the offline analysis, we have accessto all trigger flags and can therefore choose only a subset of events with anactive TR10 flag. There is no such possibility for Monte Carlo simulations.

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3. Event reconstruction

3.1 Energy Calibration3.1.1 Forward Range Hodoscope energy calibrationWe will discuss here the energy calibration of the FRH detector since the en-ergy deposited in this detector is used to determine the kinetic energy of thepassing through particles (mostly protons). All those detectors are made ofplastic scintillator. When a particle interacts inside a detector element (by ex-citation, ionization, etc.) light is emitted and collected by photo-multipliers.Therefore, the output signal is an electric pulse and, by integration, we calcu-late the collected charge Q. This value is approximately proportional to theenergy E deposited by the particle in this detector element. In order to getthe precise relation between Q and E a calibration must be performed. Thecalibration correspond to the determination of the translation parameters andsetting them on calibration cards. Non-uniformity and non-linearity are twoimportant effects that we have to take into account in the calibration process.They are treated separately and the corresponding procedure is presented inthe next two subsections.

Non-uniformity calibration

The efficiency of the light collection depends on the distance between the in-teraction region and the photo-multiplier. This distance, for each FRH layer, isrelated to the θ angle of a particle track. Moreover, the total energy depositedin one detector element depends on the track length inside this element whichis proportional to the inverse of cosθ , that is why we plot Qcosθ versus θ infigure 3.1.

The non-uniformity effect is seen as a systematical shift of the average valueas a function of scattering angle θ . By fitting a third-order polynomial weextract the corresponding four calibration parameters. This process is repeatedfor all layers and elements. For this purpose we use the following procedure:• we process a few runs1 with an special script to create a set of histograms

such as in figure 3.1 - one histogram for each detector element• the histograms are used as input for a fit to extract the non-uniformity

correction parameters.

1We can, in principle, use any run but the best precision is obtained if we take runs with anelastic trigger (TR2=frha1 or TR21=frha1&psc1) in order to select events with minimal ionizingparticles (fast protons punching through all forward detectors). The reason is that the energydeposit of those particles is almost independent of their kinetic energy and is proportional to thelength of their path inside the active detector material. This leads to a better separation of thenon-uniformity and non-linearity effects.

50

Figure 3.1. Illustration of non-uniformity in FRH (taken from [44]).

Non-linearity calibration

The relation between E and Q is not linear due to saturation and quenching ef-fects. The plastic scintillators are not a perfectly homogenous medium neitherthe photomultipliers have an ideal linear response to the collected light.

We need to correct for those deviations. The fitting procedure consists ondetermining two parameters for non-linearity correction. We consider the fol-lowing relation between Edep and Q:

Edep = Q×C01

1−Q× C1C0

. (3.1)

Those parameters need to be chosen such that the experimental data agreeswith the Monte Carlo simulations after non-uniformity correction had beenapplied. We use two-dimensional histograms of energy deposits in two adja-cent detector layers, for example ΔEFRH2 versus ΔEFRH1 (see figure 3.2).

Setting the values of C0 and C1 for one detector layer (e.g. FRH2) influencestwo histograms ΔEFRH2 vs ΔEFRH1 and ΔEFRH3 vs ΔEFRH2. We use a script thatallows to visualize both plots simultaneously together with three projectionsand we manually tune the parameters. A screenshot of the graphical interfaceof the script is shown in figure 3.3.

51

Figure 3.2. Energy deposits in FRH2 vs FRH1 (taken from [44]).

Figure 3.3. Non-linearity fit graphical interface.

52

Figure 3.4. Energy deposit in calorimeterversus q/e× momentum for η → e+e−γ .

Figure 3.5. Energy deposit in calorimeterversus q/e× momentum for η → π+π−γ .

3.1.2 The energy calibration of the Central DetectorWe have seen in section 2.2.2 that the Central Detector is composed of threeparts: the Mini Drift Chamber (MDC), the Plastic Scintillator Barrel (PSB)and the electromagnetic calorimeter (SE). The energy calibration of the CD isbased on the photons originating from π0 decays. The goal of the calibrationprocedure is to have the invariant mass distribution of all cluster-cluster com-binations peak at the correct π0 meson mass. The calibration is linear an ituses a pedestal run (data is collected without any threshold on the signals fordifferent detector elements) and one constant for gain in each module. A moredetailed description is presented in [59].

Here, we focus on the verification of the calibration of the electromagneticcalorimeter. In order to check the calibration of the SE we look at the his-togram of energy deposit in the SE versus the particle momentum multipliedby the sign of its charge. In such a plot, electrons and pions form characteristicbands due to the difference in their mass and stopping power. We expect thosebands for data and the Monte Carlo simulation to match.

First of all, we simulate two η decay channels: η → e+e−γ with only elec-trons in the final state and η → π+π−γ with only pions in the final state.Then, we fit the simulated electron and pion bands with a linear function (seefigures 3.4 and 3.5). Then we superimpose those lines on the correspondinghistogram for data (see figure 3.6).

As we see in figure 3.6 while the matching between the electron bands andthe fitted lines is satisfactory, for the pion bands it is not the case, as we observesome discrepancy. The reason of this offset will be investigated and correctedin the further studies. A simple solution would be to use a correction functionbut in case of our analysis, the pions are anyway rejected (see the selectionconditions in chapter 6).

3.2 Track reconstruction in MDCThe reconstruction of particle trajectories is crucial since it provides the four-momentum vectors that allow determination of all other observables. As a

53

Figure 3.6. Calorimeter calibration plot (lines are from the fits to the simulated distri-butions).

general rule, hits from different detector elements are combined into clusters.The clusters are then merged into tracks. There are different algorithms de-pending on the detector part. The MDC track reconstruction provides the mo-menta and angles of charged tracks.

We group hits in MDC, using pattern recognition algorithms into trackletsand parametrize them as helices. At least 7 hits are needed to form a track.Then, we use a fitting routine that refine the parameters of each tracklet. Weassume that the magnetic field is homogeneous in the whole MDC detector.This is not exactly true therefore a systematic uncertainty is introduced. Thehelices are then extrapolated to the calorimeter and matched with its clusters.

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4. Monte Carlo simulations

4.1 Pluto simulation frameworkPluto is a simulation framework for heavy ions and hadronic physics based onROOT environment [41]. It was initially developed by the HADES collabo-ration at GSI [66]. It adds a library of C++ classes providing an easy way tosimulate different reactions in particle physics. The package includes modelsfor Dalitz decays, resonance spectral functions with mass-dependent widths,and anisotropic angular distributions for selected channels. The generation ofthe homogeneous and isotropic phase space is based on the GENBOD rou-tines [50]. For elementary reactions, PLUTO uses angular distribution modelsfor selected channels based on the parametrization of the existing data. Themodels used for simulations of the reactions treated in this work will be men-tioned in the next sections.

In addition to the large predefined set of classes and models, the PLUTOframework allows the user to include new angular distributions, redefine branch-ing ratios, add new reactions or particles.

The output file of a PLUTO simulation has a ROOT tree format. It containsthe four-vectors of all final state particles as well as positions of the verticeswhere the decays take place. The PLUTO simulation doesn’t take into accountany detector effects. Some of the simplest experimental limitations such asthe geometrical acceptance can be, in principle, implemented by using the so-called filters. The output of a PLUTO simulation provides an input for theWASA Monte Carlo program that mimics the detector response.

4.2 WASA Monte CarloThe WASA Monte Carlo software is based on the GEANT3 package devel-oped in CERN [25] [27]. We use this program to determine the detector re-sponse to particles generated from PLUTO. Based on the physical models, weobtain the signals from different detector elements.

4.3 Production of the η mesonIn our experiment, the η meson is produced in p-p collisions at 1.4 GeV in-cident proton kinetic energy (corresponding to 2.14 GeV/c momentum). The

55

threshold energy for η production in proton-proton collision is 1.256 GeV.Since the cross section for η production increases with energy it would seemnatural to use the highest possible beam energy. However, we are limited bythe forward detector geometry - forward scattered particles can only be de-tected if their θ angle is in 3◦−18◦ range1. The other issue is that while the ηproduction increases with energy so are the cross sections of most backgroundreactions. We therefore set the beam kinetic energy to 1.4 GeV which is acompromise.

The η meson is, just next to the pions, the lightest non-strange particle.Its production mechanism was investigated by many studies: in π−p → η pproduction channel see, for example, references [53], [62] or [26], in pp →ppη [17], [28] or [19]. The production mechanism of η in p-p collisions isdominated by the S11 or N∗ resonance at 1535 MeV/c2. This structure sitsvery close to the η−N threshold, which means that the s-wave η−N interac-tion is strong and its effects manifest themselves in the final state interactions(FSI). This strong and attractive interaction might even lead to the formationof quasi-bound states of the η meson with a nucleus. This interesting pos-sibility, beyond the scope of this work, is explored in the WASA-at-COSYexperiment, see reference [6] for further details.

In figure 4.1 we show the pp → ppη and pp → ppη ′ production crosssections as a function of the excess energy Q =

√s− 2mp−mη where

√s is

the total center-of-mass energy. The increase of the total cross section withenergy that is apparent in figure 4.1 is mostly related to the Q2 dependence ofthe non-relativistic three-body space. However, if one modifies this with theone-pole approximation to the S-wave proton-proton final state interaction, thenear-threshold energy dependence becomes:

σT (pp→ ppη) =C(

Qε

)2/(1+

√1+Q/ε

)2, (4.1)

where C is a constant and ε is the pole position (see Ref. [56]).The η production mechanism is represented by the Feynman diagram in

figure 4.2. This picture assumes that the interaction proceeds through theemission of a meson x(π,η ,ρ, ...) from one of the nucleons followed by axN → ηN transition. For a more detailed description of this process see refer-ence [17].

Since the η meson production mechanism that we adopted in this work isbased on pure phase space generation, we need to check if this choice couldhave an impact on further selection criteria. The figures 4.3, 4.4, 4.5 and 4.6represent respectively the distributions of proton kinetic energy, proton θ an-gle, η meson θ angle and the angle between dilepton and photon (in laboratoryreference frame).

1We need protons in order to tag the η meson through the missing mass peak.

56

Figure 4.1. The cross section for pp → ppη (upper points) and pp → ppη ′ (lowerpoints) production as a function of the excess energy Q. The solid curves are arbitrarilyscaled pp FSI predictions of equation 4.1. Taken from [56].

Figure 4.2. The mechanism of pp→ ppη production (taken from [17]).

57

Figure 4.3. Monte Carlo simulation (Pluto) for 106 events: proton kinetic energy.

Figure 4.4. Monte Carlo simulation (Pluto) for 106 events: proton θ angle.

58

Figure 4.5. Monte Carlo simulation (Pluto) for 106 events: θ angle of the η meson.

Figure 4.6. Monte Carlo simulation (Pluto) for 106 events: angle between dileptonand photon in the laboratory frame.

59

The differences between those distributions are sufficiently small to neglecttheir effect in our analysis, which is focused on η meson decays and not onthe details of the production mechanism.

4.4 The decays of η meson4.4.1 The η → γγ decay channelThis channel is important as a possible background since its branching ratiois 39.41%. There is a possibility that one of the photons interacts with thedetector material (e.g. the beam pipe) creating an e+e− pair. The final stateis the same as in Dalitz decay and the invariant mass of the three particlescombines to the η meson mass thus it cannot be rejected easily. Nevertheless,we can use the fact that the leptonic pair is produced outside the interactionpoint, often at the beam pipe. This suppression of the external conversion isdiscussed in section 6.4. The PLUTO model of this decay is an isotropic twobody decay of a spin-less particle.

4.4.2 The η → e+e−γ decay channelThis reaction is our main channel in this analysis. We would like to extractη → e+e−γ event candidates from our data sample. On the one hand, it consti-tutes the principal background for the search of a light dark boson that decaysinto e+e− pair. On the other hand, a large sample of the η meson Dalitz decayevents allows the extraction of the transition form factor of the η meson (di-viding the invariant mass of e+e− spectra from data by the spectra from pureQED simulation).

In order to select the η Dalitz decay channel and to reduce other backgroundcontributions, we need to apply well-chosen cuts on different variables such asangles, energies, invariant masses and also multiplicities of tracks etc. There-fore, we need to generate Monte Carlo simulations of those different channelsand observe which distributions of variables allow us to best separate the back-ground contributions from the η → e+e−γ decay.

The figure 4.7 shows the variables used in the PLUTO generation of theDalitz processes. In addition to the virtual photon invariant mass mγ∗ =me+e− ,its momentum pXRF

γ∗ , the polar θ XRFγ∗ and the azimuthal φ XRF

γ∗ emitting angles inthe rest frame of the decaying particle (in our case η) there are two anglesrelated to the photon decay into e+e− pair: the θe helicity angle and φe socalled Treiman-Yang angle (see [41]).

Since the pseudoscalar mesons are spin-less, no alignment information iscarried from the production mechanism to the decay, so θ XRF

γ∗ , φ XRFγ∗ and φe are

isotropic. The helicity angle distribution for pseudoscalar mesons is given by(1+cos2(θe)) (see reference [24]), which is included in PLUTO by default. In

60

Figure 4.7. Geometrical variables involved in the description of a Dalitz process inPLUTO (taken from [41]).

order to describe the invariant mass distribution, PLUTO generates the Dalitzdecays with a form factor. The latter can be set equal to unity in order to getthe pure QED distribution.

Figure 4.8 represents the simulated invariant mass of e+e− for a pure QED(transition form factor equals unity) and a more realistic simulation wherethe form factor is computed assuming the VMD (Vector Meson Dominance)model.

4.4.3 The η → π+π−π0 decay channelThe branching ratio of this η decay channel is 22.92%. The probability forthis channel to mimic the Dalitz decay is rather low due to the fact that itwould need a simultaneous misidentification of two pions and a loss of one ofthe photons from π0 → γγ decay.

PLUTO simulation of this channel is based on the parameterization of thematrix element (i.e. the deviation from the constant value of the Dalitz plot)used in PLUTO is based on the measurement performed by the Crystal Barrelcollaboration (see reference [13]).

4.4.4 The η → π0π0π0 decay channelAlthough this is a very common decay of the η meson with branching ratioequal to 32.7% its final state contains mostly photons from π0 → γγ decaysand therefore it can be efficiently rejected during the selection process.

61

]2 [GeV/c-e+Invariant mass of e0 0.1 0.2 0.3 0.4 0.5

-810

-710

-610

-510

-410

-310VMD

pure QED

Figure 4.8. Effect of the transition form factor on the invariant mass of e+e− spectrum.

4.4.5 The η → π+π−γ decay channelThis decay has a branching ratio of 4.2%. The final state contains two chargedtracks and one neutral particle, as the Dalitz decay.

For this decay, PLUTO uses the simplest gauge invariant matrix elementincluding p-wave interactions of the pions according to

∣∣M2∣∣ = k2q2 sin2 θ ,

where k is the photon momentum in the rest frame of the η meson, q and θare defined as the momentum of one of the pions and the angle between π+

and γ , both defined in the rest frame of the two pions.

4.5 Direct production of π mesons4.5.1 The pp→ ppπ+π− reaction channelIn this reaction no photons are produced. Nevertheless, this channel can be anissue because it has a high cross section and if a neutral track is mistakenlyadded to the event the final state would have the same topology as the η Dalitzdecay. An additional neutral track photon could originate from another event(pile-up effect) or from a misinterpretation of an electromagnetic cascade split-off.

4.5.2 The pp→ ppπ0π0 reaction channelThis is an important channel since its cross section is high with respect to theη meson production. We have to consider the case where one pion decays intotwo photons and the other one through Dalitz decay. The process of the direct

62

two pion production mostly occurs via a two-Delta resonance state, thereforewe will use this mode of the production in our simulations.

4.5.3 The pp→ ppπ+π−π0 reaction channelIf one of the photons from the decay of the neutral pion is lost, we have thesame event topology as the signal. We use the contribution from this reactionto the missing mass of two protons in our fitting procedure (see section 6.9) toreject the non-η background (see figure 4.9).

]2missing mass of two protons [GeV/c0 0.1 0.2 0.3 0.4 0.5 0.6

Numb

er of

Ent

ries

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Figure 4.9. Missing mass of two protons for pp→ ppπ+π−π0.

63

5. Data set

5.1 Initial data sample reductionOur analysis is based on data taken during the beam-time period that lastedfrom February 2012 to April 2012 (effectively around 6-7 weeks). In the ex-periment, collisions between proton beam with kinetic energy of 1.4 GeV andfrozen hydrogen pellets were used. At this energy the η meson productionchannel pp→ ppη is open.

The total amount of data collected during this period is around 115 TB.Even after initial rejection of non relevant data, for example due to other trig-gers or detector adjustment periods, we have to deal with 70 TB of stored datafiles.

Therefore, we need to further reduce this amount of data keeping onlyevents that fulfill some basic criteria. A RootSorter-based program calledPPSEL was written that selects a certain class of events and saves them forfurther processing. Our analysis focuses on charged decays of the η me-son such as η → e+e−γ or η → e+e− therefore we look for events with atleast two charged tracks in the central detector. Additionally, we also need tohave access to the dominant neutral modes of η decay such as η → γγ andη → π+π−π0 for normalization and background studies. These two classesof events were stored separately.

A common condition for both classes is a requirement of at least two proton-like tracks in the forward detector. This condition allows us to calculate themissing mass assuming the two particles are protons (see figure 5.1). For thereaction pp → ppη we expect to observe a peak at the η meson mass. TheForward Detector is thus used to tag the events with η meson production.

Table 5.1. Effect of the selection conditions.

Condition Number of events

All events 4·106

Trigger 5.57·105 (13.9%)

≥ 2 FD ch.tr. ≥ 10 MeV 4.94·105 (12.35%)

≥ 1 pair FD ch.tr. inside 10 ns time window 4·105 (10%)

proton identification 2.76·105 (6.9%)

64

Figure 5.1. Missing mass of two proton-like tracks in the Forward Detector.

Figure 5.2. Time difference in ns between charged tracks in the Forward Detector.

65

The table 5.1 shows the effect of the selection criteria applied to data samplefor a typical data file. All those cuts are applied for both charged and neutralη decay classes. The common selection steps are the following:• the trigger condition is re-checked - there are at least two tracks of

matching clusters in the Forward Detector and at least two clusters inthe Central Detector above threshold,

• at least two charged tracks in the Forward Detector with energy depositabove 10 MeV,

• at least two charged tracks in the Forward Detector inside 10 ns timewindow (see figure 5.2),

• two proton-like tracks in the Forward Detector (particle identification inFD).

After this primary selection process, the data sample is divided into two streamsthat are saved for further analysis. The stream with charged decays of the ηmeson is selected by the following set of conditions:• at least 14 hits in the Mini Drift Chamber - the event reconstruction

program needs at least 7 hits in MDC in order to build a charged trackand we want two reconstructed charged tracks,

• at least two hits in the Plastic Scintillator - only charged particles shouldprovide a signal in the PS detector.

The events with neutral decays are chosen by requiring:• no hits in the Plastic Scintillator,• at least two neutral tracks reconstructed in the Central Detector with

energy deposit above 10 MeV.The table 5.2 shows the number (absolute and relative) of events that remainafter all selection conditions for both classes.

Table 5.2. Events in the final sample.

Condition Number of events

All events 4·106

Events from charged decays 1.4·105 (3.6%)

Events from neutral decays 6.9·104 (1.7%)

The selection performed by the PPSEL program reduce the size of the datasample suitable for our analysis to around 2.7 TB thus a reduction factor of26 was achieved. Figures 5.3 and 5.4 show the missing mass of two protonsrespectively for the events from the charged and the neutral stream. Figure 5.5shows the missing mass of two protons versus the invariant mass of two pho-tons for the neutral selection.

66

Figure 5.3. The missing mass of two protons for the selection of charged decays of η .

Figure 5.4. The missing mass of two protons for the selection of neutral decays of η .

67

Figure 5.5. The missing mass of two protons versus the invariant mass of two photonsfor the selection of neutral decays of η .

5.2 Normalization using η → γγ channelThe η → γγ channel is extracted from the neutral decays selection in order toestimate the number of η mesons produced. The selection process starts withthe condition used in the initial selection of the neutral decays - the require-ment of at least two reconstructed neutral tracks in CD with deposited energyabove 10 MeV and a veto on PS signal. This is redundant for preselected dataevents but necessary for the comparison with the Monte Carlo simulations.The selection of the η → γγ channel constrains the processed events with thefollowing conditions:• at least two neutral tracks with −25ns < |tN− tP|< 5ns, where tN is the

time of the neutral signal (CD) and tP is the mean proton time (FD),• in case there are more than two neutral tracks, we choose the pair with

the largest mutual angle calculated in η rest frame1 Ωγ1γ2ηRF ,

• we require Ωγ1γ2ηRF > 140◦,

• time difference between neutral clusters is less than 10 ns,• the energy deposit of each neutral track must be above 100 MeV,• the missing mass of two protons is in the range 530−570 MeV/c2.

The invariant mass of γγ for events that pass this selection is represented infigure 5.6 (black squares).

The number of η mesons can be estimated using the η → γγ channel withthe formula:

1The η four vector is obtained by taking the difference between the initial beam and target fourvector and the two protons four vectors.

68

Figure 5.6. The invariant mass of γγ: data and η → γγ Monte Carlo simulation(WMC).

Nη =Nγγ

BRγγ ·Aγγ, (5.1)

where Nγγ is the number of η → γγ events, BRγγ is the branching ratio for thischannel and Aγγ its acceptance that takes into account the effect of the detectorgeometry, losses from the reconstruction procedure and the reduction due tothe selection criteria.

If we assume that only η → γγ channel remains in data after our selection,we can estimate the number of η mesons from formula 5.1 by fitting the datawith a Monte Carlo simulation of η → γγ (after simulating also the WASAdetector response with WMC). This is shown on figure 5.6. The number ofη → γγ events extracted is ∼ 10.5 · 106. The branching ratio for η → γγis 39.41% and the mass integrated acceptance (for our selection process) is14.5± 1.3% thus the number of η meson equals ∼ 183.8 · 106. This value isoverestimated - the fit quality is very poor since we have neglected all possiblebackground sources.

In order to correct the number of η → γγ event candidates and consequentlythe number of the produced η mesons we fit the selected data with the sum ofMonte Carlo simulations that passed through the same selection process η →γγ , pp→ ppπ0 (→ γγ)π0 (→ γγ)2 and a polynomial of degree N. The doublepion production cross section in proton-proton collisions is 324±21systematic±58normalization μb [4] compared to η meson production cross section 10 μb.The acceptance for pp→ ppπ0 (→ γγ)π0 (→ γγ) channel is 0.0250±0.0016%.

2BR(π0 → γγ)∼0.98.

69

The fit is based on the following bin per bin decomposition of the data his-togram:

Ndata(x) = Nη

(BRη→γγ ·Aη→γγ(x)+

σ(pp→ ppπ0π0)

σ(pp→ ppη)·A2π0(x)

)+PolyN

= p0(Xη→γγ(x)+X2π0(x)

)+

N

∑i=0

pixi, (5.2)

where x represents bins in the invariant mass of γγ and the acceptances A∗(x)=Nf in(x)Nini(x)

are the ratio (for each bin) between the invariant mass histogram afterand before selection cuts. For this decomposition, the parameter p0 representsthe number of η mesons Nη .

In order to estimate the number of the η mesons, we repeat the fitting pro-cedure while varying three external parameters: the binning of the fitted his-togram, the fit range and the degree of the polynomial. Figure 5.7 representsthe resulting fits for 10 MeV/c2 bin width, 350− 700 MeV/c2 fit range and athird order polynomial.

Figure 5.7. The invariant mass of γγ: data and the fit result with individual contribu-tions.

Figures 5.8 illustrate the fits when the bin width is changed while the rangeis set to 400−700 MeV/c2 and we add a third order polynomial to the simu-lations.

Figures 5.9 illustrate the fits when the fit range is varied while the bin widthis set to 10 MeV/c2 and we add a third order polynomial to the simulations.

Tables 5.3, 5.4 and 5.5 contain the number of η meson (/106) extractedfrom fits performed with a second order, third order polynomial, and no poly-nomial at all.

70

Figure 5.8. The invariant mass of γγ: fit with different binning 5 MeV/c2, 10 MeV/c2

and 20 MeV/c2.

71

Figure 5.9. The invariant mass of γγ: fit with different fit ranges 300−700 MeV/c2,350−700 MeV/c2 and 400−700 MeV/c2.

72

Table 5.3. Number of η mesons (/106) extracted from the fits with second orderpolynomial.

��RangeBin width

5 MeV/c2 10 MeV/c2 20 MeV/c2

300−700 MeV/c2 148.68 149.24 148.60350−700 MeV/c2 147.05 147.31 146.36400−700 MeV/c2 147.54 147.80 146.85

Table 5.4. Number of η mesons (/106) extracted from the fits with third order poly-nomial.



300−700 MeV/c2 149.08 149.49 148.67350−700 MeV/c2 145.04 145.44 145.87400−700 MeV/c2 137.66 138.00 137.58

Table 5.5. Number of η mesons (/106) extracted from the fits without additionalpolynomial.



300−700 MeV/c2 162.20 161.86 161.80350−700 MeV/c2 163.57 163.40 163.01400−700 MeV/c2 163.47 163.49 163.22

Given the table 5.5 neglects any source of accidental background we useonly tables 5.3 and 5.4 to extract a mean value and the standard deviation (asa systematic effect):

Nη = (145.90±0.08stat ±3.97sys) ·106.Another source of systematic uncertainties for the number of η mesons arethe selection criteria used in our analysis of the η → γγ channel. To takethem into account we repeat the analysis with slight modifications of temporal(time between the two photons), angular (angle between the two photons in ηmeson rest frame) and missing mass (of the two protons) conditions. The fit isdone with a fixed bin width of 10 MeV/c2, in a fixed 350−700 MeV/c2 rangeand with the addition of a third order polynomial. The results are shown intable 5.6

We observe that for the stricter conditions, we get more η mesons usingthe described fitting procedure. This is due to the fact that the acceptance forη → γγ channel is reduced but the relative content of this decay increases.Again, we extract the mean value and the uncertainties of those fits:

Nη = (145.84±0.07stat ±4.20sys) ·106.We see that the two mean values are consistent with each other. We thereforemerge all fit results and got the final number of η mesons:

73

Table 5.6. Number of η mesons (/106) extracted from the fits with third order poly-nomial.

Condition Nη/106 η → γγ % in data∣∣tγ1 − tγ2

∣∣< 12ns 142.14 82.84%∣∣tγ1 − tγ2

∣∣< 8ns 151.71 83.17%∠γ1γ2

ηRF > 130◦ 142.81 85.60%∠γ1γ2

ηRF > 150◦ 148.90 78.78%525MeV/c2 < MM2P < 575MeV/c2 141.66 81.77%535MeV/c2 < MM2P < 565MeV/c2 147.83 83.66%

Nη = (145.89±0.08stat ±3.93sys) ·106.

5.3 Analysis of the main trigger.During the data acquisition we use a set of hardware triggers. For an event tobe saved it has to activate at least one such trigger. During the offline analysis,we have access to the list of all triggers and we can see which triggers wereactive in each. An example of such a list, showing the set-up for a series ofruns, is presented in figure 5.10. The trigger we use in this work (number 10 inthe trigger list) is active if a coincidence of the following conditions fhdwr2,frhb2 and seh2 is present. The two first conditions concern the Forward De-tector and particles passing through it - fhdwr2 means that there should be atleast two matching tracks in the Forward Trigger Hodoscope, Forward Win-dow Counter and Forward Range Hodoscope. This means that a signal (energydeposit above threshold) was detected in the modules for each of those detec-tors corresponding to the same φ angle . The second condition, frhb2 is setwhen there are at least two signals in the second layer of the Forward RangeHodoscope. The last condition, seh2 is related to the Central Detector, it ac-tivates when there are at least two clusters with energy deposits above a highthreshold (around 100 MeV). With the above criteria, the trigger should acceptcharged and neutral η meson decays such as η → e+e−γ or η → γγ .

5.3.1 Trigger efficiencyIn order to estimate of the efficiency of a trigger i, we will compare it to an-other trigger j. Both triggers should be sensitive to the same channel. Let Nbe the total number of events of a selected channel and Ni (respectively Nj)the number of those events that activate trigger i (respectively j). We thencalculate the efficiency in the following way:

Ni = N ·Pi Nj = N ·Pj Ni j = N ·Pi j. (5.3)

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Figure 5.10. List of triggers for the experiment with the prescaling factor (fifth col-umn).

75

In this equation, Ni j is the number of events that activate both triggers i andj. Pi is the probability that trigger i accepts a signal event, we therefore usethis value as an estimate of the trigger i efficiency. The probability that bothtriggers will fire is Pi j = P(i∩ j). Making the assumption that trigger i and jare independent this probability becomes simply Pi j = Pi ·Pj and the efficiencyis:

Pi =Ni j

Nj. (5.4)

In reality the triggers are rarely independent, thus we must consider the natureof their dependency. In this case Pi j = P(i∩ j) = P(i| j) ·P( j) and we shouldcompare P(i| j) and P(i). As stated in the beginning of this section, both trig-gers are sensitive to the same channel therefore P(i| j)> P(i). This means thatour estimate of the efficiency gives a lower limit.

Trigger efficiency: η → γγ channel

Here we compare triggers 10 (TR10) and 26 (TR26) with respect to the neutraldecay η→ γγ . The only difference between those triggers is that TR26 acceptsonly neutral tracks but with a lower energy threshold and adds a veto on thePS detector. Trigger TR26 is less strict with respect to η → γγ channel sincethis decay does not have any charged track that could be rejected by the vetoon PS while the SEC energy threshold is higher for TR10. We have selectedthe η → γγ decay by applying the selection criteria described in section 5.2.

Table 5.7. Neutral selection trigger combinations.

Triggers Number of eventsTR10(¬TR26) 8248(¬TR10)TR26 81040

TR10TR26 169741

The number of events for different combinations of those triggers is shownin figure 5.7. Using formula 5.4 for those values we get a relative efficiency of(67.7±0.2)%.

Trigger efficiency: η → e+e−γ channel

The same method is applied to the η → e+e−γ channel. The two comparedtriggers TR10 and TR29 differ only by the following conditions:• TR10 - requirement of two clusters in SEC above a high threshold,• TR29 - requirement of one cluster in SEC above a low threshold and two

hits in PS.The requirement of two hits in PS is irrelevant since the initial data selectioncontains this condition. Therefore, all signal events that trigger TR10 should,a fortiori, activate TR29.

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Table 5.8. Charged selection trigger combinations.

Triggers Number of eventsTR10(¬TR29) 139(¬TR10)TR29 308

TR10TR29 204

The number of events for different combinations of those triggers is shownin table 5.83. Using formula 5.4 we get a relative efficiency of (39.8±3.3)%.

5.3.2 Trigger stabilityWe use the term of stability to characterize the changes, over the data takingperiod, of the ratio between the number of the events selected by two differenttriggers, one of them being TR10 that we use in the data selection. This pa-rameter, for TR10 and TR17, is shown in figure 5.11. Trigger 17 selects eventswith at least one signal from the forward part of the PS (thin plastic scintillatorin the CD - sensitive to charged particles) and one signal in the center part ofthe PS.

Figure 5.11. The trigger stability.

We can observe that this value changes through the duration of the experi-ment within 10% limits.

3NB: The selection was based on the analysis described in chapter 6.

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6. Analysis of the η → e+e−γ channel

The main goal of this work is either to find evidence for the decay of a darkmassive boson U to e+e− or to put an upper limit for such a process. Thesignificance of the test is directly related to the available statistics. Such a de-cay could be observed as a peak in the spectrum of the e+e− invariant massesthat could not be explained by any known contribution. We look for such asignature in the relevant spectrum of the η → e+e−γ decay. Additionally inthis decay we can extract, from a large set of η Dalitz candidates, the transi-tion form factor of the η meson (see 1.4.1). Therefore, the first step consistsof collecting the largest possible sample of η → e+e−γ events candidates. Toreach this goal, we have to implement a set of conditions that select a sig-nal enhanced data sample. The relative content of the η → e+e−γ channel issupposed to increase at each step of this process.

The rate at which the proton-proton collisions occur is about 106 s−1. Onone hand, this gives an idea about the difficulty of the task for the data acquisi-tion system to efficiently store most of the relevant information. On the otherhand, events saved by the DAQ often contain information generated from mul-tiple overlapping collisions. In order to select a particular reaction, we need toget rid of the background channels by implementing suitable conditions in theanalysis program.

6.1 Multiplicity conditionsThe multiplicity conditions consist of selecting events based on the numberof the charged and/or neutral tracks (after reconstruction). In principle, oursearch should only focus on events with one neutral track, the photon, and twooppositely charged tracks, the e+e− pair. In reality, we must proceed morecarefully. It would be wrong to set up the selection criteria on the number oftracks strictly requiring C = 2 and N = 1. For charged tracks, we are interestedonly in electrons, therefore the particle identification and pion rejection mustcome first. Also, to enhance the resolution we deal only with particles thatgive a reasonably high signal, thus set a threshold on the energy deposit ofeach track (in our case 20 MeV). Last but not least, in order to reject randombackground (pile-ups), we must consider only those tracks that are within agiven time window - this will be discussed in section 6.2.

Nevertheless, for an illustration only, we present the histograms of the in-variant mass of the two charged particles system (assuming electron mass) for

78

different channels simulation (PLUTO and WMC). Figures 6.1, 6.2 and 6.3represent the η decays while figures 6.4, 6.5 and 6.6 show the simulation ofthe direct pion production channels. The figure 6.7 shows the simulation of106 pp→ pΔ+→ ppe+e− events.

Figure 6.1. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 5 ·106 η → e+e−γ events.

Figure 6.2. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 8 ·105 η → π+π−γ events.

79

Figure 6.3. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 1 ·107 η → γγ events.

Figure 6.4. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 14 ·106 pp→ ppπ+π− events.

80

Figure 6.5. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 1 ·106 pp→ ppπ+π−π0 events.

Figure 6.6. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 1 ·107 pp→ ppπ0 (→ γγ)π0 (→ e+e−γ) events.

81

Figure 6.7. Invariant mass of two charged particles in CD (with the assumption ofelectron masses): 106 pp→ pΔ+→ ppe+e− events.

6.2 Time conditionsThe signals produced by the particles that originate from the same event, forexample a meson decay, should occur within some reasonable time window.The width of this window is determined by several factors. On one hand,the physical characteristics of the particle such as its velocity or momentum,charge, mass and its interaction properties all have an influence the particlepath inside the detector, thus on the time of the signal. On the other hand, thedetector itself plays an important role - the physical properties of the activematerials, cable lengths etc. The time of a track is determined by the TDCresponse of a given detector element. For charged particles in the central de-tector - mostly electrons, pions and protons - it corresponds to a signal fromeither the plastic scintillator barrel or the electromagnetic calorimeter. Thetimes of all neutral particles are taken from SE and the time of particles in FD(mostly charged protons or pions) is set by the response of the forward triggerhodoscope. In figures 6.8 and 6.9 we show time differences between parti-cles in the CD, neutral (N) or charged (C), and protons in the forward part ofthe WASA detector. Based on those spectra, we apply the following selectioncriteria in order to reduce most of the overlapping background:• -21 ns < timeCDN - timeFD < 5 ns,• -16 ns < timeCDC - timeFD < -6 ns.

In figures 6.10 and 6.11 we observe the indirect effect of those cuts on thespectra of time differences between charged and neutral tracks in CD. Fur-ther in the analysis, after the selection of an electron pair and a photon that,we assume, originate from the same Dalitz decay event, we apply one morecondition on the time difference between the electrons and the photon:

82

Figure 6.8. Time difference between neu-tral tracks and protons.

Figure 6.9. Time difference betweencharged tracks and protons.

Figure 6.10. Time difference betweencharged and neutral tracks in CD beforetime cuts relative to protons in FD.

Figure 6.11. Time difference betweencharged and neutral tracks in CD aftertemporal cuts relative to protons in FD.

• -20 ns < timeCDC - timeCDN < 20 ns.

6.3 Particle identification method6.3.1 ΔE-p method - basicsThe pions are produced copiously in the η meson decays as well as directly inproton inelastic collisions. The cross section for η meson production is around12 μb while the direct pion production is of the order of 30− 40 mb, thusthree orders of magnitude higher. Our ability to distinguish between electronsand pions is therefore one of the most important aspects of the analysis. Themethod used to identify the particles is based on the difference of the patternobserved in the histogram that relates the energy loss of a track in the CDversus the momentum of this track multiplied by its charge. In figure 6.12constructed from a data sample, one can see four partially overlapping bands,a pair for each charge state. The pions are in the lower band and the electrons(positrons) are located in the higher band. We can proceed in two ways:• we apply an arbitrary graphical cut,• we apply a graphical cut using a well-defined procedure.

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Figure 6.12. Energy deposit in SEC versus charge × momentum plot - data.

Given the two bands are rather well separated in the plot (even if they overlapfor large momenta), we have finally used the first method to disentangle pionsand electrons. This cut is presented in figure 6.13.

Figure 6.13. Energy deposit in CD versus charge × momentum plot - graphical cut.

6.3.2 ΔE-p method - development of a systematic procedureNow, let’s discuss the other method that was developed as an attempt to opti-mize the electron selection. We have seen that the identification of the chargedparticles is one of the most important and most difficult issues in this analysis.One of the main objections that one can make with respect to the presentedgraphical method is that it is arbitrary. At higher momenta (energies), electron

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and pion bands are overlapping and the applied graphical cut adds uncertain-ties that are difficult to estimate. Here, we propose a well-defined procedure,which uses the plot data, to separate electrons and pions.First, we generate Monte Carlo simulations (WMC) of η meson decays: onesample of events with electrons in the final state and one sample with pions inthe final state. Therefore, we simulate one million events for both η → e+e−γand η → π+π−γ channels. We construct the same histograms as used in thelast section 6.3.1 i.e. the energy deposited in the central detector versus theparticle momentum multiplied by the particle charge. We then fit a line tothose electron and pion bands (e.g. by using a least-squares fit algorithm onan area of the histogram limited to those bands). Given the problem with theenergy calibration mentioned in section 3.1.2, the pion band fit has to be basedon data rather than on the simulation. The result of such procedure is shownin figure 6.14. Then, we use the data sample to compute and plot the distancesbetween each data point on those histograms and the fitted line that corre-sponds to one of the fitted bands (in this case - the electron band). Finally,we can draw such plots for different energies (horizontal slices): 60 MeV, 100MeV, 300 MeV and 500 MeV. In figure 6.15 we can observe two peaks: onecentered at zero, identified as electrons, and a second broader peak, pions,centered at some distance from zero (energy dependence). In figure 6.15 we

Figure 6.14. Particle identification fit.

can draw a line (corresponding to one point in figure 6.14) that maximizes afigure of merit. The latter depends on the considered reaction. If the electronsrepresent our signal with Ne being their number on one side of the line and pi-ons are the background (with Nπ being the number of pions on the same sideof the line) we can maximize the ratio Ne√

Ne+Nπ. When our signal is very weak

85

Figure 6.15. Normalized distances from the e+ fitted line. The vertical red line repre-sents the position of the π+ line.

(compared to background) this reduces to Ne√Nπ

. Finally, if our reaction is abun-

dant it could make sense to use NeNπ

ratio. If we repeat this process for a fewprojections (energy slices) and for each charge, we can construct two obliqueseparation lines in figure 6.14 and use them as an identification condition forour analysis.

This alternative identification method was developed and is available forfuture use. In this work we will not use it for two reasons. First of all, forconsistency the proposed method would require a precise energy calibrationfor pions (see section 3.1.2). Secondly, we will show that the simple graphicalcut identification described in section 6.3.1 is sufficient to reduce and controlthe pion content in our data.

6.4 Rejection of the background from external pairproduction

Photons passing through the detector material can interact with nuclei and pro-duce e+e− pairs. This mechanism is called pair production or external photonconversion. Although the WASA experiment is designed to limit as much aspossible this effect, it still represents a large contribution to the e+e− spectra.The main source of this background is the beam pipe. When electrons areproduced off vertex, on the beam pipe, their tracks are wrongly reconstructed.The reconstruction algorithm in the MDC assumes that those tracks originatefrom the interaction point (0,0,0) rather from their true origin located on thebeam pipe (see figure 6.16).

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Figure 6.16. Momenta vectors at the beam pipe for electrons from η → e+e−γ (left)and from photon conversion in the beam pipe material (right) (figure from [21]).

Figure 6.17. WASA Monte Carlo of η →e+e−γ .

Figure 6.18. WASA Monte Carlo of η →γγ .

If we perform calculations assuming that the interaction point is somewhereon the beam pipe, the invariant mass (IMeeBP) of e+e− pairs that were createdthere should be zero - their momenta vectors should be parallel to each other.The leptonic pairs coming from the η meson Dalitz decays are generated atthe (0,0,0) interaction point and are drawn apart by the magnetic field so theirtracks are no longer parallel when they reach the beam pipe and thus theirinvariant mass on the beam pipe is non zero.

The second feature that allows us to separate those two contributions is theradius of the closest approach (CA) of the e+ and e− tracks in the XY plane.The projection of the beam pipe can be schematically described as a thick cir-cle with inner radius of 30 mm. Therefore, we can expect that the positions ofthe closest approach of leptonic pairs from external photon conversion are lo-cated around 30 mm while those originating from η Dalitz decay are closer to0. Finally, we plot CA versus IMeeBP and perform a graphical cut to separatethe signal from the external conversion background. Figures 6.17 and 6.18show this plot for the simulated signal η → e+e−γ , background η → γγ whilethe figure 6.19 represents the data and the cut we apply.

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Figure 6.19. Radius of closest approach versus the invariant mass of e+e− at the beampipe - data with graphical conversion cut.

6.5 Selection based on total missing energy versus totalmissing mass plot

We use the mass difference between pions and electrons (139 MeV/c2 versus0.5 MeV/c2) to select electrons and reject pions by graphically cutting on thehistogram of total missing energy versus total missing mass. In figure 6.20,

Figure 6.20. Total missing energy versus total missing mass for η → π+π−γ simula-tion.

we see the histogram of the total missing energy versus the total missing massfor η → π+π−γ decay channel. By comparison with the same histogram forthe η → e+e−γ channel (6.21), we see that excluding the region with posi-tive missing energy and missing mass above 75 MeV/c2 rejects only a smallfraction of the signal but strongly reduces the contribution from η → π+π−γchannel (and other pionic channels - see figures 6.22 and 6.23).

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Figure 6.21. Total missing energy versus total missing mass for η → e+e−γ simula-tion.

Figure 6.22. Total missing energy versus total missing mass for pp→ ppπ+π− sim-ulation.

Figure 6.23. Total missing energy versus total missing mass for pp → ppπ+π−π0

simulation.

89

Figure 6.22 shows the total missing energy versus the total missing massfor pp→ ppπ+π− decay channel and the effect of the cut on this reaction.

Figure 6.23 shows the total missing energy versus the total missing massfor pp→ ppπ+π−π0 decay channel and the effect of the cut on this reaction.

6.6 Angular distributions for the η → e+e−γ channel6.6.1 Angle between η and the beam direction in laboratory

frameThe differences in the angular distributions of particle tracks that come fromdifferent reactions allow us to disentangle between different channels. There-fore we can clean our sample by applying appropriate angular conditions. Firstof all, we are interested in a decay of the η meson that is produced in the col-lision of protons pp→ ppη . Therefore, we can reconstruct the missing four-vectors for two protons such as: Pmiss = Pb +Pt−Pp1

−Pp2where Pi on the

right side of the equation stands for the four-vector related to the beam, targetand the two reconstructed (scattered) protons. The left side of the equationrepresents the missing four-momentum. If an η meson is produced its four-momentum would be denoted Pη . The figure 6.24 represents θLAB (angle withrespect to the beam or Z axis in the WASA reference frame) of this missingvector for two different Monte Carlo simulations, one with η meson produc-tion pp→ ppη (blue), the other with direct pion production pp→ ppπ+π−π0

(black). We see that for reactions with pion production, the tail of this distri-bution is larger, thus it motivates a cut on 30◦ (red line). Figure 6.25 showsthe same distribution for the data sample.

6.6.2 Angle between γ∗ and γ in η meson rest frameThe other important angular distribution, directly related to the selection ofη → γ∗γ → e+e−γ channel, is the angle between the virtual photon or dileptonγ∗ and the true photon γ in the rest frame of η meson. Given the momentumconservation, this angle should be equal to 180◦. In reality, due to uncertain-ties and errors introduced by the detection and the reconstruction processesthe distribution has shape shown in figure 6.26. The figure 6.27 represents theangular distributions for η → e+e−γ and three background channels normal-ized to the maximal value in order to emphasize the differences in shape. Toimprove the signal-to-background ratio, we reject events with angle betweenγ∗ and γ in η meson rest frame lower than 140◦.

90

Figure 6.24. Theta angle of η meson in the laboratory reference frame - Monte Carlosimulations.

Figure 6.25. Theta angle of η meson in the laboratory reference frame - data.

91

Figure 6.26. Angle between γ and γ∗ in η meson rest frame - simulation of η →e+e−γ .

Figure 6.27. Angle between γ and γ∗ in η meson rest frame - simulation of differentchannels.

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6.7 Summary of the η → e+e−γ selectionThe table 6.1 summarizes the effects of the η → e+e−γ selection criteria onη → e+e−γ and different background channels.

Table 6.1. η → e+e−γ and background reactions.

Reaction Nsim final selection Acceptance Nexpected

η → e+e−γ 5 ·106 121447 0.0243 ∼ 10900pp→ ppπ0 (→ γγ)π0 (→ e+e−γ) 107 492 4.9 ·10−5 ∼ 1200

η → γγ 107 176 1.8 ·10−5 ∼ 450η → π+π−γ 4 ·106 2 5 ·10−7 ∼ 1

pp→ ppπ+π−π0 3 ·106 1 3.3 ·10−7 ∼ 15pp→ ppπ+π− 1.4 ·107 0 - -

pp→ pΔ+→ ppe+e− 106 0 - -

6.8 Radius of closest approach after selectionThe only background channels that survive the selection process are η → γγand pp→ ppπ0 (→ γγ)π0 (→ e+e−γ). In order to prove this statement, fig-ure 6.28 shows the distribution of the radius of the closest approach betweene+e− pairs, i.e. the projection on the XY axis of the vector constructed byjoining the center of the shortest segment between the two helices to the ori-gin (interaction point).

Figure 6.28. Radius of closest approach.

Pileup contribution doesn’t appear on this plot since it is distributed ran-domly, often outside the considered range. We observe that the simulation of

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η → e+e−γ and the two background channels describes data with good accu-racy.

6.9 Method for non-η background suppressionIn the previous sections, we made several selection cuts in order to extract oursignal, the η → e+e−γ decay. The purpose of those conditions is to rejectas much as possible of the background channels. Nevertheless, we know thatthere is some remaining background that, for non-η decays, should be smoothin the vicinity of the η meson peak. We will try to estimate and reject thisbackground using the following procedure.

We start with a bi-dimensional histogram (see figure 6.29), the missingmass of two protons versus the invariant mass of e+e− for the selected dataset. The histograms that are shown in figures 6.31-6.38 are projections of 50MeV/c2 slices in e+e− invariant mass, from 0 to 400 MeV/c2 on the missingmass axis. For each histogram, we perform a fit using the sum of the sim-ulated pp → ppπ+π−π0 background (to mimic a multiparticle phase spacebehavior) multiplied by a third order polynomial and a Lorentzian function(that describes the η → e+e−γ signal). Figure 6.30 presents the global fit tothe whole 0− 400 MeV/c2 e+e− invariant mass range. For each slice in thee+e− invariant mass, we extract the signal content i.e. the number of events inthe lorentzian function needed to obtain the best agreement with data.

The result of this fit together with a Monte Carlo simulation of η → e+e−γsignal, is shown in figure 6.39. This procedure removes most of the back-ground with the exception of events from η meson decays. The distribution ofevents in figure 6.29 is nonuniform, the number of events strongly decreaseswith increasing e+e− invariant mass. It is therefore natural to apply differentbinning in different mass ranges. It would also allow to test the fit stabilityand serve as a check for systematical uncertainties. This will be done in sec-tion 8.1.2 where we present the results of this procedure applied to the case ofthe η form factor extraction.

94

Figure 6.29. Missing mass of two protons versus invariant mass of e+e− (data).

Figure 6.30. Global fit to the missing mass of two protons.

95

Figure 6.31. Fit to the missing mass oftwo protons for IMe+e− in (0,50) MeV/c2

range.

Figure 6.32. Fit to the missing massof two protons for IMe+e− in (50,100)MeV/c2 range.







96

Figure 6.39. Invariant mass of e+e− - extracted η content and VMD simulation withform factor corresponding to the ρ meson mass.

6.10 Combinatorial backgroundThere are two sources of combinatorial background. One is the wrong com-binations of tracks originating from the same reaction (event) and the otherare random coincidences. The first case arises when many e+e− pairs are pro-duced in a reaction, for example in the so-called double Dalitz decay η →e+e−e+e−, or when a charged pion is misidentified and treated as a lepton(electron or positron). Since the reconstruction algorithm is not 100% effi-cient, some of the electron tracks are not reconstructed. Sometimes a particleis not detected due to the detector geometry, when, for example, it escapesthrough the beam tube. The analysis criteria could then lead to the wrongchoice of a charged particle. This type of background can, in principle, beassessed directly by simulating the relevant channels.The second source of the combinatorial background, called random (or acci-dental) coincidences or “pile-ups”, is more difficult to handle. It is due to theoverlapping tracks from different events and reactions1. The temporal resolu-tion of the data acquisition is finite, therefore we can expect situations wherethe particles coming from two different reactions are merged into one event.The selection criteria implemented in previous sections reject only a part ofthe pile-up contribution. The simulation of this background is possible but

1For example, an overlap between pp → pp(η → e+e−γ

)and a large cross section channel

like pp→ pnπ+.

97

it would require to generate a Monte Carlo cocktail of all possible reactionsweighted by their relative occurrence in the sample. In the next section, wepropose a simpler method allowing to estimate the distribution of this back-ground.

6.10.1 Estimate of the number of random coincidencesOur estimate of the combinatorial background is based on the following ap-proach. We assume that the main sources of combinatorial background are theevents with three charged tracks. An example of such event is η → e+e−γand an additional pion track (e.g. from a direct pion production) if the pion ismisidentified as an electron. There are two classes of such events:• class N21 with N+ = 2 and N− = 1,• class N12 with N+ = 1 and N− = 2.

We neglect events with two charged pairs. This class could contribute to thecombinatorial background but with much lower probability2. In any case forour data four track events are very rare as shown in figure 6.40.

Figure 6.40. Charged tracks multiplicity after particle identification.

We know that the N21 class has one wrong +− combination, one correct+− pair and one (obviously wrong) ++ combination. Therefore, N21

+− =N++where N21

+− represents the number of combinatorial background eventsfrom N21 subset. Analogically, we have N12

+− = N−− combinatorial back-ground events from N12 class events. Summing up the contributions we es-timate the number of the combinatorial background events:

NCB+− = N21

+−+N12+− = N+++N−−. (6.1)

2If the random coincident particles are pions then the PID must fail at least once and at leasttwo correct tracks must be rejected by the selection process.

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In order to extract N21++ and N12−−, we process our data sample using our stan-

dard selection cuts and require exactly two identically charged particles af-ter identification. We find that N++ = 42 and N−− = 16. This leads usto the estimation of the number of combinatorial background in our sampleNCB+− = 42+16 = 58.

6.10.2 Invariant mass distribution of random coincidencesThe sum of the e+e− invariant mass distributions of N++ and N−− eventsprovides us with the expected distribution of the e+e− invariant mass of thecombinatorial background. Given the low statistics, we fit a third order poly-nomial to the binned (40 MeV/c2) histogram, then we create a histogram basedon the fitted function (see figure 6.41). This smooth histogram will be usedfurther in this work.

Figure 6.41. Invariant mass of e+e− distribution for the combinatorial background.

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7. Analysis of η → e+e− channel

As it was already described in section 1.5.2 the η → e+e− is a very rare two-body decay. A massive η particle disintegrate into an electron-positron pair.This reaction has simple kinematics, especially in the η meson rest frame. Theopening angle between e+e− in this frame is exactly 180◦. The correspondingangle in the laboratory frame, neglecting detector and reconstruction smear-ing, is shown in figure 7.1 for an incident proton kinetic energy of 1.4 GeV.The invariant mass of η → e+e− system is equal to the η meson mass, 548MeV/c2, and each lepton carries on average 300 MeV that are almost com-pletely deposited in the calorimeter (see figure 7.2). Those features, amongmany other, are used to select η → e+e− candidates from the data sample.They are presented in section 7.1.

Figure 7.1. Angle between e+e− pair (laboratory frame): η → e+e− simulation.

7.1 Summary of η → e+e− selection criteriaThe analysis of this channel is based on the same 2012 data set as for theη → e+e−γ reaction. The sample after the data reduction described in section

100

Figure 7.2. Energy of e+ and e−: η → e+e− simulation.

5.1 is suitable for this analysis as it contains events with two charged tracks inCD without any condition on the neutral tracks. The criteria that were used toselect the η → e+e− candidates are the following1:• special trigger that select events with two clusters with energy deposit

above threshold in CD,• exactly two charged tracks in CD and in FD, no neutral CD tracks,• particle identification such as 0.8 <

∣∣∣momentumenergy

∣∣∣ < 1.1 for each chargedtrack,

• sum of energy deposits of the two charged tracks > 550 MeV,• energy deposit for each track > 320 MeV,• angle between electrons in space > 89◦,• angle between electrons on OXY plane > 135◦,• 500 MeV/c2 < invariant mass of e+e− < 700 MeV/c2,• 500 MeV/c2 < missing mass of two protons < 600 MeV/c2,• missing polar angle of ppe+e− < 6◦,• missing momentum of ppe+e− >−500 MeV/c,• missing energy of ppe+e− >−200 MeV,• missing mass of ppe+e− > 1750 MeV/c2,• mean time of protons - time of each electron < 10 ns,• mean time of protons - mean time of electrons < 8 ns,

1The missing values are calculated by taking the difference between four-vectors. For example,the missing polar angle of ppe+e− is the polar angle of the four-vector Pbeam +Ptarget−Pp1−Pp2−Pe+−Pe−.

101

• η emission polar angle θee < 30◦.Those selection conditions are based on the work of dr. Marcin Berłowskidescribed in his PhD thesis [21]. The main features of this new analysis are:• we use a larger data set from another run period (2012),• we include the whole range of θ angles by using all calorimeter modules

(forward, central and backward),• we use a different trigger - at least two clusters above threshold (≈ 100

MeV) in CD for each track,• we broaden the condition on the time difference between tracks in FD

and charged tracks in CD from 10 ns to 20 ns,• we use a different particle identification (graphical cut).

7.2 Selection of η → e+e− event candidatesThere are several sources of background for the η → e+e− reaction. Wewill focus on two reactions that give the most important contribution to thisbackground due to their high cross section and/or the difficulty to disentan-gle their final state from our signal. Those channels are pp→ ppπ+π− andη → e+e−γ .

7.2.1 pp→ ppπ+π− backgroundThe importance of this channel is due to its large cross-section (∼ 0.6 mb)with respect to η production (∼ 10 μb) coupled with the same final state - twocharged particles in CD. This channel has no peak in the η mass region butrather contributes to the continuous background. In order to suppress it wealso exploit the fact that while most of the electrons deposit all their energy inthe electromagnetic calorimeter (via electromagnetic showers), the pions passthrough the SEC and leave only a part of their energy2.

7.2.2 η → e+e−γ backgroundAlthough most of the e+e− pairs from this reaction have low invariant masses,we need to consider the case when the virtual photon takes most of the decayenergy, leading to large e+e− masses. Then, the real photon has low energyand can easily go undetected due to the threshold values in the detector ele-ments. In this case, the final state is the same as in η→ e+e− and the kinematicis very similar.

2With the exception of those pions that decay inside the detector or undergo nuclear interactions.

102

Figure 7.3. Angle between electrons onOXY plane and applied cut: data.

Figure 7.4. Angle between electrons onOXY plane and applied cut: η → e+e−simulation.

Figure 7.5. Sum of energy deposits of thetwo electrons and applied cut: data.

Figure 7.6. Sum of energy deposits of thetwo electrons and applied cut: η → e+e−.

7.2.3 Baryonic resonance backgroundThe main background from baryonic resonances could originate from pp→pΔ+ (→ pe+e−) channel. However, the cross section for Δ+ production isof the order of 1 mb while BR(Δ+ → pe+e−) = 4 · 10−5 we can neglect thischannel in comparison to η → e+e−γ . With 106 simulated Monte Carlo eventsnot any survives the selection.

7.2.4 Combinatorial backgroundUsing the same approach as described in section 6.10 we find N++ = 11 andN−− = 5. Therefore we estimate the combinatorial background contributionto be of the order of 16±4 events.

7.2.5 Data selectionFigures 7.3, 7.4, 7.5, 7.6, 7.7 and 7.8 illustrate some of the selection criteriapresented in section 7.1 by comparing data with the Monte Carlo simulationof the signal reaction η → e+e−.

103

Figure 7.7. Invariant mass of ee and ap-plied cut: data.

Figure 7.8. Invariant mass of ee and ap-plied cut: η → e+e− simulation.

7.3 Summary of η → e+e− selectionTable 7.1 shows the simulated acceptances and expected number of events fordifferent channels.

Table 7.1. η → e+e− and background reactions.

Reaction Nsim final selection Nexpected

η → e+e− 8.4 ·105 2.1 ·103 -η → e+e−γ 5 ·106 1 (4.36) 90% C.L. ∼ 1

pp→ ppπ+π− 1.4 ·107 0 (2.44) 90% C.L. ∼ 680pp→ ppπ+π− [21] 107 - ∼ 156+248

−27combinatorial background - - ∼ 16

The ratio between σ (pp→ ppπ+π−) and pp→ ppη at 1.4 GeV is around600 μb10 μb = 60. The number of expected pp → ppπ+π− events, 680, is prob-

ably overestimated due to the limited statistics of the simulation, e.g. if wegenerated 10 times more pp→ ppπ+π− events and obtained zero events afterselection, the number of expected events would be 68.

The pp → ppπ+π− simulation in reference [21] was performed in sucha way that IMπ+π− = Mη , therefore we consider the resulting number of ex-pected pp→ ppπ+π− events to be a better estimate for this background chan-nel.

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8. Results and Discussion

8.1 Determination of the transition form factorWe have seen in section 1.4.1 that the transition form factor can be extractedby comparing (dividing) the experimental spectrum of e+e− invariant massfrom η → e+e−γ decay with its theoretical pure QED contribution. First, weneed to select a low background sample of η → e+e−γ events. In this work,we have used two different approaches.

8.1.1 Background subtraction using η → e+e−γ selection cutsThe data sample used here is produced by directly applying a set of criteriato the initial data. Those are described in the sections 5.1, 6.1, 6.2, 6.3, 6.4,and 6.6. The final distribution is obtained by subtracting the background chan-nels η → γγ and pp → ppπ0 (→ γγ)π0 (→ e+e−γ). In order to extract thetransition form factor, we divide the resulting data histogram by the simulatedpure QED η → e+e−γ decay. As input we use three sets of data and simula-tions resulting from a modification of the missing mass condition:• Sample A: missing mass of two protons in [520,580] MeV/c2 range,• Sample B: missing mass of two protons in [530,570] MeV/c2 range,• Sample C: missing mass of two protons in [535,565] MeV/c2 range.

The resulting histograms are represented in figures 8.1, 8.2, 8.3, 8.4, 8.5 and8.6 using the combinatorial background contribution consistent with the analy-sis described in section 6.10 (58 events). We show those plots for two differentbin widths, 20 MeV/c2 and 40 MeV/c2. The fit function used is the square ofthe slightly transformed equation 1.9:

[F(q2)

]2=

(1

1− q2

Λ2

)2

=

(1

1−bη ·q2

)2

, (8.1)

with the fit parameter bη = Λ−2.Table 8.1 presents the results of different fits.The uncertainties that appear in the table 8.1 are statistical only. In order

to estimate the systematic uncertainty we choose a reference value for Λ−2.In our case, it will be the measurement with the lowest χ2/nd f value and15 pileup events (3.05±0.15) GeV−2. The systematic uncertainty is thencalculated using the following formula:

105

Figure 8.1. Transition form factor fit (20 MeV/c2 bin width): Sample A.

Figure 8.2. Transition form factor fit (20 MeV/c2 bin width): Sample B.

Figure 8.3. Transition form factor fit (20 MeV/c2 bin width): Sample C.

106

Figure 8.4. Transition form factor fit (40 MeV/c2 bin width): Sample A.

Figure 8.5. Transition form factor fit (40 MeV/c2 bin width): Sample B.

Figure 8.6. Transition form factor fit (40 MeV/c2 bin width): Sample C.

107

Table 8.1. Results of the form factor fits.

Sample Bin width CB events Λ−2 [GeV−2] χ2/nd f

Sample A20 MeV/c2 58 3.38± 0.12 1.9

280 2.89± 0.21 1.1

40 MeV/c2 58 3.54± 0.12 2.3280 3.10± 0.21 1.5

Sample B20 MeV/c2 58 3.05± 0.15 1.3

280 2.29± 0.28 1.1

40 MeV/c2 58 3.18± 0.14 1.4280 2.44± 0.29 1.7

Sample C20 MeV/c2 58 3.01± 0.15 1.4

280 2.14± 0.31 1.3

40 MeV/c2 58 3.13± 0.15 1.7280 2.27± 0.31 2.2

σsys =1N

√∑

i

(xre f − xi

)2, (8.2)

where xre f is the reference value and xi the N measurements. Equation 8.2leads to the following result: Λ−2 = (3.05±0.15stat ±0.15sys) GeV−2.

We see that we have a systematic discrepancy for masses above 280 MeV/c2

which is more pronounced for sample A (less strict missing mass of two pro-tons cut). This could be explained by the presence of non-η background eventswith high invariant masses that were not rejected by selection criteria of ouranalysis. An attempt to solve this problem was described in section 6.9 andthe results are presented in the next section.

8.1.2 Background subtraction using the fit to two proton missingmass distribution

The procedure used here is described in section 6.9. We use the same dataselection as in section 8.1.1 and remove the non-η background by a fittingprocedure on the proton missing mass. As the result we get a distribution ofe+e− invariant mass. In order to extract the form factor, we need to dividethis distribution by the η → e+e−γ simulation with the form factor equal to1. Then, analogically to section 8.1.1 we fit the obtained data points to extractthe transition form factor. As an example, the result for a uniform 50 MeV/c2

binning, is presented in figure 8.7.This fit leads to the form factor parameter value of Λ−2 = (1.97±0.29)

GeV−2 while the fit χ2 = 0.48. We treat this measurement as our referencevalue. We perform some systematical checks by varying numerous parametersthat influence the fitting. First, we change the bin width, independently in the0−100 MeV/c2 and in the 100−400 MeV/c2 e+e− invariant mass range. For

108

Figure 8.7. Form factor resulting from fit based on background subtraction.

example, we change the bin width in the 0− 100 MeV/c2 mass range to 25MeV/c2, such plot is shown in figure 8.8.

This fit χ2 = 1.36 and the form factor parameter is Λ−2 = (1.91±0.29)GeV−2. In this highly populated low mass region the fit is rather stable,the main issue is to check the stability of the large mass region (above 100MeV/c2). The fit and consequently the form factor value depends mostly ofthe behavior of the fit function for large masses where the statistic is low. Itis therefore important to check the fit stability for IMee > 100 MeV/c2. Thefigure 8.9 shows the fit for a 30 MeV/c2 bin width.

Here, the form factor parameter is Λ−2 = (1.97±0.30) GeV−2 while the fitχ2 = 1.15.

Table 8.2 summarizes the results of the fits for different sets of parameters.The first column indicates the data sample. The sample C was already usedin section 8.1.1 while the sample D has more stringent time cuts (see alsosection 6.2 and figures 6.8 and 6.9). Here tFD is the mean time of two protonschosen in the Forward Detector while tCDC and tCDN , are the times of chargedand neutral tracks in the CD, respectively.

Sample C• -16 ns < tCDC - tFD < -6 ns,• -21 ns < tCDN - tFD < 5 ns.

Sample D• -14 ns < tCDC - tFD < -4 ns,

109

Figure 8.8. Form factor fit based on background subtraction (bin width modified for0−100 MeV/c2).

Figure 8.9. Form factor fit based on background subtraction (bin width modified for100−400 MeV/c2).

110

• -19 ns < tCDN - tFD < 3 ns.

The second column contains the bin width used for the fitting of the signalcontent. The third column describes the range of the signal content fit. Thefourth column indicates the bin width used for the form factor fit in the 100-400 MeV/c2 range1. The last two columns contain the Λ−2 value and theassociated χ2.

Table 8.2. Results of the form factor fits.

Sample MM2P fit bin MM2P fit range FF fit bin Λ−2 [GeV−2] χ2/nd f

Sample C

2 MeV/c2

480-580 MeV/c2100 MeV/c2 1.88± 0.28 0.360 MeV/c2 2.21± 0.29 1.250 MeV/c2 2.50± 0.28 1.9


4 MeV/c2



Sample D

2 MeV/c2



4 MeV/c2



We take the mean of all results with χ2/nd f ≤ 2.706 - it corresponds toa 90% confidence level. We then calculate the systematic uncertainty in thesame way as in section 8.1.1. The result is Λ−2 =

(1.92±0.30stat±0.18sys

)GeV−2. Finally, taking into account a systematic uncertainty interval of [1.74;2.1]our reference value becomes:

1The bin width value for the 0-100 MeV/c2 range is set to 100 MeV/c2 (only one bin) since ithas low influence on the form factor fit.

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1.97±0.29stat+0.13sys−0.23sys

. (8.3)

The figure 8.10 shows this result in comparison to CB/TABS (see reference[7]) as well as to the vector meson dominance model and the pure QED (point-like) approach.

Figure 8.10. Form factor comparison between this work and other measurements.

Table 8.3 summarizes the results from different experiments and basic in-formation about those.

Table 8.3. FF fits from various experiments.

Experiment Source of η η → e+e−γ candidates Λ−2 [GeV−2]

CB/TABS γ p→ η p 5.4 ·104 (1.97±0.11tot)Reference [46] pd → 3Heη 5.2 ·102

(2.27±0.73stat±0.46sys

)Reference [45] pp→ ppη 3.1 ·103 (1.9±0.33stat)

This thesis pp→ ppη 1.1 ·104 1.97±0.29stat+0.13sys−0.23sys

8.2 U− γ couplingAs it was already mentioned in section 1.4.4, the signature of a hypotheticalmassive dark boson decaying into e+e− pair is a narrow peak, smeared by de-tector resolution and reconstruction features, superimposed on the usual Dalitzdistribution of the e+e− invariant mass spectrum. The first step consists of the

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selection of the η → e+e−γ events candidates using criteria that were pre-sented and discussed in chapter 6. Figure 8.11 shows the final e+e− spectrumfor data, Monte Carlo simulations of the three main background channels -η → γγ , pp→ ppπ0π0 with one Dalitz decay, pileups - as well as the sum. Infigure 8.12 one can observe the difference between data and the background.We do not observe any statistically significant peak and therefore we can set anupper limit on the branching ratio for a hypothetical η →Uγ decay, directlyrelated to the U− γ coupling strength parameter ε .

Figure 8.11. Invariant mass of e+e− - data and the background simulations.

In order to extract this limit we follow a similar approach to reference [5].For a given value of the U boson mass corresponding to the kth true invariantmass bin, the number of events in the ith bin of reconstructed invariant massof e+e− can be expressed by:

Ni/Ntot =1

Γb ∑j

Si jηbj νb

j +Sikηkβ . (8.4)

The j and k indices correspond to the true, unperturbed mass distribution whilei represents the reconstructed mass distribution. The experimental smearing2

is characterized by the Sik matrix (normalized such that ∑k Sik = 1, see fig-ure 8.13), the content of each true k mass contributes to a range of i recon-structed mass bins. The effect of the selection criteria is described by theacceptance function ηk presented in figure 8.14. The first term on the rightside of equation 8.4 describes the contribution of η → e+e−γ and background

2The smearing is due to the finite detector resolution and to uncertainties and approximationsof the measurement of parameters that are needed to compute the invariant mass of ee.

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Figure 8.12. Invariant mass of e+e− - difference between data and sum of backgroundMonte Carlo simulations.

channels (sum over b) while the second term corresponds to the hypotheticalη →Uγ channel and a subsequent U → e+e− decay. Ntot is the number of theη mesons produced and effectively detected. In this formalism, the parameterβ represents the branching ratio of the η → Uγ channel. We can write theequation 8.4 in a simplified form:

Ni = ∑b

Nbi +nk

i β , (8.5)

where nki = NtotSikηk represents the reconstructed distribution for a given MU

hypothesis. The corresponding histograms for different values of MU areshown in figures from 8.15 to 8.22.

We solve this equation for each k thus for each U boson true mass MU (cor-responding to the index k). The parameter β =BR(η →U(→ e+e−)γ) is fittedusing the Least Squares Method (implemented in the ROOT environment) foreach MU . The figure 8.23 shows the resulting upper limit β as a function ofMU .

In order to compare this result with other experiments, we convert the upperlimit on β into a limit on the coupling parameter ε . We use the followingtransformation:

ε2 =β

2BR(η → γγ) ·BR(U → e+e−)

(1−M2

UM2

η

)−3

|F(MU)|−2 , (8.6)

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Figure 8.13. The smearing matrix.

Figure 8.14. Acceptance of η →U(→ e+e−)γ .

115

Figure 8.15. Reconstructed e+e− in-variant mass distribution for MU = 50MeV/c2.








116

Figure 8.23. Upper limit on β as a function of U boson mass MU .

where

BR(U → e+e−) =1

1+√

1− 4m2μ

M2U

(1+

2m2μ

M2U

) (8.7)

is the branching ratio of the U → e+e− decay that decreases for M2U ≥ 4m2

μdue to the opening of a new decay channel U → μ+μ−. The form factor

F(MU) =(

1− M2U

Λ2

)is calculated at the η meson mass and Λ= 0.72 [20] [14].

Figure 8.24 shows the effect of variations of event selection criteria on ε2

calculation. The systematical uncertainties on the ε2 value due the systemat-ical uncertainty on the number of η mesons were investigated and found tobe negligible. The final variation of the ε2 limit with U boson mass (e+e−invariant mass) is shown on figure 8.25.

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Figure 8.24. Upper limit on ε2 as a function of e+e− invariant mass (U boson mass).The spread of points reflects the effect of changes in missing mass and time selection.

Figure 8.25. Upper limit on ε2 as a function of e+e− invariant mass (U boson mass).

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8.3 Search for η → e+e−

8.3.1 Initial η → e+e− analysisAfter repeating the analysis chain used in reference [21] with the data sam-ple collected in 2012, we find 191 subsisting η → e+e− candidates. We fit thetwo proton missing mass distribution with a polynomial representing the back-ground and a Lorentz function representing the signal. The latter is centered atthe η meson mass of 547 MeV/c2 and the width is limited to the 8−9 MeV/c2

range. This constraint on the signal width is determined from an η → e+e−Monte Carlo simulation (see figure 8.26). Figures 8.27, 8.28 and 8.29 show fitsfor different bin widths in a 530−560 MeV/c2 mass range, figures 8.30, 8.31and 8.32 in a 530−580 MeV/c2 mass range, while figures 8.33, 8.34 and 8.35for a 520−590 MeV/c2 mass range.

Figure 8.26. Fit to the η → e+e− two proton missing mass distribution.

From those fits we can deduce the limit on η → e+e− branching ratio. Theformula used is taken from Feldman and Cousins [39]:

BRlimit = BR+λ ·σBR =(Nev−Nback)+λ ·σev

Acc ·Nη, (8.8)

where Nev is the number of event candidates with the statistical uncertainty σev,Nback is the number of background events, Acc is the acceptance on the η →e+e− channel and Nη is the number of η mesons in data. The λ coefficientdepends on the assumed confidence level via the formula:

CL =∫ μ+λσ

−∞

1σ√

2πexp

[−1

2

(x−μ

σ

)2]

dx =12

[1+ er f

(λ√

2

)], (8.9)

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Figure 8.27. Fit to two proton missing mass (530−560 MeV/c2): 2 MeV/c2 bin width.


Figure 8.29. Fit to two proton missing mass (530− 560 MeV/c2): 10 MeV/c2 binwidth.120



Figure 8.32. Fit to two proton missing mass (530− 580 MeV/c2): 10 MeV/c2 binwidth.

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Figure 8.35. Fit to two proton missing mass (520− 590 MeV/c2): 10 MeV/c2 binwidth.122

where the mean μ and its standard deviation σ correspond, in our case, to BRand σBR. For a 90% confidence level λ = 1.28. The table 8.4 summarizes theresults of the BR(η → e+e−) limit for different fits.

Table 8.4. Results of the BRlimit fits.

Bin width [MeV/c2] BR limit [/10−5] χ2

530−560 MeV/c22 6.3 1.475 2.8 0.34

10 1.7 0.58

530−580 MeV/c22 2.1 1.375 3.6 1.87

10 1.7 0.70

520−590 MeV/c22 5.3 1.355 4.9 1.65

10 15.7 8.92

We reject the last measurement given its high χ2 value. The mean fromother fits leads us to the result:

BRlimit(η → e+e−) = 6.2 ·10−5. (8.10)

8.3.2 Optimization of the η → e+e− analysisThe result from section 8.3.1 seems to be suboptimal with respect to our dataset (≈ 150 ·106 η mesons). This is mostly due to the extremely low acceptancefor the η → e+e− channel that is 0.33%. When this number is compared to theacceptance from the analysis in [21], 0.72%, it becomes clear that the selectioncriteria should be optimized for our data sample. First of all, we need to iden-tify what criteria are responsible for most of the acceptance losses. It appearsthat the biggest inefficiency is due to the selection related to energy depositsin the electromagnetic calorimeter (SEC). Figures 8.36 and 8.37 represent E1vs E2, where Ei is the energy deposited in SEC by a charged particle (here e+

or e−), for a Monte Carlo simulation of the η → e+e− channel. Those simu-lations were generated using two different versions of the RootSorter package(RS1 and RS2).

Those figures show the distribution of the energy deposits after an initialselection on the sum of energy deposits: Esum > 550MeV . We observe a dis-crepancy between those patterns which becomes even more evident after theselection condition that each particle energy deposit is above 320 MeV -seefigures 8.38 and 8.39. We can also compare the identification plots (energy de-posit in SEC vs charge sign × momentum), see figures 8.40 and 8.41. Again,we can observe differences between distribution of events reconstructed withRS1 and RS2. The events reconstructed with RS2 lie in more compact bandsand at slightly lower deposits. In order to take those differences into account,

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Figure 8.36. Energy deposits of chargedparticles in SEC (RS1 before cut).

Figure 8.37. Energy deposits of chargedparticles in SEC (RS2 before cut).

Figure 8.38. Energy deposits of chargedparticles in SEC (RS1 after cut).

Figure 8.39. Energy deposits of chargedparticles in SEC (RS2 after cut).

optimize our acceptance and therefore improve the limit on the branching ra-tio, we have modified the threshold on the energy of charged particles (from320 to 305 MeV) and adapted the particle identification graphical cut. Thosemodifications allow us to increase the acceptance from 0.33% to 0.54%. Ana-logically to section 8.3.1, we tried to fit the missing mass to two protons withdifferent binning and ranges, the resulting fits are shown in figures 8.42 to8.50.

Table 8.5. Results of the BRlimit fits.

Bin width [MeV/c2] BR limit [/10−5] χ2

530−560 MeV/c22 3.6 1.285 4.0 -

10 23 -

530−580 MeV/c22 1.7 1.075 2.4 7.2

10 3.9 -

520−590 MeV/c22 4.4 1.045 0.8 2.78

10 3.4 -

The table 8.5 summarizes the results of BR(η → e+e−) limit for differentfits. The mean from fits with χ2 < 10 leads us to the result:

BRlimit(η → e+e−) = 2.81 ·10−5. (8.11)

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Figure 8.40. Particle identification plot - RS1.

Figure 8.41. Particle identification plot - RS2.

125




126




127




128

8.4 Summary and outlookUsing data collected in 2012 at COSY synchrotron in Jülich (Germany) weselected a sample of ∼ 11000 η → e+e−γ events from recorded proton in-teractions with a windowless frozen hydrogen pellet target. The analysis wasexclusive i.e. we selected events where all particles from the η → e+e−γ reac-tion were reconstructed. In order to limit the initial amount of data, we applieda preliminary selection that distinguished between the neutral and charged de-cays. Then, we have set up an analysis procedure, based on the analysis ofMonte Carlo simulations of various reactions and their interaction with theWASA detector. The η mesons were tagged by the missing mass of the for-ward scattered protons detected in the FD. For this purpose, the energy calibra-tion of the FD was performed. A particle identification method was tested andapplied to data. We have also developed a new identification procedure thatcould be exploited in further studies. We have studied the trigger efficiencyand analyzed a neutral η → γγ decay to extract the number of produced ηmesons independently. An estimation of the combinatorial background wasdone and included as an additional background source. We have reduced allbackground contributions and demonstrated that we control our data sampleby matching the experimental distributions with simulations. A method ofnon-η background subtraction by fitting the two protons missing mass spectrawas developed. The resulting e+e− invariant mass spectrum was used to ex-tract the η transition form factor and to search for a hypothetical dark photon.

The transition form factor parameter was found and systematical uncertain-ties were analyzed leading to the value of Λ−2 = 1.97±0.29stat

+0.13sys−0.23sys

GeV−2.The search for a dark photon was performed by fitting the whole range

of e+e− invariant mass distribution and taking into account the experimentalresolution (smearing matrix) and the acceptance function. With no signal froma dark boson observed, we have set an upper limit on the coupling parameterbetween this dark particle and real photons (see figure 8.25).

The data were also used to search for the very rare η → e+e− decay. Sinceno signal was observed, we used a fitting procedure to set up an upper limit onthis channel branching ratio with 90% confidence level: BR(η→ e+e−)< 6.2 ·10−5. We have made an effort to optimize the selection criteria for our datasample (different trigger and beam time period). The new resulting branchingratio limit with 90% confidence level is BR(η → e+e−)< 2.81 ·10−5.

In the future, data sets from different years can be merged. It would sig-nificantly increase the available statistics which would improve all the resultspresented in this work.

Another possible study would consist in performing an inclusive measure-ment of the e+e− invariant mass distribution. This would allow us to workwith statistics larger by about two orders of magnitude.

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9. Summary in English

The subject of the work lies in the field of experimental particle physics. Themain topic is the study of decays of a composite particle called the η meson.Those decays are interesting for many reasons, not least because we can usethem for the search of traces of dark particles. Their existence is postulated byphysicists in response to one of the greatest mysteries of modern science.

9.1 The missing mass puzzleAlready in 1783, John Michell, a British astronomer, wrote a work in which hementions the possibility that there are stars so big and massive that the force ofgravity they generate makes their observation impossible. He calculated thatif the radius of the sun was 500 times larger, all the light emitted by such abody would be forced by its own gravitation to return to the body. Therefore,such stars would be invisible to an external observer.

A few years later, Pierre Simon de Laplace performed a similar analysis andarrived to the same conclusions. Nowadays, we know that such astronomicalobjects exist and we call them black holes.

We mention those examples in order to show that considerations about un-observable matter are not a feature of modern physics.

The XXth century was a period of great success of physics. With QuantumMechanics explaining the processes at the microscopic scales and the GeneralRelativity giving a precise picture of the Universe at large scales, physicistscould be tempted to think that the only remaining piece of the puzzle was howto connect those formidable theories. Then, a series of astronomical observa-tions revealed that there is another problem.

Between 1924 and 1932, Hendrik Oort, a Dutch astronomer, made severalinteresting discoveries related to the structure of our galaxy, the Milky Way.He found that it has a gigantic halo and confirmed the hypothesis that thegalaxy is rotating. Then, in 1932, he performed an analysis of the distributionof angular velocities of some stars of the Milky Way and came to the conclu-sion that beside the usual "luminous" matter, our galaxy has to contain largeamounts of "dark" matter in order to be consistent with the observed spectra.

In 1933, a similar analysis was done by a Swiss astronomer, Fritz Zwicky.He focused on the relative motion of eight galaxies in the Coma cluster andobserved that their relative velocities are so large that without adding an invis-ible component that could contribute to the overall cluster gravity, they should

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break apart long ago. Zwicky concluded that the only possible explanationwas the presence of a large amount of invisible matter in this cluster. FritzZwicky was the first scientist to use the term "dark matter" in opposition to"luminous matter". He postulated that the problem of missing (or invisible)matter is even more general and extrapolated it to other clusters of galaxies.This hypothesis was confirmed in 1937 by Sinclair Smith observations of theVirgo cluster and, more recently, in the 60’s and 70’s Vera Rubin, an Americanastronomer, found similar effects when measuring orbital velocities of stars inthe Milky Way and Andromeda galaxy.

An increasing number of astrophysical observations provide hints for theexistence of dark matter: gravitational lensing, large scale structures forma-tion, galactical rotational curves, cosmic microwave background, those andmany other phenomena seem to indicate its presence. Of course, since there isstill no direct evidence for dark matter, many different explanations of the ob-served anomalies were proposed. Maybe the most famous example is calledthe modified Newton dynamics - MOND. The only problem with those at-tempts is that while providing an explanation for one or two anomalies theyfail to resolve other issues. Up to now, only the hypothesis of dark matter isable to account for all strange cosmological observations.

Nowadays, we estimate that there is about four times more dark matter thannormal matter. The puzzle consists not only of the question why there is somuch dark matter but also what is its nature? We know that it has to interactvia gravitation but what about the three other interactions (electromagnetic,nuclear weak and nuclear strong)? We think that either dark matter particles donot couple to any other interactions or that this coupling is extremely weak. Inthis work, we assume that this coupling exists and that normal particles interactwith dark matter via a dark mediating particle, the so-called dark photon (or Uboson).

9.2 Strategies of dark matter detectionObviously, if we hope to be able to detect dark matter in some other way thanthrough its gravitational effects we have to assume that it can interact withnormal matter. However, the probability of this interaction must be very low,otherwise one would already have been able to observe effects of dark matterin many experiments.

The search for dark matter is commonly subdivided into three categoriesrelated to the detection mode: dark matter production, direct and indirect de-tection.

Dark matter productionThe first strategy one can use to detect dark matter is by producing it by col-liding normal matter particles with very high energies. Dark matter should

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be produced in those interactions if only the coupling between dark and usualmatter is different from zero (even if it is extremely weak). This can be doneusing particle accelerators such as the Large Hadron Collider at CERN. It isthen possible, at least in principle, to detect events with dark matter produc-tion and scientists have many different methods at their disposal to make therelevant analyses.

Direct detectionThe direct detection makes use of the fact that if dark matter exists it should bein gravitational equilibrium with our galaxy and we are therefore immersed ina large flux of dark particles. In some dark matter scenarios, one can estimatethat around 100 000 dark particles flow through each square centimeter everysecond. In order to detect some of those particles, huge underground (to reduceexternal background) detectors are being built. Again, assuming that the darkmatter can interact (even very weakly) with normal matter, we expect that fromtime to time, an atom nucleus of the detector material should recoil in reactionto a closely passing by dark particle. Those are very tiny effects thereforerequiring the most refined technologies and most precise analysis tools (andskills). An example of such experimental setup is XENON1T situated underGran Sasso massif - a huge tank filled with liquid xenon.

Indirect detectionThe third strategy consists at detecting dark matter through its hypothetical an-nihilation and subsequent decays to ordinary particles such as photons, muons,electrons, etc. The difficulty of this method is how to deduce the natureof dark matter and of its properties on this basis. The annihilation of darkparticles should be more probable in regions where dark matter density isthe largest. In most scenarios, the galactic center is one such place and weshould therefore expect more indirect signals of dark matter from this area.Indeed, we observe some anomalies such as an excess of positrons at highenergies (experiment PAMELA, AMS), photon peak suggesting e+e− anni-hilation (SPI-INTEGRAL), gamma-ray excess from galactic center (FERMI-LAT), etc. Those observations could be accounted for by theoretical modelsinvolving dark matter particles and antiparticles annihilations. One of thosemodels is used in this work and will be shortly presented in the next section.

9.3 The dark photonIn order to search for a signature of dark matter we have to choose an initialmodel that describes its nature and the phenomena that it is involved in. Weconsider a relatively simple model where dark matter particles can annihilateand emit a dark photon in the process. This dark particle (sometimes also

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called U boson) is very similar to the standard photon with an important dif-ference - it has a non-vanishing mass. This massive dark photon subsequentlydecays into an electron-positron pair. This is a crucial feature since it opensthe door to the search of light dark matter. We look for e+e− pairs that havea given mass, corresponding to the mass of their parent - the dark photon. Ofcourse, there are many possible interactions between normal particles that pro-duce e+e− pairs. In other words, there are multiple background channels tothe reaction that involve dark particles we are looking for. We can disentanglevarious reactions using the differences in their typical characteristics such asangles, momenta, energies, particle multiplicities etc. Therefore we need tocarefully analyze many hundreds millions of events which in our case comefrom collisions of protons accelerated to high velocities by the COSY syn-chrotron situated at Forschungzentrum Jülich in Germany (see section 2.1).To investigate the reactions that occur we used the WASA detector (see sec-tion 2.2). The later consists of a set of dedicated detectors that allow us toreconstruct particle trajectories, their energies and momenta and therefore tochoose our data sample by setting appropriate selection conditions.

9.4 The results

Figure 9.1. Invariant mass of e+e− - data and the background simulations.

For the studies in this work around 11 000 η → e+e−γ event candidateswere selected (see figure 9.11). A dedicated analysis and a fitting procedurewere set up in order to search for an unpredicted excess of events that couldindicate a potential new particle decaying into e+e−. We found no such effect.Therefore we stated that no evidence of dark photon in mass range 0− 400

1This is Fig. 8.11 from chapter 8.

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MeV/c2 was found in our data and we were able to restrict the set of possiblecharacteristics of this hypothetical dark particle (see figure 9.22).

Figure 9.2. Upper limit on ε2 as a function of e+e− invariant mass (U boson mass).

Figure 9.3. Form factor comparison between this work and other measurements.

Our results also provided information about the electromagnetic structureof the η meson. In order to eliminate most of the background channels, wehave implemented a special fitting procedure that allowed us to extract thee+e− events coming from the η → e+e−γ decay with greater precision. Thefigure 9.33 is the result of the division of the extracted distribution of e+e−invariant masses by the theoretical, pure QED, spectrum. The slope of this

2This is Fig. 8.25 from chapter 8.3This is Fig. 8.10 from chapter 8.

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distribution when the invariant mass goes towards zero, Λ−2, is directly relatedto the electromagnetic interaction region of the η meson. In this work, thevalue of Λ−2 has been determined to:

1.97±0.29stat+0.13sys.−0.23sys

GeV−2. (9.1)

This corresponds to an interaction radius of about 0.3 fm.We also investigated the very rare decay η → e+e−. Since its theoretical

branching ratio within the Standard Model is of the order of 10−8 this decay isvery sensible to any possible background. Therefore it happens to be a perfectplace to look for new physics, such as a hypothetical dark matter contribution.After applying very stringent selection criteria, we ended up with a relativelysmall sample of events. We tried to fit a continuous background with an ad-mixture of η → e+e− events but the signal content appeared to be consistentwith zero (see e.g. figure. 8.49 from chapter 8). Therefore we have set up anupper limit with 90% confidence level on the branching ratio of the η → e+e−decay:

BRlimit(η → e+e−) = 2.81 ·10−5. (9.2)

Our results represent a contribution to the study of the electromagnetic struc-ture of the η meson and to the search for a signature of light, sub-GeV, darkmatter. They provide additional constraints for some theoretical models.

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10. Summary in Swedish - Svensksammanfattning

AbstraktHuvudämnet för detta arbete ligger inom området experimentell partikelfysikoch baseras på studier av e+e− par från η mesonsönderfall. Studierna av dessasönderfall är intressanta av flera skäl, inte minst eftersom vi kan använda demför att leta efter spår av partiklar som skulle bestå av så kallad mörk materia.Deras existens är postulerade av fysiker som svar på ett av de största mys-terierna i modern vetenskap nämligen att det behövs mer massa i universumän den hittills observerade för att kunna förklara flera astrofysikaliska obser-vationer.

Datasamplet som använts för detta arbete samlades in av WASA-at-COSYexperimentgruppen genom mätningar av proton-proton kollisioner vid 1.4 GeVkinetisk strålprotonenergi. Experimentet ägde rum vid acceleratorn och la-gringsringen COSY, på Forskningscentret Jülich i Tyskland under 2012. Enlagrad protonstråle fick växelverka med strålmålspelletar av fryst väte. Visatte upp urvalskriterier för att filtrera fram kandidater till η → e+e−γ hän-delser. Dessa utgör en sällsynt variant av elektromagnetiskt sönderfall av ηmesonen som sker med en sannolikhet kring 6.9 ·10−3. Det resulterande sam-plet av händelsekandidater låg till grund för tre olika analyser.

I den första extraherade vi η-mesonens övergångsformfaktor som är enfunktion av mesonens inre kvark- och gluonstruktur. Vi implementerade enspeciell metod för att minska bakgrunden från direkt pion-produktion.

Den andra analysen utgjordes av sökandet efter en smal struktur fördel-ningen av den invarianta massan av elektron-positron-paren i det framtagnaurvalet av η → e+e−γ kandidater. Många teoretiska modeller och några astro-och partikelfysikaliska mätningar tyder på förekomsten av en ny boson, ävenkallad den mörka fotonen som kopplar till både “mörka” partiklar och stan-dardmodellpartiklar. Denna boson skulle kunna sönderfalla till ett e+e− parmed väldefinierad massa och därför kunde den ha upptäckts vid sökandet eftersmala toppar i e+e− parens invarianta masspektrum. Eftersom ingen statistisktsignifikant signal observerades, kan vi sätta en övre gräns på kopplingsparam-etern ε2.

Den tredje analysen syftade till att göra ett urval av η → e+e− händelsekan-didater. Detta är ett mycket sällsynt sönderfall och därför också känsligt förfysik bortom standardmodellen. Ingen signal från η → e+e− observerades,därför kan vi bara sätta en övre gräns på sannolikheten för detta sönderfall.

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10.1 Mysteriet med den saknade massanDen brittiske astronomen John Mitchell författade redan 1783 ett arbete därhan nämner möjligheten att det finns stjärnor som är så stora och massiva attden gravitation de genererar gör det omöjligt att observera dessa. Han haderäknat ut att om solens radie var 500 gånger större så skulle allt ljus som denutsände tvingas tillbaka av sin egen gravitation. Därför skulle sådana stjärnorvara osynliga för en extern observatör.

Några år senare gjorde Pierre Simon de Laplace en liknande analys ochkom till samma slutsats. I dag vet vi att sådana astronomiska objekt existerar,nämligen de så kallade svarta hålen.

1900-talet var en tid då fysiken gjorde stora framsteg. Kvantmekanikenkunde förklara processer på den mikroskopiska skalan och den allmänna rel-ativitetsteorin gav en precis bild av universum på den stora skalan. Fysikernakunde frestas tro att den enda pusselbit som fattades var den som knöt sam-man dessa två fantastiska teorier. Sedan kom en serie av astronomiska obser-vationer som visade att det fanns ytterligare ett problem.

Mellan 1924 och 1932 gjorde den nederländska astronomen Hendrik Oortett antal intressanta upptäckter relaterade till vår egen galax, Vintergatan. Hanfann att Vintergatan har en gigantisk halo och verifierade hypotesen att galaxenroterar. Han gjorde sedan, 1932, en analys av vinkelhastigheten för ett antalstjärnor i Vintergatan och kom till slutsatsen att det måste finnas stora mängderav “mörk” materia utöver den vanliga “luminösa” materien för att förklara deobservationerna.

1933 gjordes en liknande analys av den schweiziska astronomen Fritz Zwicky.Han fokuserade på den relativa rörelsen mellan åtta galaxer i Coma-klustretoch fann att de relativa rörelserna var så stora att de skulle separerat för längesedan om det inte fanns ytterligare en stor mängd osynlig materia närvarande.Fritz Zwicky var första vetenskapsman som använde termen “mörk materia”.Han postulerade att problemet med den felande (mörka) materien var allmänoch extrapolerade detta till andra galaxkluster. Denna hypotes bekräftades1937 av de observationer som Sinclair Smith gjorde av Virgo-klustret.

Senare, under 60- och 70-talet, fann den amerikanska astronomen Vera Ru-bin liknande effekter vid studier av stjärnors banrörelser i Vintergatan ochAndromedagalaxen. Fler astrofysikaliska observationer pekar mot närvaronav mörk materia: gravitationslinser, galaktiska rotationsmönster, formatio-nen av storskaliga strukturmönster och kosmisk mikrovågsbakgrund. Än idag finns det inget direkt bevis för existensen av mörk materia, så alterna-tiva förklaringar av dessa observationer har förts fram. Den mest kända avdessa är handlar om att den Newtonska dynamiken modifieras för stora avstånd(MOND). Problemet med de alternativa förklaringarna är de bara kan förklaravissa av de observerade effekterna, men inte alla samtidigt. Fram tills i dagär hypotesen om den mörka materien den enda som kan förklara alla observa-tioner.

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Det uppskattas att det finns ungefär fyra gånger mer mörk materia än densom är direkt observerbar. Gåtan består inte enbart i varför det finns så mycketmörk materia men också i vad den består av. Vi vet att den måste växelverkavia gravitation, men vi vet inte vad som gäller för de tre andra krafterna (elek-tromagnetism, svag och stark växelverkan). Vi tror att om den mörka materienöverhuvudtaget växelverkar med dessa tre krafter så måste dess koppling tillvanlig materia vara mycket svag. I detta arbete har vi antagit att det finns ensvag koppling och att normala partiklar växelverkar med mörk materia via enmörk utbytespartikel, en så kalla mörk foton (även kallad U-boson).

10.2 Strategier för att detektera mörk materiaDet är uppenbart att om vi ska kunna detektera mörk materia på annat sätt änvia gravitationseffekter så måste vi anta att den kan växelverka med normalmateria. Denna sannolikhet måste dock vara mycket liten, annars skulle mörkmateria redan upptäckts i experiment. Sökandet efter mörk materia delas van-ligtvis upp i tre kategorier som är relaterade till dess detektion: produktion avmörk materia, direkt eller indirekt detektion.

Produktion av mörk materiaEn strategi som man kan använda för att detektera mörk materia är att produc-era denna vid kollisioner mellan normala materiepartiklar vid höga energier.Mörk materia kommer att produceras vid dessa kollisioner om kopplingenmellan mörk och vanlig materia är skild från noll (även om den är mycketliten). Detta kan ske vid partikelacceleratorer som Large Hadron Collider(LHC) vid CERN. Där är det möjligt, i princip, att detektera kollisioner därmörk materia produceras och ett flertal analysmetoder har utvecklats för attfiltrera fram sådana händelser.

Direkt detektionVid direkt detektion använder man det faktum att om mörk materia finns såär den i gravitationell jämvikt i vår galax. Vi är i så fall utsatta av ett stortflöde av mörk materia. Man kan uppskatta detta flöde av dessa partiklar i vissamodeller och kommit fram till att ett flöde i storleksordningen av 100 000 avdessa passerar varje kvadratcentimeter och sekund. För att försöka detekteradessa partiklar byggs stora detektorer djupt ner i jorden. Detta för att minskabakgrunden från kosmisk strålning. Under antagandet att mörk materia kanväxelverka med normal materia kan vi förvänta oss att det då och då händer atten atomkärna i detektormaterialet rekylerar vid denna process. Signalen fråndessa processer kommer vara väldigt liten och det krävs ytterst avanceradetekniker och analysmetoder för att upptäcka detta. Ett exempel på ett sådantexperiment är XENON1T som är uppbyggt under Gran Sasso-massivet ochbestår av en stor tank med flytande xenon.

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Indirekt detektionEn tredje strategi bestå av att detektera mörk materia genom att anta att denkan annihilera och att vanliga materiepartiklar som fotoner, elektroner, my-oner m.m. emitteras vid denna process. Svårigheten här är att härleda denmörka materiens egenskaper utifrån dessa annihilationsprodukter. Annihila-tion av partiklar av mörk materia bör vara mer sannolik i de regioner där dessdensitet är högst. De flesta modellerna utgår ifrån att detta är i centrala delarav galaxer och man förväntar sig därför en starkare signal från centrum av vårgalax. Vissa anomalier har faktiskt observerats som exempelvis ett överskottav positroner med hög energi (PAMELA, AMS), en fotontopp som kan tolkassom resultat av e+e− annihilationer (SPI-INTEGRAL), ett överskott av gam-mastrålning från galaxcentrum (FERMILAT) etc. Dessa observationer kanförklaras av modeller för annihilation av mörk materia. En av dessa modellerhar använts i detta arbete och presenteras kortfattat i nästa sektion.

10.3 Den mörka fotonenFör att kunna söka efter en signal från mörk materia måste man utgå från enmodell som beskriver dess egenskaper. Vi har använt oss av en relativt enkelmodell där partiklar av mörk materia annihilerar varandra och utsänder en s.k.mörk foton utsänds vid denna process. Denna mörka foton (även kallad U-boson) liknar den vanliga fotonen, men med en viktig skillnad - den har enmassa. Modellen förutsäger att denna massiva mörka foton sedan sönderfallertill ett e+e− par som har den mörka fotonens massa. Detta är av avgörande be-tydelse eftersom detta ger en möjlighet att söka efter mörk materia. Det finnsemellertid många andra processer som producerar e+e− par och man måstedärför göra en noggrann analys av data från hundramilliontals händelser. I vårtfall kommer dessa data från kollisioner mellan protoner med hög energi somhar accelererats i synkrotronen COSY vid Forschungcentrum Jülich i Tysk-land (se sektion 2.1). Data från dessa reaktioner har samlats in med hjälp avWASA-detektorn (se sektion 2.2).

10.4 ResultatFör detta arbete utvaldes cirka 11 000 händelser somvar kandidater till ηmesoners sönderfallskanaler med e+e− - par. Data för dessa händelser har spe-cialanalyserats för studier av mycket sällsynta processer. Till normaliseringenav data har η mesonens sönderfall till e+e− och en foton använts. Det sistnäm-nda sönderfallet är väl känt och står för en kontinuerlig bakgrundsfördelningvid sökandet efter en smal topp i fördelningen av e+e− parens invarianta massa(se figur 10.1). Närvaron av en sådan topp skulle kunna signalera existensen aven mörk foton och speciella analysmetoder har utvecklats för detta ändamål.

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Ingen sådan signal upptäcktes vid analysen. Därigenom kan en övre gräns påkopplingen mellan en mörk foton och en foton, ε2, bestämmas i massområdet0-400 MeV/c2 (se figur 10.2).

Figure 10.1. Den invarianta massfördelningen för e+e− - par från experimentet.

Den ovan nämnda bakgrunden från sönderfallsprocessen η → e+e−γ är avintresse i sig eftersom den innehåller information om η mesonens elektro-magnetiska struktur. Denna information erhålls genom att jämföra den ex-perimentella fördelningen av e+e− - parens invarianta massa med vad somförväntas om η mesonen vore en punktformig partikel. Resultatet, den s.k.övergångsformfaktorn, är den kontinuerliga fördelningen av e+e− - parens in-varianta massa som visas i figur 10.3. Lutningen på denna fördelning där deninvarianta massan går mot noll, Λ−2, är direkt relaterad till η mesonens elek-tromagnetiska växelverkansområde. I detta arbete har värdet på Λ−2 bestämtstill:

1.97±0.29stat+0.13sys.−0.23sys

GeV−2. (10.1)

Detta motsvarar en växelverkansradie på 0.3 fm.Ett sökande efter den mycket sällsynta process där η mesonen sönder-

faller till ett e+e− - par utan ytterligare partiklar har också gjorts i detta ar-bete. Sannolikheten för denna process kunde bestämmas till att vara mindreän 2.81 ·10−5, att jämföras med den teoretiska förutsägelsen på 10−8.

140

Figure 10.2. Uppmätt övre gräns för den mörka fotonens koppling till gamma-kvantai detta arbete.

Figure 10.3. η mesonens formfaktor bestämd i detta arbete tillsammans med andradata och teoretiska beräkningar.

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11. Acknowledgements

I would like to express my gratitude to all people without whom this thesiswould not have been possible.

First of all, I would like to thank my “Polish” supervisor Prof. JoannaStepaniak for hours spent discussing about all the small and bigger problemsthat came out during the realization of this work. Hers deep understanding ofphysics concatenated with an incredible amount of patience were my majorassets during all those years of scientific investigations. Thank You very muchProfessor for the pleasant atmosphere You create everywhere You are.

I am also sincerely grateful to my “Swedish” supervisor dr hab. AndrzejKupsc for his wise suggestions and enormous knowledge that he was alwayswilling to share. Thank you for all your support during many years and foryour kindness, in particular during my stays in Uppsala. Thanks to you Ialways felt welcome in Ångström.

Special thanks to Marcin Berłowski, for his incredible insight, critical mindand huge help in solving my many frustrating technical issues.

I would like to express my sincere thanks and gratitude to Hans Calén forhis precision and patience while correcting my thesis till the last moment andto Tord Johansson for his help with the swedish translation of the summary.

I would also like to thank all the smart people that I had the chance to dis-cuss with during my PhD studies, Andrzej Pyszniak, Lena Heijkenskjöld, Wo-jciech Krzemien, Carl-Oscar Gullström, Józef Złomanczuk, Magnus Wolke,Patrick Adlarson, Volker Hejny, Paweł Marciniewski, Kjell Eriksson, FlorianBergmann and many others.

A very special thanks to my grandmother for the delicious meals that keptme in good health during most difficult moments...

Thank you, my dear friend, Bartłomiej Fałkowski for the joyful momentsthat we shared during nightlong discussions about physics and the mysteriesof the universe.

Last but not least, I want to thank my parents, my brother and my belovedwife for their continuous support and especially for the extraordinary patiencethey shown while I was working on this thesis.

This doctoral dissertation was prepared at the National Centre for Nuclear Re-search in Warsaw and at the Department of Physics and Astronomy of the Up-psala University, conferred by prof. Joanna Stepaniak and dr Andrzej Kupsc.

142

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