PAWEŁ PODKOPAŁ 
INVESTIGATIONS OF THE CHARGE 
SYMMETRY CONSERVING REACTION 
dd → 3Henη0 WITH THE WASADETECTOR 

ADOCTORAL DISSERTATION 
PERFORMED IN THE RESEARCH CENTRE JÜLICH, GERMANY 
AND SUBMITTED TO THE FACULTY OF PHYSICS, ASTRONOMY 
AND APPLIED COMPUTER SCIENCE 
OF THE JAGIELLONIAN UNIVERSITY 

THESIS SUPERVISOR: 
PROF.DR. HAB. ANDRZEJ MAGIERA 

CRACOW 2011 
PAWEŁ PODKOPAŁ 
BADANIE REAKCJI dd → 3Henη0 ZACHOWUJ ˛ACEJ SYMETRI ˛E ŁADUNKOW ˛
A PRZY U˙


ZYCIU DETEKTORA WASA 
DYSERTACJA DOKTORSKA 
WYKONANA W CENTRUM BADAWCZYM JÜLICH, NIEMCY 
I PRZEDSTAWIONA RADZIE WYDZIAŁU FIZYKI, ASTRONOMII 
I INFORMATYKI STOSOWANEJ 

´
UNIWERSYTETU JAGIELLONSKIEGO 
PROMOTOR: 
PROF. DR HAB. ANDRZEJ MAGIERA 

KRAKÓW 2011 

Contents 

1 Introduction 1 
2 Data and Theory Overview 3 
3 Experimental Setup 7 
3.1 AcceleratorSystem ............................. 7 

3.2 PelletTarget................................. 9 

3.3WASA DetectorSetup ........................... 10 

3.3.1Forward Detector .......................... 11 

3.3.1.1ForwardWindow Counter ................ 11 

3.3.1.2 Forward Proportional Chamber . . . . . . . . . . . . . 12 
3.3.1.3 ForwardTrigger Hodoscope . . . . . . . . . . . . . . 13 
3.3.1.4ForwardRangeHodoscope ............... 13 


3.3.1.5 Forward Range Intermediate Hodoscope . . . . . . . . 14 
3.3.1.6ForwardRange Absorber ................ 14 

3.3.1.7ForwardVetoHodoscope ................ 15 


3.3.1.8 Light Pulser Monitoring System . . . . . . . . . . . . 15 
3.3.2 Central Detector .......................... 15 

3.3.2.1 MiniDrift Chamber ................... 16 

3.3.2.2 Superconducting Solenoid . . . . . . . . . . . . . . . 16 
3.3.2.3 Plastic Scintillator Barrel ................ 17 


3.3.2.4 Scintillator Electromagnetic Calorimeter . . . . . . . . 19 
3.4 Data AcquisitionSystem .......................... 20 

3.4.1 Read-Out Electronics ....................... 21 

3.4.2 Trigger ............................... 21 

4 Event Reconstruction 23 
4.1 AnalysisFramework ............................ 23 

4.1.1 Event Generator .......................... 24 

4.1.2 Quasi-Free Process ......................... 25 

4.1.3 Event Generator BasedOnPartialWave Decomposition . . . . . 28 
4.1.4WASAMonteCarlo ........................ 29 

4.1.5 RootSorter ............................. 30 

4.2 Detector Calibration ............................. 31 

4.2.1 Plastic ScintillatorsInForward Detector . . . . . . . . . . . . . . 31 
4.2.2 Kinetic Energy ReconstructioninForward Detector . . . . . . . . 35 
4.2.3 Calibration of Scintillator Electromagnetic Calorimeter . . . . . . 37 
4.3Track Reconstruction ............................ 38 

4.3.1 Track Reconstructionin theForward Detector . . . . . . . . . . . 38 
4.3.2 Track Reconstruction in the Central Detector . . . . . . . . . . . 41 
4.4Particle Identification ............................ 42 

5 Phenomenological Models 45 
5.1 Choiceof IndependentVariables ...................... 45 

5.2 Cross Section ................................ 47 

5.3 Quasi-Free ReactionModel ......................... 49 

5.4 PartialWave Expansion fora Three-Body Reaction . . . . . . . . . . . . 50 
5.4.1 Momentum DependenceofPartial Amplitudes . . . . . . . . . . 52 
5.4.2 Cross Section for dd → 3Henη0 Reaction ............. 53 

6 Analysis of thedd → 3Henη0 Reaction 57 
6.1 RunSummary ................................ 57 

6.2 Event Selection ............................... 59 

6.3 KinematicFit ................................ 61 

6.3.1 Fit Constraints ........................... 62 

6.3.2 ErrorParametrization ........................ 62 

6.3.3 Probability Distribution ...................... 64 

6.4 Comparison of Simulation and Experimental Data . . . . . . . . . . . . . 67 
6.5 ReconstructionEfficiency ......................... 72 

6.6 Acceptance Correction ........................... 77 

6.7 Luminosity Determination ......................... 79 

7 Results and Conclusions 83 
7.1 TheTotal Cross Sectionofdd → 3Henη0.................. 83 

7.2 Systematic Uncertainties .......................... 83 

7.3 Differential Cross Section Distributions 
8 Summary and Outlook 
A Appendix 
B Appendix Bibliography Acknowledgements 

.................. 85 

91 93 97 97 103 











1 Introduction 
Investigations of the charge symmetry breaking (CSB) in strong interaction is one of the most challenging topics in hadron physics. The charge symmetry is an invariance of a system under a rotation by 180◦ around the second axis in the isospin space. In quantum chromodynamic (QCD) the charge symmetry requires the invariance under exchange of up and down quarks. However,this quarks havedifferent masses and charges, therefore the charge symmetryisnota strict symmetryoftheQCD Lagrangian.Itmaybeexpectedthat this elementary sources of the CSB will show up also on the hadronic level. In this way CSB studies may help to connect quark-gluon dynamics to hadronic degrees of freedom, allowing in particular to access the mass difference of up and down quarks. Once those contributions canbe treated theoretically,a studyof CSBinlow energy hadronphysicsis a unique window to the quark masses and thus to fundamental parameters of the standard model. Desire to find such interlink motivated extensive investigations in which a lot of attention was paid to the experimental and theoretical studies of CSB [1]. 
Manystudies comprise investigations of various nuclear systems and reactions. The first evidence of CSB comes from the difference of the low energy nucleon-nucleon scattering lengths of n-p and p-p systems [2] in the same spin singlet state after necessary corrections for electromagnetic effects. The experimentally determined (Coulomb corrected) 71 keV difference of the binding energies of 3H and 3He may arise from the CSB[2]. Unfortunatelyin both discussed casesthe results are strongly influencedbynonnegligible theoretical uncertainties due to the Coulomb corrections. Such problems do not arise for neutron-proton elastic scattering where the effect of electromagnetic inter-action is negligible. In these studies the CSB manifests as non vanishing difference of analysingpowers for neutron and proton.Very small differenceof analysingpowerswas observed [3–6] indicating CSB in nucleon-nucleon scattering, however still large controversies exist about the origin of CSB in nucleon-nucleon scattering [7–10]. From all such studies it is known that at the nuclear level charge symmetry is broken due to the presence of the electromagnetic effects and due to mass difference in isomultiplets of nucleons and mesons. Net effect of CSB on strong interaction is strongly obscured when investigating nuclear systems. There are, however, some nuclear processes in which CSB is dominated by properties of strong interaction, the best candidates being charge symmetry forbidden dd → 4Heη0, which was searched for since manyyears. 
Only recently the first observation of dd → 4Heη0 reaction was reported [11] for the beamenergiesveryclosetothe reaction threshold.Atthesametimethe informationonthe CSB in np → dη0 manifesting as the forward-backward asymmetry becameavailable [12]. Those new data triggered also advanced theoretical calculations, which provides the opportunitytoinvestigatethe influenceofthequark massesin nuclearphysics[13].In order to access such information the advanced calculation within effective field theory are necessary. This becomes possible with the use of chiral perturbation theory (ChPT) [14, 15], which has been extended to pion production reaction [16]. The first steps toward theoretical understanding of dd → 4Heη0 reaction have been taken [17, 18]. It was found that the existing data are not sufficient for the precise determination of the parameters of the ChPT and the new data are required. These new data should comprise the measurement of charge symmetry forbidden dd → 4Heη0 reaction and charge symmetry conserving dd → 3Henη0 channel. The measurement of first reaction should be performed at beam energy higher than used in Ref. [11], preferentially with determination of the polarization observables. This would enable to study the contribution of the higher partial waves allowing to extract relevant parameters of ChPT. The measurement of the second reaction is necessary in order to study the relevance of the initial and final state interaction, which strongly influence the results for dd → 4Heη0 reaction. 
The goal of this thesis was experimental investigation of dd → 3Henη0 reaction. These studies are an important part of the general program of investigation of CSB in the dd → 4Heη0 reaction forbidden by charge symmetry. As there are no experimental data whatsoever for the dd → 3Henη0, already the information on the total cross section may contribute significantly to the understanding of the CSB sources in the forbidden re-action. The measurements of the full differential cross section for dd → 3Henη0 reaction may delivereven stronger constraints for the theoretical analysisof thedd → 4Heη0 reaction. The measurement presented in this thesis is also a first step toward the experimental investigation of dd → 4Heη0 reaction including the polarization observables, which are planned at COSY in the future. 
In this thesis the first results for the dd → 3Henη0 reaction are presented. The experimentwas performed withtheWASA detection systemat COSY accelerator. Chapter 2is dedicated to the presentation of the present knowledge on the dd→ 4Heη0 reaction including the existing data and their theoretical analysis. The problems arising from this analysis are presented, which call for the new data on the charge symmetry conserving reaction. In chapter3theWASA detection system at the Cooler Synchrotron in Jülich is introduced with extended information on the components used in the present experiment. Chapter4 describes methods used in the data analysis and the first steps necessary for identification of the investigated reaction. Chapter 5 is devoted to the phenomenological model necessary for the acceptance correction of the experimental data. This model wasalsousedtoextractphysical informationaboutthe reaction mechanism.Inchapter6 the analysis of the data is presented. Final experimental distributions and the comparison of the obtained experimental results with the phenomenological model are discussed in chapter7. Finally,the summaryand conclusions resulting from thiswork are presentedin chapter 8. 

2 Data and Theory Overview 
The most promising data allowing to access the CSB effects in the strong interaction for hadronic systems are the forward-backward asymmetry measurement in np → dη0 reaction [12] and cross section measurement for dd → 4Heη0 reaction [11]. In those investigated systems the electromagnetic effects are negligible, allowing direct observation of CSB in strong interaction. 
The measurement of np → dη0 reaction was performed at TRIUMF with a 279.5 MeV neutron beam.Thisbeam energy correspondstoonly2MeVexcess energy. Therefore, with the use of magnetic spectrometer, the whole angular distribution was measured by detecting scattered deuterons. CSB manifests as a non-vanishing forward-backward asymmetry: 
dλ(ϕ) − dλ(η − ϕ)
Afb = (2.1)
dλ(ϕ)+ dλ(η − ϕ) 
where ϕ is the c.m. deuteron scattering angle. If charge symmetry holds the forwardbackward asymmetry should be zero. This is shown schematically in Fig. 2.1 where the considered reaction is drawn also when exchanging up and down quarks (which corresponds to the exchange of proton and neutron). In order to achieve better accuracy, only angle integrated forward-backward asymmetry was extracted with the final value of Afb =[17.2± 8.0(stat) ± 5.5(syst)] · 10−4.The reached accuracyisnotveryhighwiththe statistical deviation of only two standard deviations from zero. 
Figure 2.1: Schematic drawing of the np → dη0 reaction demonstrating the symmetry of the process when exchanging up and down quarks. 
Amongvarious processesin whichCSBmaybeinvestigatedthe reactiondd → 4Heη0 isverywellsuitedforsuch studies.Since deuteronand 4Hehaveisospinequaltozeroand η0 has isospin equals to one, it is obvious that this reaction is forbidden without isospin symmetry breaking. In self-conjugate systems (which have third isospin component equal zero) a charge symmetric Hamiltonian cannot connect states which differ in total isospin. Therefore the reaction dd → 4Heη0 is also forbidden if charge symmetry holds. The only possibilityofCSBdueto electromagnetic interactionis isospin1admixturetothe ground state of 4He nuclei. Such an isospin impurity was estimated to be very small [19] even if not discovereduptonow isospin1 excited stateexists. Thereforethe observationofthe dd → 4Heη0 reaction gives the clear evidence of CSB in the strong interaction only. 
Attempts to measure dd → 4Heη0 reaction have been undertaken since manyyears. The early measurement of this reaction yields only the upper limit for the differential cross section(see Ref.[20]and references therein).In oneexperimentonlyat deuteron incident energy of 1100 MeV this reaction was observed [21] with the differential cross section of dλ(ϕc.m.=107o)/dΔ = 0.97 ± 0.20 ± 0.15 pb/sr. However, this result was questioned even by some participants of the experiment (Ref. [22]). 
The situation was clarified by measurement of the total cross for dd → 4Heη0 reaction at beam energies very close to the reaction threshold [11]. The measurement was performed at two beam energies corresponding to 1.4 MeV and 3.0 MeV excess energy. The background was substantially reduced by coincidence measurement of all reaction products. Outgoing 4He nuclei were detected with the magnetic spectrometer and η0 was detected with neutral particle calorimeter via its two δ decay. The total cross section reported is 12.7± 2.2pb for lower beam energy and 15.1± 3.1pb for higher beam energy. 
Those two independent observations of CSB effects in the np → dη0 and the dd → 4Heη0 reactions should be analysed within an appropriate theoretical framework in order toextractthe informationaboutthe microscopic sourcesofCSBsuchase.g.upanddown quarks mass difference.Atheory collaboration group, aiming at calculation of the CSB effects in hadronic processes has been formed and the work on the theoretical frontier is in progress. The most appropriate theory for such a goal is the Chiral Perturbation Theory[15].The formalism usesthefactthethe interactionofpionswith matterislargely controlled by the approximate chiral symmetry of QCD, with pions being the Goldstone bosons resulting from the spontaneous breakdown of this symmetry. In this effective field theory quark and gluon degrees of freedom are replaced by hadronic degrees of freedom. The most general Lagrangian with symmetries the same as the QCD Lagrangian may be constructed with the use of the power expansion in the momenta small to the scale of about1GeV.Uptonow mostwork within ChPTwas doneforthetwopion system,the ηNsystem and, more recently, the NN system [14, 23]. In addition, a promising scheme was derived to also analyze pion production in nucleon-nucleon collisions [24–26], and recentlyto meson productioninvarious reactions[16].Atthe presentstageChPTmaybe also applied to light nuclei and other many-body systems [27]. Since chiral perturbation theory allows for a systematic analysis of hadronic reactions, also the breaking of QCD symmetries can be addressed quantitatively, one example being the isospin. In this way ChPT has become an ideal tool for the theoretical analysis of CSB in np → dη0 and dd → 4Heη0 reactions. 
The first calculations conducted for np → dη0 using ChPT [28] lead to the predicted effect much larger than observed experimentally resulting from charge symmetry violation in η0-nucleon interaction. The recent calculations [29] are successful in describing total cross sections, the shape of angular distributions and the analysing power for pion production reactions np → dη0 and pp → dη+. However, they fail in reproducing the forward-backward asymmetry induced by CSB overestimating the data by 2.4 standard deviations. Therefore further theoretical analysis is required and more precise data on CSB in this reaction are necessary. 
The first calculations using ChPT were performed for the dd → 4Heη0 reaction [17] using very simplified model what allowed to identify the most important ingredients necessary for more precise calculations. It was found that at the leading order (LO) only charge symmetry violation in pion re-scattering contributes, there is no next-to-leadingorder (NLO) contribution and some next-to-next-leading-order (NNLO) contributions were identified. The diagrams for LO contributions are shown in Fig. 2.2. Diagram (a) in this figure contributes for η0− 4He relative angular momentum equals0(s-wave) and equals1(p-wave). Diagrams (b) and (c) may contribute for the p-wave only. Calculations were performed for s-wave using plane wave approximation in the entrance channel and simplified 4He nucleus wave function. It was found that the contribution from the LO term becomes negligibly small due to spin-isospin selection rules and the symmetry of 4He nucleus wave function. The NNLO terms result in the cross section by one order of magnitude smaller than theexperimental one.A value closer to theexperimental cross section can be obtained only for surprisingly large value of the graph used to estimate the influence of the short range physics. More reliable calculations were performed [18] using realistic two-and three-nucleon interactions together with the recent advantages of the four-body theory [30–32]. That allowed to properly treat effects of deuteron-deuteron interaction in the initial state and to use realistic 4He bound-state wave function. This calculations confirmed that for s-wave the LO contribution is negligible and at NNLO the cross section is of the same order as the value determined experimentally. One of the most important issue was the identification of dramatic influence of initial-state interactions. It necessitates in the new independent measurements providing information on pion-production reactions with the same initial state. 
At NNLO new terms with unknown strength contribute to s-wave pion production in the dd → 4Heη0 reaction. Their strength can be fixed by the combined analysis of forward-backward asymmetry observed in the np → dη0 reaction and the dd → 4Heη0 reaction. However, in order to get a non-trivial prediction of CSB in pion production an additionalobservableis needed.ThismissingobservablemaybeprovidedbyCSB p-wave pion production in dd → 4Heη0 reaction. In this case the coupling strengths are given by the leading CSB η-nucleon amplitude (diagram (a) in Fig. 2.2) and the leading CSB ηnucleon-nucleon vertex (diagram (b) and (c) in Fig. 2.2). However, similarly as for the s-wave pion production, the contribution of diagram (a) is suppressed as a consequence Figure 2.2: Leading-order diagrams inducing strong CSB for dd → 4Heη0 reaction. Diagram (a) occurs for s-and p-waves, while diagrams (b) and (c) contribute only for p-wave. The crosses indicate the occurrence of CSB, the dots represent a leading-order charge invariant vertex. Dashed lines denote pions, single solid lines denote nucleons and double solid lines denote L. 
of selection rules. Therefore p-wave pion production provides direct access to the CSB η-nucleon-nucleon coupling constant. 
The presented overview of the existing data and the status of the theory demonstrate the necessity of the new measurements that would allow to complete the program of the CSB studies for hadrons. In order to successfully carry out this program and especially to isolate the isospin violating matrix elements of interest, more information on the related isospin conserving interactionsis needed.For thedd → 4Heη0 reaction a close relative is given by dd → 3Henη0. Especially since the initial state is the same, from this reaction important constraints will follow for the initial state interaction to be used in the analysis of the isospin violating channel. In addition, experimental information on this reaction will at the same time help to show how well in general the isospin conserving part of the four nucleon system is understood. In turn the dd → 4Heη0 reaction should be measured at higher beam energy, preferentially with the polarization observables which allow to extract unambiguously the p-wave contribution directly from the experimental data. This defines the whole experimental program which can be realized at the COSY accelerator. The results of this thesis are the first important step towards the successful execution of the CSB studies. 

3 Experimental Setup 
Theexperimentwhichis describedinthisthesiswas carriedoutattheInstitutefor Nuclear PhysicsoftheForschungszentrum Jülich, Germany.Forthe measurementthe Coolersynchrotron COSY together with theWASA detection systemwas used.WASA(Wide Angle Shower Apparatus) was originally installed at the CELSIUS storage ring at the TSL in Uppsala, Sweden [33] and was operated until the shutdown of the accelerator in 2005. Aftertheendoftheexperimental program[34]thefacilitywas shippedto Jülich,Germany.Themovewas motivatedbyseveral significantfactors which allowedto continue and enhance the physics program foreseen for CELSIUS. Major advantages of use of the WASA detector in combination with COSY are: 
• 	
significantly higher beam momentum up to 3.7 GeV/c(at CELSIUS beam momentum was limitedupto2.1GeV/c) 

• 	
polarized and phase space cooled proton and deuteron beams. 


Combining COSY andWASA together we obtained multi-purpose detection system, focused on the investigation of the properties and interactions of nucleons in the strongly nonpertubative region of QCD [35]. After successful installation and first commissioning runsin thefallof 2006WASA has been taking data since April 2007. 
In this chapter a technical overview of the accelerator and the detector systems is given, moreover, the unique pellet target and the data acquisition system is described. The chapter closes with a section where a short overview is given on the different software tools which were used throughout the analysis. 
3.1 Accelerator System 
The acceleratorand storagering COSY (COoler SYnchrotron)[36]attheForschungszentrum Jülich can provide high quality polarized and unpolarized, proton and deuteron beams in the momentum range from 295 MeV/c up to 3.7 GeV/c, corresponding to an energy range between 45 MeV and 2.94 GeV for protons, and from 67 MeV to 2.23 GeV for deuterons. The COSY operatesincycles.In eachcycle, first theH− orD− ions are preaccelerated in the cyclotron JULIC and injected into the storage ring via a charge exchanging stripper carbon foil. In the standard operation up to 1011 particles can be stored in the ring, yielding typical luminosities of 1031 cm−2s−1 for internal experiments. After filling the ring the ions are accelerated until theyachieve desired energy, then the magneticfieldiskeptstableandparticlesarestoredfora certaintime.Thispartofthecycleis called flat top. At the end of the flat top, the beam is dumped and the dipole magnets are ramped down to injection level, so that a new cycle can begin. 
The synchrotron is equipped with two cooling systems. Electron cooling is applied up to 645 MeV/c, while in the higher momentum regime the stochastic cooling [37] is used. They guarantee high quality beams with small emittance and momentum spread which canbe usedfor internal(oneof whichistheWASA detector)endexternalexperiments.In additiontothecoolingofthebeama barrierbucketcavity[38]canbeused,to counteract the energyloss inducedbythe interactionofthebeamwiththetarget.The methodisquite efficientin caseofa target thickness ≥ 1015 atoms/cm2 as usedbytheWASAfacility. The layoutofthefacilityisshowninFig. 3.1.The technical parametersof COSY aregathered inTable 3.1. 
Figure 3.1: The view of the COSY–facility. 
3.2PelletTarget 
The COSY storage ring  
circumference number of magnets momentum range cooling momentum resolution number of particles stored  184 m 24 dipoles, 54 quadruples 0.295 GeV/c – 3.7 GeV/c electron (pbeam <645 MeV/c) stochastic (pbeam >1.5 GeV/c) 10−3 without cooling 10−4 with cooling 1011 (uncooled, unpolarized)  


important prerequisite for4η detection. Figure 3.2 showsa schematicof the PelletTarget system. The pellets (hydrogen or deuterium) are produced in the pellet generator where a high purity liquid jet is broken up into uniformly sized and spaced micro spheres by means of acoustic excitation of the jet nozzle. The droplets freeze by evaporation in the droplet chamber and form pellets which are injected through capillary into the vacuum chamber.Here,thepelletbeamis collimatedbythe skimmerandtravelsdownthe narrow pellet tube to the scattering chamber. After passing the interaction point with the COSY beam, the pellets are collected in the cryogenic pellet dump situated below the detector. Some characteristic featuresof the target are listedinTable 3.2. 
The Pellet Target system  
pellet diameter pellet frequency(at interaction vertex) pellet -pellet distance pellet stream diameter at vertex pellet velocity effective target thickness  ∼ 35 µm 5-12 kHz 9-20 mm 2-4mm 60 -100 m/s >1015atoms · cm−2  

Table 3.2:Performance of the Pellet Target. 
During the winter shutdown in 2007, first deuterium pellets were produced at COSY, however,atthattimetheproblemwiththeblockingoftheglassnozzle occured.Afteralot ofinvestigationstheproblemwas identifiedwiththehelpofthe infrastructure established for glass nozzle manufacturing in the Central Department of Technology (ZAT) at the Forschungszentrum Jülich. It was found that blocking the nozzles was due to debris from sinter filters at the gas input side of the nozzle. This material problem was eventually solved prior to the experiment described in this thesis. In that way a reduction of time necessaryforthetargetregenerationfrom approximately12toonly3hourswasachieved. Witha stable pellet beama deuterium target thickness closeto4·1015 atoms · cm−2 and pellet rates up to 104 pellets/s were obtained. 

3.3 WASA Detector Setup 
The designoftheWASA detectoris optimisedtotagareactionbymeasuringthe forwardgoing recoil nucleon and nuclei by an array of plastic scintillators and straw tube layers and to identify decay patterns of produced mesons by a straw tube tracker in a solenoidal magnetic field,a barrelof plastic scintillatorsanda closeto4η electromagnetic calorimeter (see Fig. 3.3). Performing exclusive or semi-exclusive measurements allows an effective discriminationof background channels anda cleanevent identification. 
The position(x,y,z)intheWASA detectorisgiveninaright handed rectangular coordinate system with origin positioned close to the intersection of the pellet beam with the 
3.3WASA Detector Setup 
circulating COSY beam. The Z-axis is directed along the beam. The X-and Y-axes are in the plane orthogonal to the beam. The X-axis is directed outwards from the beam ring in the horizontal plane while the Y-axis is directed upwards.Two angles can be defined in this coordinate system; the angle measured towards the Z-axis is the polar or scattering angle ϕ, the anglein the(X,Y)plane is the azimuthal angleφ. 
Figure 3.3: Schematic layout of the WASA detector as installed at COSY. The abbreviations used for different detector are explained in the subsequent section. 
3.3.1 Forward Detector 
The main purpose of theForward Detector (FD) is the detection of scattered projectiles and charged recoil particles like protons, deuterons and He nuclei. It comprises a set of plastic scintillator detectors allowing for particle identification by means of energy loss andforthe determinationofthetotalenergy.In additionitconsistsofastrawtubetracker whichprovidesaprecise measurementof angles. Combining these information altogether, the forward detector is capable to reproduce the complete 4-vector of a particle. All FD plastic scintillators may supply information for the first level trigger logic. Some propertiesof the forward detector are summarizedinTable 3.3. 
3.3.1.1 ForwardWindow Counter 
The first subdetector of theForward Detector in beamdirection is theForwardWindow Counter (FWC). It is located directly behind the scattering window. The FWC consist of two layers made of 3 mm thick BC408 plastic scintillator material (Fig. 3.4). Each element in first layer is inclined by 20◦ relative to the plane perpendicular to the beam to followas closeas possiblethe conicalshapeoftheexit windowofthe scattering chamber. The second layer is perpendicular to the beam axis and is rotated in φ directionby half an element with respect to the first layer. This geometry provides an effective granularity of 48 elements which coincides with the granularity of the subsequent scintillator detector. This allows for a track reconstruction of forward going particles. 
The FWC plays crucial role in the first level trigger. It allows for a very effective selectionofeventsbasedonthe multiplicityofcharged tracks.Hitswith similarpositionand time information in subsequent detectors which are pointing to the target region are considered as a good candidates to be accepted by trigger, while the events not fulfilling this condition are rejected. This method significantly suppresses the amount of background caused by secondary interactions in the beam pipe or in the flange at the entrance to the FD.In ourexperimentitwas essentialto incorporatetheFWCinthe triggerfor inducinga high thresholdon deposited energiesin orderto separate 3He from protonsand deuterons. More detailed information concerning this detector can be found in [40]. 
The Forward Detector  
number of scintillators  340  
scattering angle coverage  2.5◦ -18◦  
scattering angle resolution  ∼ 0.2◦  
amount of sensitive material  50 g/cm2  
-in radiation lengths  ≈ 1g/cm2  
-in nuclear interaction lengths  ≈ 0.6 g/cm2  
maximum kinetic energy for stopping (Tstop)  
η± ,p,d, 4He  170 /300 /400 /900 MeV  
hit time resolution  ≤ 3ns  
relative energy resolution  
-stopped particles  1.5% -3%  
-particles with Tstop <T<2Tstop  3% -8%  
particle identification  LE-E, LE-LE  

Table 3.3:Basic properties of the Forward Detector. 

3.3.1.2 Forward Proportional Chamber 
TheForward Proportional Chamber(FPC)isastrawtubetracker.Itisusedfor accurate reconstruction of track coordinates and for precise determination of the angles of charged particles originating from the target region. It is composed of four modules, each with four staggered layers of 122 proportional drift tubes. The modules are rotated by 90◦ in the plane perpendicular to the beam with respect to each other. The straws are cylindrical drift tubes made of 26 µm thick, aluminized Mylar foil with the anode wire located in 


3.3.1.3 ForwardTrigger Hodoscope 
TheForwardTrigger Hodoscope(FTH)showninFig.3.6 consistsofthreelayersof5mm thick, BC408 plastic scintillators. The first layer is segmented into 48 straight, wedgeshaped elements. The second and third layer are assembled from 24 elements, shaped as an Archimedean spirals, oriented clockwise and counter-clockwise, respectively (see Fig. 3.6). The radius of the active surface of FTH is 580 mm, with an inner hole of 48 mm diameter for the beam pipe. The unique geometry of this detector results in a pixel structure obtainedbyoverlapoftwoorthree scintillation detectors.Inthatwaythemultihit situations canbe resolved and the noise contributionis reduced significantly. Thefast readout signals are used in the first level trigger in coincidence with the FWC and FRH to settle the charge multiplicity conditions. The FTH detector can be also used for identificationof particleslike 3Heor 4He with kinetic energy below350MeV,which are stopped before first layer of FRH. Due to the radiation damage and aging effects of the scintillator material, the FTH detector have been refurbished in 2008 [42]. 

3.3.1.4 Forward Range Hodoscope 
TheForward Range Hodoscope (FRH) is positioned downstream of the FTH. It consists of5layersmadeofplastic scintillatorBC400.Thefirstthreeplaneshaveathicknessof11 cm, whereasthelasttwohavea thicknessof15cm.Eachplane incorporates24 elements, read out individually by XP2412 photomultiplier tubes. The main purpose of the FRH is to measure the energies of the forward going particles. The multi-layered structure of the FRH allows many LE-Ecombinations which are used for particle identification. In addition, FRH supplies the information about the azimuthal angle and the multiplicity, for the first level trigger logic. In the near future, also the usage of the energy will be possible to have as a final trigger output, the information about the missing mass of an event. 
Figure 3.6: The three layers of the Forward Trigger Hodoscope hit by two particles (left), pixel structure obtained from the intersection of the hit modules, projected onto one plane (right). 

3.3.1.5 Forward Range Intermediate Hodoscope 
TheForward Range Intermediate hodoscope (FRI)is placed between the third and fourth layer of the FRH. It consists of two layers of 5.2 mm thick bars of plastic scintillator. The modules are rotated by 90◦ with respect to each other, forming rectangular pixels. Its purpose is to deliver precise time and two-dimensional position information. More detailed information about the design and performance of this detector is presented in [43]. 

3.3.1.6 Forward Range Absorber 
TheForward Range Absorber (FRA)isa passive iron absorber positioned directlydownstream, after the last layer of FRH. The thickness of this absorber can be adjusted so that protons originating from ε production are just stopped in the absorber, whereas higher energy protons produced in background reactions, e.g. multi pion production, penetrate the FRA and induce signal in the FVH which can be used in the trigger to reject those events. 
3.3WASA Detector Setup 
Figure 3.7: TheForward Range Ho-Figure3.8: SchemeoftheForwardVeto 
doscope, diameters of layers are given Hodoscope. 
in mm. 


3.3.1.7 ForwardVeto Hodoscope 
TheForwardVeto Hodoscope (FVH) shown in Fig. 3.8 is the most downstream detector planeintheWASA setup.It consistsof12 horizontal plastic scintillator bars witha thicknessof2 cmanda widthof13.7cm, equippedwith photomultipliersonboth sides.The hit position along a bar may be reconstructed from signal time information. In the first level trigger the signals are used for rejection (or selection) of particles punching through the FRH and FRA. 

3.3.1.8 Light Pulser Monitoring System 
The Light Pulser Monitoring System provides reference light pulses via light fibers to all scintillation countersin orderto monitor theirgain duringtheexperiment. Since both organic and inorganic scintillators are used, two types of light sources were designed.A xenon flash tube from Hamamatsu is used for the CsI elements of the calorimeter and three LED-based light sources for all plastic scintillators. From those four sources the lightsignalsare transmittedtoindividual elementsviaanetworkoflight fibers.A more detailed description of the LPS can be found in [44]. 


3.3.2 Central Detector 
The central detector (CD) surrounds the interaction point and is designed mainly for detection and identification of photons and charged particles produced directly or originating from light meson decays. It consists of an inner drift chamber (MDC) for precise tracking of charged particles,asolenoid (SCS) providing magnetic field for momentum reconstruction,thinplastic scintillatorsinacylindricalgeometry(PSB)forparticle identificationand 

3.3.2.2 Superconducting Solenoid 
The magnetic field needed for momentum reconstruction in MDC is provided by the Superconducting Solenoid (SCS). The SCS is capable of producing a magnetic field up to 

The PSB was calibrated using pp → dη+ events selected from data collected with a proton beamofEkin = 600 MeV.Forevent selection the following criteria were applied: -a trigger requiring one or more hitsin theForward Range Hodoscope; -a corresponding hit in the calorimeter (azimuthally matching the hit in PSB); 
-one charged trackin theForward Detector; 
-cuts on the 2-body angular correlation shown in Fig. 3.11. 

In the calibration procedure weexploitfact that for binary reaction the scattering angle and energy are strongly correlated. It implies that η+ scattered at a certain angle has well defined energy. Therefore, whole PSB detector can be divided into several angular bins, andforeachbin,experimentally measuredlightoutputcanbecomparedto corresponding energy deposit obtained fromMC simulation.Typical light output asa functionof scattering angle is presented in right panel of Fig. 3.11. It can be noted that particles which hit the scintillator close to the PMT produce higher light output then particles which are emitted at small polar angle (larger distance to the PMT). This behavior can be attributed to the light attenuation effect (see 4.2.1). 
The procedure for obtaining calibration and nonuniformity constants is sketched in Fig. 3.12.For MC simulation and data the mean energy deposit in each bin is obtained by fitting Gaussian function to the deposited energy distribution. This information is used to calculate theaveraged ratio between real data and Monte Carlo. Subsequently, the ratiois normalizedtounityatthepoint(in thiswork70◦)for which the linear calibration constant was extracted. In order to derive the non-uniformity corrections the obtained dependence (seeFig.3.12)is fittedwithanexponential function.This proceduremakesuseofthefact that the simulation does not account for the light attenuation along the scintillator bars. Having a consistent set of parameters, after the calibration is applied, the real energy can be computed as: 
Edep = QDC· Calibcons · NU(ϕ) (3.1) 
where NU(ϕ) is the non-uniformity correction dependent on scattering angle of η+ , Calibcons is the individual calibration constant and QDC denotes the measured charge (in QDC channels). More details about energy calibration method as well as about performance of PSB detector can be found here [47, 48]. 
Figure 3.12: Calibration coefficients and non-uniformity of two modules in central part of PSB detector. The light attenuationis parametrizedby fitting anexponential function. 
= 
Figure 3.13: Schematic view of the Scintillator Electromagnetic Calorimeter. The Mini Drift Cham-ber tubes and both forward and backward Plastic Scintillator Barrel end caps are indicated. 

Figure 3.14: Coverage of the polar angle in calorimeter. The shape and size of the CsI crystals for 24 layers are shown, from backward (left) to forward (right) part of the SE. The numbers correspond to the number of crystals in each ring. 



3.4 Data Acquisition System 
All componentsof theWASA detector are servedby the upgraded data acquisition system(DAQ)designedtocopewiththehigh luminosities.ThecurrentlyusedDAQsystem evolved fromthe second generationofDAQsystems[50] usedinexperimentsat COSY anditwasoptimizedtoreachthehighestpossibleevent rates.Themajordevelopmentwas focused on implementation of new synchronization system and digitization modules. The 
Figure 3.15: Structure of the new DAQ system of WASA detector. The figure is adapted from [50]. 
newDAQsystemoftheWASAexperimentis hierarchically structuredintolayers,asillustratedbyFig. 3.15. Signals from the front-end electronics (preamplifier,splitters, discriminators) are connected to the digitizing modules in the 14 crates. Each crate is equipped with an optimizedLVDS1 systembus and so-called system controller (SC) which is responsible for the readout of the digitizing modules and the transfer of data to the readout computerfarm.Theevaluationofthe digitized signalsis doneby FPGAs.The digitized informationisstoredinthe internalbuffersuntilthevalidtriggerarrives.Thisallowstrigger delays up to2 µs and avoids delaying the data signals. When the trigger appears an FPGA selects the digitized signals inside a predefined time interval. 
All crates are connected to the readout computer farm via optical links. An event builder collects the data streams from the individual readout systems and writes them in the cluster formattoa disc array. Each cluster can contain data from more than oneevent, thus the decoding of the cluster format into events is necessary. The conversion can be doneby the EMS software [51]. 
1LowVoltage Differential Signaling 
3.4 Data Acquisition System 
3.4.1 Read-Out Electronics 
The main purpose of the read-out electronics is to convert the analog signals of the detector components into digital information. In case of theWASAfacility all analog signals from the plastic scintillator detectors are transmitted via coaxial cables, to the active or passive splitters. There, the signals are distributed into two branches. One branch feeds thesignalstothe continuous-sampling front-endFastQDCsmodulesequippedwith12-bit ADC converters running at 160 Msamples/s while the other branch delivers the signals to the leading edge discriminators. The logic signals from discriminators are passed to the long-range TDCs for hit time measurement. The 64 channel module based on the GPX ASIC is multi hit capable, this means time information for several hits within certain time window relative to the trigger can be stored. Additionally, all logic signals are also passed to the trigger logic described in next section. 
Due to the different shape of the pulses derived from the calorimeter, special kind of SlowQDCs modules with a sampling rate of 80 MHz have been developed. In both types of QDCs the charge integration is done in FPGA. In addition, double pulse detection, baseline information, pedestal subtractionand time stampingbyzero crossing detectionis performed.Toextract time information from calorimeter an algorithm for time-stamping providedby FPGAis used.In caseoffaster signals from the plastic scintillatorsin forward detector the TDCs channels are required, which are connected to the corresponding discriminator outputs. 
Forthe measurementofthedrifttimeofthestrawtubesintheWASA detectorsystem, a 64 channel TDC module based on the F1 ASIC was used, which already had been developed for the ANKE [52]experiment. Before the signals reach theLVDS inputsof the TDCs, they are amplified and discriminated by CMP-16 chips. 

3.4.2 Trigger 
One of the most crucial challenges of the experiment is the proper set-up of the trigger which allows to retrieve the information related to the events of interest. It is obvious that data acquisitionisnotfastenoughtorecordalleventswhichareregisteredinthe detectors therefore, some kind of event selection must take place. 
The WASA trigger system is based on a set of multiplicity, coincidence and track alignment conditionswhichhavetobe fulfilledinordertoaccepttheevent.IntheFig.3.16 a flow chart of the trigger system is shown. 
During data taking for this experiment there were two trigger levels, both implemented in hardware. The first trigger level is intended for triggering the hardware acquisition, and for generatinggate and control signals for the front end electronics. The signals of the plastic scintillator detectors are passed to the multiplicity units where the clustering of neighboring hits takes place (up to four clusters in each detector plane can be detected). In the next step geometrical overlap between clusters in FWC, FTH, and FRH is performed and a set of primary triggers is produced. The simple multiplicity sig
Figure 3.16: Structure of the new trigger system of WASA detector. 
nals are combinedin coincidence matricesto form more complextriggerexpressions after individual delay matching in programmable delay units. 
The purposeofthe secondlevel triggeristo increase selectivityofthe acceptedevents by using additional information about cluster multiplicity and total energy sum [53] in the calorimeter.A clusterin the SECis defined asa groupof hitsin adjacent elements, while a multiplicity of clusters can be associated with the number of possible meson decay products. Additionally,to distinguish charged clusters among neutral, the geometric overlap of the crystals and coincident hits in PSB is checked. In SEC the analog sums are obtained from groups of signals corresponding to 16 crystals in the central part of the calorimeter, 12 crystals in the forward and9in backward part. The summed signals are evaluated by a dual threshold discriminator giving logic signals for low and high energy deposits. 
The balanceoftheevent ratesisachievedbyprescalingallhighrate triggers.Thetriggersare connectedtoatrigger selectorunitwith48triggerinputs.Ithas4programmable 48 bit masks, connected to one OR output each. The masks allow to set in parallel, up to 32 different coincidence conditions which can activate the readout. The logic output signals of all trigger channels are passed to an additionalFastTDC and this information is used later in the offline analysis to determine which trigger was responsible for starting the data acquisition. 



4 Event Reconstruction 
4.1 Analysis Framework 
The software used for the analysis is split into three main packages which allow for event reconstruction, Monte Carlo detector simulation and event generation. In following section, the role of individual programs in the process of data analysis will be briefly discussed.For better understanding, the flow chart showing the consecutive steps in the analysis chain is presented in Fig. 4.1. 
Figure 4.1: Flow chart of the analysis chain. 
4.1.1 Event Generator 
In this thesisforanevent generationthe Pluto++ package[54]was used.Itis aimedatthe studyof hadronic interactionsfrompion production thresholdto intermediate energiesofa fewGeVper nucleon. Originallyitwas designed mainlyforthe HADES[55]experiment, but with the time softwareevolved and now other collaborations likeWASA or CBM2 
[56] are using Pluto++ to perform simulations. The package consists of a collection of C++ classes andis entirely based on theROOT[57] environment. After the user specifies the reaction channel, Pluto generates kinematically allowed sets of momentum 4-vectors for all final state particles. The calculation of homogeneous and isotropic phase space is based on the GENBOD [58] procedures. If there is no any specific reaction dynamics defined, kinematical weight per generated event is constant. When anydistribution in the phase space is expected to follow a certain reaction model, the aforementioned weight is multipliedbyafactor accordingtothe model parametrization. This weighthastobe considered during the analysis. Pluto++ offers several elementary reactions where the angular distribution models based on the parametrization of existing data are included, however for more complicated channels the event generator should be adapted to the corresponding physics by the user. 
In case of dd → 3Henη0, the reaction is composed of two possible reaction mechanisms: the direct production with the three-body phase space and the production in pd subsystem with a spectator neutron stemming from the deuteron beam or the deuteron target. The quasi-free reactions mean that only one of the nucleons in the deuteron is participating in the interaction while the other does not take part in the momentum transfer (see Fig. 4.2). This is to some extent the consequence of the small binding energy of the deuteron. 
Figure 4.2: Quasi-free model for a particle production reaction via dd → nsp3Heη0. 
= 
= 
= 

with P marked with subscripts being the four-momentum of the beam, target, spectator and sub-reaction, respectively. In order to assure reasonable speed of computations and 
= 

= 
= 
Figure 4.4: Fermi distribution of nucleon momentum inside the deuteron (left). Map of the probabilities for pairs(pfermi ,cosϕ). Functionp(pfermi) is used to populate the momenta of nucleon inside the deuteron according to Fermi distribution. The λ(ε) term expresses the total cross section as a function of ε (right). 
Considering the Eq. 4.2 one sees that ε is a function of the pion momentum which can be calculated as: 
√ 
2 22
pη = ((sx − m− 4m2 η0)2− m· mη0)/4sx (4.7)
3He 3He 
The following description refers to the situation where we have neutron as a spectator fromtarget.Ifonewantstofocusoncasewhere spectatorisstemmingfromthebeam,then 
4.1 Analysis Framework 
Figure 4.5: Differential cross section (black lines) obtained from Pluto for few selected beam energies. On each spectrum function 4.1 fitted to the data of Ref. [59] is superimposed (blue lines). 
additionally,Lorentz boost of the Psp to the beam system should be performed. In the main event loop the direction and Fermi momentum of the participant nucleon is determined according to the distribution presented in Fig. 4.4. In the deuteron rest frame, participant moves in opposite direction to the spectator having the same momentum. Finally,the subreactionX → 3Heη0is processed.Asthisisa2-body reaction,the momentaof 3Heand η0 inglobalc.msystemare determinedonlybythe massof subsystemX.Inordertosample events witha specified angular distribution the rejection method [62] has been applied.To make use of it, the function from Eq. 4.1 was normalised at cosϕ = 1to one and random numberN ∈ (0,1)was generated.Inthe subsequentsteptheloop wheretheX → 3Heη0 is placedis repeated till the randomvalueNis larger than the normalised cross.Events which satisfied following criterionhaveexpected angular distribution. Aftertheeventloop is finished the four momentum vectors of all particles are transformed to the laboratory frameby meansof Lorentz transformation and saved for further processing. 
To check the quality of angular parametrization in the generator we tried to reproduce cross section presented in [59]. The function 4.1 perfectly describes the data what means that if one had implemented it in the generator than obtained angular distribution should reproduce the data points in the exact way. Before the comparison, all spectra from the generator has been normalised accordingly. The agreement between function 4.1 describing the data of Ref. [59] and output from the generator is shown in Fig. 4.5. As can be seen there is no discrepancywhat implies that parametrization was successfully incorporated. 

4.1.3 Event Generator BasedOnPartialWave Decomposition 
In order to describe the data, apart from the quasi-free model the partial wave expansion for dd → 3Henη0 reaction has been developed (see Section 5.4). As a result the production amplitude was decomposed according to the angular momenta of the final state particles. This information was used to implement appropriate matrix element in 3-body generator.Taking the angular momenta dL23 of particle2in subsystem 2-3 anddL1 of particle1in globalCM we can list the seven partialwave contributions whichhave been incorporated in the generator. According to the formula for four-fold differential cross section(seeEq. 5.17in Section 5.4.2)the following matrix elements presentedinTab.4.1 were implemented in 3-body generator. The implementation of3-body generator based 
d
H0 = |T|2 σ constant sSwave(dL1 = 0,L23 = 0) 
H1 = |T|2 σ |dp|2cos2ϕp sP wave(dL1 = 0,dL23 = 1) 
d
H2 = |T|2 σ |dp|2sin2ϕp sP wave(dL1 = 0,L23 = 1) 
d
H3 = |T|2 σ |dq|2cos2ϕq pS wave(dL1 = 1,L23 = 0) 
d
H4 = |T|2 σ |dq|2sin2ϕq pS wave(dL1 = 1,L23 = 0) 
H5 = |T|2 σ |dq||dp|cosϕqcosϕp interference sP and pS 
H6 = |T|2 σ |dq||dp|sinϕqsinϕpcosφ interference sP and pS 
Table 4.1:The matrix element parametrisation based on partial wave formalism applied for dd → 3Henη0 reaction. 
on partial wave decomposition has been done as a two-step process. First the reaction d+ d→ 1+ X23 is sampled and one dimensional histogram containing probability distribution for invariant mass of two-particle intermediate system is created. In the second step the reactionX23 → 2+ 3is processed.Let’sdenotethe massof subsystemX23byM23,one can write thatM23 mayvary between the limitsM2+ M3 ≤ M23 ≤ s− M1, wheres corresponds to the square of the total energy. In order to construct the probability distribution ofM23theloopoverall possible massesis processedandeachentryinthe histogramis weighted according top2L23+1q2L1+1, momentapandq are calculated using Eq. 5.1. The kinematics of the reactiond + d→ 1+ X23 can be written in terms of four-momentum vectors as: 
Pbd+ Ptd → P1 + P23 (4.8) 
4.1 Analysis Framework 
In the generator this decay is implemented in the global CM system, thus from the momentum conservation we can write: 
∗∗ 
dp1 = −dp23 (4.9) 
Based on that, we can express four-momentum vectors in the global CM system as: 
√ 
∗
P1 ∗ =( (M12+ |dp1∗|2),dp1) (4.10) 
√ 
∗
P∗ =( (M2 p ∗ 1|2),dp1) (4.11)
23 23 + |d
whereM23 is sampled from the histogram which contains the probability distribution of having correctly distributed masses in subsystemX23. The momentum |dp ∗ 1| in the global CM system may be calculated in following way: 
s = |Pbd+ Pdt |2 (4.12) 
E∗ =(s− M2 s (4.13)
1 23 + M21)/2 √ 
√ 
|dp ∗ 1| = E∗12 + M2 (4.14)
1 
In parallel, the angular distribution for particle denoted as “1“ is sampled. The azimuthal φ angleisevenly distributedinthe interval[0,2η],while scattering angleϕ follows the sin2ϕ or cos2ϕ function if we are interested in p-wave. In case of s-wave, cosϕ is sampled from isotropic distribution. The similar operations are conducted forX23 → 2+ 3process. First the momentum of particle indicated as “2“ is calculated in the rest frame of subsystem X23 and the angles φ and ϕ are sampled isotropically. Since we are in CM system, we have complete information to construct four-vector of particle marked as “3“. What’s left to be done is to sample angular distribution for particles in subsystemX23 according to expected partial wave contributions. This has been achieved by employing the rejection sampling [62] method. After the event loop is finished the four momentum vectors of all particlesare transformedtothe laboratoryframeby meansof Lorentz transformationand savedinROOTTrees [57] format. 


4.1.4 WASA Monte Carlo 
The WMC detector simulation is based on the GEANT3 [63] software package, which was originally designed for the high energy physics experiments. The GEANT program simulates the passage of elementary particles through the matter using Monte Carlo techniques. Before starting simulation user has to provide set of parameter files which contains descriptionofthe detector geometryusedintheexperiment.Herethe definitionofshapes and materials of all active detector elements, e.g. scintillators, as well as all passive ones like scattering chamber or beam pipe should be included. 
The WMC reads the information about the particle four-vectors from the output of the external event generator. Those particles are then tracked through the detector and their interaction with the active and passive detector material is simulated by taking into ac-count the energy losses, multiple scattering, decays and other processes with known cross sections. The scintillator response and hence the light output depends on efficiency for converting ionization energy to photons. This effect is called quenching and it is a complex function of mass, charge and energy of the incident particle. The WMC compensates thiseffectby modifyingthe energy lossesusing semi-empirical Birks’s formula[64].The processes like light attenuation in the scintillators, response of photo multiplier tubes or electronic noises are not implemented in theWASA Monte Carlo. The match between simulations and the response of the detector can be obtained via smearing of the simulated observables. The parameters used for smearing are defined in proper filters which are applied in the analysis program. 
The output from the WMC is saved in a format which is comparable to the one used for experimental data and therefore can be analyzed by the same reconstruction program. The original four-vectors of the generated particles are also written to the WMC output together with vertex position. Those informations allow for study the reconstruction efficiency, resolution and detector acceptance. 

4.1.5 RootSorter 
The analysis of simulated data as well as experimental data files was performed within the event reconstruction software framework called RootSorter [65]. This is an analysis tool,basedontheROOTplatform,entirelywritteninC++usingobject orientedprogramming techniques. The modular structure of the software guarantees flexible analysis code management.The essential conceptof RootSorteristoholdtheexperimentalrawdataas well as calibrated ones and also analysis results like clusters or tracks in class objects with common interface. The specific filters reads and interprets data fromexternal sources (e.g. EMS event stream), decode these data and put proper information in the correct data ob-ject.Inthefirststepofevent reconstructionthe assignmentofdatafromconverter modules to the according subdetectors is performed, the time and energy information is combined into hit objects and saved to the so-called RawHitBanks. In case of simulations the Monte Carlo data are stored in a MCHitBanks. At this stage treatment of experimental data and the simulated data varies. Data from the experiment have to be calibrated, TDC information is converted to time and QDC signals into deposited energy. Hits with informations outside the predefined allowed regions are discarded. The necessary parameters for the calibrationarestoredinacentralMySQL database.Theyhaveuniquetagwhichspecifies for what period of data taking they are valid. The obtained calibration on hit level is not final since some calibration procedures depend on higher level reconstruction, however it can be used to proceed with cluster and track finding algorithms.To match the Monte Carlo simulation with the real data, set of smearingfactors is applied and tuned until the experimental resolution is reproduced. After this procedure simulated and experimental data are handled identically during further steps of the analysis. 
4.2 Detector Calibration 
The RootSorter framework offers a user set of low level utilities which are designed foronline monitoringofthedataduringexperiment.The implemented client-servermechanism allows to connect running analysis from anyhost and guarantees the access to all histograms. 



4.2 Detector Calibration 
4.2.1 Plastic ScintillatorsInForward Detector 
Before we focus on the calibration method let us surveysome of basic effects which occur when the scintillator material is struck by charged particle. One of the most important phenomena which can be observed during passage of the particles through the scintillation detector is their energy loss as result of atomic collisions. The consequence of the latter is the ionization or excitation of the atoms in the material. When the atoms deexcite, photons are emitted. The light is transmitted to the photomultiplier where it is converted into a weak current of photoelectrons which is then further amplified by an electronmultiplier system. Under the action of electric field, electrons are accelerated towards the anode from which the electric signal can be taken. The height of the pulse is proportional to the number of photons impinging on the photocathode. When we assume that the light collectionefficiencyof the scintillatoris one hundred percent and the photomultiplierisa linear device then the amplitude of the pulse is proportional to the energy deposited in the scintillatorbythe incoming particle. This makes the scintillatorsasuitable instruments for measurement of particle energy. However, in the reality, this linear relation does not hold and one has to take into account several effects which cause deviations. First of all the the light collection depends on the geometry of the detector and on the absorption caused by the scintillator material. The latter begin to play severe role when the dimensions of the counter are such that the total path length traversed by the photons is comparable to the attenuation length1. In general also the light yield for given energy depends on different types of particles. Moreover, for given particle type it does not always vary linearly with the energy. This effect is known as a quenching and can be associated with interactions between excited molecules created along the way of incident particle. As the result a fractionofthe energy whichwould transform into luminescenceis diminished. According to [64] the amount of fluorescent light emitted per unit path length dL as a function of 
dx deposited energy dE 
dx isgivenby the formula: 
AdE
dL 
= dx (4.15)
dx 1+ kBdE 
dx 
whereAisa scintillationefficiencyandkBisa constant that depends on the particle type and the material. Another important effect which should be considered is the non linear response of the photomultiplier. It occurs mainly for huge pulses due to space charge 
1The attenuation lengthis defined as length after which the light intensityis reducedbyfactore−1. 
effects or the high voltage fluctuations between the dynodes. Finally, one has to check if there are no any time dependent effects. This can be done by selecting particles which have well defined energy deposit in the detector and checking if there was a drift of the energy deposit peak within a long time scale. If anyof these effects is recognized in the data then it has to be parametrised and corrected on the calibration level. 
In this experiment in order to do the particle identification via LE-LEtechnique, the energy calibration of the plastic scintillator detectors in both layers of FWC and first layer of FTH has been performed. The calibration procedure starts with the parametrization of the light output of protons stemming form the quasi-free elastic scattering dd → ppnn reaction as a function of the scattering angle. Due to the kinematics those protons have fairly constant specific energy loss independent from the scattering angle. It implies that anydeviations from a constant light output can be associated to a non uniform light collection efficiency. The light output of the one module in FWC detector as a function of scattering angle is shown in Fig. 4.6. Here, in order to get rid of the dependence on the 
Figure 4.6: The light output for protons originating from quasi-free elastic scattering as a function of the scattering angle in one module of the FWC detector (left). In the right panelthe same dependenceis plottedbut additionalythe polynomial(red curve)of second order which was used for parametrisation of the nonuniformity is presented. 
path length traveled by protons the ADC content is multiplied by cosϕ. In case of FWC detector the light output distribution was parametrized with a polynomial of second order as it is shown in the right panel of Fig. 4.6. What one sees there, slightly contradicts a naive intuition that the light collection efficiencywill raise while increasing polar angle. This mannerofthinking assumesthatiftheparticlehitsthemoduleatasmallpolarangle, the emitted photons are less likely to reach the PMT then in case where element is struck at a large angle close to the photocathode. It is true in case of scintillators which have a shape of rectangular bars. Since the FWC modules are pizza-like this makes geometrical focusing effects, hence the light collection is slightly higher near the tip of the element which correspondstothe smallpolar angles.To correctforthe non-uniformityofthelight output in the first layer of the FTH detector similar procedure has been applied. The only difference was that for the parametrization, higher orders of polynomial have been used. 

itisshowninFig. 4.8.Byfitting polynomialof second orderwe are ableto find correction factors which are used for alignment.With this method we can proceed iteratively for the rest of the modules in the FWC1 detector. The outcome of the procedure is illustrated in the right panelof Fig. 4.8.To perform relative calibration for FWC2 and FTH1 detectors an analogous technique has been applied. 
The conversion of the signal amplitude to the energy deposit LEin a given detector was achieved by comparing the measured data to the Monte Carlo simulations. Based on assumption that the relative normalisation equalized possible non-linearities of different modules, we used for the comparison whole layers instead of doing the calibration element-wise. In orderto ensure well defined energy loss special data sample containing Figure 4.8: The polynomial of the second order used to equalize the light outputs of two neighboring elements in FWC1 detector (left). Relatively aligned light output in the FWC1 detector (right). 
3He from reaction dd → 3Henη0 and dd → 3Hen was preselected. The light output of one layer (e.g FWC2) after correction for non-uniformity is plotted versus the light output of the first layer in FTH detector. This plot is then compared to the same correlations obtained from simulation. In left panel of Fig. 4.9 several distinctive points for which comparisonwas performed are markedby red circles. The calibration function whichwas 
used to align the Monte Carlo and the data is following:  
dE = a· ADC≈ (1− (b/a) · ADC≈)  (4.16)  
where parameters a and bcan be expressed as:  
a = Y0 c· X0  (4.17)  

Y0
b= · (1− 1/c) (4.18)
c· X20 
The Y0 and X0 correspond to mean value of deposited energy of 3He from Monte Carlo and the real data, respectively. The parameter cdescribes the deviation from linearity. This form of calibration function has the advantage over a polynomial, because the number of parameters is reduced, moreover, one knows which calibration constant should be modifiedin ordertogeta defined changein calibration.To control interactivelythe alignment of LE-LEbands specialROOTmacro has been written. This tool makes possible to tune calibration constants c,Y0 and X0 by making slices from LE-LEplots and superimposing the Monte Carlo distributions on the experimental ones till they match. Since the nonuniformity parametrization has been carried out for six angular bins in the range from ϕ = 2.5◦ to ϕ = 18◦ also the constants for absolute energy calibration wereextracted separately for each angular bin. To avoid discontinuity of calibrated energy between neighboring bins, the linear interpolation was applied. The right panel of Fig. 4.9 shows LE-LEband after applying absolute energy calibration. 
4.2 Detector Calibration 
Figure4.9:The non-uniformity correctedlight outputsof 3Heinthe secondlayerofFWC versus the first layer of FTH detector. Three distinctive points which were used for absolute calibration have been marked by the red circles (left). Non-uniformity and nonlinearity corrected LE-LEplot. The presented energy loss band corresponds to the3He originating from dd → 3Henη0 (right). 

4.2.2 Kinetic Energy ReconstructioninForward Detector 
In case of particles which are stopped in the detector, whole deposited energy is equal to the true kinetic energy. In reality, this behaviour does not hold due to the energy loss in the passive material of the detector and the inefficiencies in the scintillation process. To compensate for all these effects, the kinetic energy is reconstructed with the use of special lookup tables containing parameterized functions which allow recalculate Ekin from energy deposited in the detector. During preparation of individual tables one has to take into account the type of the particle, the detector plane where the particle was stopped, the scattering angle, and the number of layers used for energy reconstruction. All this informations have been derived from Monte Carlo simulations. In the first step single tracks for 3He with kinetic energies varying between 0 GeV and 0.5 GeV were simulated. An analytical parametrization of energy loss as a function of kinetic energy hasbeen obtainedbyfitting histogramsshowedinFig.4.10.Here,the correlation between deposited energy in specific detector layers versus the kinetic energy are presented. Since the energy loss depends on the scattering angles, the parametrization have been prepared separately for six angular bins.Tokeep the smoothnessof reconstructed energy between neighboring bins, the linear interpolation was applied. The 3He particles stemming from dd → 3Henη0 have kinetic energy in the range 125 MeV -245 MeV therefore they are punching through the FWC and then are stopped mostly in the second and third layer oftheFTH detector. Thisexcludesthe possibilityof usetheForward Range Hodoscope which usually serves for energy reconstruction in most experiments carried out with the Figure 4.10:The parametrizationof deposited energyasafunctionof initial kinetic energy performed for selected angular bin. The fitted functions are shown in blue. Different cuts on energy loss allow to distinguish between cases where 3He particles are either stopped or they are punching through. 
WASA detector. The studies of performance of FTH2 detector described here [66] have shown that signals from most modules in the second and third layer were strongly position dependent, moreover, for some of the elements it was not possible to separate signals from the background though the voltage applied to the photomultiplier was at maximum. When one adds to this the complicated shape of modules (Archemedian spirals) than finallyit turnedoutthatisnot possibleto calibratethesetwolayersin plausibleway.This limits the number of detectors which could be used for kinetic energy reconstruction to the FWC1, FWC2 and first layer of FTH1. For the latter, one has to separate the case where 3He are stopped and when they are punching through. These situations can be disentangleby checkingif helium reached or not the second layerofFTH. The blue lines inFig. 4.10showthe fitted parametrizations,fortheFWC polynomialof fourth orderwas used while for FTH power functions and polynomials of lower order have been applied. Thevaluesofthefit parameterstogetherwiththe formulaswhichwereusedforfittingcan be found in Appendix B. Having accomplished parametrization we defined Ψ2 function in the following manner: 
n 
(LEmk − LEtk(Ekin))2 
Ψ2 = ∝ (4.19)
λ2 
k=1k 
where n is the total number of layers traversed by the particle and λk denotes uncertainty of the energy deposit in layer k. The numerator in equation 4.19 is calculable as difference between measured energy deposit LEmk andexpected energy loss LEtk(Ekin) determinedby an analytical parametrization.To find the kinetic energyin eachevent the minimization of Ψ2 is performed. This is realized by a simple algorithm which executes quickly and returns those values of the parameter Ekin which give the lowest value of Ψ2. The kinetic energyof 3Heis assumedtobe properly reconstructedifthe calculated Ψ2 value is smaller then Ψ2 =5.Ifthe Ψ2 is exceeded, then one can presume that 3He was misidentified 
max max 
2During this experiment the old FTH detector was used. Due to the radiation damage and aging effects of the scintillator material,itwas replacedby the new onein 2008. 
4.2 Detector Calibration 
Figure 4.11: The difference between the reconstructed kinetic energy and the one obtained fromtheevent generator.Byfittingthis distributionwithgauss functionthe inaccuracyof the energy determination can be estimated (left). The evaluated from MC inaccuracyof the reconstructed energyversus 3He kinetic energy (right). 
and such an event is discarded. The evaluation of reconstruction error was obtained by comparing reconstructed kinetic energy from Monte Carlo data with the ”true” one, i.e. the kinetic energygivenby theevent generator (see Fig. 4.11). 

4.2.3 Calibration of Scintillator Electromagnetic Calorimeter 
In order to determine the energies of photons originating from neutral pion decay, the calibration of Scintillator Electromagnetic Calorimeter has to be performed. Prior to the installation of SEC at the COSY ring all crystals in the the calorimeter were tested and a pre-calibrated using radiativesources and cosmic muons [67]. This calibration is sufficient for online monitoring, although it can not be used for photons since the mechanism of energy deposition is different. Thus the further calibration was carried out with the use of photonsfrom neutralpiondecay.Forthat purposeeventswithexactlytwo neutral tracks in the Central Detector are selected.Two neutral tracks correspond to two neutral clusters (see Section 4.3.2) produced by photons which traverse the crystals in the calorimeter. The Lorentz invariant mass for each pair of photons can be calculated as: 
√ 
Mδ1,δ2 = 2Eδ1Eδ2(1− cosϕ1,2) (4.20) 
where Eδ1 and Eδ2 are the measured energies of the photons based on the preliminary calibration and ϕ1,2 is the opening angle between the photons momenta.For each η0 candidate, the invariant mass is assigned to the crystals with the largest energy deposit in the cluster. If the invariant mass distribution of two photons is shifted with respect to the η0 mass then for each crystal of the calorimeter the calibration correctionfactor is applied according to: 
≈ mη0
E= (4.21)
δ1,δ2Eδ1,δ2
Mδ1,δ2 
Figure 4.12: Two photon invariant mass distribution versus the detector number in the calorimeter (left). The mass of neutral pion is marked with red dashed line. In the right panel the invariant mass is shown. The peak corresponding to the η0 mass is clearly seen. 
where E≈ is the uncorrected energy of δ1 and δ2, Eδ1,δ2 is the energy after applying 
δ1,δ2 
corrections, mη0 is mass of neutral pion andMδ1,δ2 is the invariant mass of two photons. This procedure is repeated until the invariant mass distributions of all modules in the SEC is centered at the exact mass of neutral pion. In the left panel of Fig. 4.12 the invariant mass distributionversusthe module numberin calorimeteris presented.Therightpanelof Fig. 4.12 shows the invariant mass of two photons stemming from dd → 3Henη0 reaction. 


4.3 Track Reconstruction 
IncaseoftheWASA detectorwecan distinguishtwomajorpartswhere particlescanbe detected,namelytheforwardandthe centralpart. DependingonwhichpartoftheWASA detector a particle traversed, two different algorithms for the tracks reconstruction are available in RootSorter framework. In general, both of these reconstruction procedures attempt to reproduce the particle trajectories with properly assigned four-momentum vectors. First, the algorithm has to determine which hits originate from the same particle. Those hits are merged into clusters which are the base components of tracks. In the following sections the description of the algorithms used for the reconstruction of 3He and η0 tracks willbegiven. 
4.3.1 Track Reconstructionin theForward Detector 
As already mentioned in the previous section most helium originating from dd → 3Henη0 reaction are stopped in the second layer of the FTH detector. This implies that in the track 
4.3Track Reconstruction 
reconstruction routine following detectors can be involved: two layers of FWC, straight layer of FTH andForward Proportional Chamber.To minimize the frequencyof occurrenceofrandomhitsthe reconstructionalgorithmworkson pre-calibrated HitBanks.First, all hits from both layers of FWC detector are put into two temporary arrays. The index of an element in such an array corresponds to the number of hit module in the detector. This unique pattern allows to select only hits which come from overlapping elements. If there is more then one such a combination additional constraints are applied e.g. the smallest time difference between hits or the energy deposit thresholds. The overlap between elements in FWC defines the information about the azimuthal angle for the track candidate. Based on that, the track is extrapolated to the next detector which is the FPC. Here, hit straws from four independent modules are combined into clusters.To form the cluster the geometricaloverlap betweenthehit sensing wiresis checked.Ifwe assumedan ideal case whena clusterwould containonly hits stemming fromthe impactof one particle thendue to the orientation of straws (see section 3.3.1.2) we can expect in total four overlapping clusters in the FPC detector. In reality, this number can be increased due to electronic noise or artifacts in the reconstruction. The exclusion of additional clusters is realized by comparing the azimuthal overlap. Algorithm starts with cluster in first plane and tries to match it with another one from an adjacent layer. If succeeded then the search of third best matching cluster from anyunused layer is performed. After all possible combinations have been checked,overlapping clusters are merged into so-called detector track.To determine the coordinates of clusters the position of the sensing wires is used. The crossing point of all wires defines the coordinates of the track, assuming its origin in the origin of theWASA coordinate system. In Fig. 4.13 the residuals and efficiencyfor modules used in track finding procedure are presented. The residuals are defined asa difference between measured and calculated position of cluster in given module. Let us denote four possible clusters which can form trackin FPCbyX,Y,U,V(each cluster represents one module). If,for instancewewantto estimate residualfor clusterXthenwehavetocheckwhatwas the measured position and compare it to the expected position calculated from clusters Y,U,V.Inan analogous mannerwe can determinethe residualsfor remaining modules. The residuals centered at zero mean that geometry of FPC used in the track reconstructionis compatible withthe one measuredexperimentally.To determinetheefficiencyof given module we have to prepare sample of FPC tracks without requesting this particular detector plane. In this work the efficiencyof reconstructing FPC clusters have been studied on 3He sample stemming from dd → 3Hen reaction. The ratio of the number of reconstructed tracks with a cluster in the investigated detector plane to the total number of reconstructed tracks defines the detection efficiency. In the right column of Fig. 4.13 efficiencyfor individual modules as a function of cluster position in particular module is plotted. As can be seen, efficiencyfor all planes is close to 100%. The dip around zero corresponds to the holes in each module which are necessary to provide the room for beam pipe. Subsequently, the track segments from the FWC and FPC are checked for an azimuthal overlap. If the condition is fulfilled then theyare assigned to the same track. In the last step the track finding procedure searches for the hits in the straight layer of FTH and again if there is an azimuthal overlap with already formed FWC-FPC track, those Figure 4.13: The distribution of residuals as a function of measured cluster position for individual modules in FPC detector (left column). Efficiencyfor four modules used in the track reconstruction procedure for 3He originating fromdd → 3Hen reaction. 
4.3Track Reconstruction 
Figure 4.14: The resolution of polar (left) and azimuthal (right) angles. The angles reconstructedby meansofFPChavebeen comparedtothe initial onesgivenby generator. 
hits are merged into this track. The sizes of the overlaps, time differences and a minimum amount of deposited energy for clusters contributing to the track are defined as set of external parameters.Thepresentedschemeoftrack reconstructionallowsto connectagroup of modules from different detectors which were hit by one particle. At this stage also the spatial information of the track is known. What is missing to get four-vector for given particle is the energy. It can be calculated using the procedure described in section 3.2.2. The algorithm for kinetic energy reconstruction operates on the energy deposits from the modules of the detectors which are assigned to the track. In each track the time information is available which is defined by the time of the hit stemming from FTH1 detector being the part of the track. In Fig. 4.14 the averaged angular resolution for reconstructed 3He track is demonstrated. 

4.3.2 Track Reconstruction in the Central Detector 
The track reconstruction in the central detector makes use of an information provided by three different types of detectors, namely Mini Drift Chamber, Plastic Scintillator Barrel and Electromagnetic Calorimeter.For each of them separate routines for cluster reconstruction have been implemented. The role of track reconstruction procedure is to assign those clusters which belong to the same particle. The track finding algorithm is capable to reproduce charged as well as neutral tracks. In this work it was employed for the neutral tracks only, since we were interested in the measurement of the neutral pion.To reconstruct the neutral tracks, it was sufficient to possess the information about the clusters from PSB and SEC. The MDC detector was not of importance here, it plays crucial role for the charged particlesbut for the neutral can be omitted in the track finding routine. Here, a neutral track is defined by the presence of a cluster in the SEC with no matching cluster in PSB detector. When photons traverse the crystals in the calorimeter theyproduce an electromagnetic shower which develops over a group of adjacent crystals. The size of the shower depends on the energy of the photons. Modules involved in an electromagnetic shower form a cluster. It is a task of the cluster finding routine to identify and merge all crystals belonging to one electromagnetic shower originating from one primary particle. The algorithm used for that purpose is an iterative procedure which operates on the two-dimensional space. In first step routine searches for the crystal which contains hit with the highest energy deposit. This element becomes the center of the new cluster. Adjacent neighbors are scanned and joined to the cluster. The required criterion is that eachhit includedinto cluster shouldhave energy depositofat least2MeV. Moreover, an appropriatehitsshouldnotbeseparatedintimefrom referencehitbymorethan50ns.Algorithm pursues search for new crystal with highest energy deposit, which is not included in the previous cluster and the procedure to find neighboring elements is repeated. This process is iterated until all hits in neighboring crystals have been used. Furthermore, the routine rejects clusters witha total energy below10MeV. This condition reduce so-called split-offcontributions which occur due to the interactions of hadrons and leptons with the active materials in the calorimeter. These interactions canfake electromagnetic showers whichmightbe consideredastrue neutral candidatesbythe reconstruction algorithm.The totalenergydepositofaclusterisevaluatedbysumminguptheindividualenergydeposits from crystals which have been assigned to the cluster. The time information of the cluster is defined by the time extracted from the element with the highest deposited energy. The position dX of the cluster is calculated as the weighted sum of positionsdxi of individual modules that form the cluster: 
dX= ∝iwidxi , (4.22)∝iwi 
wheretheweightswiare associatedwiththeenergydepositedEiinthesinglecrystals: 
wi = MAX{0,W0+ ln Ei }. (4.23)∝iEi
Thevalueofthe constantW0hasbeenfoundtobeequalto4[68].To distinguish neutral clusters from the charged ones in the calorimeter the information from the Plastic Scintillator Barrel is mandatory. The neutral cluster in SEC is defined by the absence of a corresponding cluster in PSB which can be formed only when charged particles traverse the detector.A clusterin the Plastic Scintillator Barrel canbe formedbya single hit or two hits. The latter case occurs when particle impinging the overlap region of two adjacent elements in the detector. The hits from neighboring modules are merged into cluster iftheyare separatedin timebyless than10nsand eachof themhasa minimum deposited energy of 0.5 MeV. The time information of the cluster is calculated as the mean value of the time associated to the contributing hits. The deposited energy assigned to a cluster is extracted from the hit with the highest energy deposit. The assignment of clusters from SEC and PSB described in the following section allows to classify particles measured in the central detector as the neutral one. 


4.4 Particle Identification 
The identification of particles in the Forward Detector has been done by means of the specific energy loss which is described by Bethe-Bloch formula [69]. The used LE− LE 
4.4Particle Identification 
technique relies on the comparison of the deposited energy in different layers of the FD detector. When plotting LE− LE, a distinct correlation for different particles types can be observed.IntheleftpanelofFig.4.15theenergylossinonelayerofFWC detectorversus the energy loss in the first layer of FTH detector is presented. The scatter plot is dominated 
Figure 4.15: The energy lossin theForwardWindow Counterversus energy lossin the first layer of theForwardTrigger Hodoscope. The obtained energy patterns allow to distinguish between different particles types. The graphical cut indicated by black line representstheregion usedto select 3He candidates (left).Two photoninvariant mass distribution, the peak corresponding to theη0 mass is clearly visible (right). 
by3He stemming fromdd → 3Henη0. As expected, the energy resolution is good enough to clearly distinguish between bands formed by different particles types. The descending band inside the graphical cut indicates 3He particles stopped in the straight layer of FTH while the lower ascending band corresponds to helium punching through the detector. Below the marked region the energy loss patterns for deuterons and protons originating mainly from breakup reactions are visible. Due to the threshold (see Section 6.1) in FWC only the branch where stopped protons are grouped survived, the rest with lower energy losses was cut out by the trigger. 
After identification of 3He in the forward detector two neutral tracks in the central detector are requested and the corresponding invariant mass is constructed. This combination provides a very efficient way for the selection of the dd → 3Henη0 channel. In the right panel of Fig. 4.15 the two photon invariant mass distribution is demonstrated. More details about the coincidence technique used for particle selection willbegivenin section where the experimental trigger is described. 


5 Phenomenological Models 

At present there is no microscopic theoretical model for description of dd → 3Henη0 re-action. However, some model describing the data at least qualitatively is necessary for the acceptance corrections. Therefore simple model based on physical assumptions was constructed. It will be used not only for acceptance correctionsbut it deliver also some physical information about the reaction process. Therefore the comparison of such model with experimental data deliver already some constraints on the microscopic calculations that will be performed in the future. 
In this section first the independent variables necessary for the presentation of threebody reaction data are briefly discussed and the expressions for some differential cross sections are given. Than the components of quasi-free reaction process are presented. Finally the partial-wave decomposition for three-body reaction is introduced. All these together are the ingredients of the phenomenological model used in the analysis of present data. 
5.1 Choiceof IndependentVariables 
Generallyfora reactionoftwo particles leadingto n particles in the final state there are 3n degrees of freedom corresponding to variables necessary to completely describe the final state. The four momentum conservation reduces this number to 3n − 4. Additionally,if the particles in the initial state are not polarised, the initial state have axial symmetry. Therefore there is one trivial variable corresponding to rotation around the beam axis direction. Than finally for the complete description of a three-body reaction 3n − 5 non-trivial independent variables are necessary. Therefore for the investigated reaction dd → 3Henη0 with unpolarised particles in the initial state and three particles in the final state there are four independent variables fully describing the reaction kinematics. 
The natural coordinate system used commonly for the description of the three-body reactions is given by the Jacobi-coordinates. The momenta dP,dpanddqin this system are defined by 
dP=dp1+dp2+dp3 
dp= 1 (E3dp2− E2dp3) (5.1)
E2+E3 
dq= 1 [(E2+ E3)dp1− E1(dp2+dp3)]
E1+E2+E3 
where the outgoing particles momenta are denoted by dp1,dp2,dp3 and the total energies of particles areE1,E2,E3.Usingthecyclic permutationof indicesitis possibleto obtain other two equivalent sets of Jacobi momenta. For the description of the reaction it is most convenient to use the center-of-mass reference frame. Then the Jacobi momenta have simple interpretation visible in Fig. 5.1. In the following this coordinate system will be used with the z-axis along the beam momentum direction. Since the description is in center-of-mass system than dP= 0and one obtain the conditiondp2+dp3= −dp1. Solving Eq. 
5.1 with this condition and applying Lorentz transformation to the subsystem of particles 2-3 one obtains expressions for momentadpanddq 
Figure 5.1: Jacobi momenta and angles definitions for a three-body reaction. 
1
dp=(E3dp2− E2dp3)= dp≈ 2,
E2+E3 1
dp= − (E3dp2− E2dp3)= −dp≈ 3, (5.2)
E2+E3 
dq=dp1. 
5.2 Cross Section 
Therefore the interpretation of Jacobi momenta is following: dqis the momentum of a particle1 in the global center-of-mass system,dpis the momentumdp≈ 2 of particle2 in center-of-mass of the subsystem 2-3. 
The momentum vectorsdpanddqfully describe the kinematics of the three-body reaction.Howeverfor three-body reaction there areonly four independentvariables necessary for the full description. In the following theywill be chosen as:M23 -invariant mass of the 2-3 subsystem, cos ϕq -cosineoftheangle betweenthe momentumvectordqand beam momentum (z-axis) in global center-of-mass system, cos ϕp -cosine of the angle between momentum vectordpand beam momentum (z-axis) in the global center-of-mass system and φ -relative angle between planes defined by dqand beam momentum and bydpand beam momentum. 


5.2 Cross Section 
The total cross sectionisa normalised integralofthe transition probabilityoverthe whole available phase space. According to [70] the total cross section for 3-body reaction may by expressedby 
()
∫ 
33
1 d3pi
λ = √ ∞α4Pa + Pb− ∝Pj|T|2 , (5.3) 2 γ(s,M2a,Mb2)(2η)5i=1 2Ei j=1 
where s is the square of invariant mass in the entrance channel,Ma,Mb are the masses of beam and target particles,Pa,Pb andPj denote four-momenta of the beam, target and outgoing particles,pi are componentsof momentum vectorsof outgoing particles. The transition matrixTisa functionof four independentvariables which areexpressedby the momenta of all particles involved in the reaction. The γ function is defined as 
22
γ(x,y,z)= x+ y+ z2− 2xy− 2yz− 2zx . (5.4) 
Using the γ function, the entrance channel center-of-mass momentum may be expressed as 
√ 
γ(s,M2a,M2b) P∗ a = P∗ = √ . (5.5)
b 
2s 
Similarly the absolute value of the Jacobi momenta may be calculated 
√√ 
γ(s,M21,M2 γ(M2 2,M32) q= √ , p= , (5.6)
23) 23,M2 
2 s 2M23 
whereM1,M2 andM3 are the massesof the outgoing particles. 
In the expression 5.3 certain integrations may be performed up to the moment when only four independentvariables remain.For these integrationsitissufficientto calculate phase space only since |T|2 is a function of four independent variables only. The threebodyphasespaceR3mybe calculatedfromthe formula[70] 
∫ 
R3 = ds23R2(s,M21,s23)R2(s23,M22 ,M23) , (5.7) 
where s23 = M2 23.Asthe2-bodyphasespaceR2onemayusethe formulae[70] 
∫∫
qq
R2(s,M21,s23)= √ dΔq = √ dcosϕqdφq,
4s 4s 
∫∫ (5.8) 
pp
R2(s23,M22,M23)= √ dΔp = √ dcosϕpdφp,
4s23 4s23 
wheredΔp anddΔq are the solid angles, ϕp and ϕq are polar angles, φp and φq are azimuthal angles of momentum vectors dpanddqin global center-of-mass system. Inserting equation 5.8 to expression 5.7 one obtains 
1 ∫ ds23
R3 = √ pq√ dcosϕqdφqdcosϕpdφp. (5.9)
16s s23 
For the unpolarised beam and target one may integrate over one of the polar angles living only the relative polar angle φ = φp− φq.Onemayuseinvariant massM23 = √ s23 of the subsystem 2-3 instead of using s23. Using these substitution equation 5.9 may be rewritten in the form 
∫
2η
R3 = √ pqdM23dcosϕqdcosϕpdφ . (5.10)
8s 
The choice with equations 5.8 and further steps define the independent variables that maybeusedfor descriptionof three-body reactionandtheyareM23,cosϕq, cosϕp and φ. Up to now the integrations were performed over dependent variables therefore the result given by equation 5.10 for the phase space is valid also for the cross section. Inserting equation 5.10 into equation 5.3 and using equation 5.5 the four-fold differential cross section may be expressed in the form 
d4λ 1 
= pq|T|2 . (5.11)
2ηdM23dcosϕqdcosϕpdφ 32(2η)5sP∗ 
a 
The integrationover someof independentvariables leadstotheexpressionsfor single differential cross sections as e.g.dλ/dM23,dλ/2ηdcosϕq,dλ/2ηdcosϕp,dλ/dφ. 
5.3 Quasi-Free Reaction Model 


5.3 Quasi-Free Reaction Model 
A neutron and a proton are loosely bound in a deuteron. Therefore in high momentum transfer reactions in which deuteron is involved, the reaction can proceed via interaction withasingle nucleonofthe deuteron,whilethesecond nucleonisa spectator.Theimportance of such reaction mechanism is limited only to some regions of the phase space, in which the spectator nucleon have the same momentum as in the deuteron. This remains the main characteristic of the quasi-free reaction mechanism and is used to check the validityof this model. Experimentallythe quasi-free approximationwas checked comparing data for proton-proton induced reactions with reactions induced by proton-deuteron initial state with a neutron spectator. The comparison was performed for pion production reactions [71, 72] and for ε production [73,74].Very good agreementof the total cross sections measuredwiththesetwomethods demonstratedvalidityofthe quasi-free approximation.Inpresentanalysisthe quasi-free reactionmodelwasusedasapartofthe theoretical approach. The two-body reaction pd → 3Heη0 was considered as the quasi-free part of the three-body dd → 3Henη0 reaction. Since in investigated reaction two deuteron’s are involved the reaction may proceed with projectile or target neutron spectator. The quasi-free reaction have been shown schematically in Fig. 4.3 in Section 4.1.2. It is seen that full description of the quasi-free contribution to dd → 3Henη0 reaction can be obtained using the cross section forpd → 3Heη0 in the proper energy range convoluted with the momentum distribution of the proton in the deuteron. The calculations of the cross section for pd → 3Heη0 reaction are rather difficult. Therefore, instead of the theoretical calculations the existing data parametrisation was used. The available data for the differential cross section forpd → 3Heη0 reaction cover the energy range from the threshold up to 10 MeV above threshold [75]. There are also measurements at higher energies corresponding to excess energy above threshold of 40, 60 and 80 MeV [76]. Use of the parametrised experimental cross section for pd → 3Heη0 reaction have additional advantage. It may be expected that in such treatment the quasi-free contribution will be absolutely normalised. The momentum distribution of the nucleons in the deuteron can be obtained using deuteron wave-function. In the present analysis an analytically parametrised deuteronwave function based on theParis potential [60]was used.In the data that we have collected not whole distribution of the proton momentum in deuteron is accessible. The deuteron beam momentum fordp → 3Heη0 reaction (with target proton at rest) correspondingtothe reaction thresholdis 1.284 MeV/c.Inthe presentexperimentthe deuteron beam momentumwas1.2 MeV/c. Thereforein present datathedp → 3Heη0 reaction will occur as quasi-free process only involving protons which sufficiently large momenta in the deuteron target. In order to reach the threshold for dp → 3Heη0 reaction the proton in the deuteron target must have the momentum of at least 35 MeV/c. 

5.4 	PartialWaveExpansionforaThree-Body Reaction 
Partial wave expansion is a commonly used method for the analysis of nuclear and elementary particle reactions [77, 78]. This method is very useful since due to short range of strong interaction only limited number of partial waves (and accordingly limited number of expansion coefficients) is sufficient for description of the reaction. Each partial wave correspondstothe angular momentum withinthe system.Ifthe energyavailablefor the reaction is small than only small angular momenta are allowed. In a special case of a reaction with the energy close to reaction threshold it is usually sufficient to consider only processes with s-wave(angular momentum equal0)and p-wave(angular momentum equal 1). 
In the textbooks as e.g. [77] usually the formulae for two-body reaction are given. Here the general formulae for the three-body case are presented, which are more complicated since more couplings of spins and angular momenta are required. In following these couplings are directly related to the choice of independent variables used in the description of the three-body reaction as discussed in section 5.1. 
Denote spin of the incident particles by dsa,dsb and for outgoing particles by ds1,ds2 andds3. The orbital angular momenta are dLi -entrance channel, dL1 -particle1 in global center-of-mass system, dL23 -particle2in subsystem 2-3. The total angular momenta are dJ -entrance and exit channel (they are equal due to angular momentum conservation), dj1 -particle1 in global center-of mass system,dj23 -particle2 in subsystem 2-3. As the 
quantisation axis (z-axis) it is most convenient to use the beam direction. 
For the entrance channel one may define total spindsi =dsa +dsb, than the total angular momentum is dJ=dsi+dLi. When adding spins and/or angular momenta one has to use Clebsch-Gordan coefficients which contain the projections onto z-axis (ma, mb, mLi, mJ for spins, orbital angular momentum and total angular momentum correspondingly). 
Forexit channel one may define spin for subsystem 2-3ds23 =ds2+ds3 and total angular momentum for subsystem 2-3dj23 =ds23 +dL23. Similarly one may define total angular momentum for particle1dj1 =ds1+dL1. These total angular momenta are coupled to entrance total angular momentum dJ=dj1+dj23. For each sum of spins and/or angular momenta one has to use their projections onto z-axis (denoted by m1, m2, m3 for spins, mL1, mL23 for orbital angular momenta and mj1, mj23 for total angular momenta) and corresponding Clebsch-Gordan coefficients. 
The coupling may be performed in a different order, first adding all spins to the total exit channel spin, then adding all orbital angular momenta to total angular momentum in the exit channel, and then adding total spin and total orbital angular momentum to total angular momentum J. Such coupling is equivalent to the coupling described above. One obtain the same number of amplitudes, however, they are labeled by different set of the quantum numbers. There is a unique relation between the amplitudes obtained with these two couplings. 
5.4PartialWave Expansionfora Three-Body Reaction 
With such decomposition one obtains amplitudes aΠ which are labeled by quantum numbers denoted by Π =(si,Li,s23,j23,L23,L1,j1,J). Using these amplitudes one my obtain total transition amplitude which is labeled by projections of all spins involved. This transition amplitudeis obtainedasa sumoverall possible Π of the product of corresponding Clebsch-Gordan coefficients, amplitudes aΠ and spherical harmonics forL23 andL1. 
mL23 mL1
The spherical harmonicsY(pˆ) andY(qˆ) astheargumenthavethe directionofver
L23 L1 
sorsˆpandˆqwith respect to the z-axis. 
Additionally the parity conservation constraints should be inserted into the formula. Internal parity of the particles participating in the reaction is denoted by Ia, Ib, I1, I2, I3. Then the entrance channel parity is Ii =(−1)LiIaIb and the exit channel parity is If =(−1)L1+L23I1I2I3.Parity conservation requires that Ii = If therefore one should include in the formula for amplitude the Kronecker delta function αIi,If. 
For identical particles in the entrance channel additional constraints on the Li+ si should be included. For identical bosons the symmetry of the entrance channel wave function requires that Li+ si should be even, while for fermions the antisymmetry of the wave function requires that Li+ si should be odd. Therefore for identical bosons the identity function is αidentity = αLi+si (mod2),0 and for identical fermions it is 
αidentity = αLi+si (mod2),1. 
Finally the transition amplitude labelled with all spin projections may be written in the form 
Tma,mb 
= ∝ (sa,ma,sb,mb|si,ma + mb)(Li,0,si,ma + mb|J,ma + mb)
m1,m2,m3 
si,Li,s23,j23, 
L23,L1,j1,J 

(s2,m2,s3,m3|s23,m2+ m3)(s1,m1,L1,mL1|j1,mj1) 
(s23,m2+ m3,L23,mL23|j23,mj23)(j1,mj1,j23,mj23|J,ma + mb)
√ 
mL23 mL1
αIi,Ifαidentity asi,Li,s23,j23,L23,L1,j1,J 2Li+ 1YL23 (pˆ)YL1 (qˆ) (5.12) 
with following substitution mj1 = m1+ mL1, mL23 = ma + mb− mL1 − m1− m2− m3, 
mj23 = ma + mb− mL1 − m1. 
The transition probability is obtained by averaging over spin projections in the en-trance channel and sum over all spin projections in the exit channel of the squared transition amplitude 
1 
|Tma,mb |2
|T|2 = ∝ m1,m2,m3. (5.13)
(2sa + 1)(2sb+ 1) 
ma,mb, m1,m2,m3 
5.4.1 Momentum DependenceofPartial Amplitudes 
The transition probability |T|2 in equation 5.13 is a function of four independent variables: M23, ϕp, ϕq and φ. The dependence on the angles is explicitly contained in the spherical harmonics in equation 5.12. There the amplitudes aΠ may depend only onM23 or equivalently on momentapand q. This dependence cannot be found without microscopic calculations, however, one can make some predictions under simple assumptions. 
Generally when deriving the transition amplitude f the integral of some transition operatorVmultipliedbythe initial Γi and final Γf state asymptotic wave functions have to be calculated. The wave function Γi depend on the initial momentum and the wave function Γf for the three-body depend on momentapandq. 
fσ (Γf(p,q)|V|Γi(P∗ )) . (5.14)
a
If there are no long range interactions (as e.g. Coulomb interaction) the wave functionscanbe approximatedbytheplanewaves.Theplanewavefor momentumQ canbe expanded in terms of partial waves and asymptotically can be expressed by the Bessel functionjL(QR), whereRisa rangeof interactionandLisan angular momentum[78]. Expanding the Bessel function into powers of QR leads to the first term proportional to QL. Therefore in good approximation each plane wave ΓPW can be approximated by 
ΓPW(Q) σ QL . (5.15) 
In the investigated dd → 3Henη0 reaction all outgoing particles are neutral. Therefore it is well justified to replace the final state wave function by product of two plane wave functions,whichcanbeexpressedbythefirsttermoftheexpansionoftheBessel function. Finallyitmaybeexpectedthatthe momentum dependenceofthe transition amplitudefor the three-body reaction can be well approximated by 
fσ pL23qL1 . (5.16) 
Therefore additional momentum dependence of amplitudes aΠ in equation 5.12 was takeninthe formpL23qL1. This leadstotheexplicit momentum dependenceofthe cross sectionof the formp2L23q2L1. 
It should be pointed out that this approach neglect other possible momentum dependencies of the transition amplitude. They may appear e.g. due to final state interaction or due to the presence of the resonances in anysubsystem of two final particles. The pions have usually small final state interaction with nucleons, therefore it may be neglected for 3He− η0 and n − η0. The final state interaction 3He− n can be strong. However, for dd → 3Henη0 reaction the isospin symmetry requires the 3He− n system to be in the state withthetotalisospinequalto one.Thefirst 4Heexcitedlevelswithisospinonewhichmay be responsible for final state interaction are at about 20 MeV and are quite broad [79]. Therefore no strong final state 3He− n interaction is expected. 
5.4PartialWave Expansionfora Three-Body Reaction 
assignment number  particle 1  particle 2  particle 3  subsystem 2 − 3  
1  3He  n  η0  n− η0  
2  n  η0  3He  η0− 3He  
3  η0  3He  n  3He− n  
Table 5.1: Possible independent assignments of the particles for dd→ 3Henη0 reaction. The permutation of the particles in subsystem 2-3 leads to the same description of the reaction. 



5.4.2 Cross Sectionfor dd → 3Henη0 Reaction 
Using MATHEMATICA [80] the partial wave expansion for dd → 3Henη0 reaction was performed limiting to the s-and p-wave in the exit channel. In the choice of independent variables described in section 5.1 as well in the partial wave expansion there is a freedom to assign particle denotedby1andtwo remaining particles forming subsystem 2-3.In the present case the three independent assignments shown inTab. 5.1 are possible.For each assignment it is possible to perform the partial wave expansion. It results in the partial wave amplitudes aΠ which are labelled by quantum numbers specific for a chosen assignment and therefore theyhavedifferent interpretation. However,it is always possible to express amplitudes aΠ foragiven assignmentby such amplitudes for other assignment with the proper re-coupling of spins and angular momenta. An example of the resulting amplitudes aΠ for assignment2fromTab. 5.1is presentedin AppendixA. 
With these amplitudes and taking into account explicitly their dependence on the momenta the four-fold differential cross section may be generally represented as: 
[
d4λ pq 
= A0+ A1q2+
2ηdM23dcosϕpdcosϕqdφ 32(2η)5sP∗(2sa + 1)(2sb+ 1)
a
1 ()1 ()
222
A3p+ A2q1+ 3cos2ϕq+ A4p1+ 3cos2ϕp+
44
]
A5pqcosϕpcos ϕq+ A6pqsinϕpsin ϕqcos φ . (5.17) 
The term withA0 corresponds to sS partialwaves, the terms withA1 andA2 correspond to pS partialwaves,the termswithA3andA4 correspondto sP partial waves and terms withA5 andA6 correspond to pS and sP waves interference. If there are no additional dependencies ofAi on the outgoing particles momenta the equation 5.17 should fully describe the cross section for a three-body reaction. 
The general form of the cross section 5.17 is the same for all possible assignments of the particlesgivenin table 5.1.However,thephysical meaningof amplitudesAi depend onthe specific assignment. Thereisno direct relation between amplitudesAifordifferent particle assignments similar to that which can be found for amplitudes aΠ. 
The cross section 5.17 can be integrated over some of the independent variables in order to obtain single differential cross section used in further analysis. The integration over angles may be performed analytically, while the integration over M23 have to be performed numerically. The corresponding integralsoverM23 willbe denotedby 
√
∫ ( s−M1)2 
IsS = pqdM23 , (5.18) 
(M2+M3)2 
√
∫ ( s−M1)2 
IpS = pq3dM23 , (5.19) 
(M2+M3)2 
√
∫ ( s−M1)2 
IsP = p3qdM23 , (5.20) 
(M2+M3)2 
√
∫ ( s−M1)2 2
IpS+sP = pq2dM23 , (5.21) (M2+M3)2 
The constantfactorin equation 5.17 willbe denotedby 
11
C= (5.22)
32(2η)5sP∗ (2sa + 1)(2sb+ 1)
a 
The integration over cos ϕp, cosϕq and φ gives theinvariant massM23 distribution 
dλ []
22
= 16η2CpqA0+ A1q+ A3p. (5.23)dM23 
The integrationoverM23, cos ϕq and φ gives the angular distribution for ϕp 
dλ [1 ()]
= 4ηCA0IsS + A1IpS + A3IsP + A41+ 3cos2ϕpIsP, (5.24)
2ηdcosϕp4
and similarly the integrationoverM23, cosϕp and φ gives the angular distribution for ϕq 
[()]
dλ 1 
= 4ηCA0IsS + A1IpS + A3IsP + A21+ 3cos2ϕqIpS. (5.25)
2ηdcosϕq4
Finally the distribution for φ is obtainedby integrationoverM23, cos ϕp and cos ϕq 
5.4PartialWave Expansionfora Three-Body Reaction 
[]
dλη2 
= 8ηCA0IsS + A1IpS + A3IsP + A6IpS+sP cos φ. (5.26)
dφ 16
In the single cross section formulae for four the independent variables the term with amplitude A5 does not appear. This term describes the correlation between ϕp and ϕq angles.Itis possibleto constructtheexpressionforthe cross section containing termA5 using different variables as e.g. cos ϕp− cos ϕq or cos ϕp+ cos ϕq. The differential cross section in these variables has a form: 
[
dλ 1 
= 4ηCA0IsS + A1IpS + A3IsP ∓ A5IpS+sP−
2ηd(cos ϕp± cos ϕq) 3
1()

A0IsS + A1IpS + A3IsP + A2IpS + A4IsP ± A5IpS+sP|cos ϕp− cos ϕq|+
2
3(A2IpS + A4IsP)|cos ϕp− cos ϕq|2−

4
()]
11
A2IpS + A4IsP ∓ A5IpS+sP|cos ϕp− cos ϕq|3. (5.27)
43



6 Analysis of the dd→
3
Henη0 

6.1 Run Summary 
The data presented in this work have been collected during ten days of production run in the endof 2007.In theexperiment the deuteron beam with kinetic energyEkin = 0.350 GeV (pbeam = 1.2GeV/c) impinging on a deuterium pellet target was used to initiate the reaction dd → 3Henη0.In theTab. 6.1 anoverviewofexperimental conditions during the data taking period is presented. In the left panel of Fig. 6.1 the beam intensity as mea
beam energy 0.350 GeV beam momentum 1.2 GeV/c pellet rate 4-8kHz events per pellet 2-4 deuterons in flat top 6· 109 typical luminosity ≈ 1030 cm−2s−1 integrated luminosity ≈ 350 nb−1 magnetic fieldof the solenoid 1T main trigger fwHea1|fwHeb1|Vfrha1|seln1 cycle length 54 s data taking within cycle 43 s dutyfactor 0.8 DAQlife time 88% effective data taking time 420000 s 
Table 6.1:List of the parameters valid for production run performed in November 2007. 
suredbya beam current transformer (BCT)over the timeincyclesis shown.A typical beam cycle obtained during the experiment had a length of 54 s. The visible drop in the intensity is due to the beam target interaction. The flat top period when the beam energy is kept constantandthedataareregisteredbytheDAQsystemwas approximately43s.The variation of the trigger rates over the cycle are illustrated in the right panel of Fig. 6.1. By measuring the ratio of issued and accepted trigger signals, the life time of the data acquisition system was estimated to be 88%. 
Figure 6.1: The beam intensity over the time in cycles (left). The ratio of accepted to the incoming triggers as a function of time in cycle (right). 
During the experiment all detector components of WASA-at-COSY were fully operating. The newly installedForwardWindow Counter showed anexcellent performance. All modules revealed very uniform (i.e. position independent) response, which was crucial for triggering. This made possible to achieve a high selectivity and efficient background suppression alreadyonthe triggerlevel.For triggeringondd → 3Henη0 a coincidence betweena high energy depositin both layersof theForwardWindow Counter, one or more low energy(E>20MeV) neutral clustersinthe calorimeterandaveto conditiononthe first layerof theForward Range Hodoscopewas used. The latter criterionwas imposed bysmall energy of helium ejectiles which are stopped latest in the third layer of the FTH. The threshold in FWC was chosen such that protons (mainly from deuteron break-up) were discriminated while tritons were still in trigger. With these settings the rate from main trigger occupied only up to 20% of theDAQcapabilities (i.e. less than 2000 trigger/s).Thus,itwassufficientroomto includeintothetriggeralsootherisospin conserving channels as dd → 3Hepη−, dd→ tpη0 and dd → tnη+ . 
To verify if any severe changes of the beam momentum have occurred over the accelerator cycle the behavior of the neutron missing mass in function of time has been checked. The distribution shown in left panel of Fig. 6.2 indicates that beam momentum waskept stableover the timeincycle. Thisis provenby the positionof the missing mass which is centered at the exact mass of the neutron. In order to monitor the performance of the central detector during the experiment, the position of invariant mass of two photons as a function of run number was plotted. The obtained dependence is shown in right panel of Fig. 6.2. The data points are positioned at the pion mass within the statistical fluctuations. 
6.2 Event Selection 
Figure 6.2:The behaviorofthe neutron missing mass reconstructed from 3Heand η0 over the timeincycle.The slicesofthetwo dimensional distribution were projected ontoYaxis and fitted withgauss function. The meanvalue obtained from the fit for each sliceis indicated by gray horizontal line (left). The invariant mass of two photons as a function of run number (right). 

6.2 Event Selection 
This chapter contains the preselection criteria that were designed to preferentially select dd → 3Henη0eventsover background processes.In thisworktwostagesofevent selection can be distinguished. First is based on hardware triggers while the second makes use of characteristic signatureof thephysics process we are interested in. 
The main trigger fwHea1|fwHeb1|Vfrha1|seln1 set up during the experiment was based on a logic AND of the following trigger patterns: 
fwHea1|fwHeb1 -a high threshold on signals from geometrically overlapping elementsin both layersofForwardWindow Counter detector; 
Vfrha1 -veto on anyhitsin theForwardVeto Hodoscope; 
seln1 -one or more neutral clusters in the calorimeter. 
The data picked up by this trigger still consists of a large collection of unwanted events. In order to keep only events of interest and to reduce computation time a preselection has been performed. The data were presorted by imposing set of relatively loose requirements listed inTab. 6.2. The most influential cut applied on light output in FWC1 detector reduced the initial amount of data by approximately 80%. The optimal value for this condition has been found by varying the level of cut and checking how the invariant mass of the two photons is influenced. The multiplicity condition set for theForward Detector means that we requireexactly one charged particle track.In order to be able to reconstruct the angles of particles detected in FD the demand on number of clusters in FPC detector is imposed. Additional selection criteria are based on the information from Central Detector.To selectevent candidates for the decayof η0 → δδ only those events are accepted which contain two neutral clusters in the calorimeter. The MC simulations have shown that photons originating from dd → 3Henη0 have energy above 20 MeV and opening angle larger than 30◦. Those restrictions were set during the preselection to suppress signals not stemming from the channel of interest. The time difference between the detection of products of pion decay has to be smaller than 40 ns (see Fig. 6.3). It is noteworthy that all conditions applied for the Central 
Forward Detector 
Track multiplicity =1 Cluster multiplicity in FPC ≥ 2 Light output in FWC1 ≥ 3800 ch The reference signals from the LPS ON 
Central Detector 
Cluster multiplicityin PSB =0 Cluster multiplicityinSE =2 Time difference between clusters in SE ≤ 40 ns Opening angle between clusters in SE ≥ 30◦ Cluster energy in SE ≥ 20 MeV The reference signals from the LPS ON 
Table 6.2:The criteria of events preselection. 
and the Forward detectors have to be fulfilled in coincidence. To ensure that 3He and photons originatefromthe sameevent,acutonthetime correlationshowninFig.6.3has been performed. In order to correct for thegain drifts, information from the light pulser systemwas includedduringthe preselection process.To minimizeedgeeffectsduetothe geometry of the detector, an additional constraint was applied to the scattering angles of the reconstructed tracks.In theForward Detector tracks with scattering angles from3◦ to 18◦ were considered while in the Central Detector only photons with ϕ angles limited to the range between 20◦ -169◦ have been accepted. The validation of the trigger condition was achieved by a request that each event must have energy deposit in both layers of the FWC detector and in the first layer of the FTH as well as one or more neutral clusters in calorimeter. Moreover, there should be no hit in FVH. 
6.3 Kinematic Fit 
For the selected events the four-vectors of the reconstructed particles were used to calculate the missing mass according to formula: 
√ 
MM(3Heη0)= Pb+ Pdt − P3He − Pη0 . (6.1)
d 
The right panel of Fig. 6.3 presents the missing mass,built of helium and two photons obtained during presorting. The distribution shows the expected peak at the mass of the neutron. 
Utilizing presented cuts, a relatively pure sample of dd → 3Henη0 candidates have been obtained. This preselected data were refinedbyimproving the energy calibration and applying the kinematic fit. The precise calibration guarantees the separation of 3He from lighter particles what yields to complete background elimination. On this level of analysis energy resolution and thresholds for individual detectors are computed. These values are used to smear Monte Carlo data in order to reproduce the experimental resolution. 
Figure 6.3: The correlation between mean time of two neutral cluster in calorimeter and time detectionof 3He measuredby FTH1 detector (left).Timedifference betweentwo neutral tracks in Central Detector. The tracks are accepted within a time window denoted byred dashed lines (middle). Missing mass reconstructed from 3He detectedintheForward Detector and two photons measured in the Central Detector (right). 


6.3 Kinematic Fit 
The kinematic fit is a least-square fit with constraints based on the Lagrange multipliers technique.The detailed descriptionofthe method canbe foundin Ref.[81,82].The basic idea behind the use of kinematic fit is to improve the resolution of measured kinematical variables. All quantities we measure, e.g. energy or angles, have uncertainties which can manifests itself in the population of kinematically forbidden regions of the phase space. The purpose of kinematic fit procedure is to vary measured observables within the uncertainty until certain kinematic constraints are fulfilled. Limits for variation of measured parameters are determined by the experimental resolution. The constraints are based on physical propertiesof studied processesand are usedto testhypothesesforgiven reaction. As a result of the kinematic fit the corrected values of energies and angles as well as a value of the correspondingΨ2 is returned. The latter is used to validate fit results. If the properhypothesisis selected and all resolutions used as input for the kinematic fit are Gaussian shaped then the Ψ2 probability function withNdegreesof freedom defined as: 
∫ θ
1 − 11
P(Ψ2|N)= √ e2tt2N−1dt (6.2) 2N (1N) Ψ2 
2
shouldhave flat distribution between0and1. 
6.3.1 Fit Constraints 
The kinematic fit for the reaction dd → 3Henη0 makes useoffiveconstraints.Fourof them are related to the overall energy and momentum conservation. Additionally the demand on mass of η0hasbeen applied. Afterthe definitionof reactionhypothesisthe information about measured and unmeasured parameters are passed to the fitting routine. In our case we measure3×3parameters namely the energy,polar and azimuthal angles of helium and two photons. The neutron is undetected thus we have3unmeasured parameters. Having that information the number of degrees of freedom can be calculated as: 
N= 4+ nc − u (6.3) 
where 4 stands for the four conditions due to the four-momentum conservation, nc is the number of additional constraints and u denotes the number of unmeasured variables. Inserting appropriatevaluestotheEq.2.2 resultsinN =2. 

6.3.2 ErrorParametrization 
Proper parametrization of uncertainties of the measured physical quantities is one of the most critical part when preparing input for kinematic fit. This is because the fitting routine minimises the Ψ2 using the constraints supplied by varying the experimentally measured parameters. The error description can be determined from MC simulation, by comparing the reconstructed observables to the initial values given by the event generator. This procedure can be applied only when experimental data are well reproduced by simulation. Such a prerequisite is realized by smearing of the simulated observables until they match experimental resolutions. The errors for the kinetic energies, polar and azimuthal angles are computed as the absolute difference of the reconstructed and true values from generator. 
6.3 Kinematic Fit 
Figure 6.4: The error parametrization of the reconstructed variables ϕ,φ andEkin. 
LEkin = Erec − Egen , 
Lϕ = ϕrec − ϕgen, 
Lφ = φrec − φgen . 
The distributions are fitted with a Gaussian function and the obtained width λ is treated as an error of given variable. Systematic studies of the error distributions have been performed by checking if λ(LEkin), λ(Lϕ) or λ(Lφ) depend on kinetic energy and scattering angle.For that purposetwo dimensional parametrizationwas applied.Thekinetic energy range 130 -220 MeV for 3He was divided into seven slices of 12 MeV each and the scattering angle range 15◦ wasdivided into fifteen intervalsof1◦ size.In thatway whole two dimensional space can be separated into close to one hundred cells.For each cell the errors ofEkin, ϕ and φ are computed. The variation of errors as function of kineticenergyand scatteringanglefor 3HearepresentedinupperrowinFig.6.4.Here,the left plot shows that errors of polar angles of 3He decrease exponentially with the kinetic energy. In the middle picture of the same row, the resolution of the 3He azimuthal angle is shown. As can be seen the error is independent of the energy and rises approximately linearly with increasing polar angle. The last plot in upper row illustrates the behavior of the error of kinetic energy. One can notice there is almost no dependence on scattering angle and the error is dominated by changes in kinetic energy itself. The resolution gets worseforhigh energetic 3He.Worseningof resolutioncanbe attributedto smaller energy deposit of 3He in the detector, what affects the energy reconstruction procedure. In the lower row of Fig. 6.4 similar considerations were conducted for photons. Left plot shows that the error of polar angle can depend on the part of the calorimeter where photons were detected.For the forward and backward end-caps where the sizes of crystals are smaller than in the central part the errors are larger. This situation holds also for description of the error of azimuthal angle. In the most right plot the errors of kinetic energy are displayed. Theydo not depend on scattering angle and get larger with the increase of the energy. 
For the parametrization of errors of the kinematic observables which reveal only dependence on one variable, an analytic functions were used. In case of λ(LEkin) for 3He and λ(Lϕ) for photons, where variation of errors behave in more complicated way two dimensional tables were prepared.Forgivenvalueof kinetic energyand scatteringangle we can do a table lookup and extract corresponding error. 
Before applying the error parametrization some additional studies of systematic deviations have been performed. One noteworthy example is the azimuthal angle of 3He measuredin theForward Detector. The procedure which reconstructs the angles assumes that particlesinFD followsa straight track from thevertex,butin reality partof the track lies in the magnetic field of the solenoid. This results in curved trajectories in the region closetovertex.To compensatefor thiseffectthe corrections accordingto[83]have been applied: 
0.3· L· z· B
Lφ = (6.4)
P· cosϕ wherezis the chargeof the particlein unitsof electron charge,Bis the magnetic field in Tesla, L is the longitudinal component of the particle trajectory in meters, P is the momentum of the particle in GeV/c and ϕ is the polar angle. 

6.3.3 Probability Distribution 
The measured parameters of two photons and helium which are given as an input to the kinematic fit are modified within the error limits of that parameter. These modified values can be interpreted in terms of probabilities and Ψ2 distribution. 
In Fig. 6.5 the Ψ2 and the corresponding probability distribution for all fitted events in the data as well as Monte Carlo simulation are shown. The distributions have been obtained for the dd → 3Henη0hypothesis in the kinematic fit. Events which have highΨ2 values are located in the region close to zero in the probability distribution. Those events are not of interest because theymost likely do not well satisfy the constraints applied to the fitting procedure. On this level of analysis the data are almost background free (see the right panel of Fig. 6.3) so the rise in the probability distribution for large Ψ2 values can notbe attributedto background contributions. Since sucha trendis also visibleinMC 
6.3 Kinematic Fit 
Figure 6.5: Ψ2 distribution (left) and the probability distribution (right) obtained from the kinematic fit. The gray histograms represent the data while in red the MC distributions are shown. 
it indicates this can be the issue of the error estimation. Least square fit needs Gaussian shaped resolutions as input. Of course, in real life measurement errors rarely follow this distributionexactly and usuallyhave some significant non-Gaussian tails.Toverify the correctness of the obtained errors, pull distributions were constructed. The pull value is defined as the difference between measured πrec and fitted πfit values obtained by the kinematic fit, normalized by the quadratic error difference. 
πrec − πfit
Pull = (6.5)
λ2 − λ2 
rec fit 
The minus sign in the denominator comes from the correlation between the measured and fitted observables. If the errors are correctly estimated and there are no systematic shifts, then the pull quantity will be distributed like a Gaussian centered at zero with λ equal to 
1.Theexamplesofpull distributionsforexperimental data areshowninFig. 6.6.One can observe that pulls for energy approximately obeya Gaussian distribution. In case of pulls constructed for polar angles, deviation from a Gaussian shape is noticeable. The discrepancyin the width from unity demonstrates the underestimation of the measurement error. On the other hand the reason for sucha behavior can also liein thefact that kinematic fit improves mainly the energy resolution, but has almost no effect on the angular observables. This leads to the situation where in the denominator two comparable quantities appear what can result in some numerical problems. Under investigation was also influenceof thefact that neutronwas treatedin the kinematic fit as an unmeasured particle. 
Figure 6.6: Example pull distributions of kinetic energy and polar angle for photons and helium. Plots were madeforexperimental data. Pullshave been fitted with Gaussian function (red lines). The results of the fit indicate there is no systematic shift between the reconstructed and true values, however errors for some variables after the fit are underestimated. 
For such a case it is important to provide reasonable starting values for the unmeasured variables. In this approach the kinematic constraints are linearized at each iteration step and Newton’s method is used to find the minimum. Thus to avoid a risks of non convergent solution or finding some other local minimum the starting values for neutron were extracted from the missing four momentum vector. 
For further analysis events with a probability value below 0.1 have been excluded. Those events do not well satisfy the constraints applied in kinematic fit, either because theywere wrongly reconstructed or theystem from background. The criteria used to select the limit for the probability cut are presented in section 7.2. 
6.4 Comparison of Simulation and Experimental Data 



6.4 	Comparison of Simulation and Experimental Data 
Before comparing results of simulations with the experimental data the symmetrization procedure has been applied to the data. This step was performed in order to reproduce events which were lost due to the acceptance limitations. When looking on angular distributions of 3He in center-of-mass system (see left panel of Fig. 6.7) one can observe the lack of events close to cosϕHe = 1 and for cosϕHe < 0. The first appears because of geometrical boundaries of FD detector while the latter is due to the energy threshold. The 
Figure 6.7: The distribution of scattering angle in CM system plotted for 3He. All events located above cosϕ > 0(denoted by dashed, blue line) undergo of symmetrization (left). The same plot after symmetrization procedure (right). 
symmetrization was realized in the following manner. First, 3He four momentum vector is calculated in the global center of mass system and only events which fulfill condition cosϕHe > 0are considered.Ineachsuchaneventfor helium, neutronandpionthePz componentof momentumvectoris replacedby −Pz. The symmetrized angular distribution of 3He in center-of-mass system is shown in right panel of Fig. 6.7. 
After applying all the selection criteria described in section 6.2, the experimental data can be presented in the Jacobi-coordinates system (see Section 5.1). The corresponding distributions for three independent assignments (seeTab. 5.1) of the particles in dd → 3Henη0 reaction are displayedin Fig. 6.8 -Fig. 6.10.To describe theexperimental distributions shown in Fig. 6.8 -Fig. 6.10 two phenomenological models presented in Section5wereemployed.Letusfirst concentrateonquasi-free reactionmodel.Inthisapproach, reaction dd → 3Henη0 may proceed with a spectator neutron stemming from the deuteron beam or the deuteron target.For the latter case we have constructed generator (description in 4.1.2 and 5.3) in which any produced spectrum can be absolutely normalized. Such a treatment allows to compare normalized data directly with the generator output passed through the WMC (see 4.1.4) and judge about the quasi-free contribution. For the quasi-free reaction, sample of one million events have been prepared. Half of the events were generated according to the case where neutron spectator comes from the Figure6.8: Experimentaldifferential distributions(no acceptance corrections)invariables (see Section 5.1) chosen for description of 3-body reaction dd → 3Henη0. The plots are made for “assignment number“ equals to two (seeTab. 5.1). 
6.4 Comparison of Simulation and Experimental Data 
Figure6.9: Experimentaldifferential distributions(no acceptance corrections)invariables (see Section 5.1) chosen for description of 3-body reaction dd → 3Henη0. The plots are made for “assignment number“ equals to three (seeTab. 5.1). 
Figure 6.10: Experimental differential distributions (no acceptance corrections) in variables (see Section 5.1) chosen for description of 3-body reaction dd → 3Henη0. The plots are made for “assignment number“ equals to one (seeTab. 5.1). 
6.4 Comparison of Simulation and Experimental Data 
Figure 6.11: In the left panel the momentum distribution of the neutron stemming from dd → 3Henη0 for data (black line) and quasi-free model (red line) is shown. In the right panel the scattering angle of pion in center of mass of subsystem 3He− η0 is presented. The distribution containsonlyeventsforwhichthe neutron momentumis smallerthen90 MeV (the cut is indicated by blue dashed line in left panel). The data and model calculations are not corrected for acceptance. 
target and the other half corresponds to the situation where the neutron spectator originates from the beam. The obtained sample was used as an input for MC simulation. After a full detector simulation has been conducted, MC sample was analyzed with the same conditions as used in the analysis of experimental data. Before experimental distributions were compared to the corresponding MC spectra, the absolute normalization had been applied.The quasi-free reactionwas normalizedusingthe cross section presentedin Section 
4.1.2. In order to normalize data, the integrated luminosity calculated in section 6.7 was exploited. 
To verify the agreement between the absolutely normalised data and the MC simulations the distribution of the neutron momentum has been checked. In the left panel of Fig. 6.11differential cross-sectionasa functionof neutron momentumisshown.One can notice that in the region of neutron momentum below 90 MeV the data are dominated by the quasi-free reaction. It is clearly visible when comparing the scattering angle of the pion in center-of-mass of subsystem 3He − η0. This distribution, obtained after applying cut (indicated by blue dashed line) on neutron momentum is presented in right panelof Fig. 6.11. Theexperimental shapeisfairly well reproducedby quasi-free model with a spectator neutron stemming from the deuteron target. This result allows to fix the contribution of quasi-free process in the data. As Monte Carlo studies have shown, the contribution of quasi-free reaction with neutron spectator originating from the deuteron beamisnegligible. Thisis causedby 3He’s energydueto which mostof them are stopped before first layer of FWC detector, therefore those events can not be reconstructed. 
The second step of comparison of MC simulation and data is the inclusion of model based on partial wave expansion for three-body reaction. As described in Section 5.4.2 in this approach the total cross section can be expressed by seven terms corresponding to different partialwaves. Those termsin formof matrix elements |T|2 (definition in Section 4.1.3) were usedasa weights duringtheevent generation. Basedon thatseven samplesof 106 events have been simulated. 
The comparisonoftheexperimental dataand Monte Carlo simulations representedin the Jacobi-coordinates was done by fitting the sum of the simulated distributions to the experimental distributions.The ideal situationwouldbetofitthe datain four-dimensional space definedby independentvariablesM23,cosϕq,cosϕp and φ. However, this solution is not feasible due to the statistical limitations. If each fitted histogram would be divided into ten bins then our four-dimensional space would contain 104 cells. The total statistics achieved in the analysis yields to about 170000 events what gives less then twenty events per cell. Therefore, for fitting the projections of four-dimensional space were used. BesidesM23,cosϕq,cosϕp,φ histograms two additional spectra have been included for fitting: cos ϕp− cos ϕq and cos ϕp+ cos ϕq.For the fit the following Ψ2 function was defined: 
Fmc = 	A0H0+(A1+ A2)H1+(A1− A2/2)H2+(A3+ A4)H3 +(A3− A4/2)H4+ A5H5+ A6H6+ A7H7 (6.6) 
nbin 
(Fexp − Fmc)2 
Ψ2 = ∝ 	(6.7)
λ2 + λ2 
k=1 exp mc 
Theexpression forFmc is transformed formula 5.17 derived from partialwave decomposition with added term A7H7 corresponding to quasi-free process. In the expression Eq. 6.6 the fit coefficients appear asA0,..,A7, whileH0,..,H7 are different MC contributions as definedinTab. 4.1.To obtain the fit parametersA0,..,A7 which would minimize thedeviationsof theMC predictions from theexperimental points representedbyFexp, the Ψ2 minimization method was used. It is noteworthythat during the minimization the parameter A7 which corresponds to quasi-free contribution was fixed and equal to one. The results of the fit for three different assignments (see Tab. 5.1) of the particles are presented in Fig. 6.12 -Fig. 6.14. From the comparison we can conclude that the model used for Monte Carlo simulations describes the experimental data very well. Of course the final conclusion canbedrawn afterthedifferential cross-sectionforexperimental data are acceptance corrected. Then the resulting distributions can be compared directly to the theoretical predictions. 

6.5 Reconstruction Efficiency 
Before we proceed with the evaluation of corrections for the detector acceptance some studies related to reconstruction efficiencywill be shown. During the estimation of overall reconstructionefficiencyfollowingfactors shouldbe considered: the geometric accep
6.5 Reconstruction Efficiency 
Figure 6.12: The distributions of independent variables calculated according to the quasi-free and partial wave decomposition of dd → 3Henη0 reaction. The six spectra compares the experimental data (red dots) with the results from the MC (black solid line). The data are absolutely normalisedbut not corrected for acceptance, similarly as MC results. The MC curve corresponds to the sum of quasi-free and seven terms from partial wave decomposition. The plots are made for “assignment number“ equals to two (seeTab. 5.1). 
Figure 6.13: The distributions of independent variables calculated according to the quasi-free and partial wave decomposition of dd → 3Henη0 reaction. The six spectra compares the experimental data (red dots) with the results from the MC (black solid line). The data are absolutely normalisedbut not corrected for acceptance, similarly as MC results. The MC curve corresponds to the sum of quasi-free and seven terms from partial wave decomposition.Theplotsaremadefor “assignment number“equalstothree(seeTab.5.1). 
6.5 Reconstruction Efficiency 
Figure 6.14: The distributions of independent variables calculated according to the quasi-free and partial wave decomposition of dd → 3Henη0 reaction. The six spectra compares the experimental data (red dots) with the results from the MC (black solid line). The data are absolutely normalisedbut not corrected for acceptance, similarly as MC results. The MC curve corresponds to the sum of quasi-free and seven terms from partial wave decomposition. The plots are made for “assignment number“ equals to one (seeTab. 5.1). 
tance,the detectionefficiencyandtheefficiencyof algorithms usedforevents reconstruction. The influence of those effects on data and MC sample can be tested by comparison the number of events on different stages of the analysis. If MC simulations reproduce experimental distributions then anycut applied during the analysis chain should affect data and MC in comparable way. This is valid under assumption that real data and simulations are analyzed with exactly the same software program. 
To determine purely geometric acceptance for the dd→ 3Henη0 it is sufficient to use information fromevent generator after imposing restriction on scattering angles according to the geometry of detection setup. In Fig. 6.15 the kinematical distribution for helium and photons are displayed, wherethe blue lines indicatethe acceptanceoftheWASA detector. The geometric acceptance calculated under assumption that 3He and two photons are detectedin coincidenceinsidethe sensitiverangesoftheWASAsystemisfoundtobe82%. In ordertoevaluatethe reconstructionefficiency,theWASA Monte Carlo simulationwas 
Figure 6.15: The correlation between the kinetic energy and the scattering angle for 3He measured in FD detector (left), and photons detected in CD detector (right). The blue lines indicate the geometrical acceptanceof theWASA detector. 
used.Asampleof7 · 106 dd → 3Henη0 events according to the approach based on partial wave expansion have been generated. Additionally the MC input was supplemented by 106 events for the quasi-free reaction. Both samples have been processed by the MC simulationandtheoutputwas analyzedwiththe same programastheexperimentaldata.The reconstruction efficiency was computed as the ratio of events which remain after specific cutsin the analysis and the total numberof generatedevents.InTable 6.3 theefficiencies of the selected cuts applied for data and MC simulation aregathered. The notations used forcut descriptionisfollowing:C1 -numberofeventsafter selectionof candidatesforthe dd → 3Henη0, C2 -for each event the kinetic energy of 3He has to be reconstructed, the missing mass of neutron should fulfill condition:0.925 ≤ mn ≤ 0.950 [GeV], C3 -only events with cut on probability distributionP(Ψ2 ,N) > 0.1are accepted. As can be seen fromTable6.3thelargestdropofefficiencyisduetotheevent candidate selection. After this step data are expected to be background free, what is visible in Fig. 6.3. Therefore, 
6.6 Acceptance Correction 
Cut  MC Phase Space  MC Quasi− free  MC Sum  Data  
- 7· 106  106  8· 106  
C1  1726284  147722  1874006  288135  
C2  1323112  121635  1444747 (77%)  221473 (77%)  
C3  1049958  97068  1147026 (79%)  169824 (77%)  
Table 6.3: The overall reconstruction efficiencyfor the reaction dd→ 3Henη0. In fourth and fifth column the efficiencies of the cuts applied for combined Monte Carlo samples and data are shown. The efficiencies for the subsequent cuts are computed relative to the previous one. 


any further selection criteria (kinematic fit) which aimed mainly for increasing resolution should affect data and MC simulation in roughly the same way. This is reflected by the resultsgivenin fourthand fifth columnofTable6.3.Thefactthatcuton probability distribution of the kinematic fitP(Ψ2 ,N) > 0.1 rejects instead of 10% almost 23% can be attributed to non-Gaussian error distribution. This makes that probability distribution isnotflatbutpeakedtowardsthe smallvaluesofP(Ψ2 ,N). The influence of the overall reconstruction efficiency on the different distributions will be discussed in Section 7.2. 

6.6 Acceptance Correction 
Before anyphysics conclusion in the interpretation of the data can be drawn, all experimental distributions have to be corrected for the overall efficiency. If the data covered full acceptance range than anymodel would be applicable to perform acceptance and inefficiencies corrections. In our experiment that was not the case. Therefore, it was crucial to provide model which describes experimental data as good as possible. This should also help to minimize effects like finite resolution or limited acceptance which causes that event migration from the histogram bins they are supposed to populate according to their original kinematics to the neighboring ones. 
In principle, two methods can be applied to perform acceptance corrections. First is based on multidimensional corrections. In this case acceptance is expressed as a function of independentvariables which describe the studied reaction unambiguously.For 3-body unpolarized reaction four independentvariables canbe identifiedto correct datainamodel independentway.In this approach, eachexperimentaleventis weightedby4-dimensional function. There were attempts to employthe method for this work, however, due to the statistics limitations and acceptance holes a multidimensional corrections did not provide satisfying results. Instead, one-dimensional corrections have been used. The application of this type of corrections requires model that resembles the data. As can be seen from Fig. 6.12 -Fig. 6.14 the model presented in Section 5 meets those requirements. The 
Figure 6.16:Overallefficiencycorrectionsfor independentvariables cosϕq,cosϕp,M23,φ. The acceptance correction functions were obtained using mixture of quasi-free process and model based on partial wave expansion for three-body reaction (see Section 6.4). 
acceptance correctionfactors for distributions of interest were prepared in form of onedimensional histograms according to following formula: 
77 Feff−acc = ∝ AkRk/ ∝ AkHk (6.8) k=0k=0 
whereHk denotesthe histograms obtainedfromthe generatorwhileRk referstothe reconstructed histograms from MC simulations. The number of terms in the sum corresponds to the number of MC contributions fitted to the data (see Eq. 6.6) withAk being the fit coefficients. The acceptance correction distributions derived for exemplary differential cross-sections are shown in Fig. 6.16. 
6.7 Luminosity Determination 


6.7 Luminosity Determination 
In order to calculate the cross section for dd → 3Henη0 the luminosity has to be determined. The relation between luminosityLand the cross section λ can be written as: 
Nexp 
L= (6.9)
λβ 
whereNexp isthe numberofdd → 3Hen events measured in the experiment, β is the efficiencyand acceptance correction function and λ is the cross section of reference reaction measured at the same beam energy as dd → 3Henη0. 
In this work to determine luminosity the measurement of dd → 3Hen has been performed. Theevents were collected with the minimum bias trigger (prescaled withafactor of 100) which requires exactly one hit in first plane of FRH detector. In order to extractonlyeventsof interestasetof selection criteriahasbeen applied.To startwith,one chargedtrackinFDis required,thatgivesasignalintheFPCand reachestheFRH1plane. The latter condition is exploited for comparison of the energy deposited by different particles. Since 3He stemming from dd → 3Hen are mostly stopped in FRH1 they form distinct peak which is clearly separated from the protons and deuterons stemming from break-up reactions. Applying cut on energy deposit LEFRH1 ≥ 100MeV we were able to reduce background contribution to negligible level. It is also noteworthythat events were 
Figure 6.17: The correlation between energy deposited in FWC2 versus FTH1 for 3He stemming from binary reaction (left).In the right panel the 3He missing mass distribution is shown. Pronounced peak at the mass of the neutron can be observed.AGaussian fit is representedby red line. 
selected without anyrequirement on reconstruction of the neutron. The quality of background suppression is demonstrated in Fig. 6.17. Left panel shows the correlation between energylossof 3HeinfirstlayeroftheForwardWindow Counterversus energylossinthe first layer of theForwardTrigger Hodoscope. Right panel displays helium missing mass 
calculated according to the formula: 

√ 
MM(3He)= Pb+ Pt (6.10)
dd− P3He 
The resulting missing mass distributionrevealsa background freepeakatthe massofthe 
Figure 6.18: In left panel dependencedλ/dΔcm(cosϕcm) for selected momentum (1.387 GeV/c) is fitted with function 6.12. Fit function is shown in red. The blue bars correspond to confidence intervals calculated for fitted function. Right panel shows dλ/dΔcm asa function of momentum. The dependence was fitted by a polynomial of second order. The value of cross section for our beam momentum (1.2 GeV/c) is displayed in green. The width of the bar corresponds to the error calculated as confidence interval. 
neutron. This spectrum is fitted with a Gaussian function and from the fit parameters the total number Nexp of detected dd → 3Heneventswasextracted.To determine the cross section we used the data presented in Ref. [84]. Authors measured the dd → tp reaction for several beam momenta between 1.09 GeV/c -1.78 GeV/c and dd → 3Hen for beam momenta in the range of 1.1 GeV/c -2.5 GeV/c. Moreover, theyshowed that cross sections for both channels measured at the same beam momentum (1.65 GeV/c) are almost identical. Based on that assumption we employed the dd → tp reaction to perform interpolation and calculate λ for our beam momentum.Wehave chosen this channel insteadof dd → 3Hen because it delivers more points for interpolation in the region close to the 1.2 GeV/c.The tritonscoverthe c.m. angular range between0and65◦ what correspondsto0◦ -22◦ interval in the laboratory system. Those boundaries coincide with the region where we measure 3He from binary reaction. In order to perform interpolation, data for beam momenta of 1.109 GeV/c, 1.387 GeV/c, 1.493 GeV/c and 1.651 GeV/c were exploited. 

In the next step for each momentum,dλ/dΔcm is plotted as a function of cosϕcm and fitted (see left panel of Fig. 6.18) with the empirical parametrization [84]: 
3 
Ωicos ϕcm
f(ϕcm)= ∝Πie(6.12) i=1 
Using the Πi and Ωi constants from the fit we can nowchoose ϕ1,...,ϕn points and calculate the correspondingvaluesof the functionf(ϕ1),...,f(ϕn), the errors λ(f(ϕ1)),...,λ(f(ϕn)) are computed as the confidence intervals for fitted function. After that for each ϕi the momentum dependence of the differential cross section was fitted and interpolated to pbeam = 1.2 GeV/c. Here, we tried to fit either polynomial of second order using three points for interpolation (see right panel of Fig.6.19) or linear function using two points in the neighborhood of interpolated value. The results of parametrisation, depending on whichtypeof functionwasused,are presentedinrightpanelofFig.6.19.The acceptance corrections were determined using a GEANT-based simulation program for which the parametrised cross section was used as an input distribution. In left panel of Fig. 6.19 the output from the simulation is compared to the data. Based on that comparison we select the range cosϕcm ∈ (0.76 − 0.96) and for this interval, the integrated total cross section was computed using formula: 
∫ 2η ∫ 0.96 
dλ
λ = dχ dcosϕcm . (6.13) 
00.76 dΔcm 
As a result the following values for the integrated cross section of dd → 3Hen were obtained: 
λ1 = 49.8± 0.33 µb 
λ2 = 53.6± 0.17 µb (6.14) 
where indices ’1’ and ’2’ refer to the type of function (polynomial or linear) used for fitting distribution presented in right panel of Fig. 6.19. The uncertainties of cross sections were computed numerically during the integration of distribution shown in left panel of Fig. 6.19. In this spectrum the information about the error propagation from all steps of parametrization method is included. 
Taking into accountNexp = 174062 of the detected (in the range cosϕcm ∈ (0.76 − 0.96))dd→ 3Heneventsand prescalingfactorof100,we obtained following resultfor integrated luminosity: 
Lint =(350.1± 1.8stat. ± 24.8sys.) nb−1 (6.15) 
Inthe calculationofLintforthe cross sectionthevalueof λ1 has been used. The systematic error is connected to the difference between cross sections calculated using either linear or polynomial parametrization.The statistical erroris relatedtothe numberof detected 3He. The total uncertainty of the normalisation can be obtained by adding the uncertainties of Ref. [84] data which is 7% in the absolute normalisation. 


7 Results of the analysis 
Inthischaptertheresultsofthecomparisonofthe collecteddatawithphysicalmodelsare discussed. The total cross section as well as possible reaction mechanisms for the reaction dd → 3Henη0 investigated at beam momentum of 1.2 GeV/c are presented. 
7.1 TheTotalCross Sectionof dd → 3Henη0 
Asalready mentionedinpreviouschapterthetotal cross sectionforgiven reactioncanbe calculated according to the following relation: 
Nexp 
λ = (7.1)β Lint 
withLint being the integrated luminosity, β -theoverallefficiencyandNexp the total numberof reconstructedevents. After applyingall selection criteria describedin chapter6and takinginto accountthe symmetrizationofeventsas presentedinbeginningofsection6.4, wehave obtainedNexp = 331206eventsof interest. This information combined withthe valueofthe luminositygivenbyEq.6.15and acceptance correctionfactorof24.8%,leads to the following result for the total cross section: 
λtot =(3.81± 0.01stat. ± 0.42sys.) µb (7.2) 
In the next section the main sources of systematic uncertainties are discussed. 

7.2 Systematic Uncertainties 
Along with the statistical fluctuations associated with a sample of limited size there are a number of effects that could lead to systematic uncertainties on the final result of the total and differential cross section. In this work, one of the dominant source of systematic error originates from the luminosity evaluation. The error estimated from the luminosity estimation procedure, was found to be ∼ 7%. Onto this result, additional 7% from the overall uncertainty in the absolute normalization of data [84] used for luminosity evaluation should be propagated. 
Another important component of systematic error is associated with the cut on the probability distribution.Inthe kinematicfit procedure,hypothesis thatevent canbe considered as dd → 3Henη0 was confirmed by requiring that the P(Ψ2|N) distribution is flat. In Section 6.3.3 the probability distribution was considered flat for probabilities P(Ψ2|N) ≥ 0.1. In order to estimate howthis cut affects the results, different regions of the probability distribution were tested.InFig.7.1different conditions appliedtotheP(Ψ2|N) 
Figure 7.1: The probability distribution obtained from the analysis of the selected data. The gray histograms represent the data while in red the MC distribution is shown. The black arrows show the different regions ofP(Ψ2|N) distribution which were used for the systematic error studies. 
distribution are indicatedbyblack arrows.For each markedregion,the analysishave been repeated and the number of reconstructed events in the experiment, overall efficiencyand the total cross section were determined. The corresponding results aregathered inTable 
7.1. The cross section values obtained for different selection region were compared to the nominal value calculated for confidence level ≥ 10%. From the comparison the contribution of uncertainty was estimated to be below 5%. This number corresponds to the cut ≥ 80% on probability distribution for which the deviation of cross section with respect to the reference value P(Ψ2|N) ≥ 0.1is maximal. 
Systematical errors due to the acceptance corrections were checked by varying the startingvalues for the fit parametersA0,..,A7 in relation 6.6. Subsequently,the acceptance corrections maps (see Fig. 6.16) for slightly different fit coefficients were compared. The deviations turned outtobe smaller then0.1%. 
7.3 Differential Cross Section Distributions 
P(Ψ2|N) ≥ X [%]  Nrec  Overall Eff.[%]  λtot [µb]  
0.05  355883  26.6  3.82 ± 0.27  
0.1  331206  24.8  3.81 ± 0.27  
0.2  285876  16.2  3.87 ± 0.28  
0.35  225836  16.4  3.91 ± 0.28  
0.5  170118  12.3  3.94 ± 0.28  
0.65  114410  8  3.96 ± 0.28  
0.8  65662  4.68  4.0 ± 0.28  
0.1 -0.8  220214  16.6  3.8 ± 0.27  
Table 7.1: The number of reconstructedevents, the overall efficiencyand the total cross section calculated forvarious cuts (see Fig. 7.1) on the probability distribution. The errors given for cross sections include only uncertainties from luminosity estimation. 


Additionally,seriesof cross-checksregardingcutonenergydepositof 3Hehavebeen performed. This condition was very important because it allowed to completely eliminate background.Variation of the limits for this cut did not show anysignificant influence on thevalueof the cross section.Taking into account all aforementioned sourcesof uncertainties the the total systematic error for the cross section was estimated to be 11%. 

7.3 Differential Cross Section Distributions 
In order to obtain the final differential cross section, the distributions shown in Fig. 6.12 Fig. 6.14 have been corrected for the acceptance. These distributions were compared with a combination of two purely phenomenological models, namely the quasi-free approach and the model based on partial wave expansion for three-body reaction. As discussed in 
6.4 the absolute value of the quasi-free contribution was fixed by use of the parametrized cross section for the experimental data for pd →3 Heη0 reaction. The integrated quasi-free (with neutron being spectator either from target or beam) cross section contributing to the investigated reaction amount to2×580 nb, what corresponds to 30% of the total cross section for dd →3 Henη0 reaction. The rest of the observed cross section should be attributed to some other processes in which all entrance channel nucleons and all exit channel particles are involved. The properties of these other processes were investigated using partial wave expansion and fitting various contributions to the experimental differFigure 7.2: The correlation between parameters A0 and A1 obtained from the fit ofM23 distribution. The allowed region for those parameters was determined based on assumption that theyboth should be positive and on the Ψ2 value of the fit. 
ential distributions. Exploiting thefact that reaction dd → 3Henη0 have been measured close to the threshold, the partial wave expansion was limited to the processes with total angular momentum not larger than one in the final state. This implies that six independent parameters (see Appendix A) should be used for full description of the differential cross section given by equation 5.17. The fixed value of the quasi-free cross section was used while amplitudes A0,...,A6 were fitted to the differential distributions with the previously described setofvariables:M23, cos ϕp, cosϕq, φ and cos ϕp± cos ϕq. 
To obtain all termsAi occurring in equations 5.17, the differential distributions were fitted simultaneously using relations 5.23 -5.27. After the fit has been performed, the covariance matrix of fit parameters was inspected in order to determine whether the variables are dependent on one another. As can be seen from Table 7.2 the fit parameters A0, A1, A3 are almost fully correlated and for the rest parameters the correlation was not observed.Areasonforsuch correlationisthattheshapeofM23 spectrumisonlytheone, which determinesthevaluesofA0,A1andA3 parameters.We triedtogetridof correlation by splitting the fit into two steps. In first step all differential distributions except ofM23 were fitted simultaneously. In the fit functions 5.24 -5.27 the linear combination A0IsS + A1IpS + A3IsPwas replacedby one parameterA013. Than only four parameters A2,A4,A6,A5andA013 werefixedbythefit, which appearedtobenot correlated.The parameterA3was then calculated as 
1 ()
A3 = A013 − A0IsS − A1IpS(7.3)
IsP 
and replaced into equation 5.23.In thatwaythefit functionforM23 containedonlytwo 
7.3 Differential Cross Section Distributions 
A0  A1  A2  A3  A4  A5  A6  
A0  1.000  -0.913  0.013  -0.895  0.000  0.000  0.014  
A1  -0.913  1.000  0.038  0.648  -0.002  -0.032  -0.056  
A2  0.013  0.038  1.000  0.006  -0.046  -0.069  0.074  
A3  -0.895  0.648  0.006  1.000  0.007  0.029  0.115  
A4  0.000  -0.002  -0.046  0.007  1.000  0.031  -0.137  
A5  0.000  -0.032  -0.069  0.029  0.031  1.000  0.009  
A6  0.014  -0.056  0.074  0.115  -0.137  0.009  1.000  
Table 7.2: The covariance matrix obtained for fit parameters. The fit parametersA0,..,A6 are defined in Section 5.4.2). 


free parametersA0 andA1. Subsequently, theM23 distribution have been fitted requiring thatA0,A1 andA3 are positive, since all these parameters areexpressed viaa2 Π amplitudes (see equations A.1, A.2 and A.4). Unfortunately, the fit results forA0 andA1 parameters turnouttobestill correlated.Hence,wecheckedthis correlationbyfixing parameterA1 andallowingtovaryA0inthefitofM23 distribution.Inthis procedureitwas requiredthat the Ψ2inthefit cannotbelarger than2·Ψmin2 . The Ψmin2 is thevalue obtained from the fit for case where no anycondition on fit parameters was imposed. The resulting dependence of the fixedA1 parameter asa functionofA0 parameter are shownin Fig. 7.2. The observed correlationmaybe representedby A1 = 16.0· 1012 − 17.6× A0 dependence. The limits of the correlation presentedinFig.7.2 wereusedto computethecoverageof parameterA3. Thefinal resultsforallcoefficientsAiaregatheredinTable7.3.Itisseenthat uncorrelated parametersA2,A4,A5 are determined withthe accuracybetter than3%andA6 parameter have uncertainty of about 10%.For the parametersA0,A1,A3 which are correlated only their ranges were determined.The parametersA0andA3maybevariedbyfactor3within given range, however the uncertainty for their certain values are about few percent. The valueof parameterA1isbadly determinedanditcanbevariedinaverybroadrange. 
The comparison of the experimental differential distributions with quasi-free contribution, fitted partial wave based model and their sum for “assignment number“ equals to two (seeTab. 5.1) are presentedin Fig. 7.3. Similar results were obtained for the comparison of the experimental differential distributions with model prediction for “assignment number“ equal one and three (seeTab. 5.1). Contributions from quasi-free model, partial wave decomposition and their sum are shown in blue, green and red, respectively. In the left upper panel of Fig. 7.3 the cosine of angle ϕp between 3He momentum in the center-of-mass of subsystem 3He− η0 and beam momentum (z-axis) is displayed. The shape of the quasi-free contribution and the partial wave model are quite similar and reveal the shape of the experimental distribution and their sum very well describes the data. 
Figure 7.3: The experimental distributions (black) corrected for acceptance are compared to theoretical predictions (red line) based on phenomenological models which were described in details in Chapter 5. The plots are made for “assignment number“ equals to two (seeTab. 5.1). Individual contributions from partialwave decomposition and quasi-free model are shown in green and blue, respectively. 
7.3 Differential Cross Section Distributions 
Fit results  
sS wave  A0 [nb/sr2] (3.23 -9.27)·1011± 1.12·1010  
pS wave  A1 [nb/GeV2sr2] 2.53·109− 1013  A2 [nb/GeV2sr2] 4.76·1013± 3.03·1011  
Sp wave  A3 [nb/GeV2sr2] (2.72 -7.80) ·1013 ± 2.11· 1012  A4 [nb/GeV2sr2] 6.07·1013± 1.53·1012  
pS+ Sp interference  A5 [nb/GeV2sr2] -1.24 ·1014± 1.12·1012  A6 [nb/GeV2sr2] 8.74·1012± 9.05·1011  
Table 7.3: Amplitudes obtained in the fit of experimental differential cross sections with the model based on partial wave decomposition. The uncertainties correspond to one sigma deviations of the fitted parameters (there is no uncertainty for parameterA1 since itwas fixed on certainvalues).For the parametersA0,A1 andA3 which are correlatedin the fit only their range is given. 


While the quasi-free contribution is of about 30% of the observed cross section the observed agreement with the data demonstrate the importance of the p-wave in 3He− η0 subsystem. The right upper panel of Fig. 7.3 shows the cosine of angle ϕq between neutron momentum in global center-of-mass system and beam momentum. One can notice that distribution rises steeply at the borders. This shape could not be reproduced only by p-wave contribution alone. It is seen that for the observed rising of the differential cross section at small and large ϕq angles the quasi-free process is responsible. It is most likely thatinthe quasi-free contributionhigherpartialwavesareinvolvedwhat causestheshape is more steeper then the quadratic dependence on cosϕq expected for p-wave contribution.For ϕq angles around 90◦ the experimental cross section is completely dominated by model basedon partialwave decomposition.Verygood agreementofthe model with the experimental differential cross section as function of cos ϕq demonstrate the validity of the applied model and the importance of the neutron p-wave contribution to the reaction mechanism. In the middle row (left panel) of Fig. 7.3 the relative angle φ between planes defined by 3He momentum and beam momentum and by neutron momentum and beam momentumis presented.Theexperimental points arefairly well describedby sum of quasi-free and partial wave contributions. As it is seen the observed anisotropymay arise for quasi-free reaction and when pS and sP interference is present. The contribution of pS and sP interference dominates the experimental differential cross section. In the right panel of the middle row the invariant mass of the 3He− η0 subsystem is plotted. It is seen that quasi-free reaction contributes mainly forlow 3He− η0 invariant mass, where it accounts for about 50% of the observed cross section. Higher invariant masses are very well reproduced by the part of the model based on partial wave decomposition. One can notice, that theoretical curve shows small deviation from experimental distribution. One of the reason for that can be the assumption that transition amplitudes are proportional to pL23qL1. Such dependence is only some approximation and it may happen that this approach is not strictly fulfilled. Additional effects which are not implemented in the model as e.g. final state interaction or presence of the resonances in anysubsystem of two final particles may also influence the 3He− η0 invariant mass distribution. In the lower row of Fig. 7.3 cosϕp+ cosϕq and cosϕp− cosϕq distributions are presented. They were used for extraction of the term with the amplitudeA5 which does not appear in any of the single cross section formulaeforvariables described before.TheA5 termis assignedtothe correlation between ϕp and ϕq angles and it is present only for pS and sP interference. 


8 Summary and Outlook 
In this thesis the reaction dd → 3Henη0 measured atpd = 1.2 GeV/c beam momentum have been investigated using theWASA at COSYfacility.For the first time the informations on total cross section and differential distributions of dd → 3Henη0 reaction were obtained. The measured total cross section equals to λtot =(3.81± 0.02stat. ± 0.42sys.) µb was measured with accuracyof about 11%. The various differential distributions exhibit rich structures indicating important contributions of s-and p-partial waves. 
For the comparison of the experimental differential distributions to the theoretical expectations based on the phenomenological approach the combination of quasi-free model and partial wave expansion model for three-body reaction were used. The contribution of the quasi-free model was fixed based on the available data for pd → 3Heη0 reaction. The partialwaveexpansionupto the final state angular momenta not higher than1 was used. The amplitudes of this expansion were fitted to the differential experimental distributions, demonstrating the importance of the p partial waves in the final state. The overall agreement of the applied model with the experimental differential distributions is very good. Therefore the present phenomenological model may be considered as a guidance for the microscopic description within Chiral Perturbation Theory, which is under construction. The microscopic model should include the quasi-free reaction mechanism, which accounts for 30% of the total cross section and dominates in specific regions of differential distributions especially for some variables. The large part of the microscopic model should include processes in which all entrance and exit channel particles are involved. These processes have to proceed to the final states described by s and p partial waves. The importance of the observed pS and sP interferences should put additional constraints to the microscopic model. Depending on the results delivered by the microscopic model the measurements of the dd → 3Henη0 reaction may be extended by studies of polarization observables. The experimental data on differential analysing powers should allow to separate the contributions of s and ppartial waves, what was not possible with present data. 
Apart from providing rich amount of data for description with microscopic model, the measurement allowed to study the experimental conditions for experimental studies of the dd → 4Heη0 reaction whichwas identifiedas oneofthekeyissuesofthephysics programoftheWASA-at-COSY collaboration.Inordertostudythis reactiontheknowledge of the cross section for dd → 3Henη0 is a very valuable information because this is the main background channel. Presently obtained cross section for dd → 3Henη0 reaction isbyfactor of5·104largerthantheexpected cross sectionfordd → 4Heη0 reaction at the same beam momentum. This indicates the necessity of very strong background reduction in future investigations of dd → 4Heη0 reaction. Based on the results and the experiences gained during the dd→ 3Henη0 beam time the two-week measurement of dd → 4Heη0 at pd = 1.2 GeV/c beam momentum have been performed. The obtained data are currently under analysis [85]. Depending on the results, it will be proposed to perform this measurement with higher statistics and polarized beam. This will allow to extract p-wave contributions to the Charge Symmetry Breaking amplitude and to fix some parameters of the Chiral Perturbation Theory terms responsible for this symmetry breaking. 

A Appendix 
Anexampleof partialwavedecompositionfordd → 3Henη0 reaction assigning neutron as particle1, η0as particle2and 3Heas particle3is presented.The calculations were limited to final state maximum angular momentum equal 1. The resulting amplitudes presented in table A.1 may be grouped to sS (amplitudes a1 and a2), sP (amplitudes a3 and a10) and pS-wave (amplitudes a11 and a18).In this notation first index correspondto particle1 angular momentum in global center-of-mass system and the second index correspond to the relative angular momentumof particles2and3in subsystem2-3.Thesiandli arespin and angular momentumin the entrance channel,s23,L23 andj23 denote spin, angular momentumand total angular momentumin subsystem2-3,L1andj1 are angular momentum and total angular momentum of particle1andJcorrespond to the total angular momentum of the system in initial and final state. The spherical harmonics were normalized by
∫ 
Yl,m(ϕ,χ)Y∗ l,m(ϕ,χ)dΔ = 1. 
A0 = a12+ 3a2 (A.1)
2 2 2 2 2 222 2 
= 5a(A.2)
A1 12 + 5a17 + 5a18 + 3a14 + 3a15 + a11 + a13 + 5a16 
√
15 253
2 222 ∗∗
A2 = − a17 + a18 − a15 + 5a16 − 70υ(a12a17)+ 6υ(a12a14)−
147 2
√ 
√√ √
12 10
∗∗∗ ∗
32υ(a12a15)+ 22υ(a12a13) − 5υ(a17a18) − 3 υ(a17a14)+
77
√√ √ 
552 12
∗∗ ∗∗
3 υ(a17a15) − 4 υ(a17a13) − 12 υ(a18a14)+ √ υ(a18a15)+
7777 
√√
12 
∗∗ ∗
√ υ(a18a13) − 32υ(a14a15)+ 2 10υ(a11a16) (A.3)
7 
222 22222
A3 = 5a4+ 5a9+ 5a6+ 3a7+ a3+ 5a8+ a(A.4)
10 + 3a5 
amplitude  si  li  s23  L23  j23  L1  j1  J  transition  
a1 a2  1 1  1 1  1 2 1 2  0 0  1 2 1 2  0 0  1 2 1 2  0 1  3P0 →1 S1 1S1 3P1 →1 S1 1S1  sS  
a3  0  0  1 2  1  1 2  0  1 2  0  1S0 →1 S1 1P1  
a4  2  0  1 2  1  3 2  0  1 2  2  5S2 →1 S1 1P3  
a5  2  2  1 2  1  1 2  0  1 2  0  5D0 →1 S1 1P1  
a6  2  2  1 2  1  1 2  0  1 2  1  5D1 →1 S1 1P1  sP  
a7  2  2  1 2  1  3 2  0  1 2  1  5D1 →1 S1 1P3  
a8  0  2  1 2  1  3 2  0  1 2  2  1D2 →1 S1 1P3  
a9  2  2  1 2  1  3 2  0  1 2  2  5D2 →1 S1 1P3  
a10  2  4  1 2  1  3 2  0  1 2  2  5G2 →1 S1 1P3  
a11  0  0  1 2  0  1 2  1  1 2  0  1S0 →1 P1 1S1  
a12  2  0  1 2  0  1 2  1  3 2  2  5S2 →1 P3 1S1  
a13  2  2  1 2  0  1 2  1  1 2  0  5D0 →1 P1 1S1  
a14  2  2  1 2  0  1 2  1  1 2  1  5D1 →1 P1 1S1  pS  
a15  2  2  1 2  0  1 2  1  3 2  1  5D1 →1 P3 1S1  
a16  0  2  1 2  0  1 2  1  3 2  2  1D2 →1 P1 1S1  
a17  2  2  1 2  0  1 2  1  3 2  2  5D2 →1 P3 1S1  
a18  2  4  1 2  0  1 2  1  3 2  2  5G2 →1 P3 1S1  
Table A.1: The amplitudes for lowest partial waves fordd→ 3Henη0 reaction. 


√
15 253
2 222 ∗∗
A4 = − a9+ a10 − a7+ 5a8− 70υ(a4a9)+ 6υ(a4a6)−
147 2
√ 
3 √ 2υ(a4a ∗ 7) + 2 √ 2υ(a4a ∗ 5) − √ √  12 7 √ 5υ(a9a ∗ 10) − 3 √  10 7 υ(a9a ∗ 6)+  
3  5 7υ(a9a ∗ 7) − 4  5 7υ(a9a ∗ 5) − 12  2 7υ(a10a ∗ 6) +  12 √ 7 υ(a10a ∗ 7)+  
12 √ 7 υ(a10a ∗ 5) − 3 √ 2υ(a6a ∗ 7) + 2 √ 10υ(a3a ∗ 8)  (A.5)  
A5  = 10υ(a12a ∗ 4) − √ 70υ(a12a ∗ 9) + 6υ(a12a ∗ 6) + 3 √ 2υ(a12a ∗ 7)−  
2 √ 2υ(a12a ∗ 5) − √ 70υ(a17a ∗ 4) + √ √  55 7 υ(a17a ∗ 9) − √  12 7 √ 5υ(a17a ∗ 10)−  
3  10 7 υ(a17a ∗ 6) − 3  5 7υ(a17a ∗ 7) + 4 √  5 7υ(a17a ∗ 5) −  12 7 √ 5υ(a18a ∗ 9)+  
120 7 υ(a18a ∗ 10) − 12  2 7υ(a18a ∗ 6) −  12 √ 7 υ(a18a ∗ 7) − √  12 √ 7 υ(a18a ∗ 5)+  
6υ(a4a ∗ 14) − 3 √ 2υ(a4a ∗ 15) + 2 √ 2υ(a4a ∗ 13) − 3 √ √ √  10 7 υ(a9a ∗ 14)+  
3  5 7υ(a9a ∗ 15) − 4  5 7υ(a9a ∗ 13) − 12  2 7υ(a10a ∗ 14) +  12 √ 7 υ(a10a ∗ 15)+  
12 √ 7 υ(a10a ∗ 13) + 6υ(a14a ∗ 6) + 3 √ 2υ(a14a ∗ 7) − 3 √ 2υ(a15a ∗ 6)− 3υ(a15a ∗ 7) − 2υ(a11a ∗ 3) + 2 √ 10υ(a11a ∗ 8) − 2υ(a13a ∗ 5)− 2 √ 10υ(a16a ∗ 3) + 20υ(a16a ∗ 8)  (A.6)  

√ 

35 

3

∗
υ(a12a 
∗
∗∗ 
7)+ = 10υ(a12a 
9) − 3υ(a12a 155 
υ(a12a
A6
√

6) −
4)+ 
2 

2

√ 

√

√

35 

2
√ 
6

∗∗
∗ )+10
2υ(a12a5)+ 
υ(a17a υ(a17a 5υ(a17a
4)+ 9)+ 
14
√ 
7

√ 

√

5 

14

35 

5 

7

6

∗
υ(a17a7) − 2
∗
∗
υ(a17a υ(a17a 
∗ 
5)+ 5υ(a18a9)+
3 

6)+ 
27

7

√ 

45 

7 

2 

7

6 

6√ 

7 

υ(a18a10)+ 6
∗
∗∗ 
7)+ 
υ(a18a 
∗ 
5)−
υ(a18a υ(a18a
√

6)+ 
7 

√ 

√

3 

5 

14

∗∗ 
15) −
∗3)+ 13 
∗ 
14)−
3υ(a4a υ(a4a 2υ(a4a υ(a9a
√

14)+ 
2
√
√ 

√ 

35 

27

5 

7

2 

7

6

υ(a9a15)+ 2
∗
υ(a9a13)+ 6
∗
∗∗ 
15)−
υ(a10a υ(a10a
√

14) − 
7 

6 

√ 

9 

9

∗
9 

6)+ 
2
∗∗∗ 
7)−
υ(a10a υ(a14a υ(a15a υ(a15a
√

√

7) −13)+ 
7 

2 

2

√−

∗ )3 
∗
10υ(a11a 
∗
2υ(a11a 
∗ 
√
8) − 2υ(a13a5)+ 10υ(a16a 
∗ 
3)+ 5υ(a16a8) 
(A.7) 


B Appendix 

ϕ[deg]  p0  p1  p2  p3  p4  
0-3 3-6 6-9 9-12 12 -15 15 -18  0.00986 0.01811 0.01888 0.02050 0.02107 0.02171  0.02559 -0.08890 -0.09442 -0.11273 -0.11375 -0.11453  -0.30381 0.27422 0.29539 0.384692 0.38188 0.37251  0.81094 -0.4288 -0.46711 -0.65619 -0.64270 -0.59934  -0.69083 0.26453 0.29103 0.43539 0.42276 0.37311  
Table B.1: In FWC1 detector for kinetic energy parametrization polynomial of fourth order was used. In the table values of the fit parameters obtained for six angular bins are presented. 


ϕ[deg]  p0  p1  p2  p3  p4  
0-3 3-6 6-9 9-12 12 -15 15 -18  0.02078 0.02211 0.02262 0.02326 0.02366 0.02410  -0.123175 -0.14430 -0.14961 -0.15656 -0.15807 -0.16000  0.40310 0.52482 0.54760 0.57956 0.57895 0.57981  -0.62650 -0.91648 -0.96025 -1.02582 -1.01374 -0.99792  0.36965 0.61103 0.64219 0.69181 0.67775 0.65239  
Table B.2: In FWC2 detector for kinetic energy parametrization polynomial of fourth order was used. In the table values of the fit parameters obtained for six angular bins are presented. 


ϕ[deg]  p0  p1  p2  
0-3  -0.29746  3.86830  -11.8258  
3-6  -0.09547  0.83669  -0.51588  
6-9  -0.08032  0.60609  0.32362  
9-12  0.03639  -1.10801  6.54220  
12 -15  0.03260  -1.05575  6.28564  
15 -18  -0.04486  0.07679  2.07313  

Table B.3:For particles stopped in FTH1 detector polynomial of second order was used. In the table values of the fit parameters obtained for six angular bins are presented. 
ϕ[deg]  p0  p1  p2  
0-3  0.00587  0.12212  0.35640  
3-6  0.00562  0.10549  0.40415  
6-9  0.00559  0.10823  0.40520  
9-12  0.00562  0.109261  0.40716  
12 -15  0.00569  0.10921  0.41013  
15 -18  0.00579  0.10900  0.41361  

Table B.4:For particles punching through the FTH1 detector power function of the form: Ekin(Edep)= p0/(Edep − p1)p2) was used. In the table values of the fit parameters obtained for six angular bins are presented. 

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Acknowledgements 
During writing my thesisihave lostfaith thatiwill reach this chapter so manytimes it wouldbehardto count.Fortunately,there werealwayspeoplewho wereableto resurrect thesparkofhopeinmethati candothat(well sometimesthatihavetodothat ©). This isthe chapter whereI wouldliketoexpressmy gratitudeto those people. 
Firstand foremostI wouldliketo thankthe supervisorofmyPhD studies, Prof.Andrzej Magiera who helped me to find an interesting topic and provided ceaseless help in both structuring and writing this thesis. His knowledge andphysics intuition wereinvaluable during countless discussionwehave had.Withouthis guidanceImostlikelywould have been lost and discouraged. 
Iexpressmy gratitudetoProf.Hans Ströherforagreat opportunitytoworkin research fieldof NuclearPhysicsatthe InstitutefürKernphysikinForschungszentrum Jülich. 
I am grateful to Prof. Bogusław Kamys for possibility to prepare this dissertation in theFaculty of Physics, Astronomy and Applied Computer Science of the Jagellonian University. 
I am very grateful to Dr. Volker Hejny, my tutor during the stay in Jülich. I have approached you with thousands of questions, and you always knew the answers. Thank youfor teachingmeanelegantwayof programmingandforbeingextraordinarily patient withevery mistakeImadeorideaI misunderstood.Itwouldnothavebeen possibleto write this doctoral thesis without your help. 
Iwould like to thank the “boss” ofWASA collaboration Dr. MagnusWolke for creating friendly atmosphere during my stay in Uppsala at the very begin of my way in the worldof particlephysics. 
Iamhighly indebtedto AleksandraWro´
nska and ChristianPauly for several important suggestions. 
My sincere thanks are due to Marek Jacewicz andTytus Smoli ´
nski for assistance duringthe assemblingofPSC detector.Iwould alsoliketo thank othersWASA-at-COSY collaborators who have contributed to this work. A warm thank you to all my friends 
´ 
(especially Michał Smiechowicz)and colleaguesinthe Institutetokeepmein shape. 
FinallyIwouldliketoextendmy deepest gratitudetomyFamily.