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 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.