Jagiellonian University Marian Smoluchowski Institute of Physics Leading Modes of the 3π0 production in proton–proton collisions at incident proton momentum 3.35 GeV/c Benedykt R. Jany Doctoral Dissertation prepared at the Nuclear Physics Department Supervisor: Prof. Dr. hab. Zbigniew Rudy Cracow 2011 Poland ∞ Ex Nihil Multum Abstract This work deals with the prompt pp→ pp3π0 reaction where the 3π0 do not origin from the decay of narrow resonances like η(547),ω(782),η ′ (958). The reaction was measured for the proton beam momentum of 3.35 GeV/c with the WASA-at-COSY detector setup. The dynamics of the reaction is investigatedby DalitzandNyborgplotsstudies. Thereactionisdescribedby the model assuming simultaneous excitation of two baryon resonances Δ(1232) and N∗(1440)where resonances are identiﬁed by their unique decays topology on the missing mass of twoprotons MMpp dependent Dalitz andNyborg plots. The ratio R = Γ(N∗(1440) → Nππ)/Γ(N∗(1440) → Δ(1232)π → Nππ)=0.039±0.011(stat.)±0.008(sys.)is measured for the ﬁrst time in a direct way. It shows that the N∗(1440)→ Δ(1232)π → Nππ decayis aleading mode of 3π0 production. Itis also shown that the MMpp is very sensitive to the structure of the spectral line shape of the N∗(1440)resonance as well as on the interaction between the Δ(1232) and N∗(1440) resonances. The multipionspectroscopy –aprecisiontool todirectly accessthepropertiesof baryon resonances is considered. The pp → ppη(3π0) reaction was also measured simultaneously. It is shown that the η production mechanism via N∗(1535)is 43.4±0.8(stat.)± 2.0(sys.)%of the total production, for the η momentum in the CM system qηCM =0.45 − 0.7 GeV/c. First time momentum dependence of the η angular distribution is seen, the strongest eﬀect is observed for the cos(θηCM ) distribution. Streszczenie Pracatajestdedykowana reakcjibezpośredniej produkcji pp → pp3π0 gdzie 3π0 niepochodzą z rozpadu wąskich rezonansówjak η(547),ω(782),η ′ (958). Reakcja została zmierzonadlapędu wiązkiprotonowej 3.35 GeV/c przy pomocy systemu detekcyjnewgo WASA-at-COSY. Dynamika reakcji jest studiowana przy pomocy wykresów Dalitza i Nyborga. Reakcja jest opisywana przez model zakładający równoczesne wzbudzenie dwóch rezonansów barionowych Δ(1232) i N∗(1440) gdzie rezonansy są identyﬁkowane dzięki niepowtarzalnej topologii ich rozpadów na zależnych od masy brakującej dwóch protów MMpp wykresach Dalitza i Nyborga. Stosunek rozgałęzien R = Γ(N∗(1440) → Nππ)/Γ(N∗(1440) → Δ(1232)π → Nππ) =0.039 ± 0.011(stat.)±0.008(sys.)zostałpo razpierwszy wyznaczony wbezpośredni sposób; oznacza to że gałąź rozpadu N∗(1440) → Δ(1232)π → Nππ jest członem wiodącym produkcji 3π0 . Pokazanejest że MMpp jest bardzo czuła ze względu na strukturę linii spektralnej rezonansu N∗(1440) oraz na od-działywanie między resonansami Δ(1232) i N∗(1440). Spektroskopia wielopionowajakoprecyzyjne narzędziedobezpośredniegodostępudo własności rezonansówbarionowychjest rozważana. Przekrój czynny na reakcję został wyznaczony σpp→pp3π0 =123±1(stat.)±8(sys.)±19(norm.)µb. Równoczesnie została zmierzona reakcja pp → ppη(3π0). Pokazane zostało,że mechanizmprodukcji mezonu η przez rezonansN∗(1535)jestrówny 43.4±0.8(stat.)±2.0(sys.)%całkowitejprodukcji, został onwyznaczony dla pędu mezonu η w środku masy równego qηCM =0.45 − 0.7 GeV/c. Po raz pierwszy zaobserwowano zależność pędową rozkładów kątowych dla mezonu η, najsilniejszy efektjest widocznydla rozkładu cos(θηCM ). Contents 1 Introduction 1 2 The Experimental Setup 5 2.1 CoolerSynchrotronCOSY.................... 5 2.2 TheWASA atCOSYdetector.................. 8 2.2.1 ThePelletTarget..................... 9 2.2.2 TheForwardDetector .................. 11 2.2.3 TheCentralDetector. . . . . . . . . . . . . . . . . . . 12 3 Physics of 3π0
production 19 3.1 Theory and data status -Physics Motivations . . . . . . . . . 19 3.2 Choiceof theObservables .................... 27 4 Analysis of the experimental data 31 4.1 TheExperimentalConditions .................. 31 4.2 TheEventSelection ....................... 34 4.3 TheDetectorResponse. . . . . . . . . . . . . . . . . . . . . . 43 4.4 TheKinematicFitting ...................... 49 4.4.1 Theerrorparametrization ................ 49 4.4.2 The diagnostics after the Kinematic Fit . . . . . . . . . 52 5 Results and error discussion 71 5.1 The pp→ pp3π0
reaction. . . . . . . . . . . . . . . . . . . . . 71 5.1.1 Themodeldescription .................. 71 5.1.2 Themodel validation . . . . . . . . . . . . . . . . . . . 107 5.1.3 TheCrossSectionextraction. . . . . . . . . . . . . . . 117 5.1.4 The Acceptance and Eﬃciency Correction . . . . . . . 124 5.2 The pp→ ppη(3π0)reaction ................... 141 6 Summary and Conclusions 165 Appendix A Kinematics of ﬁve particle phase space 175 Appendix B WASA-at-COSY Detector Calibration 179 Appendix C The WASA-at-COSY Monte-Carlo Simulation 187 Appendix D Track Reconstruction in WASA-at-COSY 193 Appendix E The Kinematic Fit 197 Appendix F Bayesian Likelihood energy reconstruction 207 Appendix G Data Tables – Results 221 References 243 Acknowledgments 255 1 Introduction Nucleon–Nucleon reactions typically lead to the abundant pion production. It is due to the fact that this pseudoscalar meson [1] has the lowest mass from allmembers ofpseudoscalar meson nonet and carries no exoticquantum numbers. Isospinofpionis 1, this was used in suitable deﬁnition of G parity. Unstable particles, which appear at ﬁrst stage of nuclear reaction, usually decay into pion channels; it means that investigation of pion spectra is one of the techniques reﬁned for the analysis of the unstable particles or states. Asanexample, so-calledABC eﬀect[2]isobservedinthereactionswith two pion production. Similarly, simultaneous detection of three pions was used e.g. in the investigation of η meson produced in pp → ppη → 3π0 reaction; luckily the η meson has narrow width and technically the analysis was not that diﬃcult[3]. In this dissertation the properties of the prompt 3π0 production in the proton-proton collisions attheincidentproton momentum ofPbeam =3.35 GeV/c where the threepionsdo not originfrom thedecays of the narrow resonances like η(547),ω(782),η ′ (958))aredescribed. Thedynamics of thisprocess was never studied in details neither experimentally nor theoretically, the cross section is also unknown. The reaction was measured withtheWASA-at-COSYdetector setup[4] locatedin theInstitutef¨urKernphysik of theForschungszentrum J¨ulichGermany at the Cooler Synchrotron COSY. Using the unique capabilities of the WASA-at-COSY installation to detect the charged and neutral multiparticle coincidences with a large acceptance, all ﬁnal state particles were reconstructed from the signals in the detectors. This provided a data set of high statistics for the later analysis. The studiespresentedinthis work concentrate onthe extraction ofthe reactiondynamicsinthe modelindependent way using onlythebasicprinciples likeenergy and momentumconservation –applyingkinematiccalculationsin the framework of Monte-Carlo model simulations. This is realized in a systematic wayby studying theinvariant masses ofthe subsystems using missing mass of twoprotonsdependentDalitz andNyborgplots. The analysisleads to conclusions that the reactionproceeds via simultaneous excitations of two baryon resonancesΔ(1232)and N∗(1440).Thedevelopedapproachﬁrsttime allowed to extractthe spectroscopicinformations(branching ratio, spectral line shape) of the N∗(1440)resonance in a direct way. In parallel also the pp → ppη(3π0)reaction was measured. The η meson production was successfully described by assuming two mechanisms: the resonant production via excitation of N∗(1535)and a non resonant part. 1 1 INTRODUCTION For the ﬁrst time also the momentum dependence of the η angular distribution was seen. As a consequence one concludes that the multipion spectroscopy may be treated as a precision tool to directly access the properties of baryon resonances. This dissertation is divided into four main mainstream parts. First part Section 2 describes the experimental setup. The Cooler SynchrotronCOSYisdescribed. Theproperties oftheWASA-at-COSYdetector setup are overviewed. Theproperties ofthedetector components(Forward and Central Detector) as well as the pellet target are given. SecondpartSection 3isdedicated to thephysics of 3π0 production. Theory and data status together with physics motivations are given. Later the selected choice of the observables is presented. ThirdpartSection4deals withthe analysis ofthe experimentaldata. The experimental conditions are listed. Next, the selection of events is discussed. Later, the experimental detector resonance is compared with the detector response simulations. Finally the procedure of the kinematic ﬁtting of the events together with the used error parametrization is presented. Forth part Section 5 contains the results and their errors. The pp → pp3π0 reaction is studied. The Monte-Carlo model description is proposed together with the supporting arguments. Next, the parameters of the model are derived from the experimental data by studying the missing mass of two protons dependent Dalitz and Nyborg plots, the overall model of the reactionispresented. Theparameters of the model arediscussed; the possibleinteractionbetween theΔ(1232) and N∗(1440)resonances or theinﬂuence of the N∗(1440)spectral line shape is taken into consideration. The modelis validatedby detailed statistical analysis and otherprocesses contribution to the model are veriﬁed. Later, the cross section is extracted. The Dalitz and Nyborg plots are corrected for the detection eﬃciency and geometrical acceptance,the absolute normalization of the spectraisperformed. The pp → ppη(3π0) reaction is studied in parallel. The accessibility of the phase space is checked and the observables are deﬁned. Next, the production mechanism is studied and described by assuming two mechanisms: the resonant production via excitation of N∗(1535) and a non resonant part. Later, the angular distributions are investigated and the angular anisotropy is extracted. A summary with conclusions is given in Section 6. The obtained results arediscussed and compared with other experimental measurements as well as available theoretical models. The multipion spectroscopy – a high precision tool to directly access the properties of baryon resonances is considered. Jagiellonian University 2 Benedykt R. Jany 1 INTRODUCTION There are also six appendixes A, B, C, D, E, F, which are toolboxes, covering the technical aspectsbehind thedata analysispresented in the main parts. In the Appendix G the results of this work i.e. the acceptance and eﬃciency correctedDalitzandNyborgPlotsand theangulardistributionsof the η meson are presented as tables of numbers. Jagiellonian University 3 Benedykt R. Jany 2 The Experimental Setup The WASA detector setup had been operated since 1998 in The Svedberg LaboratoryinUppsala(Sweden). Ithasbeenbuilt with theaimonstudying the decays of η-mesons in nuclear reactions. In 2003 it has been decided to shut down the CELSIUS ring, to stop the operation of WASA at CELSIUS. Shortly after that announcement the idea came up to continue the operation of WASA at COSY. The reasons are obvious: The combination of WASA and COSY would be of advantages for both communities. COSY oﬀers a higher energy than CELSIUS, allowing the extension of the studies into the η ′ sector. The WASA detector has an electromagnetic calorimeter as a central component and, thus, the ability to detect neutral decay modes involvingphotons – such adeviceismissing atCOSY.Theproposal[4]for movingWASAtoCOSY[5]was acceptedby theCOSYPACin2004. The WASA detector was dismounted during summer 2005 and shipped to J¨ulich (Germany). The ﬁnal installation and commissioning took place in the end of 2006. 2.1 Cooler Synchrotron COSY Figure 1: View at the Cooler Synchrotron COSY. The Cooler Synchrotron COSY (Fig. 1) is located in the Institute f¨ur 5 2.1 Cooler Synchrotron COSY 2 THE EXPERIMENTAL SETUP Kernphysik of the Forschungszentrum J¨ulich Germany. It delivers phasespace cooled polarized or unpolarized protons (deuterons) of momentum from p = 300 MeV/c up to p = 3700 MeV/c. The ring has a circumference of 184 m and can be ﬁlled with up to 1011 particles. When using the internal cluster target the luminosity of 1031cm−2s−1 can be reached. Two cooling methods can be applied during accumulation of the beam to reduce the phase-space volume, electron cooling at injection energies and stochastic cooling at higher energies. In the electron cooling method an electron beam, moving withthesameaveragevelocity asprotonbeam(acting asa coldgas),is mixed with theprotons(hotgas),by mixing thetwogases with diﬀerent temperatures the averagekinetic energydrops. The stochastic cooling is the process in which the deviations of nominal energy or position of particles in a beam are measured and corrected. The electron cooling system in COSY is applied at injection momentum p=300 MeV/c and reaches up to p = 600 MeV/c, and the stochastic cooling from p = 1500 MeV/c to 3700 MeV/c. Both, proton and deuteron beams, can be provided unpolarized as well aspolarized. AtCOSYinternal and external targetpositions are in operation(Fig.2). Forfurtherdetails see[6]. Jagiellonian University 6 Benedykt R. Jany 2 THE EXPERIMENTAL SETUP 2.1 Cooler Synchrotron COSY Figure2: The accelerator complex withthe cyclotron, theCOSYring andthe experimental installations. The place of WASA is within one of the straight sectionsofCOSY.InthepresentedFigurethebeamcirculatesclockwise. For more details please visit http://www.fz-juelich.de/ikp/cosy. Jagiellonian University 7 Benedykt R. Jany 2.2 The WASA at COSY detector 2 THE EXPERIMENTAL SETUP 2.2 The WASA at COSY detector Figure 3: Cross section of the WASA-at COSY detector. Beam comes from theleft. TheCentralDetector(CD)build aroundtheinteractionpoint(at theleft). Thelayers oftheForwardDetector(FD)are shown on the right. Othersymbols(MDC,PSB,FPC, ...) willbeexplainedintextof thethesis. As mentionedbeforetheWASAdetector– Wide Angle Shower Apparatus (Fig. 3), was designed to study various decay modes of the η-meson. This is reﬂected in the detector setup. The η-mesons are produced in reactions of the type pp −→ ppη. Due to the kinematics boost, the two protons are going into the forward direction, while the light decay products of the η are distributed into 4π. In order to detect the protons, a φ-symmetric(0−360 deg)forward detector for θ ≤ 18◦ is installed. The particles are identiﬁed and reconstructed by means of dE measurement and track reconstruction using drift chambers. A trigger, which is set only on the forward detector, can be used to select events independently from the decay mode of the η-meson registered by the Central Detector. Particles comingfrom mesondecays(e±,µ±,π± and γ), are detected in the centralpart ofWASA(θ ≥ 20◦). Momentum reconstruction for charged particlesisdonebytrackinginamagnetic ﬁeld and theenergy of theneutral particles is measured using an electromagnetic calorimeter. By including the central detector in the trigger also very rare decay modes can be studied Jagiellonian University 8 Benedykt R. Jany 2 THE EXPERIMENTAL SETUP 2.2 The WASA at COSY detector Figure 4: 3D View of the WASA detector[4]. while using a very high luminosity of up to 1032 cm−2s−1 . A 3D view of the detector setup(Fig.4). 2.2.1 The Pellet Target Thepellettarget system(Fig.5) was a specialdevelopmentforWASA.The ”pellets” are frozen droplets of hydrogen or deuterium with a diameter between 25 µm and 35 µm. The advantages of using pellet target compared with a standard internal gas target are the following: • high targetdensity, allowshighluminosities necessaryfor studying rare decays • thin tube delivery through the detector, 4π detection possible • precisely localized target, small probability of secondary interactions inside the target The central part of the system is the pellet generator where a stream of liquid gas (hydrogen or deuterium) is broken into droplets by a vibrating nozzle. The droplets freeze by evaporation into a ﬁrst vacuum chamber forming a pellet beam. The beam enters a vacuum-injection capillary where itis collimatedandisfedthrough a 2m longpipeintothe scatteringchamber (Fig. 6). An eﬀective beam thickness for hydrogen of 3 ∗ 1015 atoms/cm2 Jagiellonian University 9 Benedykt R. Jany 2.2 The WASA at COSY detector 2 THE EXPERIMENTAL SETUP has been achieved with a beam diameter 2 − 4 mm, a frequency of pellets 5−10 kHz, and an average distance between the pellets of 9−20 mm. Figure5: ThePelletTarget system[4].

Figure 6: Photography of generated pellets. Jagiellonian University 10 Benedykt R. Jany 2 THE EXPERIMENTAL SETUP 2.2 The WASA at COSY detector 2.2.2 The Forward Detector TheForwardDetector(FD)(Fig. 3) tags mesonproductionby measuring the energies(dE-E) and angles offorward scatteredprojectileslikeprotons, deuterons, neutrons and charged pions. The produced mesons are then reconstructed using the missing mass technique. Detector covers angles from 3◦ to 18◦ . It consists of several layers of detectors described below: • FWC The Forward Window Counters The FWC is the ﬁrst detector in the Forward Detector. It consists of 12 plastic scintillators of 5 mm thickness. It is used to reduce the background from scattered particles originated from the beam pipe or the exit ﬂange. • FPC The Forward Proportional Chambers The next detector is the FPC. It consists of 4 modules each containing 122 straw tubes detectors. The modules are rotated relatively to each other by 45◦ . The FPC is used as a precise tracking device. • FTH The Forward Trigger Hodoscope Close to FPC the FTH (“J¨ulich Quirl”) is installed. It consists of 3 layers of plastic scintillators, one with straight modules, two with bended ones. Each layer has a thickness of 5 mm. It is used for a rough determination of the hit position on the higher level and as a starting value for the track reconstruction, see Appendix D. • FRH The Forward Range Hodoscope Thekinetic energy of theparticlesismeasuredby theFRH.It consists of 5 layers of cake-piece shaped plastic scintillators of 11 cm thickness. There are 24 scintillators pro layer. It is also used for particle identiﬁcation by the dE-E technique. • FRI The Forward Range Interleaving Hodoscope Betweenthird andforthlayer ofFRH twolayers ofplastic scintillators areinstalled(FRI).Eachlayeris made of 32strips of 5.2 mm thickness. The FRI is used to determine the scattering angles of neutrons. • FVH The Forward Veto Hodoscope The last layer of FD is FVH. It consists of 12 horizontally oriented scintillatorstripswithphotomultipliersonboth sides. Thehitposition is determined from the time diﬀerences of the signals. It is used to identify particles which are not stopped in the FRH. Jagiellonian University 11 Benedykt R. Jany 2.2 The WASA at COSY detector 2 THE EXPERIMENTAL SETUP 2.2.3 The Central Detector The Central Detector (CD) surrounds the interaction point and is constructed to identify energies and angles of the decay products of π0 and η mesons, with close to 4π acceptance. It consists of: • SCS The Superconducting Solenoid TheSCSproducesanaxial magnetic ﬁeld necessary formomentumreconstruction usingtheinnerdrift chambers. As superconductorNbTi/Cu is used cooled down by liquid He at 4.5K. The maximal central magnetic ﬁeldis 1.3T.Thereturnpathforthe ﬁeldisdoneby ayokemade of 5 tons of pure iron with low carbon content. • MDC The Mini Drift Chamber TheMDCisbuild around thebeampipe anditisusedfor momentum and vertexdetermination(Fig.7). It consists of 17 layers with in total 1738straw tubesdetectors. It covers scattering anglesfrom 24◦ to 159◦ [7]. For the resolution refer to Table 1. particle p [MeV/c] resolution △p/p electrons 20−600 < 1% pions, muons 100−600 < 4% protons 200−800 < 5% Table 1: MDC resolution • PSB The Plastic Scintillator Barrel The PSB surrounds the MDC inside the SCS. It consists of 146 pieces of 8 mm thick strips that form a barrel like shape. It is used together with MDC and SEC, and acts as a dE-E and dE-momentum detector and as well as a veto for photons. • SEC The Scintillator Electromagnetic Calorimeter The SEC is the heart of the WASA detector and maybe the most important part. At CELSIUS it has been used to measure electrons and photons up to 800 MeV. However, using a diﬀerent setting the energy range can be extended taking into account the higher energy available at COSY. It consists of 1012 CsI(Na) crystals shaped like a truncatedpyramids(Fig.8). It covers anglesfrom 20◦ to 169◦ . The crystals are placed in 24 layers along thebeam(Fig.9). Thelengths of the crystals vary from 30 cm (centralpart),25 cm (forwardpart)to 20 cm (backward part). The forward part consists of 4 layers each 36 Jagiellonian University 12 Benedykt R. Jany 2 THE EXPERIMENTAL SETUP 2.2 The WASA at COSY detector Figure7: MDCandBepipe. ThefullyassembledMDCinsideAl-Be cylinder (upper left)[7]. crystals, covering the range of 20◦ −36◦ . The central part consists of 17 layers with 48 elementseach, covering therangebetween 36◦−150◦ , and thebackwardpart with 3 layers, two with 24 crystals and one with 12. Thegeometricaldistribution of the crystals(Fig.10) and(Fig.11). The calorimeter consists ofsodiumdopedCsI crystals. Theyarepainted withtransparent varnishfor moistureprotection and wrappedin 150 µm teﬂon and 25 µm aluminized mylarfoil[8]. For moreinformationrefer toTable2. Detailedproperties aredescribedin[9]. More detail information on WASA Detector components and electronics canbefoundin[10–12]. Jagiellonian University 13 Benedykt R. Jany 2.2 The WASA at COSY detector 2 THE EXPERIMENTAL SETUP amount of active material 16 X0
(radiation length [9]) geometric acceptance: polar angle: azimuthal angle: 96% 20◦ −169◦ 0◦ −180◦ relative energy resolution: Cs(137)662keV 30% (FWHM) maximal kinetic energy for stopping: pions/protons/deuterons 190/400/500MeV Table 2: SEC parameters Figure 8: CsI(Na) crystal fully equipped with light guide, photomultiplier tube andhousing[8]. Jagiellonian University 14 Benedykt R. Jany 2 THE EXPERIMENTAL SETUP 2.2 The WASA at COSY detector Figure9:SECplanar map(arrowindicatesbeamdirection)[4]. Figure10: Schematic view ofthe calorimeterlayout. Itconsists oftheforward part(yellow, ontheleft), centralpart andthebackwardpart(red, onthe right)[4]. Jagiellonian University 15 Benedykt R. Jany 2.2 The WASA at COSY detector 2 THE EXPERIMENTAL SETUP Figure11: Photography of theSEC(forwardpart to theleft,beam comes from the right). Jagiellonian University 16 Benedykt R. Jany 3 Physics of 3π0 production 3.1 Theory and data status -Physics Motivations Figure12:The existing experimentaldata[13]forthetotal cross sectionfor pp→ pp3π0 and pp→ ppπ+π−π0 reaction versusprotonbeamkinetic energy Tbeam,theproposed models(from[13–15]) for cross section scaling[16–18]. The beam kinetic energy at threshold TbThr. and beam momentum at thresh old PThr. b for pp→ ppX reaction where X = η(547) (TbThr. =1.255 GeV,PbThr. =2.670 GeV/c) X = ω(782) (TbThr. =1.892 GeV,PbThr. =2.670 GeV/c) X = η ′ (958) (TbThr. =2.405 GeV,PbThr. =3.208 GeV/c) X = Φ(1020)(TbThr. =2.593 GeV,PbThr. =3.404 GeV/c) respectively indicated. The ﬁeld of 3π0 production in the proton-proton collisions in the proton beam kinetic energy region between Tbeam =1 − 3 GeV where the three 19 3.1 Theory and data status -Physics Motivations 3 PHYSICS OF 3π0 PRODUCTION pionsdonotoriginfromthedecays ofthenarrowresonances(like η,ω,η ′ ) is unexplored both experimentally and theoretically. The dynamics of this process was never studied in details. The total cross section was measured recentlyforfewdatapointsforlowenergies(Fig.12)[13]. andthesimple modelsforthe cross section scaling wereproposed[16–18]. The statistical (Phase Space) model prediction is based on the phase space considerations[16]. TheFSIFaeld-Wilkin modelpredictionisbased of the assumption that there exist aproton-protonFinalStateInteraction(FSI) which couldbe modeled as discussed in[17]. The FSI Delof model prediction is also based on the assumption that there exist aproton-protonFSI which couldbe modeled asdiscussedin[18](for the pprelative momentaintheCenter ofMassgreaterthan 300 MeV/c pure Phase Space is used). It shouldbe noticed thatthetwoproposed models[17,18]areconsistent with existing 3π0 data, but for the higher beam kinetic energy Tbeam ∼ 2.5 GeV their predictions diﬀer by a factor 2. Asproposedby[13]toget someinformations aboutthedynamics of the reaction one can study also the cross section ratio σ(pp→ ppπ+π−π0) (1) σ(pp→ ppπ0π0π0) (assumingisospin conservation one canperform calculationsfordiﬀerent reaction scenarios see Table. 3). Using the available experimentaldata(Fig.12) itispossibleto calculate the ratio(Eq.1) only at energy 1.36 GeV [13]: �σ(pp→ ppπ+π−π0) =6.3±0.6(stat.)±1.0(syst.) (2) Exp. σ(pp→ ppπ0π0π0) T=1.36 GeV beam Itisseenthatthecalculated ratioisconsistent withthePhaseSpace(statistical model) value Table 3. Butsuch a studies cannot substitute thefull study ofthedynamics(which might be very complicated) in terms of invariant mass studies in the subsystems and assumptions about variousbaryon resonances excitations(i.e. excited states of the nucleon[1,20,21]). Such a studies, in a simpliﬁed form, were performed for the case of the pp → ppπ+π−π0 reaction[22–24],for muchhigherbeam momenta 5 GeV/c (Fig. 13),5.5 GeV/c (Fig. 14)and 10 GeV/c (Fig. 15). The diﬀerent invariant mass systems were considered, in all of the cases strong signal from the Jagiellonian University 20 Benedykt R. Jany 3 PHYSICS OF 3π0 PRODUCTION 3.1 Theory and data status -Physics Motivations Assumed Reaction Scenario σ(pp→ppπ+π−π0) σ(pp→ppπ0π0 π0) Phase Space (statistical model) 8 [13, 16] Δ(1232)N∗(1440) Δ(1232) → pπ N∗(1440)→ pππ 4 [13] Δ(1232)N∗(1440) Δ(1232) → pπ N∗(1440)→ πΔ(1232) 7 [13, 19] Δ(1232)N∗(1520) >> 8 [13] Table 3: Diﬀerent scenarios of pp→ ppπππ reaction. Description in text. Δ(1232) was seen in pπ invariant masses. Also an evidence for N∗(1440)signal was seen in pπ+π− and pπ+π0 invariant masses for the case of 5.5 GeV/c beam momentum. From a pure theoretical point of view, there exist no microscopic model for the three pion production in contrast to the NN → NNππ reactions where complete microscopic model based on the excitations and decays of variousbaryon resonances exists[25]. In addition,in case ofthe reaction at3.35 GeV/c beam momentum(beam kinetic energy 2.541 GeV)one can calculatetheDeBrogliewavelength λ of the incoming proton[26, 27]: h λ = =0.3701 fm (3) p where h is the Planck’s constant and p is a particle momentum. When one comparesthis value with theprotondiameter[28]: 1.754±0.014 fm (4) One concludes that the λ ismorethanfourtimesmalleri.e. theincoming proton“feels”theinnerstructureof thetargetproton. Onecanalsocompare this value with the range of the gluon induced interaction (in case of η ′ nucleon interaction), estimated by the two-gluon eﬀective potential, which Jagiellonian University 21 Benedykt R. Jany 3.1 Theory and data status -Physics Motivations 3 PHYSICS OF 3π0 PRODUCTION is in order of ∼ 0.3 fm [29]. Probing such small distances, the quark-gluon degrees of freedom may play a signiﬁcant role in the production dynamics. This fact will make the diﬃculty in the interpretation of the particle interactionsby common existing microscopic models[30–37] mimic interaction by exchange of various light mesons like π,η,ρ,ω and are more applicable for lower energies (below ∼ 2 GeV beam kinetic energy, equivalent to ∼ 2.8 GeV/c beam momentum)[38]. Itmightbe moreplausible to use rather the microscopic approachesbased ontheQuantumChromoDynamics(QCD)(excitations ofquark-gluondegrees offreedom)[39–43]for thefuture modelof 3π production-which might be very diﬃcult theoretical task. Taking all these facts into account, nowadays the experimental analysis of the 3π production shouldbe concentratedon the extraction of the reaction dynamics in the model independent way using only the basic principles like energy and momentum conservation. This might be done is a systematic way by studying the invariant masses of the subsystems. Such a approach is presented later in this work. The dynamics of the pp → pp3π0 reaction is not understood and it was neverinvestigatedindetails, this makes the reaction a veryinteresting object for studies itself and also for the following reasons: In heavy ion collisions the multiplepionproduction oﬀers apossibility to look at aproperties of thebaryon resonancesinthe nuclear matter[44–53]. It is also well established that in these reactions pions are mostly produced by thebaryon resonances excitations[44–46]. As shown above, the physics of pp→ pp3π0 at T =2.54 GeV is very closely related to the baryon resonances. There is enough energy to excite various baryon resonances-specially Δ(1232)and N∗(1440)(seenTable5onpage75 andFig.52 onpage76). One can study thedynamics of theproducedbaryon resonances in similar way like it was attempted for the pp → ppπ+π−π0 re-action [22–24]. Using this elementary reaction one can also get the spectroscopic informations about baryon resonances states [1, 20, 21]. Like in the heavy ion collisions in nuclear medium [44–53], one may think about themultipionspectroscopy –atool todirectly acesthepropertiesofbaryon resonances. The knowledge about the reaction is very important for transport codes like BUU/HSD [41–43, 54, 55], INCL [56], QMD [57, 58] used intensively in the modeling of the nucleon–nucleus or nucleus–nucleus interactions, since they usethe elementary reactions as aninputfortheir calculations. Thepions are the most abundantlyproduced mesonsin these reactions and theydeliver Jagiellonian University 22 Benedykt R. Jany 3 PHYSICS OF 3π0 PRODUCTION 3.1 Theory and data status -Physics Motivations theinformationaboutthe ﬁrst stageof thereaction. All of themodelsfailin thedescription of thepion spectra[59–61]. Inclusion of the new pp→ pp3π0 reaction channel mighthelpin thebetter understanding ofthepiondynamics in these models. It is essential also to know the cross section and the dynamics of the pp → pp3π0 reaction, specially in the beam kinetic energy range 1.8 − 2.8 GeV (see Fig. 12). It forms one of the most severe background for ω(782),η ′ (958),Φ(1020) mesons: • hadronic decays e.g. η ′ → 3π0 -isospin forbidden, ω → 3π0 -C parity forbidden • leptonic and semileptonic decays also, since the π0 can undergo Dalitz ++− +− decay π0 → ee−γ e.g. ω → ee, Φ → π0ee These decays are of special interest of many collaborations like WASA-atCOSY[4,5]orHADES[62]since using them one can study the symmetries (e.g. isospin symmetry) and symmetrybreaking(C, CP parity). Jagiellonian University 23 Benedykt R. Jany 3.1 Theory and data status -Physics Motivations 3 PHYSICS OF 3π 0 PRODUCTION Jagiellonian University 24 BenedyktR.Jany 3 PHYSICS OF 3π0 PRODUCTION 3.1 Theory and data status -Physics Motivations Jagiellonian University 25 Benedykt R. Jany 3.1 Theory and data status -Physics Motivations 3 PHYSICS OF 3π 0 PRODUCTION Jagiellonian University 26 BenedyktR.Jany 3 PHYSICS OF 3π0 PRODUCTION 3.2 Choice of the Observables 3.2 Choice of the Observables One considers reaction a + b → 1+2+3+4+5 (5) − → with masses mi and momenta pi in the center of mass frame, where total √ energyinthe center of massframeis s. Theprobabilitythatthe momentum of the ith particle will be in the range d3pi canbe expressed as[63]: d15P = d15V|M|2 (6) where M statesfor theinvariant matrix element for theprocess and d15V is the Lorentz invariant phase space element available for the reaction. One can rewrite(Eq.6) tothefollowing form:   55 √ 2 − d15P 1 � → = d3 p1d3 p2d3 p3d3 p4d3 p5 δ3  pj δ3  Ej − s  |M|(7) 32E1E2E3E4E5 j=1 j=1 → J−2 where Ei = pi 2 + mi are energiesin theCMframe(units are chosen so the c =1). The eventdistribution(Eq.7)couldbe expressedin terms oftheinvariant masses deﬁned by: 2 →→ M2 −(− i...j =(Ei + ...Ej)pi + ... −pj)2 (8) The threeparticleinvariant masses and thefourparticleinvariant masses are connected with the two particle invariant masses by: M2 M2 (222) ijk = ij + Mik 2 + Mjk 2 −mi + mj + m (9) k (222 2) M2 = M2 ik + M2 jk + M2 + M2 mmj + mml (10) ijkl ij + M2 il + M2 jl kl −2i + k + Such a representation is also very convenient to analyze the reaction in terms of the resonances in the subsystems (one assumes that M depends only on this invariant masses). One canintegrate(Eq.7) over spatial orientations ofthe entire system. This integration can be done assuming that the created system “has forgotten” aboutdirection of thebeamparticle(3 dimensions less). Next, one can integratefurther,leavingexplicitdependence only on E1,E2,E3,E4,p12,p123,φ(12)3 ,φ(123)4 (this canbedone as systemhastofulﬁllenergy and momentum conservation, 4 dimensions less). Jagiellonian University 27 Benedykt R. Jany 3.2 Choice of the Observables 3 PHYSICS OF 3π0 PRODUCTION Finally, one is left with 8 dimensions only[64]: π2 d8P 2 = dE1dE2dE3dE4dp12dp123dφ(12)3 dφ(123)4 |M|(11) 4 →−−− −−− →→→ →→→ where −p12 = p1 +p2 , p123 = p1 +p2 +p3 , φ(12)3 -anglebetween the surface →−→ → deﬁned by the −p1 ,p2 vectors and the −p3 vector, φ(123)4 -angle between the →−→ → surface deﬁned by the p−12,p3 vectors and the −p4 vector ( see Fig. 16). pi → plays for the length of the vector −pi. Ifone assumes that Mdoes notdependon φ(12)3,φ(123)4 one mayintegrate overit and obtain[64]: d6P = π4dE1dE2dE3dE4dp12dp123 |M|2 (12) In our case we are studying the reaction π0π0π0 pp→ pp(13) 12534 Jagiellonian University 28 Benedykt R. Jany 3 PHYSICS OF 3π0 PRODUCTION 3.2 Choice of the Observables One would like to study the dynamics for this reaction: • The correlations between the pions One would need to study the M2(π0π0)i.e M2 and M2 12 25 • The interactions between the protons and between the pions One would need to study the M2(pp)and M2(3π0)i.e M2 and M2 34 1235 • The possible resonances in proton pion and proton two pions systems One would need to study the M2(pπ0)and M2(pπ0π0)i.eM2 and M2 45 123 It would be also very convenient to study in addition the event distribution as function of the missing mass of the two protons (√ )2 − →→ MM2 = s −E3 −E4+(−p3 + p4 )= M2 (14) pp 125 as a parameter, since the MMpp deﬁnes the maximal available kinetic energy for the 3π0 system in its rest frame (Q3π0 ) and simultaneously the maximal available kinetic energy for the pp systeminits restframe(QMax ): pp QMax 3π0 = MMpp −m1 −m2 −m5 = MMpp −3mπ0 (15) √ √ QMax pp = s −MMpp −m3 −m4 = s −MMpp −2mp (16) where mπ0 -mass of the π0 meson, mp -mass of the proton. Selecting the missing mass one selects how the energy is distributed between the three pion system and a two proton system. It will be shown in Appendix A how to express the event distribution in appropriate variables. Jagiellonian University 29 Benedykt R. Jany 4 Analysis of the experimental data The purpose of the analysis is to obtain physical spectra from the collected data and validate it by the Monte-Carlo simulation. The schematic view of theevents ﬂowand analysischainispresentedin(Fig.17). Onecandivided this process into the two phases: 1. The events collection phase • The raw data collection The data are collected in the WASA-at-COSY experiment (see Section 2.2), later the raw data have to be calibrated (see Appendix B). • The Monte-Carlo simulation First the events are generated and later the detector response is simulated(see Appendix C). 2. The data processing phase Thispartisdone using theRootSorter analysisframework[65] which isbased onROOT[66]. Firstfromthe collecteddatathetracks are build, which correspond to the physical particles (see Appendix D). Later the statistical hypothesis test is done on tracks by the kinematic ﬁt procedure(see Appendix E). 4.1 The Experimental Conditions The aim of the experiment was to measure the pp→ pp3π0 → pp6γ reaction at proton kinetic energy T =2.541 GeV, which corresponds to the momentum of 3.350 GeV/c and excess energy Q=598 MeV (center ofmass energy √ s =2.879 GeV/c2). The MonteCarlo simulation, based onhomogeneously andisotropicallypopulatedphase space(seeAppendix C), wasperformed to study thekinematics ofthe reaction. The results(Fig.18) show that most of the protons are going to the FD detector and most of the photons to the CD detector. Assuming that two protons are registered in FD detector and six photons in CD detector,using Monte-Carlo simulation, the geometrical acceptance of Geom.Acc. =14.24% (17) has been obtained. The experimental data were collected during one week run in May 2008, using pellet proton target and COSY phase space cooled proton beam at incident momentum of 3.35GeV/c. The data were collected under following trigger condition: 31 4.1 The Experimental Conditions 4 ANALYSIS OF THE EXPERIMENTAL DATA Jagiellonian University 32 BenedyktR.Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.1 The Experimental Conditions Photons θ [deg] 180 160 400 140 350 100 120 CD 250 300 80 200 60 150 40 100 20 50 00 0.2 0.4 0.6 0.8 1 1.2 0 Ek [GeV] (a) Protons θ [deg] 60 300 50 250 40 200 30 150 20 100 10 FD 50 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 Ek [GeV] (b) Figure18: Thekinematics of the pp→ pp3π0 → pp6γ reaction at 3.35GeV/c -Monte Carlo simulation homogeneously and isotropically populated phase space. Fig. 18(a) photon scattering angle versus kinetic energy, geometrical boundaries of the CD detector marked. Fig. 18(b) proton scattering angle versus kinetic energy, geometrical boundaries of the FD detector marked. Jagiellonian University 33 Benedykt R. Jany 4.2 The Event Selection 4 ANALYSIS OF THE EXPERIMENTAL DATA • more then one hit in the FD detector -time overlap with FWC and FTHdetectors(to ensure that thehits comefromthe sameinteraction) • morethen onehitinthethirdlayer of theFRHdetector(this ensures that one has at least two high energetic particles) • more then one neutral group(veto on overlapped element of PS detector) inthe calorimeter withlowthreshold onthegroup(∼ 50 MeV) (this implies that at least two photons are registered) • vetoonthesignal “atleastonehitinPSdetector” (oneensuresthat the charged particles are excluded) the trigger was not prescaled. 4.2 The Event Selection The event selection was optimized to select from the data pp3π0[6γ] ﬁnal state. First the FD tracks (see Appendix D) with a minimum energy deposit of the track of 20 MeV were selected to suppress detector noises. After this selection charged tracks multiplicity in FD detector was compared with Monte-Carlo simulation pp → pp3π0 assuming homogeneously and isotropicallypopulatedphase space(Fig.19). Next the events with two or more charged tracks tracks in FD detector were chosen, time correlationbetween the trackspairs was checked(Fig.20). The cut on the time diﬀerence between two charged tracks was performed, the events in prompt peak were selected and accepted for Time Diﬀerence from −4 ns to 10 ns)(Fig. 21), to create a pair oftwo charged tracks. After selection of two charged tracks in FD correlated in time, event was analyzed further. The tracks multiplicities in CD detector were examined and compared withMonte-Carlosimulation(Fig.22). IntheCD track ﬁnding procedure (see Appendix D) the signals of PS detector were used as a selection criteria for charged(neutral) tracks, the 1 MeV values was chosen as a PS cluster threshold. The veto on charged CD tracks was put. Time correlation was checked between mean time of two charged tracks in FD and a neutralCDtrack(Fig.23). The cut onthetimediﬀerencebetween mean time of two charged tracksinFD and a neutralCD track wasperformed, the eventsinpromptpeak were selected(timediﬀerencefrom −10 ns to 20 ns )(Fig. 24). The neutral tracks multiplicity was checked Fig. 25 and compared with Monte-Carlo simulation. The events with six neutral CD tracks correlated with mean time of the two charged tracks in FD were chosen. Jagiellonian University 34 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.2 The Event Selection (a) Experimental Data. (b) Monte-Carlo simulation Jagiellonian University 35 Benedykt R. Jany 4.2 The Event Selection 4 ANALYSIS OF THE EXPERIMENTAL DATA TimeFD1 [ns] 2040 103 2020 2000 102 1980 10 1960 1940 1 1940 1960 1980 2000 2020 2040Time2 [ns] FD (a) Time of the ﬁrst charged track in FD versus the second one. 90000 80000 70000 60000 50000 40000 30000 20000 10000 0 -60-40-20 0 20 40 60 TimeFD1 -TimeFD2 [ns] (b) Time diﬀerence between ﬁrst charged track in FD and the second one. Figure 20: Time dependences charged tracks FD detector. Jagiellonian University 36 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.2 The Event Selection (a) Time of the ﬁrst charged track in FD versus the second one after the cut. (b) Time diﬀerence between ﬁrst charged track in FD and the second one after the cut. Jagiellonian University 37 Benedykt R. Jany 4.2 The Event Selection 4 ANALYSIS OF THE EXPERIMENTAL DATA Jagiellonian University 38 BenedyktR.Jany (a) Charged Tracks Multiplicity, Experimental Data. (b) Neutral Tracks Multiplicity, Experimental Data. (c) Charged Tracks Multiplicity, Monte-Carlo simulation. (d) Neutral Tracks Multiplicity, Monte-Carlo simulation. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.2 The Event Selection Around 4.4 millions of such events were selected for later analysis. Total reconstruction eﬃciency deﬁned as Tot.Rec.Eff = Rec.Eff ∗ Geom.Acc (18) where Rec.Eff -reconstruction eﬃciency of the applied cuts, Geom.Acc geometrical acceptance(Eq. 17). It was estimated viaMonte-Carlo,based on homogeneously and isotropically populated phase space, to be: Tot.Rec.Eff =3.90% (19) Jagiellonian University 39 Benedykt R. Jany 4.2 The Event Selection 4 ANALYSIS OF THE EXPERIMENTAL DATA TimeMean [ns] FD 2040 103 2020 2000 102 1980 10 1960 1940 1 1940 1960 1980 2000 2020 2040 2060 Time [ns] CD (a) Meantime of thetwocharged tracksinFD versusthetime of the neutral track in CD. ×103 100 80 60 40 20 0 -60-40-20 0 20 40 60 TimeMean-Time [ns] FDCD (b) Time diﬀerence mean time of the two charged tracks in FD and the time of the neutral track in CD. Figure 23: Time dependences between mean time of the two charged tracks in FD and the time of the neutral track in CD. Jagiellonian University 40 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.2 The Event Selection (a) Meantime of thetwocharged tracksinFD versusthetime of the neutral track in CD after the cut. ×103 -60-40-20 0 20 40 60 TimeMean-Time [ns] FDCD (b) Time diﬀerence mean time of the two charged tracks in FD and the time of the neutral track in CD after the cut. Figure 24: Time dependences between mean time of the two charged tracks in FD and the time of the neutral track in CD with the cut on the time diﬀerence. Jagiellonian University 41 Benedykt R. Jany 4.2 The Event Selection 4 ANALYSIS OF THE EXPERIMENTAL DATA (a) Experimental Data. (b) Monte-Carlo simulation. Jagiellonian University 42 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.3 The Detector Response 4.3 The Detector Response Afterthe selection of two charged tracksinFD(protons) correlatedintime with six neutral tracksinCD(photons) thedetector response was compared with the Monte-Carlo simulation of pp → pp3π0 assuming homogeneously and isotropically populated phase space. (a) Experimental Data. (b) Monte-Carlo simulation. The response of the CD detector was checked by plotting invariant mass oftwophotonpairs(Fig. 26). The π0 signal is seen on the combinatorial Jagiellonian University 43 Benedykt R. Jany 4.3 The Detector Response 4 ANALYSIS OF THE EXPERIMENTAL DATA background. Thespectrawere ﬁtted with asumofGaussian(describing the peak) and the polynomial of the forth order(describing the combinatorial background). The ﬁtted curve describing the background was subtracted from thedatapoint and simulation, the spectra were normalized to the same area and compared (Fig. 27). It is seen that the Monte-Carlo simulation describes the experimental data very well. The maximum of the peak is at Mπ 0 ≈ 0.135 GeV/c2 indicatedby redlinein(Fig.27). The response of theFDdetector wasalsocheckedby comparingdiﬀerent dE−E plots usingdiﬀerentlayers oftheFRHdetector(Figs.28293031). It is seenthatthe mostofthe chargedtrackstravelthroughthe wholeFDdetector. The experimental dE −E plots are good described by the Monte-Carlo simulation, which proves that the measured energy deposits are consistent with protons energy deposits. Jagiellonian University 44 Benedykt R. Jany 0.12V] [Ge 102 Edep FRH10.1 0.08 10 0.06 0.04 1 0.02 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (a) Experimental Data. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.3 The Detector Response Jagiellonian University 45 BenedyktR.Jany 0.12 Edep FRH1 [GeV]Edep FRH1 [GeV] 0.080.08 10 0.06 0.06 10 0.04 0.04 1 0.02 0.02 1 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (b) Monte-Carlo simulation. (c) Monte-Carlo simulation, with no hadronic interactions. Figure 28: Energy deposits in FRH detector, layer 1 versus whole FRH, dE−E plots, of two charged tracks. Good agreement between experimental data and Monte-Carlo visible. It is seen that the most of the charged tracks travel through the whole FD detector. 0.12 102 0.1 0.1 102 102 4.3 The Detector Response 4 ANALYSIS OF THE EXPERIMENTAL DATA Jagiellonian University 46 BenedyktR.Jany 0.18 0.16 0.14 0.12 10 0.1 0.08 Edep FRH1+FRH2 [GeV] 0.06 10.04 0.02 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (a) Experimental Data. Edep FRH1+FRH2 [GeV] 0.18 0.16 0.14 0.12 0.1 10 0.08 0.18 0.16 0.14 0.12 0.08 10 0.06 0.06 Edep FRH1+FRH2 [GeV] 102 102 0.1 0.04 1 0.04 0.02 0.02 1 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (b) Monte-Carlo simulation. (c) Monte-Carlo simulation, with no hadronic interactions. Figure 29: Energy deposits in FRH detector, layer from 1 to 2 versus whole FRH, dE − E plots, of two charged tracks. Good agreement between experimental data and Monte-Carlo visible. It is seen that the most of the charged tracks travel through the whole FD detector. 0.25 102 0.2 0.15 Edep FRH1+FRH2+FRH3 [GeV] 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.3 The Detector Response Jagiellonian University 47 BenedyktR.Jany 10 0.1 1 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (a) Experimental Data. 0.25 0.15 0.05 0.05 1 10 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] Edep FRH1+FRH2+FRH3 [GeV] (b) Monte-Carlo simulation. (c) Monte-Carlo simulation, with no hadronic interactions. Figure 30: Energy deposits in FRH detector, layer from 1 to 3 versus whole FRH, dE − E plots, of two charged tracks. Good agreementbetweenexperimentaldataandMonte-Carlovisible. Itisseenthatthemost of thecharged tracks travel through the whole FD detector. Edep FRH1+FRH2+FRH3 [GeV] 0.25 102 0.2 0.15 0.2 102 10 0.1 0.1 10 0.3 4.3 The Detector Response 4 ANALYSIS OF THE EXPERIMENTAL DATA Jagiellonian University 48 BenedyktR.Jany 0.25 102 0.2 Edep FRH1+FRH2+FRH3+FRH4 [GeV] 0.15 10 0.1 0.05 1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (a) Experimental Data. 103 Edep FRH1+FRH2+FRH3+FRH4 [GeV] 0.3 0.25 0.15 0.05 0.3 0.25 0.15 Edep FRH1+FRH2+FRH3+FRH4 [GeV] 0.05 102 10 0.2 1020.2 0.1 0.1 10 1 0 0 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] Edep FRH1+FRH2+FRH3+FRH4+FRH5 [GeV] (b) Monte-Carlo simulation. (c) Monte-Carlo simulation, with no hadronic interactions. Figure 31: Energy deposits in FRH detector, layer from 1 to 4 versus whole FRH, dE − E plots, of two charged tracks. Good agreement between experimental data and Monte-Carlo visible. It is seen that the most of the charged tracks travel through the whole FD detector. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting 4.4 The Kinematic Fitting TheMonte-Carlo simulations(seeAppendixC) areextremely useful toolfor understanding of complexdata. Thefollowing convention concerningMonte-Carlo simulation will be used: • The true value: value assumed in the simulation, used as an input • Thereconstructed(measured) value: valueaftertrackspropagationand reconstruction(see Appendix D) • The ﬁtted value: value after the kinematic ﬁtting Thekinematic ﬁtting(seeAppendixE)wasusedtoimprovetheresolution of the variables and to balance the overall four-momentum to keep only the kinematic complete events. The hypothesis of the reaction pp → pp6γ → pp3π0 was tested on the experimental data using kinematic ﬁt. The overall four momentum was balanced. The 6γ were combined into the 3π0, the π0 mass constraint was used. The energy, polar angle θ and an azimuthal angle φ of the neutral tracks in CDdetector(thephotons) was ﬁtted. ForthetwotracksinFDdetector(the proton candidates) the polar angle θ and an azimuthal angle φ was ﬁtted. Theenergy of thetrack wasputasanunknownforthe ﬁt,sincetheprotons energies are to high to reconstruct them via dE − E energy losses in FRH detector in a standard way as well as in the developed Bayesian Likelihood approach(seeAppendixF). Most ofprotontracks arepassing throughthe wholeFRHdetector(seeSection4.3Figs.28293031). 4.4.1 The error parametrization The errors oftheparameters weredetermined usingWASA-at-COSYMonteCarlosimulation,by simulating singlephotontracksinCDdetector and single proton tracks in FD detector respectively. The errors of the variables were derived by the ﬁtting of the Gaussian function to the true minus reconstructed(measured) values in steps of the variable from which this error depends. The discrete values were put into the histograms, later the linear interpolation between the values was used to get the error value. Example of the procedure is presented in(Fig. 32). The variable dependencies of the error for the photons in the CD detectorwere checked(Fig.33). Oneseesthattheerrorsdonotdepend onthe azimuthal angle φRec. It is seen that the error of the photon energy depends only on the energy. The error of the photon polar angle θRec depends on Jagiellonian University 49 Benedykt R. Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA (a)True minus reconstructedvalue ofthe θRec angle ofthephoton as a function of the photon energy ERec. (b)True minus reconstructedvalue ofthe θRec angle ofthephoton fortheparticularphotonenergy. LineindicatestheGaussian ﬁt. Jagiellonian University 50 Benedykt R. Jany 0.3 0.16 Rec Rec Rec σ(E )/E Rec σ(E )/E Rec )/E Rec 0.25 0.14 0.25 σ(E 0.12 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting Jagiellonian University 51 BenedyktR.Jany 0.2 0.2 0.1 0.15 0.15 0.08 0.06 0.1 0.1 0.04 0.05 0.05 0.02 0 0 0 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 160 -150 -100 -50 0 50 100 150 E [GeV] θ [deg] φ [deg] Rec RecRec (a) Relative Error of energy versus en-(b) Relative Error of energy versus θRec.(c) Relative Error of energy versus φRec.ergy. 2.4 2.2 2 1.8 2 1.8 σ(θ ) [deg] Rec 2.2 2 1.8 1.6 σ(φ ) [deg] Rec σ(θ ) [deg] Rec σ(φ ) [deg] Rec σ(θ ) [deg] Rec 1.6 1.4 1.4 1.6 1.2 1.2 1 1 1.4 0.8 0.8 0.6 0.6 1.2 0.4 0.4 0.2 0.2 10 0 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 160 -150 -100 -50 0 50 100 150 E [GeV] θ [GeV] φ [deg] RecRec Rec (d) Error of θRec versus energy.(e) Error of θRec versus θRec.(f) Error of θRec versus φRec. 3 ) [deg] 3 6 2.5 σ(φRec 52.5 2 4 2 1.5 3 1 2 1 1.5 0.5 1 0 0 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 160 -150 -100 -50 0 50 100 150 E [GeV] θ [deg] φ [deg] Rec RecRec (g) Error of φRec versus energy.(h) Error of φRec versus θRec.(i) Error of φRec versus φRec. Figure 33: Dependences of the errors for the photons in the CD detector. It is see that the errors do not depend of the azimuthal angle φRec. Results of the Monte-Carlo simulations. 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA the photon energy (the cluster size dependence) and on the θRec (the effect of the detector), the two dimensional error parametrization is needed. The error of thephoton azimuthal angle φRec depends on the photon energy (the cluster size dependence) and on the θRec (the eﬀect of the detector), the two dimensional error parametrization is needed. The appropriate error parametrization for photons, which were needed for kinematic ﬁt, were prepared(Fig. 34). FortheprotoninFDdetectorthe error of theprotonpolar angle θRec as a function of the θRec and the error of the proton azimuthal angle φRec as a function of the θRec were used for kinematic ﬁt as an error parametrization (Fig. 35). The used errorparametrization(Figs.33,34,35)are results oftheMonte-Carlo simulations using single tracks. 4.4.2 The diagnostics after the Kinematic Fit The statistical relations The χ2 distribution (Fig. 36) and corresponding probability distribution (Fig. 37) (see Appendix E) for the kinematic ﬁt of pp → pp3π0 → pp6γ reaction were checked and compared with the Monte-Carlo predictions of the pp → pp3π0(blue curve) and pp → ppη (red curve) reaction, also the theoretical χ2 distributionforthe(Number ofDegrees ofFreedom)NDF =5 ispresented. ThediﬀerencebetweenthedataandMonte-Carloisvisibleand itis related to the contributionin thedata ofdiﬀerent kind ofprocesses then pp → pp3π0 reaction also the non Gaussian error response of the detector plays here an important role. Nevertheless all of the events related to these factorspopulatelowprobability area(near 0)and they could be rejected by selecting higher probability region where the probability function is getting ﬂat and is consistent with Monte-Carlo. The region of the Prob > 0.2 where the probability function(Fig. 37) is getting ﬂatwasselectedforthelateranalysis. Thisadditional cut(Prob > 0.2),introducedin ordertopurifythedata,lowerstheTotalReconstruction Eﬃciency(asdeﬁnedinEq.18). ViaMonte-Carlo simulations withhomogeneously and isotropically populated phase space one can calculate Total Reconstruction Eﬃciency including the above cut: Tot.Rec.Eff =2.26% (20) The following statistical relation for the kinematic ﬁt based on the χ2 methodforthe residuals shouldhold(seeAppendixE) : σ2 −σ2 σresidual = mf (21) Jagiellonian University 52 Benedykt R. Jany σ(E )/E Rec Rec 0.3 0.25 0.2 0.15 0.1 0.05 00 0.2 0.4 0.6 0.8 (a) Error of photon energy. RecE [GeV] 1 5 4.5 4 σ(θ ) [deg] σ(φ ) [deg] Rec σ(φ ) [deg] 2.6 2.4 2.2 2.8 2.6 2 2.4 2.2 1.8 2 1.8 1.6 1.6 σ(θ ) [deg] Rec 5 4.5 3.5 4 3.5 3 3 2.5 2.5 1.4 2 1.4 1.2 1.5 1 2 1.5 0.8 1.2 1 0.1 0.1 0.2 0.2 160 0.3 160 0.3 0.4 1 0.4140 1400.5 0.5120 1200.6 0.6100 1000.7 0.780 800.8 0.8 0.9 0.8 0.9 1 60 60 40 40 20 1 20 1 (b) Error ofphoton θRec angle. (c) Error of photon φRec angle. Figure 34: The error parametrization for photons in the CD detector used in the kinematic ﬁt. Results of the Monte-Carlo simulations. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting Jagiellonian University 53 BenedyktR.Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA (a) Error ofproton θ angle. (b) Error ofproton φ angle. Figure35:The errorparametrizationforprotonsintheFDdetector usedin the kinematic ﬁt. Results of the Monte-Carlo simulations. Jagiellonian University 54 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting (a) Logarithmic scale. (b) Linear scale. Jagiellonian University 55 Benedykt R. Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA (a) Logarithmic scale. (b) Linear scale. Figure37:Theprobabilitydistributionforthekinematic ﬁtof pp→ pp3π0 → pp6γ reaction. Jagiellonian University 56 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting and residual = m −f (22) where m -measured value(before kinematic ﬁt) of energy, polar angle θ, azimuthal angle φ; f -ﬁtted value(after kinematic ﬁt) of energy, polar angle θ, azimuthal angle φ; σresidual -the standard deviation of the residual value distribution;σm -the standard deviation of the measured value (the error of the measured value); σf -thestandarddeviationofthe ﬁtted value(the error of the ﬁtted value). The only possibility to calculate the σm and σf is via Monte-Carlo simulations. Usually σm and σf might be very similar, the correction of the kinematic ﬁt could be very small. This could lead to the very high numerical inaccuracy of the diﬀerence σm 2 −σf2 . To avoid this problem instead of calculation directly this diﬀerence and comparing it with the σresidual it could bedone other way by changing the(Eq. 21) to the form: σf = σ2 −σ2 (23) m residual Now one compares the Jσ2 −σ2 with the σf, here always σm will m residual be diﬀerent from σresidual. Therelation(Eq.23)wascheckedforthekinematic ﬁtof thedata. The σm wasextractedbyﬁtting trueminusmeasured(reconstructed) valuedistribution withtheGaussianfunction usingMonte-Carlo simulation. Theσf was extracted by ﬁtting true minus ﬁtted value distribution with the Gaussian function usingMonte-Carlo simulation. Theσresidual wasderivedfromthe experimentaldatabyﬁttingGaussianfunctiontothemeasured(reconstructed) minus ﬁtted value distribution. The estimation of the parameters was done forallﬁtted variablesi.e. energy of thephotons,polarangle θ ofphotons and protons and azimuthal angle φ of photons and protons. This task was done by meansof theMonte-Carlosimulations. Thedistributionsusedfortheparameter extraction(Figs.38,39,40,41,42) are all centered at 0 value. The measured minus ﬁtteddistribution,(rightplotsonFigs.38,39,40,41,42), shows good agreement between experimental data and Monte-Carlo simulation. The results of the comparison are presented in Table 4. It is seen, in the last column of the table, that the σf diﬀer from the calculated one Jσ2 −σ2 by a few percent, the ratio is almost equal 1. This proves m residual thecorrectnessof thekinematic ﬁtprocedureaswell astheproperlyderived error parameterization. Jagiellonian University 57 Benedykt R. Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA Figure 38: Photon Energy. left: Monte-Carlo Simulation, true minus reconstructed value divided by the reconstructed value.center: Monte-Carlo Simulation, true minus ﬁtted value divided by the ﬁtted value. right: Reconstructed minus ﬁtted value divided by the ﬁtted value,gray area corresponds to the experimental data, blue line to the Monte-Carlo simulation. Black line corresponds to the ﬁt of the Gaussian function. Figure 39: Photon polar angle θ. left: Monte-Carlo Simulation, true minus reconstructed value.center: Monte-Carlo Simulation, true minus ﬁtted value. right: Reconstructed minus ﬁtted value,gray area corresponds to the experimental data, blue line to the Monte-Carlo simulation. Black line corresponds to the ﬁt of the Gaussian function. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA Figure 40: Photon azimuthal angle φ. left: Monte-Carlo Simulation, true minus reconstructed value.center: Monte-Carlo Simulation, true minus ﬁtted value. right: Reconstructed minus ﬁtted value,gray area corresponds to the experimental data, blue line to the Monte-Carlo simulation. Black line corresponds to the ﬁt of the Gaussian function. Figure 41: Proton polar angle θ. left: Monte-Carlo Simulation, true minus reconstructed value.center: Monte-Carlo Simulation, true minus ﬁtted value. right: Reconstructed minus ﬁtted value,gray area corresponds to the experimental data, blue line to the Monte-Carlo simulation. Black line corresponds to the ﬁt of the Gaussian function. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA Figure 42: Proton azimuthal angle φ. left: Monte-Carlo Simulation, true minus reconstructed value.center: Monte-Carlo Simulation, true minus ﬁtted value. right: Reconstructed minus ﬁtted value,gray area corresponds to the experimental data, blue line to the Monte-Carlo simulation. Black line corresponds to the ﬁt of the Gaussian function. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting from MC σm from MC σf from Exp. σresidual Jσ2 m −σ2 residual σf√ σ2 m−σ2 residual Ephoton 0.1081 0.0770 0.0744 0.0785 0.9812 θphoton [deg] 1.5270 1.4995 0.2030 1.5138 0.9906 φphoton [deg] 1.8920 1.8200 0.2198 1.8792 0.9685 θproton [deg] 0.0996 0.0999 0.0087 0.0992 1.0070 φproton [deg] 0.4908 0.5068 0.1143 0.4773 1.0618 Table 4: The check of the statistical relation for the kinematic ﬁt. The π0’s and photons To see whatis thehadronic split oﬀlevel for thephotonin the electromagnetic calorimeterintheCDdetector,the energy of thelowest energeticphoton from a photon pair versus opening angle between the two photons were plotted for two probability regions Prob < 0.2 and Prob > 0.2, the experimental data were compared with a Monte-Carlo simulation of pp → pp3π0 (Fig. 43). The area of the small energies and small angles, where one should expect the hadronic split-oﬀs1
, is not populated. This means that there is low level of the hadronic split oﬀs for both probability cut regions. The Monte-Carlo describes well the experimental data. The identiﬁcation and reconstruction of π0 mesons from photons in CD detectorby thekinematic ﬁt waschecked. Theinvariant massof twophotons forming π0 which were selected by the combinatorics was plotted, for the probability region Prob > 0.2 both for experimental data and Monte-Carlo (Fig. 44). The peak position corresponds to the theoretical π0 mass. The experimental spectrum agrees with the Monte-Carlo simulation. The protons The identiﬁcation of protons in FD detector by the kinematic ﬁt was also checked. The energy loss in FHR detector versus the kinetic energy derived from the kinematic ﬁt for the proton for Prob > 0.2 was plotted both for experimental data and Monte-Carlo (Fig. 45). The experimental data are consistent with theMonte-Carlosimulation, whichprovesthatthemeasured particles are protons. Thereconstructionofprotonkineticenergybythekinematic ﬁtwasfound out using Monte-Carlo simulation by plotting true kinetic energy versus the kineticenergyderivedfromthekinematic ﬁtforproton(Fig.46(a)),which shows the correlationline. From the width of thediﬀerencebetween the true 1 secondary interaction of the photon leading to spurious cluster in calorimeter Jagiellonian University 63 Benedykt R. Jany 2400 2200 800 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA Jagiellonian University 64 BenedyktR.Jany Energy [GeV] Energy [GeV] Energy [GeV] Energy [GeV] 0.5 0.5 700 0.4 600 2000 1800 1600 1400 0.4 5000.3 0.3 1200 400 10000.2 0.2 800 300 6000.1 0.1 200 400 100 0 200 0 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Opening Angle [deg] Opening Angle [deg] (a) Prob < 0.2, Experimental Data. (b) Prob > 0.2, Experimental Data. 2500 2000 800 0.5 700 0.5 0.4 0.4 600 500 1500 0.3 0.3 0.2 1000 0.2 300 400 0.1 500 0.1 200 0 0 100 20 40 60 80 100 Opening Angle [deg] 120 140 160 180 0 20 40 60 80 100 Opening Angle [deg] 120 140 160 180 0 (c) Prob < 0.2, Monte-Carlo simulation. (d) Prob > 0.2, Monte-Carlo simulation. Figure43:Energy of thelowest energeticphotonfromaphotonpairversus opening anglebetweenthetwophotons. The area of the small energies and small angles, where one would expect the hadronic split-oﬀs, is not populated. The experimental data are well described by the Monte-Carlo simulation. 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting kinetic energy and the kinetic energy derived from the kinematic ﬁt divided by truekinetic energy(Fig.46(b)) the resolution of the reconstructed kinetic energy was estimated to be ∼ 12%. The checks on the experimental data were also performed. Small kinetic energies of the protons in FD detector reconstructed by the kinematic ﬁt were selected EKin < 0.6 GeV for Prob > 0.2 and compared with the kinetic energy reconstructed using FRH detector. The correlation is visible (Fig.47(a)),thepeakindiﬀerence(Fig.47(b))conﬁrmsit. For the consistency, additional cross check was performed for protons of kinetic energy EKin < 0.6 GeV. The kinetic energy derived from the kinematic ﬁt was also recalculated to the energy loss in FRH detector by the Monte-Carlo method and compared with the measured energy loss in FRH,the correlation and apeakindiﬀerenceis visible(Fig.48). Moreover the plot of diﬀerence of energy loss in FRH and the energy loss recalculated using obtainedfromkinematic ﬁtprotonkineticenergyiscentered at 0value. It conﬁrms the correctness of kinetic energy calculation by the kinematic ﬁt. Jagiellonian University 65 Benedykt R. Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA Edep FRH [GeV] 0.35 0.25 1 0.2 0.4 10 0.3 0.15 10-1 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ekin KFIT [GeV] (a) Experimental Data. 0.4 Edep FRH [GeV] 0.35 0.25 0.2 1 0.15 0.1 0.05 10-1 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ekin KFIT [GeV] (b) Monte-Carlo Simulation. Figure 45: Energy loss in FHR detector versus the kinetic energy derived from the kinematic ﬁt for the proton for Prob > 0.2. The position of the proton band is the same for both above cases. 10 0.3 Jagiellonian University 66 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting (a) Truekinetic energy versus thekinetic energyderivedfrom the kinematic ﬁt for proton, correlation line visible. (b) Diﬀerence between the truekinetic energy and the kinetic energyderivedfromthekinematic ﬁtdividedby truekineticenergy for proton. Figure 46: Monte-Carlo simulation of pp → pp3π0 test of the proton kinetic energy reconstruction by the kinematic ﬁt for Prob > 0.2. Jagiellonian University 67 Benedykt R. Jany 4.4 The Kinematic Fitting 4 ANALYSIS OF THE EXPERIMENTAL DATA (b) Diﬀerence between the kinetic energy reconstructed using FRH detector and the kinetic energy derived from the kinematic ﬁt for proton. Jagiellonian University 68 Benedykt R. Jany 4 ANALYSIS OF THE EXPERIMENTAL DATA 4.4 The Kinematic Fitting (a) Energy loss in FRH detector versus the energy loss recalculated from the kinetic energy from the kinematic ﬁt for proton, correlation line visible. (b) Diﬀerence between energy loss in FRH detector and the energy loss recalculated from the kinetic energy from the kinematic ﬁt for proton. Figure 48: Experimental data, small kinetic energies of the protons in FD detector selected EKin < 0.6GeV, Prob > 0.2. Jagiellonian University 69 Benedykt R. Jany 5 Results and error discussion The selection involving kinematic ﬁtting of the events was performed. The complete pp3π0 ﬁnal state events are selected. The missing mass of the twoprotons(Fig.49) shows the η → 3π0 meson decay peak and a prompt 3π0 production. The Monte-Carlo simulation of pp → pp3π0 assuming homogeneously and isotropically populated phase space does not describe the experimentaldata with excluded η mesonproduction -thenonresonantpart. 5.1 The pp→ pp3π0 reaction 5.1.1 The model description Introduction -the idea of the Model The pp → pp3π0 reaction was separated from pp → ppη(3π0) reaction 71 Figure 50: Comparison of the experimental data with the Monte-Carlo phase space simulation for the MMpp < 0.5 GeV/c2 and MMpp > 0.6 GeV/c2 . Upper row: proton in the center of mass frame,Middle row: pion in the center of mass frame, Lower row: 3π0 system in the center of mass frame. From left kinetic energy, the polar angle and the azimuthal angle distribution. The experimental data are shown as a black marker, the Monte-Carlo simulationasablueline. Vertical axes -numberof events(ingivenbin) isshown. 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction by introducing a cut on the Missing Mass of two protons (Fig. 49). The region without the η meson signal was selected as MMpp < 0.5 GeV/c2 and MMpp > 0.6 GeV/c2 . Next the distribution of kinetic energy, the polar angle θ and the azimuthal angle φ fortheprotons,pions and 3π0 system, in the center of mass system was compared to the Monte-Carlo simulation of pp → pp3π0 assuminghomogeneously andisotropicallypopulatedphase space(Fig.50). The kinetic energy distributions and the polar angle distributions diﬀer strongly from the simulation: the phase space simulation does not describe this process. The following scheme of the dynamics is proposed: the pp → pp3π0 reaction is sequential two-body process pp→ R1R2 → pp3π0 (24) R1,2 -unstableparticle(BaryonResonance). Searchingforthe candidatesforR1,2 inthe mostsimpleform one maytake the two lowest lying resonances like Δ(1232) and N∗(1440), since the availablephase space volumeforthem willbethebiggest(later other arguments supportingfeasibility ofthe choice willbepresented). TheΔ(1232)candecay Δ(1232) → pπ0 and the N∗(1440)has twodominantdecaybranches, directly N∗(1440) → pπ0π0 or sequentially N∗(1440) joint reaction mechanism: → π0Δ(1232) → π0pπ0 . The pp→ Δ(1232)N ∗ (1440)→ pp3π0 (25) could be written as a two Feynman graphs (Fig. 51). (a) Model1 (b) Model2 Figure 51: Two model possibilities of pp→ pp3π0 production. Jagiellonian University 73 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION Supporting arguments The arguments that support theidea oflowest availablebaryon resonances playingrole of R1 and R2 in scheme(Eq.24) arebothqualitative andquantitative. The resonances R1,R2 should excite withlargeprobabilityinpp reaction and decay predominantly into channels with one or two pions. Δ(1232) decaysintonucleon andpion(∼ 100%) while N∗(1440)decay sequence al-most always leads to states with two pions. In Table 5 branching ratios of decays ofdiﬀerentbaryon resonances thatcouldplay role of R2 arepresented, together with Qvalue of considered R1R2 channels. It is seen that N∗(1680) and Δ(1700) arekinematically notpossible. Oneconcludesthatthebranching is considerably highest for N∗(1440), Δ(1600), Δ(1620). Moreover, the last two are known to be monopole excitations of Δ(1232) that excite extremely weaklyinpp reaction[1,67]. Also availablephase space wouldbe lower for Δ(1600), Δ(1620) as seen from the Table 5. TotalCrossSectionforpp→ pX ispresented(Fig.52); one concludesthat the highest cross section isin case of Δ(1232) and N∗(1440). Similar conclusion canbedrawnfromliteraturedata(Figs.53,54,55). Forthelowerproton momentum 2.85 GeV/c (Fig.53)than this work(i.e.3.35 GeV/c)only strong Δ(1232) signal is seen. In case of higher momentum 4.55 GeV/c (Fig. 54) the strong Δ(1232) signal and clear evidence of N∗(1440)is seen. When the momentum of the proton increases 6.2−29.7 GeV/c (Fig. 55)the Δ(1232) excitationloosesitsimportance, higherbaryon resonances appear with much bigger probability. Since experiment considered in this work was performed at 3.35 GeV/c proton momentum, from the above data (Figs. 53, 54, 55) one can conclude that the most strongly excited baryon resonances will be Δ(1232) and N∗(1440). One can present as well qualitative arguments that the simultaneous excitation of two Roper(N∗(1440)) resonances (playing role of R1R2) is also unfeasible. In the last row of Table 5 this case is considered; the Q value is negative which makes the process kinematically unfavored. In addition it is seen that the excitation of the Roper resonance is considerably lower than that of Δ(1232)(Figs.52,53,54). This makes simultaneous excitation of two Roper resonances rather unprobable in comparison to the Δ(1232)N∗(1440) system. Theproduction of3π0 via simultaneous excitation of Δ(1232)and N∗(1440) is considered as apredominant and also supportedby[13,19](seeSection3 on page 19). Jagiellonian University 74 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction R1 R2 TThr. Beam [GeV] PThr. Beam [GeV/c] Q [MeV] properties Δ(1232)P33 BR(Nπ)= 100% N∗ (1440)P11 a) BR(Nππ)= 5%−10% b) BR(πΔ(1232)) = 20%−30% 1.930 2.710 207 will be strongly excited Δ(1232)P33 BR(Nπ)= 100% N∗ (1520)D13 a) BR(Nππ)< 8% b) BR(πΔ(1232)) = 15%−25% 2.161 2.954 127 will be excited Δ(1232)P33 N∗ (1535)S11 BR(Nπ)= 100% a) BR(Nππ)< 3% 2.205 R2 spin ﬂip excitation(weak), will be b) BR(πΔ(1232)) < 1% 3.000 c) BR(πN∗ (1440))< 7% 112 weakly excited [1, 67] Δ(1232)P33 BR(Nπ)= 100% Δ(1600)P33 a) BR(πΔ(1232)) = 40%−70% b) BR(πN∗ (1440))= 10%−35% 2.400 3.203 47 R2
monopole excitation of Δ(1232)P33, will be weakly excited [1, 67] Δ(1232)P33 BR(Nπ)= 100% Δ(1620)S13 a) BR(πΔ(1232)) = 30%−60% b) BR(πN∗ (1440))= 11% 2.460 3.270 27 R2
monopole excitation of Δ(1232)P33, will be weakly excited [1, 67] Δ(1232)P33 BR(Nπ)= 100% N∗ (1680)F15 a) BR(πΔ(1232)) = 30%−40% 2.644 3.457 −33 Kinematically not possible Δ(1232)P33 BR(Nπ)= 100% Δ(1700)D33 a) BR(Nππ)= 80%−90% 2.706 3.522 −53 Kinematically not possible N∗ (1440)P11 BR(Nπ)= 55%−75% N∗ (1440)P11 a) BR(Nππ)= 5%−10% b) BR(πΔ(1232)) = 20%−30% 2.545 3.355 −1 Kinematically not possible Table 5: Possible excitations of baryon resonances(R1,2 ) in the reaction pp → R1R2, which give pp3π0 ﬁnal state, at incident proton momentum of 3.35 GeV/c. The excess energy Q,the beam kinetic energy at threshold TThr. and also the beam momentum at threshold PThr. is given. Beam Beam Jagiellonian University 75 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION Figure 52: Total Cross Section for the reactions pp → pΔ+(1232), pp → pN∗(1440)+ , pp→ pN∗(1520)+ as afunction ofbeam momentum. Thebeam momentum at threshold 1.262 GeV/c, 1.852 GeV/c, 2.081 GeV/c and beam kinetic energy at threshold 0.634 GeV, 1.138 GeV, 1.345 GeV for Δ+(1232), N∗(1440)+ and N∗(1520)+ respectively. The highest cross section in case of Δ(1232) and N∗(1440)visible. Data comefrom[68] Jagiellonian University 76 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction 2.85 GeV/c incident proton momentum, a) 2.292 deg; b) 2.911 deg; c) 3.845 deg; d) 5.288 deg; e) 6.503 deg f) 7.758 deg. The strong Δ(1232) signal visible. Theplot comesfrom[69]. Jagiellonian University 77 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION 4.55 GeV/c incident proton momentum, a) 2.292 deg; b) 2.911 deg; c) 3.845 deg; d) 5.288 deg; e) 6.503 deg f) 7.758 deg. The strong Δ(1232) signal and clear evidence of N∗(1440)visible. Theplot comesfrom[69]. Jagiellonian University 78 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction 29.7 GeV/c. The spectra are presented for the four momentum transfer −t =0.044 GeV2/c2(Upper Plot)and −t =0.88 GeV2/c2(Lower Plot). The plot comesfrom[70]. Jagiellonian University 79 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION The derivation of the model parameters from experimental data To check the validity of the hypothesis that the pp → pp3π0 reaction follows via simultaneous excitation of two baryon resonances The pp → Δ(1232)N∗(1440) reaction was simulated using kinematic calculations by PLUTO++eventgenerator[71], without anyinteractionbetweenthe Δ(1232) and N∗(1440)-homogeneously andisotropicallypopulatedphase space was assumed (see Appendix C). Two decay paths of the N∗(1440) the direct decay Model1 (Fig. 51(a)) and sequential decay Model2 (Fig. 51(b)) were taken into account. The experimental data were compared with these two simulations by comparing the following distributions: • Dalitz Plot ppX (M2(pp)versus M2(p3π0))[72–75] Theinteractionsbetweentheprotons and thethreepion system could be examined. • Dalitz Plot 3π0 (M2(2π0)versus M2(2π0))[72–75]

The correlations between the pions could be examined. • Nyborg Plot(M(pπ0)versus M(pπ0π0)[76] The resonances in the π0 −p and π0π0 −p system could be studied. Here M statesfor the invariant mass. Thedistributions(Figs.56,57,58) were studied for ﬁve diﬀerent regions of the missing mass of two protons MMpp, excluding the region of the η meson (MMpp =0.5 − 0.6 GeV/c2) (Fig.49and Table 6). The selection of the MMpp ranges implies the boundaries on the maximal available kinetic energy for the 3π0 system in its rest frame(QMax ) and simultaneously on the maximal available kinetic energy 3π0 for the pp systeminits restframe(QMax )Table 6. More details about the pp variables choice canbefoundinSection3.2 onpage27 andinAppendixA. MMpp GeV/c2 0.4−0.5 0.6−0.7 0.7−0.8 0.8−0.9 0.9−1.0 QMax 3π0 MeV ∼ 100 ∼ 300 ∼ 400 ∼ 500 ∼ 600 QMax pp MeV ∼ 600 ∼ 400 ∼ 300 ∼ 200 ∼ 100 Table6:Selected ﬁvediﬀerent regionsof themissing massof thetwoprotons MMpp. Description in text. One concludesthatthetwo modelspopulatediﬀerent areasontheplots. To check if one could describe the experimental data with a sum of the two models: Jagiellonian University 80 Benedykt R. Jany Dalitz Plot ppX 01 Dalitz Plot ppX 03 Dalitz Plot ppX 04 Dalitz Plot ppX 05 Dalitz Plot ppX 06 800 1000 900 800 400 350 300 6 2/c 4 5.2 ) GeV2/c 4 2 4.44.8 4.3 3.9 4.6 4.2 4.4 4.1 3.8 ) GeV2/c 4 2 M2(p ) GeV2/c 4 ) GeV2/c 4 ) GeV 1000 700 700 5 2 2 p 1 p 1 p 1 p 1 800 M2(p M2(p 600 600800 M 700 5.5 4.8 4.6 500 500 250 4 5 600 4.2 600 400 4.4 500 400 2003.9 3.7 4.5 4.2 400 4 3.8 300 300 150 4 400 300 3.6 3.7 3.8 200 200 100 2(pp 4 1 3.8 200 200 3.6 100 3.6 100 503.6 100 3.5 3.5 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c4 1,2 1,2 1,2 1,2 1,2 Dalitz Plot ppX 01 Dalitz Plot ppX 03 Dalitz Plot ppX 04 Dalitz Plot ppX 05 Dalitz Plot ppX 06 M2(p 800 1000 800 400 2/c 4 4.44.85.2 4.3 3.9 4.6 4.2 4.8 4.6 500 500 2505 600 4 4.2 600 400 4.4 500 400 2003.7 3.9 4.5 4.2 4400 3.8 300 300 150 4 400 300 3.6 3.7 3.8 200 200 100 M2(p ) GeV2/c 4 4 3.8 200 200 3.6 100 3.6 100 503.6 100 3.5 3.5 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c4 1,2 1,2 1,2 1,2 1,2 Dalitz Plot ppX 01 Dalitz Plot ppX 03 Dalitz Plot ppX 04 Dalitz Plot ppX 05 Dalitz Plot ppX 06 ) GeV2/c 4 2 M2(p ) GeV2/c 4 ) GeV2/c 4 6 ) GeV 1000900700 700 350 5 p 12 5.5 2 2 2 p 1 p 1 p 1 p 1 800 M2(p M2(p M2(p 600 600 300800700 4.4 4.1 3.8 800 1000 800 400 6 4.82 /c 4 5.2 ) GeVV V 4.4 V V 900700 5 4.31000 700 3.9 350 ) Ge2 /c 4 24.6) Ge2 /c 4 22 p1p1p1800 ) Ge2 /c 4 2 p1 ) Ge2 /c 4 2 p1 M 2 (p5.5 2 (pM 2 (p M 2 (p4.2 M 2 (p 4.8M600 800 600 300 700 4.4 4.1 3.8 5 500 4.6 600 4 500 250 400 4.4 500 4.2 600 3.9 400 3.7 200 4.5 300 4.2 400 4 400 3.8 300 150 4 200 3.8 4 200 300 3.8 200 3.6 3.7 200 3.6 100 100 3.6 100 3.6 3.5 100 3.5 50 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 3π0) GeV2/c 4 1,2 M2(p 3π0 ) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p Figure 56: Dalitz Plot ppX. M2(pp) versus M2(p3π0). The upper row corresponds to the experimental data, the middle row to the Monte-Carlo Model1, the lower row to the Monte-Carlo Model2 (Fig. 51). The columns from left to right correspond to the following Missing Mass of two protons bins, column 1 MMpp = 0.4 −0.5 GeV/c2,column 2 MMpp =0.6 − 0.7 GeV/c2, column 3 MMpp =0.7 − 0.8 GeV/c2, column 4 MMpp = 0.8−0.9 GeV/c2, column 5 MMpp =0.9−1.0 GeV/c2 .Theplotsaresymmetrized againsttwoprotons -each event is ﬁlled two times. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 81 BenedyktR.Jany Dalitz Plot 3π0 01 Dalitz Plot 3π0 03 Dalitz Plot 3π0 04 Dalitz Plot 3π0 05 Dalitz Plot 3π0 06 2500 1600 300 2/c4 0.35 0.3 1 π 02) GeV2/c 4 1000 1800 1400 1600 250 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 82 BenedyktR.Jany 0.6 0.7 0.6 0.5 0.4 M 0.12 0.5 0.35 2000.25 0.4 120010001500 0.3 600 0.4 0.1 1000800 1500.2 0.3 0.25 8001000 0.3 6004000.08 0.2 0.15 100600 0.2 4000.15 0.2 400500 0.1 2000.06 50 0.1 200 0.1 200 0.1 0.05 0.05 0.04 0 0 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c4 Dalitz Plot 3π0 01 Dalitz Plot 3π0 03 Dalitz Plot 3π0 04 Dalitz Plot 3π0 05 Dalitz Plot 3π0 06 232323 2323 M2(π 0 M2(π 0 1 π 0) GeV2/c 4 2 π 02) GeV2/c 4 0.5 0.45 π 02) GeV2/c 4 0.14 2(π 0 π 0) GeV 12 2000 M2(π 0 1 M2(π 0 1 1200 1400800 2500 1600 300 M ) GeV 0.5 0.6 0.7 0.35 0.14 0.45 0.60.50.3 0.4 M2(π 10 π 20) GeV2/c 4 0.12 0.5 0.35 2000.25 0.4 12001000 1500 0.3 600 0.4 0.1 1000800 1500.2 0.3 0.25 8001000 0.3 6004000.08 0.2 0.15 100600 0.2 400 0.2 0.15 400500 0.1 2000.06 50 0.1 200 0.1 200 0.1 0.05 0.05 0.04 0 0 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c4 Dalitz Plot 3π0 01 Dalitz Plot 3π0 03 Dalitz Plot 3π0 04 Dalitz Plot 3π0 05 Dalitz Plot 3π0 06 232323 2323 M2(π 01 ) GeV2/c 4 π 0) GeV2/c 4 2 π 02) GeV2/c 4 2/c4 1800 1000 1400 1600 250 2(π 0 π 0 12 π 0 2 2000 M2(π 0 1 M2(π 0 1 1200 1400800 2500 1600 ) Ge2 /c 40.6 ) Ge2 /c 40.542 /c) Ge2 /c 4 0.35 ) GeV 1800 3000.7 2 /c 4 1 V 2 π 00.5 V 2 π 0 0.45 V 2 π 0 10.3 2 π 0 1 0.14 1000 1400 2000 1 1600 V 2 π 00.6 250 1 ) Ge 0 M 2 (π 0 M 2 (π0.402 (π 0 M 2 (π1200 0 M 2 (π M0.12 1500 0.25 0.35 800 1000 0.4 1200 1400 0.5 200 0.1 0.2 600 0.25 0.3 800 0.3 1000 0.4 150 0.08 1000 0.15 400 0.2 600 0.2 600 800 0.3 100 0.06 500 0.1 200 0.15 400 400 0.2 50 0.1 200 0.1 200 0.1 0.04 0 0.05 0 0.05 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ) GeV2/c 4 3 π0 2 M2(π0 ) GeV2/c 4 3 π0 2 M2(π0 ) GeV2/c 4 3 π0 2 M2(π0 ) GeV2/c 4 3 π0 2 M2(π0 ) GeV2/c 4 3 π0 2 M2(π0 Figure57: DalitzPlot3π0 . M2(2π0)versus M2(2π0).The upper row correspondstothe experimentaldata,the middle row to theMonte-Carlo Model1,thelower rowtotheMonte-Carlo Model2 (Fig. 51). The columns fromleft to right correspond tothefollowingMissingMass oftwoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9−1.0 GeV/c2 . The plots are symmetrized against three pions -each event is ﬁlled six times. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. Nyborg Plot pπ0 vs pπ0 π0 01 Nyborg Plot pπ0 vs pπ0π0 03 Nyborg Plot pπ0 vs pπ0π0 04 Nyborg Plot pπ0 vs pπ0π0 05Nyborg Plot pπ0 vs pπ0π0 06 200020001000 1.6 π 0) GeV/c2 1400 12001800 1800 900 π 0 π 0 1 π 0 1.5 M(p 1 1 1 M(p M(p M(p 1200 1600 1600 800 1000 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 83 BenedyktR.Jany 1.7 1.7 1.6 1.51.6 1.6 1.5 1.5 1.5 1400 1400 7001.4 1000 1.4 8001200 1200 6001.4 1.4 1.4 800 1000 1000 5001.3 1.3 6001.3 1.3 1.3 600 800 800 400 1.2 400 1.2 600 600 3001.2 1.2 1.2 400 400 400 200 2001.1 1.1 200 1.1 1.1 1.1 200 200 100 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 223 223 223 223 223 Nyborg Plot pπ0 vs pπ0 π0 01 Nyborg Plot pπ0 vs pπ0π0 03 Nyborg Plot pπ0 vs pπ0π0 04 Nyborg Plot pπ0 vs pπ0π0 05Nyborg Plot pπ0 vs pπ0π0 06 π 0) GeV/c2 1 ) GeV/c2 1 M(p ) GeV/c2 1 ) GeV/c2 1 2000200010001.7 1.7 1.6 1.51.6 1.6 1.5 1.5 1.5 1400 1400 7001.4 1000 1.4 8001.4 1200 1200 600 1.4 1.4 800 1000 1000 1.3 5001.3 6001.3 1.3 1.3 600 800 800 400 1.2 400 1.2 π 0) GeV/c2 600 600 300 1 1.2 1.2 1.2 400 400 400 200 2001.1 1.1 200 1.1 1.1 1.1 200 200 100 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 223 223 223 223 223 Nyborg Plot pπ0 vs pπ0 π0 01 Nyborg Plot pπ0 vs pπ0π0 03 Nyborg Plot pπ0 vs pπ0π0 04 Nyborg Plot pπ0 vs pπ0π0 05Nyborg Plot pπ0 vs pπ0π0 06 ) GeV/c2 1 M(p ) GeV/c2 1 ) GeV/c2 1 π 0) GeV/c2 1 1.6 140012001800 1800 900 π 0 π 0 1 π 0 M(p 1 1.5 1 1 1 1200 1600 1600 800 M(p M(p M(p 1000 200020001000 M(p ) GeV/c2 π 0) GeV/c2 1 1.7 1.7 1.6 1.6 1.51.6 1.6 1.5 1.5 1.5 1.5 1400 1400 7001.4 1000 1.4 8001.4 1200 1200 600 1.4 1.4 800 1000 1000 1.3 5001.3 6001.3 1.3 1.3 600 800 800 400 1.2 ) GeV/c2 400 1.2 600 600 300 1 1.2 1.2 1.2 400 400 400 200 2001.1 1.1 1.1 200 1.1 1.1 200 200 100 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 223 223 223 223 223 Figure58: Nyborg Plot. M(pπ0)versus M(pπ0π0).The upper row corresponds to the experimental data, the middle row to theMonte-Carlo Model1,thelower rowtotheMonte-Carlo Model2 (Fig.51). The columns fromleft to right correspond tothefollowingMissingMass oftwoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = M(p ) GeV/c2 1 0.9−1.0 GeV/c2 . The plots are symmetrized against two protons and three pions -each event is ﬁlled six times. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. ) GeV/c2 1 140012001800 1800 900 π 0 1 π 0 π 0 1 π 0 1 1 1 1 1200 1600 1600 800 M(p M(p M(p 1000 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION βModel1 +(1−β)Model2 (26) one needs to know the fraction of Model1 to the Model1 + Model2 i.e. the β parameter -which actually is the fraction of the direct decay to the sum of direct and sequential decay of the N∗(1440). To estimate the β from the experimental data one needs to ﬁt the shape of the plots (Figs. 56, 57, 58) for mixture of Model1 and Model2 to the experimental data. The population of the events in missing mass of two protonsisnotapurposeof the ﬁt,andit willbeextracted aftertheshape ﬁt. Fit was performed using the chi-square method; one deﬁnes the χ2 function to minimize: [Data−(βModel1 +(1 −β)Model2)]2 χ2 = (27) σ2 + β2σ2 +(1 −β)2σ2 Data Model1 Model2 where the sum goes over each bin of the ﬁve Dalitz Plots ppX, ﬁve Dalitz Plots 3π0 and ﬁve Nyborg Plots i.e. 15 plots, all in all 19752 data point are ﬁtted simultaneously. The σData is the error of the point for experimental data, σModel1 -istheerrorof thepointfor Model1 and σModel2 -is the error of the point for Model2. For the numerical purpose of doing the ﬁt, the χ2 function(Eq.27)was redeﬁned to: [Data−(αModel1 ∗ +(1 −α)Model2 ∗)]2 χ2 = (28) σ2 + α2σ2 +(1 −α)2σ2 Data Model∗ Model∗ 12 where the models were normalized to the experimental data: Model 1 ∗ = � Data Model1 Model1 Model 2 ∗ = � Data Model2 (29) Model2 Now parameter α is the fraction of Model1 ∗ to the Model1 ∗ + Model2 ∗ , so one can write: Model1 ∗ α = (30) Model2 ∗ 1−α to get the ratio of the Model1 to Model1 + Model2 onejust redoes the normalization(Eq.29), no otherfactoris needed sincethe same amount of events for Model1 and Model2 was generated: Jagiellonian University 84 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction Model1 ∗ Model1 Model 1 == ∗ Model2 ModelModel2 2 α � Model1 = 1−α� Model2 = β = 1−β (31) 23000 22000 21000 20000 19000 18000 17000 16000 15000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 α (a) 14814.6 14814.4 Δα=0.00515 14814.2 14814 14813.8 Δχ2=1 14813.6 14813.4 14813.2 min=0.1035α 0.1 0.101 0.102 0.103 0.104 0.105 0.106 α (b) Figure 59: χ2 versus the searched parameter α for the sum of Model χ 2 χ 2 ∗ and 1 ∗ Model ﬁt. 2 The χ2 function(Eq.28)was minimizein respect toparameter α. The χ2 versus the searched parameter α (Fig. 59), the function has one minimum. Jagiellonian University 85 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION The estimated value of the α parameter is the one which gives the smallest χ2, the error of theparameterishalf of thedistancefor which the χ2 function changes by 1, this gives: χ2 14813.29 min = =0.750±0.010 (32) NDF 19751 where χ2 is the χ2 value at minimum, and NDF is the Number ofDegrees min of Freedom. α =0.1035±0.0026 . (33) It corresponds(Eq.31) to ratio: Model1 =0.0580±0.0016 (34) Model2 giving the β =0.0548±0.0015 (35) The same ﬁt was performed where the spectra for the missing mass region 0.4− 0.5 GeV/c2 were excluded, to check the sensitivity of the of the procedure, this gave the following result; � χ2 14166.21 NDF 2 = 18308 = 0.774±0.011 (36) α2 = 0.1032±0.0028 (37) min �Model1 =0.0568±0.0017 (38) Model22 β2 =0.0537±0.0015 (39) The diﬀerence between these two results was used to estimate the systematic errors: Δαsys. = |α−α2|=0.0003 (40) Δ �Model1 Model2 = sys. �Model1 = 0.0012 (41) Model1 − Model2 Model22 Δβsys. = |β −β2|= 0.0011 (42) The ﬁnal values with the systematic errors are: α = 0.1035±0.0026(stat.)±0.0003(sys.) (43) Jagiellonian University 86 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction Model1 =0.0580±0.0016(stat.)±0.0012(sys.) (44) Model2 β =0.0548±0.0015(stat.)±0.0011(sys.) (45) To verify how the Monte-Carlo simulation based on the sum of two models, with the ﬁtted parameter, describes the experimental data, the comparison was done showing the models sum for all the Dalitz and Nyborg plots (Figs. 60 61 62). It it seen that sum of the models describes very good the event populations on the Dalitz and Nyborg plots. Itis now alsopossible to calculatethe rationRofthepartialdecay widths for the decay of the Roper resonance N∗(1440): Γ(N∗(1440)→ Nππ) R == (46) Γ(N∗(1440)→ Δ(1232)π → Nππ) 4Model1 = = 6Model2 =0.039±0.011(stat.)±0.008(sys.) where 4/6 comesfromthe summing up over allisospin states[1] and Model1 Model2 (Eq. 44). The Δ(1232) and N∗(1440)were identiﬁed by their unique topologies of the events on Dalitz and Nyborg plots which were mimic by the the sum of Model1 and Model2 (Fig. 51), which were ﬁtted to the experimental data. Thepopulation ofthe events as afunction of the missing mass of twoprotons wasnotapurposeofthe ﬁt. Itisseen(Fig.63) thattheproposedprocess Model1 and Model2 (Fig.51)aswell asthehomogeneously andisotropically populated phase space populate almost the same area on the missing mass of the two protons MMpp, which is diﬀerent from the population of the experimental data. In order todescribe the eventpopulation as afunction of MMpp, the missing masspopulationfunction f(MMpp)wasderivedfrom the experimental data. First the η meson signal was subtracted from the data by ﬁtting outside the η mesonpeak thefourth orderpolynomial(Fig.64). Next the subtracted experimental data were compared with the true value of the Monte-Carlo simulation composed of the sum of Model1 and Model2 (Fig. 51);the experimental data histogram was divided by the Monte-Carlo model histogram to obtain the missing mass population function f(MMpp) Table 7(Fig. 65(b)). Jagiellonian University 87 Benedykt R. Jany Dalitz Plot ppX 01 Dalitz Plot ppX 03 Dalitz Plot ppX 04 Dalitz Plot ppX 05 Dalitz Plot ppX 06 800 6 1000 800 400 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 88 BenedyktR.Jany 4.44.8 4.3 3.9 4.6 4.2 700 4.4 4.1 3.8 500 4.6 500 2505 600 4 4.2 600 400 4.4 500 400 2003.7 3.9 4.2 4.5 4400 3.8 300 150 4 300 400 300 3.6 200 3.7 3.8 200 100 4 3.8 200 200 3.6 100 3.6 100 503.6 100 3.5 3.5 3.5 0 3.4 0 00 3.4 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 1,2 1,2 1,2 1,2 1,2 Dalitz Plot ppX 01 Dalitz Plot ppX 03 Dalitz Plot ppX 04 Dalitz Plot ppX 05 Dalitz Plot ppX 06 p) GeV2/c4 2 M2(p ) GeV2/c 4 ) GeV2/c 4 5.2 ) GeV2/c 4 2 ) GeV2/c 4 1000900 700 350700 5 2 22 p 1 4.8 M2(p p 1 p 1 M2(p p 1 5.5 M2(p 1 800 M2(p 600 300600 800 800 700 1000 4.8 ) GeV2/c 4 2 800 700 400 350 4.4 1000 4.3 ) GeV2/c4 ) GeV2/c4 6 ) GeV2/c4 2 5.2 3.9 ) GeV2/c4 2 900 5 2 2 4.6 M2(p p 1 M2(p p 1 5.5 M2(p p 1 4.8 4.2 M2(p p 1 p 1 800 M2(p 600600 300800 700 4.4 4.1 3.8 5 500 4.6 600 4 500 250 400 4.4 500 4.2 600 3.9 400 3.7 200 4.5 300 4.2 400 4 400 3.8 300 150 4 200 3.8 4 200 300 3.8 200 3.6 3.7 200 3.6 100 100 3.6 100 3.6 3.5 100 3.5 50 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 3π0) GeV2/c4 1,2 M2(p 3π0) GeV2/c4 1,2 M2(p 3π0) GeV2/c4 1,2 M2(p 3π0) GeV2/c4 1,2 M2(p 3π0) GeV2/c4 1,2 M2(p Figure 60: Dalitz Plot ppX. M2(pp) versus M2(p3π0). The upper row corresponds to the experimental data, the lower row to the Monte-Carlo model sum of Model1 and Model2 (Fig. 51). The columns from left to right correspond tothefollowingMissingMass oftwoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9−1.0 GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. Dalitz Plot 3π0 01 Dalitz Plot 3π0 03 Dalitz Plot 3π0 04 Dalitz Plot 3π0 05 Dalitz Plot 3π0 06 2500 1600 3000.5 0.45 π 0) GeV2/c 4 2 0.14 M2(π 0 π 0) GeV2/c 4 12 0.35 π 0) GeV2/c 4 2 1800 1000 1400 1600 250 2 M2(π 0 1 M2(π 0 1 2000 1200 14008000.12 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 89 BenedyktR.Jany 0.1 0.6 0.7 0.60.50.3 0.4 0.5 200 0.35 0.25 0.4 1200 1000 1500 0.3 600 0.4 M2(π 0 1 M2(π 0 π 0) GeV2/c 4 1 π0) GeV2/c 4 2 1000 0.2 800 150 0.25 0.3 0.08 1000 0.15 400 0.2 600 600 800 0.3 100 0.2 0.06 500 0.1 200 0.15 400 400 0.2 50 0.1 200 0.1 200 0.1 0.04 0 0.05 0 0.05 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 Dalitz Plot 3π0 01 Dalitz Plot 3π0 03 Dalitz Plot 3π0 04 Dalitz Plot 3π0 05 Dalitz Plot 3π0 06 2500 ) GeV2/c4 0.35 1600 3000.6 π 0) GeV2/c 4 2 0.7 0.60.5 0.5 0.35 200 0.25 0.4 1200 1000 1500 0.3 600 0.4 M2(π 0 1 π 0) GeV2/c4 2 0.5 0.45 π 0) GeV2/c 4 2 ) GeV2/c4 0.14 0.12 1800 1000 1400 1600 M2(π0 π 0 12 M2(π 0 π 0 12 0.3 250 0.4 M2(π 0 1 M2(π 0 1 2000 1200 1400800 0.1 1000 800 1500.2 0.25 0.3 0.08 1000 0.15 400 0.2 600 600 800 0.3 100 0.2 0.06 500 0.1 200 0.15 400 400 0.2 50 0.1 200 0.1 200 0.1 0.04 0 0.05 0 0.05 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 ) GeV2/c4 3π0 2M2(π0 Figure61: DalitzPlot 3π0 . M2(2π0)versus M2 (2π0).The upper row corresponds to the experimentaldata, thelower rowtotheMonte-Carlo model sum of Model1 and Model2 (Fig.51). The columnsfromleftto right correspond to the followingMissingMass oftwoprotonsbins, column1MMpp =0.4−0.5 GeV/c2,column2 MMpp =0.6−0.7 GeV/c2 , column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp =0.9−1.0 GeV/c2 . Theplotsaresymmetrized againstthreepions -each eventisﬁlled sixtimes. Fully expandableand colored version of the ﬁgure is available in the attached electronic version of the thesis. Nyborg Plot pπ0 vs pπ0π0 01 Nyborg Plot pπ0 vs pπ0π0 03 Nyborg Plot pπ0 vs pπ0π0 04 Nyborg Plot pπ0 vs pπ0π0 05 Nyborg Plot pπ0 vs pπ0π0 06 100020002000 140012001.6 M(p ) GeV/c2 M(p ) GeV/c2 1.5 1.6 1.5 Jagiellonian University 90 BenedyktR.Jany 1.7 1.6 π0) GeV/c2 1 1.7 1.6 π0) GeV/c2 1 1.5 1.5 1.5 1.5 7001400 1400 1.4 1000 1.4 8001.4 6001200 1200 1.4 1.4 800 5001000 1000 1.3 1.3 6001.3 1.3 1.3 400600 800 800 1.2 400 1.2 300600 6001.2 1.2 1.2 400 200400 400 2001.1 1.1 200 1.1 1.1 1.1 100200 200 01 0001 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 223223 223223223 Nyborg Plot pπ0 vs pπ0π0 01 Nyborg Plot pπ0 vs pπ0π0 03 Nyborg Plot pπ0 vs pπ0π0 04 Nyborg Plot pπ0 vs pπ0π0 05 Nyborg Plot pπ0 vs pπ0π0 06 M(p π 0) GeV/c2 1 1.6 π 0) GeV/c2 1 9001800 1800 π 0 1 1 1 11 1 M(pM(p M(p 1200 8001600 1600 1000 100020001.7 ) GeV/c2 2000 ) GeV/c2 1.7 14001.5 1.5 1.5 1.5 7001400 1400 1.4 1000 1.4 8001.4 6001200 1200 1.4 1.4 800 5001000 1000 1.3 1.3 6001.3 1.3 1.3 600 800 800 400 1.2 1.2 400 300600 6001.2 1.2 1.2 400 200 200 400 400 1.1 1.1 200 1.1 1.1 1.1 100200 200 01 01 0 00 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 2323 232323 22 2 22 M(p π 0) GeV/c2 1 1.6 π 0) GeV/c2 12009001800 1800 π 0 1 1.6 1.6 π 0 1 π 0 1 1 11 1 1 1 M(pM(p M(p 1200 8001600 1600 1000 Figure 62: Nyborg Plot. M(pπ0)versus M(pπ0π0).The upper row corresponds to the experimental data, the lower rowtotheMonte-Carlo model sum of Model1 and Model2 (Fig.51). The columnsfromleft to right correspond to the followingMissingMass oftwoprotonsbins, column1MMpp =0.4−0.5 GeV/c2,column2 MMpp =0.6−0.7 GeV/c2 , column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp =0.9−1.0 GeV/c2 . Theplotsaresymmetrized againsttwoprotonsand threepions -each eventisﬁlled sixtimes. Fully expandableand colored version of the ﬁgure is available in the attached electronic version of the thesis. 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction MMpp GeV/c2 f(MMpp) 0.408±0.008 3.8±1.3 0.424±0.008 3.47±0.31 0.440±0.008 3.46±0.19 0.456±0.008 3.26±0.13 0.472±0.008 3.102±0.092 0.488±0.008 3.191±0.079 0.504±0.008 3.062±0.066 0.520±0.008 2.936±0.058 0.536±0.008 2.729±0.057 0.552±0.008 2.419±0.048 0.568±0.008 2.178±0.034 0.584±0.008 1.944±0.027 0.600±0.008 1.695±0.022 0.616±0.008 1.513±0.019 0.632±0.008 1.407±0.017 0.648±0.008 1.297±0.015 0.664±0.008 1.240±0.014 0.680±0.008 1.156±0.013 0.696±0.008 1.079±0.012 0.712±0.008 0.995±0.011 0.728±0.008 0.9131±0.0099 0.744±0.008 0.8540±0.0093 0.760±0.008 0.8012±0.0088 0.776±0.008 0.7396±0.0082 0.792±0.008 0.6877±0.0078 0.808±0.008 0.6330±0.0074 0.824±0.008 0.5690±0.0069 0.840±0.008 0.5207±0.0066 0.856±0.008 0.4851±0.0065 0.872±0.008 0.4354±0.0063 0.888±0.008 0.3862±0.0061 0.904±0.008 0.3456±0.0062 0.920±0.008 0.3060±0.0066 0.936±0.008 0.2665±0.0073 0.952±0.008 0.2400±0.0091 0.968±0.008 0.202±0.012 0.984±0.008 0.159±0.025 Table 7: The missing mass population function f(MMpp). Jagiellonian University 91 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION The Overall Model One can write the overall proposed model of the pp → pp3π0 production, extracted by the Monte-Carlo comparison from experimental data: Model =[βModel1 +(1 −β)Model2]× f(MMpp) (47) The Model1 corresponds to theprocess(Fig.51(a)) and the Model2 to theprocess(Fig.51(b)), which assumes nointeractionbetweenthebaryons -homogeneously and isotropically populated phase space. The β parameteris thefraction ofModel1 to the Model1 +Model2 obtain fromthechi-square ﬁttotheexperimentaldata(Eq.45). The f(MMpp)corresponds to the missing mass population function Table7(Fig.65(b)), whichisderivedfromthe experimentaldata. Itmightbe related to thepossibleinteractionbetween thebaryons notincludedintheMonte-Carlo model andtheinternalproperties ofthe N∗(1440), which is discussed below(see page 97). The(Figs.66,67)showsthe comparison of theproposed model(Eq.47) with the experimental data. It is seen that the model describes very good the experimentaldata. Moredetailquantitativediscussion onthe validation of the developed model will be presented in Section 5.1.2 on page 107. Jagiellonian University 92 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction (a) Fit of thefourth orderpolynomial outsidethe η meson peak. (b) Subtracted η meson signal, blue markers correspond to the Monte-Carlo simulation. Jagiellonian University 93 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION (a) black: subtracted experimental data, blue: the true value of the Monte-Carlo simulation composed of the sum of Model1 and Model2. (b) The missing mass population function f(MMpp). Jagiellonian University 94 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction FD MM vs Prob 3pi0 hMM Entries 1106922 Mean 0.6508 20000 RMS 0.1155 18000 DATA 16000 Monte-Carlo PhSp 14000 Monte-Carlo Sum Monte-Carlo DN 12000 Monte-Carlo DND 10000 8000 6000 4000 2000 0.40 0.5 0.6 0.7 0.8 ppMM 0.9 [GeV/c2] 1 Figure 66: Missing Mass of two protons. Comparison of the experimental data(blackmarker) withtheMonte-Carlo simulationphase space(blueline), the Model1 (Fig.51(a))(yellowline),the Model2 (Fig. 51(b))(green line), the model sum(Eq.47)(redline). Thederived f(MMpp)(see Table 7and Fig. 65(b)) was used for the models. Vertical axes -number of events (in given bin) is shown. Jagiellonian University 95 Benedykt R. Jany 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION 160 3×10 Kinetic EnergyCMp DATA 35000θ CMp 18000φ CMp 140 MC PhSp 30000 16000 120 MC DN 25000 14000 100 MC DND 20000 12000 80 MC Sum 10000 15000 8000 60 10000 6000 40 4000 20 5000 2000 00 0.05 0.1 0.15 0.2 [GeV]CM pEk 0.25 00 20 40 60 80 100 120 140 [deg]CM p θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM p φ 150 3 ×10 Kinetic EnergyCMπ0 θCMπ0 φCMπ0 45000 22000 100 40000 18000 20000 80 35000 16000 30000 14000 60 25000 12000 20000 10000 40 15000 8000 6000 20 10000 4000 5000 2000 00 0.05 0.1 0.15 0.2 0.25 0.3 [GeV]CM 0πEk 0.35 0.4 400 60 80 100 120 140 [deg]CM 0π θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM 0π φ 150 Kinetic EnergyCM3pi0 θ CM3pi0 φ CM3pi0 40000 25000 35000 7000 30000 20000 6000 25000 15000 5000 20000 4000 15000 10000 3000 10000 2000 5000 5000 1000 00 0.05 0.1 0.15 0.2 0.25 0.3 [GeV]CM 03πEk 0.35 0.4 400 60 80 100 120 140 [deg]CM 03π θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM 03π φ 150 Figure 67: Comparison of the experimental data with the Monte-Carlo simulation for the MMpp < 0.5 GeV/c2 and MMpp > 0.6 GeV/c2 . Upper row: proton in the center of mass frame,Middle row: pion in the center of mass frame, Lower row: 3π0 system in the center of mass frame. From left kinetic energy, the polar angle and the azimuthal angle distribution. The experimental data are shown as a black marker, the Monte-Carlo simulation phase space (blue line), theModel1 (Fig. 51(a))(yellow line), the Model2 (Fig. 51(b))(green line), the modelsum(Eq. 47)(red line). The derived f(MMpp)(see Table 7and Fig. 65(b)) was used for the models. Vertical axes -number of events (in given bin)is shown. Jagiellonian University 96 BenedyktR.Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction The origin of the missing mass population function Searchingfortheoriginof themissing masspopulationfunction f(MMpp) (Table7,(Fig.65(b)))thekinematics of the pp→ Δ(1232)N∗(1440)reaction was studied in details by Pluto++ calculations, via Monte-Carlo method (see Appendix C). The following explanations are considered. The Possibility of Δ(1232)N∗(1440)interaction To check the possibility of Δ(1232)N∗(1440)interaction and its inﬂuence of the MMpp, it is very convenient to study the four momentum transfers q dependences since it gives the informations about the distance resolution of the reaction Δr (interactiondistance)accordingly to the relation[26,27,77]: Δr = (48) |q| where � is a Planck’s constant. First the four momentum transfer to the Δ(1232) as a function of the missing mass ofthetwoprotons andfour momentumtransfertothe N∗(1440) asafunctionof themissing massof thetwoprotonswasexamined(Fig.68). Itis seen thatthereis nodependencebetween thesefour momentum transfers and the MMpp. One can alsotryto relatetheMMpp withthediﬀerence offour momentum between the Δ(1232)and N∗(1440)in the center ofmassframe |q|2 (Fig.69). Itis seen that the variables are correlated. One can calculate also the average q as a function of MMpp (Fig. 70). It is now possible using the relation from (Fig. 70) to recalculate the f(MMpp) -the missing mass population function Table 7 (Fig. 65(b)) to dependence on average diﬀerence of four momentum between the Δ(1232) and N∗(1440) |q|2 (Fig. 71). It is seen that when the average diﬀerence of four momentum increases the f(MMpp) increases exponentially. If the possible interaction would be interpreted as theOneBosonExchange(i.e. by exchange ofthe π0 , f0�σ(600))than the behavior of the interaction would be ∼ 1 [30–37]. When the diﬀerence of |q|2 fourmomentumincreasestheinteractionshould stronglydecrease. Sincethe f(MMpp)behaves completelydiﬀerentlywith anincrease ofthe |q|2, as shown above, the interaction between the Δ(1232)N∗(1440)system understood as the One Boson Exchange is excluded. Jagiellonian University 97 Benedykt R. Jany 4 4800 800 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 98 BenedyktR.Jany |q|2 =|P4 -P4Δ(1232)|2 [GeV2/c4] Δ(1232) Beam |q|2 =|P4 -P4 [GeV2/c4] N*(1440)|2 N*(1440)Beam 3.5 3.5 3 700 700 3 600 600 5002.5 2.5500 2 2400 400 1.5 1.5300 300 1 1200 200 0.5 100 0.5 100 0 00 0 0.4 0.5 0.6 0.7 0.8 0.9 10.4 0.5 0.6 0.7 0.8 0.9 1MMpp [GeV/c2] MMpp [GeV/c2] (a)Four momentum transfer to the Δ(1232)asafunctionof the (b) Fourmomentumtransfertothe N∗ (1440)as a function of missing mass of the two protons.the missing mass of the two protons. |q|2 =|P4 -P4 [GeV2/c4] N*(1440)|2 N*(1440)Beam 4 1400 3.5 1200 3 1000 2.5 800 2 600 1.5 1 400 0.5 200 00 0.5 1 1.5 [GeV2/c4]Δ(1232)|2-P4 Beam=|P4 Δ(1232)|q|22 2.5 3 3.5 4 0 (c) Four momentum transfer to the N∗ (1440) versus the four momentum transfer to the Δ(1232). Figure 68: Calculations by Pluto++ via Monte-Carlo simulations. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction |q|2=|P4 -P4 [GeV2/c4] N*(1440)|2 Δ(1232) 3 800 2.5 700 2 600 500 1.5 400 1 300 200 0.5 100 0.40 0.5 0.6 0.7 0.8 [GeV/c2]ppMM 0.9 1 0 Figure69: Monte-Carlo simulation,diﬀerence offour momentumbetween the Δ(1232) and N∗(1440)as afunction of the missing mass of thetwoprotons. Calculations by Pluto++. Average |q|2=|P4 -P4 [GeV2/c4] N*(1440)|2 Δ(1232) 1.2 1 0.8 0.6 0.4 0.2 0.40 0.5 0.6 0.7 0.8 [GeV/c2]ppMM 0.9 1 Figure 70: Monte-Carlo simulation,average diﬀerence of four momentum between the Δ(1232) and N∗(1440) as a function of the missing mass of the two protons. Calculations by Pluto++. Jagiellonian University 99 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION The inﬂuence of the N∗(1440) line shape One cantryto relatetheMMpp with theinternalproperties ofthe Δ(1232) and N∗(1440); for this purpose the mass of the resonances (the realistic eﬀective spectral line shape) for diﬀerent decay modes of the N∗(1440) is plotted against the MMpp (Fig. 72). It is seen that the N∗(1440) mass is correlated with the MMpp (Figs. 72(a), 72(d)) – the correlation is seen in both cases independently on the spectral line shape. in case of the Δ(1232) (Figs. 72(a),72(c)) no correlation is visible. To see it in details, one can calculate also the average mass of the N∗(1440)as a function of the MMpp (Figs. 73(a), 73(c)). It is seen that when the missing mass MMpp of the two protonsincreases one selectsin averagehigher mass ofthe N∗(1440)from the spectralline shape. One can now transform the f(MMpp)-the missing mass populationfunctionTable7(Fig.65(b))todependence onthe average mass of the N∗(1440)(Figs.73(b),73(d)). Itis seen that f(MMpp)decreases with the average mass of the N∗(1440). One can now use the obtained f(MMpp) versus average mass of the N∗(1440)relation(Figs.73(b),73(d))and useit as an correction to the realistic eﬀective spectral line shape of the N∗(1440) calculatedby Pluto++(seeAppendixC). Itis seen(Figs.74(b),74(d))that the proposed modiﬁed spectral line shapes of the N∗(1440)are signiﬁcantly diﬀerent from the one obtained originally from the Pluto++. In particular interesting eﬀect is seen in case of N∗(1440)decaying into π0Δ(1232) (Fig.74(d)). The calculatedby Pluto++ realistic eﬀective spec- Jagiellonian University 100 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction tralline shape ofthe N∗(1440)changes, when applying correction(Figs.73(b),73(d)), tothe shape very similartotheBreit-Wignerdistribution(seeFig.74(d)). This might indicate that the proposed by Pluto++ modiﬁcation of the N∗(1440)spectral line caused by decay to Δ(1232) is not so strong as proposed. Itis also seen thatwhen modifyingthe N∗(1440)spectrallinethe Δ(1232) spectrallineis notdisturbed(Figs.74(a),74(a)). Now one can use the modiﬁed line shapes of N∗(1440) by f(MMpp) (Figs.74(b),74(d))and seehowthedata aredescribedbythe model(Eq.47), now without explicit f(MMpp). Itisseen(Figs.75,76) thattheproposed modiﬁcation ofthe modelby changingthe spectralline shape ofthe N∗(1440) results in almost the same behavior as the one with the f(MMpp)included explicitly(seefor comparisonFigs.66,67). Itis also seenthatinthis case the model describes the data signiﬁcantly better than the homogeneously and isotropically populated phase space. Concluding, two possible explanations of the origin of f(MMpp) -the missing mass population function were considered. • The possibility of Δ(1232)N∗(1440) interaction via One Boson Exchange(OBE), which was excludeddueto completelydiﬀerentbehavior of the f(MMpp)asafunctionof thediﬀerence offourmomentumthan in OBE[30–37];

• Next, it was shown that the MMpp is very sensitive to the structure of the spectral line shape of the N∗(1440). The proposed modiﬁcation of the spectral line mainly of the N∗(1440) → π0Δ(1232) accomplishes the explicitly added f(MMpp); since the main eﬀect inﬂuencing the description of the data is due to the modiﬁcation of the N∗(1440)→ π0Δ(1232) line shape – it is the leading mode of 3π0 production ∼ 95% (see Eq.47). Inparticular theproposed modiﬁcation of this spectral line is very similar to the Breit-Wigner distribution. This could indicate that theproposed by Pluto++ [71]modiﬁcation of the N∗(1440)spectral line caused by decay to Δ(1232) (unstable hadron) is not as prominent as proposed. For instance, it remains inconclusive whether the spectral line shape of N∗(1440)remains the same in case of N∗(1440)→ pπ0π0 since the contribution of this branch consists of only ∼ 5% (see Eq. 47).

Thepossibility of molecule orbound state creationof Δ(1232)N∗(1440)system as well as the excitations of the quark-gluon degrees of freedom is not excluded. Jagiellonian University 101 Benedykt R. Jany 1.7 1.8 400 1.6 250 1.7 350 1.5 300 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 102 BenedyktR.Jany MΔ(1232) [GeV/c2] MN*(1440) [GeV/c2] 200 1.6 1.3 1.4 200 250 1.5 150 1.2 100 150 1.4 100 1.1 50 1.3 50 0.41 0.5 0.6 0.7 0.8 [GeV/c2]ppMM 0.9 1 0 0.41.2 0.5 0.6 0.7 0.8 [GeV/c2]ppMM 0.9 1 0 (a) Δ(1232) line shape versus MMpp, where N∗ (1440)→ pπ0π0 (stable particles) (b) N∗ (1440)line shape versus MMpp, where N∗ (1440)→ pπ0π0 (stable particles) 1.7 450 1.6 400 1.8 1.7 MΔ(1232) [GeV/c2] MN*(1440) [GeV/c2] 250 350 2001.5 1.6 300 1.4 150 250 1.5 1.3 200 1.2 150 1.4 100 1.1 50 100 1.3 50 1 0 1.2 0 0.4 0.5 0.6 0.7 0.8 0.9 10.4 0.5 0.6 0.7 0.8 0.9 1 MMpp [GeV/c2] MMpp [GeV/c2] (c) Δ(1232) line shape versus MMpp, (d) N∗ (1440)line shape versus MMpp, where N∗ (1440)→ π0Δ(1232) (unstable particle)where N∗ (1440)→ π0Δ(1232) (unstable particle) Figure 72: Realistic eﬀective spectral line shape g(m) of Δ(1232) and N∗(1440) in the pp → Δ(1232)N∗(1440) reaction atincidentproton momentum of 3.350 GeV/c2 versus MMpp. The Δ(1232)decaysintopπ0 (stableparticles), the N∗(1440)decays into pπ0π0 (stable particles)or into π0Δ(1232) (unstable particle), when laterΔ(1232) decays into pπ0 . The correlation between the N∗(1440) line shape and MMpp visible, no correlation in case of Δ(1232) seen. Calculations by Pluto++ (via Monte-Carlo method). 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 103 BenedyktR.Jany (a) Average N∗ (1440)mass versus MMpp, (b) The f(MMpp)as a function of the average N∗ (1440)mass, where N∗ (1440)→ pπ0π0 (stable particles)where N∗ (1440)→ pπ0π0 (stable particles) (c) Average N∗ (1440)mass versus MMpp, (d) The f(MMpp)as a function of the average N∗ (1440)mass, where N∗ (1440)→ pΔ(1232) (unstable particle)where N∗ (1440)→ pΔ(1232) (unstable particle) Figure73: Calculations of pp→ Δ(1232)N∗(1440)reactionkinematics doneby Pluto++ via Monte-Carlo method. g(m) a.u. Pluto++ 30000 25000 Pluto++ and f(MM ) pp g(m) a.u. 30000 Pluto++ 25000 Pluto++ and f(MM ) pp 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 104 BenedyktR.Jany 20000 20000 15000 15000 10000 10000 5000 5000 10 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.20 1.3 1.4 1.5 1.6 1.7 1.8 m [GeV/c2] m [GeV/c2] (a) Δ(1232) line shape, (b) N∗ (1440)line shape, where N∗ (1440)→ pπ0π0 (stable particles) where N∗ (1440)→ pπ0π0 (stable particles) g(m) a.u. 35000 Pluto++ 30000 g(m) a.u. 25000 Pluto++ Pluto++ and f(MM ) pp 20000 25000 Pluto++ and f(MM ) pp 20000 15000 15000 10000 10000 5000 5000 0 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.2 1.3 1.4 1.5 1.6 1.7 1.8 m [GeV/c2] m [GeV/c2] (c) Δ(1232) line shape, (d) N∗ (1440)line shape, where N∗ (1440)→ π0Δ(1232) (unstable particle) where N∗ (1440)→ π0Δ(1232) (unstable particle) Figure74:Realisticeﬀective spectrallineshape g(m)of Δ(1232) and N∗(1440)in the pp→ Δ(1232)N∗(1440)reaction at incident proton momentum of 3.350 GeV/c2 . The Δ(1232) decays into pπ0 (stable particles), theN∗(1440) decays into pπ0π0 (stableparticles)orinto pΔ(1232) (unstableparticle), whenlaterΔ(1232) decays into pπ0 . Spectral line shapes calculated by Pluto++ (blue), proposedmodiﬁed line shapes by f(MMpp)(red)(via Monte-Carlo method). 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction Figure 75: Spectrum of the Missing Mass of two protons. Comparison of the experimental data (black marker) with the Monte-Carlo simulation phase space(blueline), the Model1 (Fig.51(a))(yellowline) , the Model2 (Fig. 51(b)) (green line), the model sum (Eq. 47) (red line). The model is calculated without the f(MMpp) population function but with proposed modiﬁed spectral line shape of the N∗(1440) (see Figs. 74(b), 74(d).) The proposed modiﬁcation of the model results in almost the same behavior as the one with the f(MMpp)included explicitly(seeFig.66). Itis seen that the model describes the data signiﬁcantly better than the homogeneously andisotropicallypopulatedphasespace. Vertical axis -numberof events(in given bin) is shown. Jagiellonian University 105 Benedykt R. Jany 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 106 BenedyktR.Jany 160 3×10 20 40 60 80 100 120 140 Kinetic EnergyCMp DATA MC PhSp MC DN MC DND MC Sum 35000θ CMp5000 10000 15000 20000 25000 30000 φ CMp2000 4000 6000 8000 10000 12000 14000 16000 18000 00 0.05 0.1 0.15 0.2 [GeV]CM pEk 0.25 00 20 40 60 80 100 120 140 [deg]CM p θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM p φ 150 80 100 3×10 Kinetic EnergyCMπ0 30000 35000 40000 45000 CMθπ0 14000 16000 18000 20000 22000 CMφπ0 60 25000 12000 20 40 5000 10000 15000 20000 2000 4000 6000 8000 10000 00 0.05 0.1 0.15 0.2 0.25 0.3 [GeV]CM π0Ek 0.35 0.4 400 60 80 100 120 140 [deg]CM π0θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM π0 φ 150 40000 Kinetic EnergyCM3pi0 25000θ CM3pi0 φ CM3pi0 35000 7000 30000 20000 6000 25000 15000 5000 20000 4000 15000 10000 3000 5000 10000 5000 1000 2000 00 0.05 0.1 0.15 0.2 0.25 0.3 [GeV]CM 03πEk 0.35 0.4 400 60 80 100 120 140 [deg]CM 3π0θ 160 180 0 -150 -100 -50 0 50 100 [deg]CM 3π0 φ 150 Figure 76: Comparison of the experimental data with the Monte-Carlo simulation for the MMpp < 0.5 GeV/c2 and MMpp > 0.6 GeV/c2 . Upper row: proton in the center of mass frame,Middle row: pion in the center of mass frame, Lower row: 3π0 system in the center of mass frame. From left kinetic energy, the polar angle and the azimuthal angle distribution. The experimental data are shown as a black marker, the Monte-Carlo simulation phase space (blue line), theModel1 (Fig. 51(a))(yellow line), the Model2 (Fig. 51(b))(green line), the modelsum(Eq. 47)(red line). The model is calculated without the f(MMpp)population function but with proposed modiﬁed spectral line shape of the N∗(1440)(see Figs. 74(b), 74(d).) The proposed modiﬁcation of the model results in almost the same behavior as the one with the f(MMpp)included explicitly(seeFig.67). Itis seenthatthe modeldescribesthedata signiﬁcantly better than the homogeneously and isotropically populated phase space. On vertical axes -number of events(ingivenbin) is shown. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction 5.1.2 The model validation The model consistency To check the Monte Carlo developed Model (Eq. 47) consistency with experimental data, the model was compared with the experimental data for all considered spectra(i.e. DalitzPlot ppX, Dalitz Plot 3π0, Nyborg Plot for diﬀerent missing mass ranges) by constructing the ratio: Experimental Data ration = (49) The Model In case of data and model agreement the ration should be equal to the value 1. The spectra of the ratio (Eq. 49) were prepared (Upper row of Figs. 77, 78, 79) The color contours correspond to relative deviation from the constant value 1. Also the relative statistical error of the ratio (Eq. 49) was computed (Lower row of Figs. 77, 78, 79). The color contours were selected to reﬂect the contours coloring of the relative deviation. It is seen that the color contours structure of the relative deviation from the constant value 1 follows the statistical error distribution. One can conclude that in the range of the statistical error the ratio is equal to constant value 1. The model is consistent with the experimental data. In addition thediﬀerence x between the experimentaldata and the model normalizedby data errors(σData)and the model errors(σData) Data−Model x = (50) J σ2 + σ2 Data Model for all data points included in the χ2 ﬁt(Eq.28)was studiedindetails. TheProbabilityDensityFunction(PDF), andtheCumulativeDistribu-tionFunction(CDF) were calculated(Figs.80(a),80(b)). Itis seenthatthe PDF is centered around mean value of 0 and it is in good agreement symmetric(the skewnessparameteris closeto 0 value). There is no systematic shift eﬀect. Since the statistical ﬂuctuations of the Data and Model points are of the Poissonian behavior ,one can show that the diﬀerence x = x1 − x2 of two statistically independently random variables x1,x2 having Poissonian distribution f1,f1 x1 −µ1 µe f1(x1,µ1)= 1 , x1! x2 −µ2 µe f2(x2,µ2)= 2 (51) x2! Jagiellonian University 107 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION where µ1,2 is the expected value ofthedistribution,isdistributedlikeSkellam distribution[78]: ��x/2 µ1 √ −(µ1 +µ2 ) f(x,µ1,µ2)= eI|x|(2 µ1µ2) (52) µ2 where I|x| is modiﬁed Bessel function of the ﬁrst kind. ThePDF(Fig.80(a))was ﬁtted withtheSkellamdistribution(Eq.52). The following values were obtained: µ1 =0.3520±0.0059 (53) µ2 =0.3584±0.0052 (54) The standard normal distribution was plotted for the comparison. The Skellam distributiondescribes thePDF andCDF spectra signiﬁcantlybetter than the standard normal distribution. Also the probability to ﬁnd events in range (−x,x) was computed(Fig.80(c)). Itis seen thattheprobability to ﬁnd the events in the range of one standard deviation x ∈ (−1,1) is equal to ∼ 0.8 i.e the Monte Carlo developed Model describes ∼ 80% of the experimental data within the statistical errors of one standard deviation. It is also seen that the probability value ∼ 1.0 is reached for the range x ∈ (−2.5,2.5) -around 100% of the experimental data is described by the MonteCarloModel withing the statistical error of ∼ 2.5standarddeviations. Concluding theMonteCarlodeveloped modelfullydescribes thedata within the statistical precision of data and model. Jagiellonian University 108 Benedykt R. Jany M2(p p) GeV2/c 4 1 2 6 M2(p p 1 2 5 M2(p p 1 2 2 ) GeV2/c4 p 1 2 10% 15% 20% 25% 30% 35% 40% 45% 50% >50% Raw Data/MC Model MM =0.4-0.5 GeV/c2 Raw Data/MC Model MM =0.6-0.7 GeV/c2 Raw Data/MC Model MM =0.7-0.8 GeV/c2 Raw Data/MC Model MM =0.8-0.9 GeV/c2 Raw Data/MC Model MM =0.9-1.0 GeV/c2 pppppppp pp Figure 77: Dalitz Plot ppX. M2(pp) versus M2(p3π0). The upper row corresponds to ratio of the experimental data divided by the Monte-Carlo model sum of Model1 and Model2 (Fig. 51), the contour colors correspondto the relative deviation from constant value 1. The lower row corresponds to the relative statistical error of the ratio. The columns from left to right correspond to the following Missing Mass of two protons bins, column 1 MMpp = 0.4 −0.5 GeV/c2,column 2 MMpp =0.6 − 0.7 GeV/c2, column 3 MMpp =0.7 − 0.8 GeV/c2, column 4 MMpp = 0.8 − 0.9 GeV/c2, column 5 MMpp =0.9 − 1.0 GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. The color contours structure of the relative deviation from the constant value 1 follows the statistical error distribution. It is seen that in the range of the statistical error the ratio is equal to constant value 1. The model is consistent with the experimental data. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 109 BenedyktR.Jany 4.4 4.85.2 4.3 3.9 ) GeV2/c 4 4.6 5.5 4.2 4.8 4.4 4.1 3.8 4.6 5 4 4.2 4.4 3.7 3.9 4.2 4.5 4 3.8 4 3.6 4 3.7 3.8 3.8 3.6 3.6 3.6 3.5 3.5 3.5 3.4 3.4 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.353.43.453.53.553.63.653.73.75 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 1,2 1,2 1,2 1,2 1,2 ) GeV2/c 4 M2(p p) GeV2/c4 1 M2(p M2(p p) GeV2/c4 1 4.4 4.85.2 4.3 3.9 4.6 5.5 4.2 4.8 4.4 4.1 3.8 4.6 5 4 4.2 4.4 3.7 3.9 4.2 4.5 4 3.8 4 3.6 4 3.7 3.8 3.8 3.6 3.6 3.6 3.5 3.5 3.5 3.4 3.4 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.353.43.453.53.553.63.653.73.75 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 M2(p 3π0) GeV2/c4 1,2 1,2 1,2 1,2 1,2 Raw Data/MC Model Errors MM =0.4-0.5 GeV/c2 Raw Data/MC Model Errors MM =0.6-0.7 GeV/c2 Raw Data/MC Model Errors MM =0.7-0.8 GeV/c2 Raw Data/MC Model Errors MM =0.8-0.9 GeV/c2 Raw Data/MC Model Errors MM =0.9-1.0 GeV/c2 pppppppp pp ) GeV2/c4 ) GeV2/c4 M2(p p) GeV2/c 4 1 M2(p 2 6 M2(p 1 p 2 5 M2(p 1 p 2 2 ) GeV2/c4 1 p 2 ) GeV2/c4 π 0 2 π 0 2 10% 15% 20% 25% 30% 35% 40% 45% 50% >50% Raw Data/MC Model MM =0.4-0.5 GeV/c2 Raw Data/MC Model MM =0.6-0.7 GeV/c2 Raw Data/MC Model MM =0.7-0.8 GeV/c2 Raw Data/MC Model MM =0.8-0.9 GeV/c2 Raw Data/MC Model MM =0.9-1.0 GeV/c2 pppppppp pp Figure 78: Dalitz Plot 3π0 . M2 (2π0) versus M2(2π0). The upper row corresponds to ratio of the experimental data divided by the Monte-Carlo model sum of Model1 and Model2 (Fig. 51), the contour colors correspondto the relative deviation from constant value 1. The lower row corresponds to the relative statistical error of the ratio. The columns from left to right correspond to the following Missing Mass of two protons bins, column 1 MMpp = 0.4− 0.5 GeV/c2,column 2 MMpp =0.6 −0.7 GeV/c2, column 3 MMpp =0.7 −0.8 GeV/c2, column 4 MMpp = 0.8−0.9 GeV/c2, column5 MMpp =0.9−1.0 GeV/c2 .Theplotsaresymmetrized againstthreepions -each eventis ﬁlled six times. The color contours structure of the relativedeviationfrom the constant value 1 follows the statistical error distribution. It is seen that in the range of the statistical error the ratio is equal to constant value 1. The model is consistent with the experimental data. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 110 BenedyktR.Jany M2 π 0 1 ) GeV2/c4 0.6 0.70.5 0.35 0.14 0.45 0.6 0.5 0.3 0.4 0.12 0.5 0.35 0.25 0.4 0.3 0.4 0.1 0.2 0.3 0.25 0.3 0.08 0.2 0.15 0.2 0.2 0.15 π 0 ) GeV2/c4 1 0.1 0.06 0.1 0.1 0.1 0.05 0.05 0.04 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 232323 2323 ) GeV2/c 4 M2 ) GeV2/c 4 0.6 0.70.5 0.35 0.14 0.45 0.6 0.5 π 0 0.3 1 0.4 0.12 0.5 0.35 0.25 0.4 0.3 0.4 0.1 0.2 0.3 0.25 0.3 0.08 0.2 0.15 0.2 0.2 0.15 0.1 0.06 0.1 0.1 0.1 0.05 0.05 0.04 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 M2(π0π0) GeV2/c4 232323 2323 ) GeV2/c4 Raw Data/MC Model Errors MM =0.4-0.5 GeV/c2 Raw Data/MC Model Errors MM =0.6-0.7 GeV/c2 Raw Data/MC Model Errors MM =0.7-0.8 GeV/c2 Raw Data/MC Model Errors MM =0.8-0.9 GeV/c2 Raw Data/MC Model Errors MM =0.9-1.0 GeV/c2 pppppppp pp ) GeV2/c4 1 ) GeV2/c4 M2 ) GeV2/c4 ( π0 2 ) GeV2/c4 M2 (M2 π 0 π0 12 (M2 (π 0 π 0 12 M2 (M2 π0 2 (π 0 M2 (M2 π 0 π 0 12 ( π 0 π 0 12 (π0 π 0 12 π 0 π0 12 ( π0 1 M(p 1 π 0 1 1 1 π 0 1 M(p 1 π0 1 π 0 1 1 ) GeV/c2 10% 15% 20% 25% 30% 35% 40% 45% 50% >50% Raw Data/MC Model MM =0.4-0.5 GeV/c2 Raw Data/MC Model MM =0.6-0.7 GeV/c2 Raw Data/MC Model MM =0.7-0.8 GeV/c2 Raw Data/MC Model MM =0.8-0.9 GeV/c2 Raw Data/MC Model MM =0.9-1.0 GeV/c2 pppppppppp Figure 79: Nyborg Plot. M(pπ0) versus M(pπ0π0). The upper row corresponds to ratio of the experimental data divided by the Monte-Carlo model sum of Model1 and Model2 (Fig. 51), the contour colors correspond to the relative deviation from constant value 1. The lower row corresponds to the relative statistical error of the ratio. The columns from left to right correspond to the following Missing Mass of two protons bins, column 1 MMpp = 0.4 −0.5 GeV/c2,column 2 MMpp =0.6 − 0.7 GeV/c2, column 3 MMpp =0.7 − 0.8 GeV/c2, column 4 MMpp = 0.8−0.9 GeV/c2, column 5 MMpp =0.9−1.0 GeV/c2 . The plots are symmetrized against two protons and three pions -each event is ﬁlled six times. The color contours structure of the relative deviation from the constant value 1 follows the statistical error distribution. It is seen that in the range of the statistical error the ratio is equal to constant value 1. The model is consistent with the experimental data. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 111 BenedyktR.Jany ) GeV/c2 1.7 1.7 1.6 1.6 1.51.6 1.6 M(p ) GeV/c2 1.5 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 2323232323 22222 M(p ) GeV/c2 ) GeV/c2 M(p π 0 1 1.7 1.7 1.6 1.6 1.51.6 1.6 1.5 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 ) GeV/c2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 2323232323 22222 Raw Data/MC Model Errors MM =0.4-0.5 GeV/c2 Raw Data/MC Model Errors MM =0.6-0.7 GeV/c2 Raw Data/MC Model Errors MM =0.7-0.8 GeV/c2 Raw Data/MC Model Errors MM =0.8-0.9 GeV/c2 Raw Data/MC Model Errors MM =0.9-1.0 GeV/c2 pppppppppp M(p ) GeV/c2 M(p ) GeV/c2 ) GeV/c2 M(p M(p 1 1 π 0 1 1 π 0 1 M(p π 0 1 1 1 π 0 1 ) GeV/c2 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 112 BenedyktR.Jany (a) Probability Density Function (b) Cumulative Distribution Function (c) Probability to ﬁnd events in range (−x,x) Data−Model σ2 +σ2 DataModel x value was calculated for all data points included in the χ2 ﬁt(Eq. 28). TheProbabilityDensityFunction was ﬁttedby theSkellamdistribution(Eq.52). The experimentaldatadistributions areplotted as ablackhistogram, the standard normal distribution is plotted as a blue dashed line, the Skellam distribution is shown a red dotted line. The Skellam distribution describes the diﬀerence distributions signiﬁcantly better than the standard normal distribution. It is seen that the probability to ﬁnd the events in the range of one standard deviation x ∈ (−1,1) is equal to ∼ 0.8 i.e the Monte Carlo developed Model describes ∼ 80% of the experimental data within the statistical errors of the data and the model. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction Veriﬁcation of other processes contribution to the model To verify a possibility of other processes contribution (Model3) to the Monte Carlo developed Model(Eq. 47)(ModelTot.)the following procedure was applied. One may assume that in order to describe the experimental data in addition to the developed Model(ModelTot.)one has other processes that could contribute (Model3), where other processes one may try to mimic by the Monte Carlo simulation of pp → pp3π0 reaction assuming homogeneously and isotropically populated phase space. This could be written as: (1−ǫ)ModelTot. + ǫModel3 (55) one needs to know the fraction of Model3 to the Model3 +ModelTot. i.e. the ǫ parameter -which actually isthefractionof theotherprocessestothe sum of the developed Model and other processes. To estimate the ǫ from the experimental data one needs to ﬁt once more the shape of the plots (Figs. 56, 57, 58) by the mixture of Model3 and ModelTot. to the experimental data. Thepopulation ofthe eventsin missingmass oftwoprotons was not apurposeof the ﬁt,itwaspreviouslyderivedfromthedataTable7(Fig.65(b)). Fit was performed using the chi-square method; one deﬁnes the χ2 function to minimize: [Data−((1−ǫ)ModelTot. + ǫModel3)]2 χ2 = (56) σ2 +(1 −ǫ)2σ2 + ǫ2σ2 Data ModelTot. Model3 where the sum goes over each bin of the ﬁve Dalitz Plots ppX, ﬁve Dalitz Plots 3π0 and ﬁve Nyborg Plots i.e. 15 plots. The σData is the error of the point for experimental data, σModelTot. -is the error of the point for the MonteCarlodeveloped Model(Eq.47) and σModel3 -istheerrorof thepoint for Model3. For the numerical purpose of doing the ﬁt, the χ2 function(Eq.56)was redeﬁned to: [Data−((1−e)Model∗ 3 )]2 Tot. + eModel∗ χ2 = (57) σ2 +(1 −e)2σ2 + e2σ2 Data Model∗ Model∗ Tot. 3 The same as in the Section 5.1.1 on page 71. The χ2 function(Eq.57) was minimizein respect toparameter e. The (Fig.81)shows the χ2 versus the searchedparameter e. The χ2 functiondoes not have a minimum. It is seen that the searched parameter e approaches asymptotically to 0for minimal χ2 value. The contribution ofotherprocesses estimated by the above procedure would be equal to 0 value. Jagiellonian University 113 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION χ 2 22000 21000 20000 19000 18000 17000 16000 15000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 e (a) 14801.34 14801.32 14801.3 14801.28 χ 2 14801.26 14801.24 14801.22 14801.2 ×10-3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 e (b) Figure 81: χ2 versus the search parameter e for the sum of Model∗ and Tot. Model3 ∗ ﬁt. The searched parameter e approaches asymptotically to 0 for minimal χ2 value. Jagiellonian University 114 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction parameter. This could be written as: Tocheckinadditionthesensitivityoftheestimateofotherprocesses contributioninrespecttotheMonteCarlodevelopedmodelparameter β (Eq.47onpage92). Theestimatewasrepeated with the β treated asafree (1−ǫ)βModel+(1 −β)ModelǫModel(58) +123 whereprocesses Modelaredeﬁnedby(Eq.47). 12,2Fitwasperformed using thechi-squaremethod; onedeﬁnesthe χfunc-2[Data−((1−ǫ)[βModel+(1 −β)Model]+ǫModel)]1232χ[22222222σ+(1 −ǫ)βσ+(1 −β)σǫσ+Data ModelModelModel123 wherethesumgoesovereachbinoftheﬁveDalitzPlots ppX,ﬁveDalitz ∗∗ tion to minimize: = J (59) Plots 3π0 and ﬁve Nyborg Plots i.e. 15 plots. The σData is the error of the point for experimental data, σModel1,2 -is the error of the point for the Model1,2 (Eq. 47)and σModel3 -is the error of the point for Model3. For the numerical purpose of doing the ﬁt, the χ2 function(Eq.59)was redeﬁned to: ]+eModel3 )]2 ∗[Data−((1−)[αModele1 [ σ2 +(1−e)2 α2σ2 Data Model +(1 −β)Model2 +(1 −α)2σ2 Model χ2 (60) = ] Model 23 e2σ2 + ∗∗∗ 1 The same as in the Section 5.1.1 on page 71.

The χ2 function(Eq.60)was minimizein respecttoparameter e and α. The (Fig. 82) shows the χ2
versus the searched parameter e and α. It is seen that the χ2 has a global minimum for the following values: χ2 14803.3 NDF = 19715 = 0.751±0.010 (61) α = 0.10±0.01 (62) e = 0.000±0.001 (63) It is seen that the searched parameter e approaches asymptotically to 0 for the χ2 approaching to the global minimum. The obtained α (Eq. 62) valueisconsistent withinthe reachedprecision with thepreviously obtained result(Eq.33 onpage86). If one considers change ofthe χ2 by 1 from the minimum at 0, one gets the ǫ ≈ 2%. Concluding, the contribution of other processes estimated here falls to the value ∼ 2%. Jagiellonian University 115 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION ×103 120 100 80 χ 2 χ2 60 40 20 10.90.8 10.70.60.5 0.40.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20.1 0 0 (a) 20000 18000 16000 14000 12000 10000 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.00 0.02 0.040.060.080.100.120.14 0.160.18 0.20 0.22 0.06 0.04 0.02 0.00 (b) Figure 82: χ2 versus the search parameter e and α for the sum of Model ∗ 1 , Model ∗ and Model ∗ ﬁt simultaneously. The χ2 function has a global mini 2 3 mum for the e =0 and α =0.1 and χ2 =14803.3. Jagiellonian University 116 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction 5.1.3 The Cross Section extraction In order to obtain the cross section for the pp → pp3π0 one needs some suitable reaction of well-known cross section to which one can do the normalization. The good choice is the pp→ ppη where the η meson decays into the same channel η → 3π0, this reaction is nicely seen in the missing mass of the two protons(Fig. 49). The cross section for the pp→ pp3π0 one can write as: N3π0 1 σ3π0 = (64) Tot.Eff.3π0 L N3π0 -number oftheidentiﬁed 3π0 events(from experimentaldata), Tot.Eff.3π0 total reconstruction eﬃciency (Eq. 18) for the pp → pp3π0 reaction (from Monte-Carlo simulation, since oneknows the modelofthe reaction(Eq.47)), L -the integrated luminosity. From pp→ ppη reaction one gets: Nη 1 L = (65) Tot.Eff.η ση Nη -number of the identiﬁed η events(from experimental data), Tot.Eff.η total reconstruction eﬃciency(Eq.18) ofthe pp → ppη reaction, from the Monte-Carlo simulation,ση -cross section for the pp → ppη reaction for the beam kinetic energy T =2.54GeV. When combining thetwo equation(Eq.64) and(Eq.65),itgives: N3π0 Tot.Eff.η σ3π0 = ση (66) Tot.Eff.3π0 Nη The cross section for the pp → ppη at T =2.54GeV was evaluated from the existing data[13,79–82]. One see that the cross section saturates athigh kinetic energy energy(around 2 GeV)(Fig. 83(a)). The constant was ﬁtted to the values above 2 GeV; the extracted cross section used for calculation of L is: ση =220±34µb (67) The total eﬃciency for the pp → pp3π0 was calculated using Monte-Carlo simulation, using developed model(Eq.47),fromthetrue number of events NTrue (i.e. generated events) and the number of events reconstructed NReco 3π0 3π0 3π0 (see Section 4.4, page 49): Tot.Eff.3π0 = 3π0NReco NTrue (68) The total eﬃciency for the pp → ppη reaction was calculated in the similar way, using the Monte-Carlo simulation, assuming that the η meson is - Jagiellonian University 117 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION (a) (b) Jagiellonian University 118 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction produced via N∗(1535)excitation andphase space(see Section 5.2,page141 (Fig. 112)): NReco Tot.Eff.η = η (69) NTrue η The error ofthe total eﬃciency Error(Tot.Eff)was calculated assuming binomial errors distribution [83]: Error(Tot.Eff) = 1 NTrue � NReco � 1− NReco NTrue � (70) To extract from the experimental data the number of the η mesons and the number of the prompt 3π0, the distribution of the missing mass of two protons was divided into four eras A,B,C,D (Fig. 84). Areas A,B are without any η meson content. Area C is the area where the η and 3π0 mesons contribute, the D area is the area below the η meson peak(only 3π0 contribute). Deﬁning it like that leads to the following relations: Nη = NC −ND (71) N3π0 = NA + NB + ND (72) NA,B,C,D -number of events in area A,B,C,D respectively. Jagiellonian University 119 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION (a) Fit of thefourth orderpolynomial outsidethe η meson peak. (b) Subtracted η meson signal, blue markers correspond to the Monte-Carlo simulation. Jagiellonian University 120 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction The only unknown is ND the number of events in area D, which is the number of events below the η peak. It was evaluated numerically by ﬁtting thefourth orderpolynomial outsidethe η mesonpeak range andprolongingit belowthepeak(Fig.85). Thenthe ND is the integral of the ﬁtted function in the η meson peak range, evaluated numerically. Also the errors of the number of η mesons and 3π0 was calculated: Error(Nη)= JError(NC)2 + Error(ND)2 (73) Error(N3π0 )= JError(NA)2 + Error(NB)2 + Error(ND)2 (74) where Error(NA,B,C,D) is the error of the number of events in diﬀerent area: Error(NA) = J NA (75) Error(NB) = J NB (76) Error(NC) = J NC (77) the Error(ND) is calculated numerically as an error of the integral of the mentioned above function. Now the statistical error of the pp→ pp3π0 cross section Error(σ3π0 )stat. was calculated using the standard error propagation procedure: Error(σ3π0 )2 = Error(σ3π0 )2 + Error(σ3π0 )2 + (78) stat. NTof.Eff. 3π03π0 + Error(σ3π0 )2 + Error(σ3π0 )2 Nη Tot.Eff.η where ∂σ3π0 Error(σ3π0 )N3π0 = Error(N3π0 ) (79) ∂N3π0 ∂σ3π0 Error(σ3π0 )Tot.Eff.= Error(Tot.Eff.3π0 ) (80) 3π0 ∂Tot.Eff.3π0 Error(σ3π0 )Nη = ∂σ3π0 ∂Nη Error(Nη) (81) Error(σ3π0 )Tot.Eff.η = ∂σ3π0 ∂Tot.Eff.η Error(Tot.Eff.η) (82) (83) The normalization error of the 3π0 cross section Error(σ3π0 )norm., related to the to error of the η meson cross section, was separated since it is not related to the statistical precision of this experiment: Jagiellonian University 121 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION Error(σ3π0 )norm. = Error(σ3π0 )ση (84) where Error(σ3π0 )ση = ∂σ3π0 ∂ση Error(ση) (85) Prob. Cut (see Fig. 37) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mean σ3π0 [µb] 138.51 132.30 128.19 124.49 122.51 119.97 112.48 110.15 123.57 Error(σ3π0 )N3π0 [µb] 0.14 0.15 0.16 0.17 0.19 0.21 0.24 0.33 0.20 Error(σ3π0 )Tot.Eff.3π0 [µb] 0.27 0.28 0.30 0.32 0.36 0.41 0.48 0.68 0.39 Error(σ3π0 )Nη[µb] 0.47 0.47 0.48 0.50 0.55 0.60 0.65 0.87 0.57 Error(σ3π0 )Tot.Eff.η[µb] 0.47 0.49 0.51 0.55 0.61 0.70 0.81 1.13 0.66 Error(σ3π0 )stat.[µb] 0.73 0.75 0.78 0.83 0.91 1.03 1.17 1.62 0.98 Error(σ3π0 )norm.[µb] 21.41 20.45 19.81 19.24 18.93 18.54 17.38 17.02 19.10 Table 8: Results of the pp→ pp3π0 cross section studies. The pp → pp3π0 cross section and equivalent errors were evaluated for the diﬀerent probability cut, the results area presented in Tab. 8. It is seen that the cross section changes systematically with the probability cut. As the estimator of the cross sectionthe mean valuefortheprobability cuthas been taken with the corresponding statistical and normalization error: σ3π0 =123±1(stat.)±19(norm.)µb (86) To estimate the systematic error of the cross section Error(σ3π0 )sys., related tothe changes of theprobability cut,thediﬀerencebetweenthe maxi-mal cross section value σMax and the minimal cross section σMin divided by 3π0 3π0 √ the 23 has been taken[84]. σMax −σMin 3π0 3π0 Error(σ3π0 )sys. = √ =8.19µb (87) 23 The cross section value for the pp → pp3π0 reaction at beam kinetic energy T =2.54 GeV is equal to: σ3π0 =123±1(stat.)±19(norm.)±8(sys.)µb (88) Now adding the errors in quadrature this gives: σ3π0 =123±21µb (89) Jagiellonian University 122 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction this correspondstotheintegratedluminosityduringthe whole experiment of; L =4.30±0.83 105µb−1 (90) The result was compared with the available data [13] and models for pp → pp3π0 production[16–18],(Fig.86). Itisseethatthedata conﬁrms thethe cross section scaling modelbased onDelofFSI[18]. Jagiellonian University 123 Benedykt R. Jany 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION 5.1.4 The Acceptance and Eﬃciency Correction Additional checks Additional checks were performed, before doing the Acceptance and Eﬃciency Correction, to see what is the inﬂuence of the probability cut on the η meson and on the 3π0 system and on the measured distributions. First η meson peak was extracted from experimental data, by ﬁtting the polynomial of the forth order to the ranges outside the peak, for diﬀerent regions of theprobability Prob =0.2−0.4, Prob =0.5−0.7, Prob =0.8−1.0 (Fig. 87). It is seen that the η meson peaks for diﬀerent probability regions are almost the same shape(the width and the peak position is almost the same), this proofs the correctness of the analysis as well as the stability of the background extraction technique. Nextthe non resonant3π0 system was studied and comparedforprobability cut Prob > 0.2 and Prob > 0.9 for experimental data(Figs. 888990 91). Jagiellonian University 124 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.1 The pp→ pp3π0 reaction It is seen that the distributions are the same independently on the selected probability. The chosen probability cut does not introduce any systematic eﬀect in the shape of the η and 3π0 distributions. Jagiellonian University 125 Benedykt R. Jany 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Figure 88: Comparison of the experimental data with Prob > 0.2 (black) and Prob > 0.9 (red), for theMMpp < 0.5GeV/c2 and MMpp > 0.6GeV/c2 . Upper row: proton in the center of mass frame,Middle row: pion in the center of mass frame, Lower row: 3π0 system in the center of mass frame. From left kinetic energy, the polar angle and the azimuthal angle distribution. The spectra are normalized to the same number of events. Figure 89: Comparison of the Dalitz Plot ppX projections for the experimental data with Prob > 0.2 (black) and Prob > 0.9 (red). The upper row corresponds to the projection to the IM2(p3π0) axis, the lower row to the IM2(pp) axis. The columns from left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5GeV/c2, column 2 MMpp =0.6 − 0.7GeV/c2, column 3 MMpp =0.7 − 0.8GeV/c2 , column 4 MMpp =0.8−0.9GeV/c2, column 5 MMpp =0.9−1.0GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. The spectra are normalized to the same number of events. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Figure 90: Comparison of the Dalitz Plot 3π0 projections for the experimental data with Prob > 0.2 (black) and Prob > 0.9 (red). The upper andlower row correspondstotheprojectiontothe IM2(2π0)axis, x and y axis of the DalitzPlot, theyareidentical. The columnsfromleftto rightcorrespond tothefollowing missing mass oftwoprotons bins, column 1 MMpp =0.4−0.5GeV/c2, column 2 MMpp =0.6−0.7GeV/c2, column 3 MMpp =0.7−0.8GeV/c2 , column 4 MMpp =0.8−0.9GeV/c2, column 5 MMpp =0.9−1.0GeV/c2 . The plots are symmetrized against two protons and three pions -each event is ﬁlled six times. The spectra are normalized to the same number of events. 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Figure 91: Comparison of the Nyborg Plot projections for the experimental data with Prob > 0.2 (black) and Prob > 0.9 (red). The upper row corresponds to the projection to the IM(p2π0) axis, the lower row to the IM(pπ0) axis. The columns from left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5GeV/c2, column 2 MMpp =0.6 − 0.7GeV/c2, column 3 MMpp =0.7 − 0.8GeV/c2 , column 4 MMpp =0.8−0.9GeV/c2, column 5 MMpp =0.9−1.0GeV/c2 . The plots are symmetrized against two protons and three pions -each event is ﬁlled six times. The spectra are normalized to the same number of events. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction 5.1 The pp→ pp3π0 reaction 5 RESULTS AND ERROR DISCUSSION The acceptance and eﬃciency correction Theacceptanceand eﬃciency correctionwastakenintoaccounttoremove the bias of the detector acceptance and the reconstruction eﬃciency. The aim is to correct the reconstructed variable VReco(it could be one or more dimensional) to obtain the corrected one VCorr (free ofthe bias). First the totaleﬃciency(Eq.18)was calculated as afunction ofthe VReco. The pp → pp3π0 using Monte-Carlo simulation was used, using developed model(Eq.47),by comparing thetruenumber of events NTrue (generated 3π0 events) with the number of events obtained from the reconstruction NReco : 3π0 NReco Tot.Eff.3π0 (VReco)= 3π0 (91) NTrue (VReco) 3π0 The error of the total eﬃciency Error(Tot.Eff3π0 )(VReco) was calculated assumingbinomial errorsdistribution[83]: � NReco � 1 NReco 3π0 Error(Tot.Eff3π0 )(VReco)= NTrue 1− NTrue (VReco) (92) 3π0 3π0 3π0 the derived Tot.Eff.3π0 (VReco)(Eq. 91)was applied to correct the reconstructed variable VReco: VReco VCorr = (93) Tot.Eff.3π0 (VReco) To obtain the absolute normalization, the corrected variable VCorr was normalizedtotheextracted crosssection(Eq.89),dividingthespectraby theintegratedluminosity value(Eq.90). Thisprocedure was applied tothe allDalitzPlots ppX, Dalitz Plots 3π0 , Nyborg Plots and theirprojections,(seeFigs.9293949596979899). Later the corrected and normalized data were compared with Monte-Carlo Simulation of homogeneous and isotropic populated phase space and the developed Monte-Carlo Model (Eq. 47), which assumes simultaneous excitation of two baryons Δ(1235) and N∗(1440)and their decay into pp3π0 ﬁnal state(SeeFigs.9293949596979899). Itis seenthattheMonteCarlodevelopedModel(Eq.47)describesthedata signiﬁcantlybetterthan the phase space model. The acceptance andeﬃciency correctedDalitz andNyborgPlots are available as tables of numbers in Appendix G. Jagiellonian University 130 Benedykt R. Jany Raw Data MM =0.4-0.5 GeV/c2 Raw Data MM =0.6-0.7 GeV/c2 Raw Data MM =0.7-0.8 GeV/c2 Raw Data MM =0.8-0.9 GeV/c2 Raw Data MM =0.9-1.0 GeV/c2 pppppp pppp ) GeV2/c 4 6 5.5 M2(p p) GeV2/c 4 1 M2(p p) GeV2/c 4 1 4.4 400 4.85.2 700 600 3000 3500 1000 4.3 3.9 350 5 4.8 4.6 3000 2500 4.2 300 250 800 4.4 2 2 2 12 2500 2000 4.1 500 3.8 p 2000 4.6 5 4 4.2 400600 4.4 200 3.9 3.7 1500 4.5 4.2 1500 4 300 3.8 150400 1000 4 3.6 4 3.7 3.8 1000 200 100 500 3.8 3.6 200 3.6 500 100 50 3.6 3.5 3.5 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 1,2 1,2 1,2 1,2 1,2 Total Efficiency MM =0.4-0.5 GeV/c Total Efficiency MM =0.6-0.7 GeV/c Total Efficiency MM =0.7-0.8 GeV/c Total Efficiency MM =0.8-0.9 GeV/c Total Efficiency MM =0.9-1.0 GeV/c 222 22 pppppp pppp 0.05 0.05 0.05 0.05 0.05 6 4.44.8 4.6 5.2 5 0.045 0.045 0.045 0.045 0.045 4.3 3.9 5.5 0.04 0.04 0.04 0.04 0.04 4.2 4.8 0.035 0.035 0.035 0.035 0.035 2 2 2 2 4.4 4.1 3.8 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 131 BenedyktR.Jany M2(pp) GeV2/c4 1 4.2 0.03 0.025 M2(pp) GeV2/c4 M2(p p) GeV2/c 4 1 1 0.03 0.025 M2(pp) GeV2/c4 M2(p p) GeV2/c 4 12 2(pp) GeV2/c 4 M2(pp) GeV2/c4 M2(p 1 1 5 0.03 0.025 M2(pp) GeV2/c4 1 4.6 0.03 0.03 0.025 4 4.4 0.025 3.9 3.7 4.5 4.2 0.02 0.02 4 0.02 0.02 0.02 3.8 4 0.015 0.015 0.015 3.7 0.015 3.6 0.015 3.8 4 3.8 0.01 0.01 0.01 0.01 0.01 3.6 3.6 3.6 3.5 0.005 0.005 0.005 0.005 0.005 3.5 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 1,2 1,2 1,2 1,2 1,2 Corrected Data MM =0.4-0.5 GeV/c2 Corrected Data MM =0.6-0.7 GeV/c2 Corrected Data MM =0.7-0.8 GeV/c2 Corrected Data MM =0.8-0.9 GeV/c2 Corrected Data MM =0.9-1.0 GeV/c2 pppp pppppp 6 10 9 5.5 8 50 45 80 70 80 50 45 40 M2(p p) GeV2/c 4 1 M2(p p) GeV2/c 4 1 M2(p p) GeV2/c 4 1 M2(p p) GeV2/c 4 4.44.8 5.2 4.3 70 3.9 5 4.8 4.6 4.4 40 4.2 4.1 60 60 7 35 35 2 2 2 2 12 3.8 4.6 50 40 50 40 5 6 30 30 25 4 4.2 M 4.4 5 25 3.9 3.7 4.5 4.2 4 20 4 20 3.8 30 30 4 3 15 3.6 15 3.7 3.8 20 20 4 3.8 2 10 10 3.6 3.6 10 10 3.6 3.5 1 5 5 3.5 3.5 0 3.4 0 3.4 0 0 0 22.5 33.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 1,2 1,2 1,2 1,2 1,2 pp pppppppp Corrected Data Errors MM =0.4-0.5 GeV/c2 Corrected Data Errors MM =0.6-0.7 GeV/c2 Corrected Data Errors MM =0.7-0.8 GeV/c2 Corrected Data Errors MM =0.8-0.9 GeV/c2 Corrected Data Errors MM =0.9-1.0 GeV/c2 6 0.2 0.18 5.5 0.16 0.2 0.18 0.2 0.18 0.2 0.2 0.18 0.16 ) GeV 2/c 4 M2(pp) GeV2/c4 1 M2(pp) GeV2/c4 1 M2(pp) GeV2/c4 1 M2(pp) GeV2/c4 4.44.85.2 0.18 4.3 3.9 5 4.8 4.6 4.4 0.16 0.16 0.14 0.16 4.2 4.1 0.14 0.14 0.14 0.14 2 2 2 2 12 3.8 p 1 4.6 5 0.12 2(p 0.12 0.12 0.1 0.12 0.1 0.12 0.1 4 4.2 M 4.4 0.1 0.1 3.9 3.7 4.5 4.2 0.08 0.08 4 0.08 0.08 0.08 3.8 4 0.06 0.06 0.06 3.7 0.06 3.6 0.06 3.8 4 3.8 0.04 0.04 0.04 0.04 0.04 3.6 3.6 3.6 0.02 0.02 0.02 0.02 3.5 0.02 3.5 3.5 0 3.4 0 3.4 0 0 0 2 2.5 3 3.5 4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 2.6 2.8 3 3.2 3.4 3.6 3.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 M2(p 3π0) GeV2/c 4 1,2 1,2 1,2 1,2 1,2 MC Model MM =0.4-0.5 GeV/c2 MC Model MM =0.6-0.7 GeV/c2 MC Model MM =0.7-0.8 GeV/c2 MC Model MM =0.8-0.9 GeV/c2 MC Model MM =0.9-1.0 GeV/c2 pppppppp pp 6 10 9 5.5 8 50 45 80 70 80 50 45 40 ) GeV2/c4 M2(p p) GeV2/c4 1 M2(p p) GeV2/c4 1 M2(p p) GeV2/c4 1 M2(p p) GeV2/c4 4.44.85.2 4.3 70 3.9 5 4.8 4.6 4.4 40 4.2 4.1 60 60 7 35 35 2 2 2 2 12 3.8 p 1 4.6 50 40 50 40 2(p 5 6 30 30 25 4 4.2 M 4.4 5 25 3.9 3.7 4.5 4 4.2 20 4 30 3.8 30 20 4 2 3 3.8 4 10 15 3.8 20 3.6 3.7 20 3.6 10 15 1 3.6 5 3.6 10 3.5 10 3.5 5 3.5 2 2.5 3 3.5 40 3.4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 0 2.63.4 2.8 3 3.2 3.4 3.6 3.8 0 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 0 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 0 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p 3π0) GeV2/c 4 1,2 M2(p Figure 92: Dalitz Plot ppX. M2 (pp)versus M2(p3π0). The rows from up to down correspond to: row 1: the experimentaldata,row2: theTotalEﬃciencyfunction(Eq.91),row3: thecorrected experimentaldata,row4: thestatistical errorof thecorrecteddata,row5:theMonte-Carlodeveloped model(Eq.47).Thecolumnsfromlefttoright correspond tothefollowing missing mass oftwoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9 − 1.0 GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. The model is normalized to the same number of events as in the experimental data. It is seen that the Monte-Carlo developed model(Eq.47)row5describesthedata signiﬁcantlybetterthanthephase space model(homogenousDalitzplot). Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. Figure 93: Dalitz Plot ppX projection to the M2 (p3π0) axis. The rows from up to down correspond to: row 1: the experimental data, row 2: the Total Eﬃciency function (Eq. 91), row 3: the corrected experimental data (black),theMonte-Carlodeveloped model(Eq.47)(red),theMonte-Carlophasespace(blue). Thecolumnsfrom left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5 GeV/c2 , column 2 MMpp =0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2 , column 5 MMpp =0.9 − 1.0 GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. The models are normalized to the same number of events as in the experimental data. It is seen that the Monte-Carlodeveloped model(Eq. 47) row3describesthedata signiﬁcantlybetterthanthephase space model. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Figure 94: Dalitz Plot ppX projection to the M2(pp) axis. The rows from up to down correspond to: row 1: the experimental data, row 2: the Total Eﬃciency function (Eq. 91), row 3: the corrected experimental data (black),theMonte-Carlodeveloped model(Eq.47)(red),theMonte-Carlophasespace(blue). Thecolumnsfrom left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5 GeV/c2 , column 2 MMpp =0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2 , column 5 MMpp =0.9 − 1.0 GeV/c2 . The plots are symmetrized against two protons -each event is ﬁlled two times. The models are normalized to the same number of events as in the experimental data. It is seen that the Monte-Carlodeveloped model(Eq. 47) row3describesthedatasigniﬁcantlybetterthanthephasespace model. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Raw Data MM =0.4-0.5 GeV/c2 Raw Data MM =0.6-0.7 GeV/c2 Raw Data MM =0.7-0.8 GeV/c2 Raw Data MM =0.8-0.9 GeV/c2 Raw Data MM =0.9-1.0 GeV/c2 pppppp pppp 2500 2000 0.5 0.45 1600 1400 0.4 1200 0.35 1000 0.3 0.6 1800 0.5 1600 1400 0.7 0.6 M2(π 0 π 0) GeV2/c 4 12 π 0) GeV2/c 4 2 M2(π 01 π 02) GeV2/c 4 M2(π 0 π 0) GeV2/c 4 12 M2(π 0 π 0) GeV2/c 4 12 0.35 0.3 300 1000 0.14 250 200 800 0.12 0.5 0.25 0.4 M2(π 0 1 1200 1000 1500 0.4 600 0.1 0.2 800 150 0.30.25 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 134 BenedyktR.Jany 8001000 0.3 600 0.08 400 0.2 0.15 100600 0.2 0.2 0.15 400 400500 0.1 0.06 200 50 0.1 200 0.1 200 0.1 0.05 0.05 0.04 0 0 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0 π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 Total Efficiency MM =0.4-0.5 GeV/c2 Total Efficiency MM =0.6-0.7 GeV/c2 Total Efficiency MM =0.7-0.8 GeV/c2 Total Efficiency MM =0.8-0.9 GeV/c2 Total Efficiency MM =0.9-1.0 GeV/c2 232323 2323 pppppp pppp 0.05 0.05 0.045 0.05 0.05 0.05 0.5 0.6 0.7 M2(π 0 π 0) GeV2/c4 12 π 0) GeV2/c4 2 M2(π 01 π 02) GeV 2/c 4 M2(π 0 π 0) GeV2/c4 12 M2(π 0 π 0) GeV2/c4 12 0.35 0.045 0.045 0.045 0.04 0.045 0.04 0.14 0.45 0.60.50.3 0.04 0.12 0.035 0.04 0.04 0.4 0.035 0.035 0.035 0.03 0.5 0.035 0.03 0.4 0.025 0.35 0.3 0.25 0.4 M2(π 0 1 0.1 0.03 0.025 0.03 0.03 0.025 0.025 0.025 0.2 0.30.25 0.02 0.02 0.02 0.02 0.3 0.02 0.08 0.2 0.15 0.015 0.015 0.015 0.2 0.015 0.015 0.2 0.15 0.01 0.1 0.01 0.01 0.01 0.01 0.06 0.1 0.1 0.005 0.005 0.005 0.005 0.1 0.005 0.05 0.05 0.04 0 0 0 0 0 0.04 0.06 0.08 0.1 0.12 0.14 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0 π0) GeV2/c 4 M2(π0π0) GeV2/c 4 M2(π0π0) GeV2/c 4 Corrected Data MM =0.4-0.5 GeV/c2 Corrected Data MM =0.6-0.7 GeV/c2 Corrected Data MM =0.7-0.8 GeV/c2 Corrected Data MM =0.8-0.9 GeV/c2 Corrected Data MM =0.9-1.0 GeV/c2 232323 2323 pppp pppppp 50000 0.14 45000 40000 0.12 35000 9000 8000 M2(π 01 π 02) GeV2/c 4 0.5 2500 0.45 2000 0.4 0.35 1500 0.3 600 60 0.6 0.7 M2(π 0 π 0) GeV2/c 4 12 π 0) GeV2/c 4 2 M2(π 0 π 0) GeV2/c 4 12 M2(π 0 π 0) GeV2/c 4 12 0.35 0.6 500 50 0.50.3 7000 0.5 0.4 6000 400 300 40 0.25 0.4 M2(π 0 1 30000 0.1 25000 5000 30 0.2 0.3 0.25 4000 20000 1000 0.3 0.08 0.15 3000 0.2 200 20 15000 0.2 0.06 10000 0.1 2000 0.15 500 100 0.2 10 5000 1000 0.1 0.1 0.1 0.04 0.06 0.08 0.1 0.12 0.140.04 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0 0.1 0.2 0.3 0.4 0.5 0.60 0.1 0.2 0.3 0.4 0.5 0.6 0.70 ) GeV2/c 4π0M2(π0 ) GeV2/c 4π0M2 (π0 ) GeV2/c 4π0M2 (π0 ) GeV2/c 4π0M2(π0 ) GeV2/c 4π0M2(π0 32 32 32 32 32 =0.4-0.5 GeV/c2Corrected Data Errors MM =0.6-0.7 GeV/c2Corrected Data Errors MM =0.7-0.8 GeV/c2Corrected Data Errors MM =0.8-0.9 GeV/c2Corrected Data Errors MM =0.9-1.0 GeV/c2Corrected Data Errors MM pp pp pp pp pp 0.2 0.2 0.18 0.2 0.2 0.2 0.5 0.6 0.7 M2(π 0 π 0) GeV2/c4 12 π 0) GeV2/c4 2 M2(π 01 π 02) GeV 2/c 4 M2(π 0 π 0) GeV2/c4 12 M2(π 0 π 0) GeV2/c4 12 0.35 0.18 0.18 0.18 0.16 0.18 0.16 0.14 0.45 0.60.50.3 0.16 0.12 0.14 0.16 0.16 0.4 0.14 0.14 0.14 0.12 0.5 0.14 0.12 0.4 0.1 0.35 0.3 0.25 0.4 M2(π 0 1 0.1 0.12 0.1 0.12 0.12 0.1 0.1 0.1 0.2 0.30.25 0.08 0.08 0.08 0.08 0.3 0.08 0.08 0.15 0.2 0.06 0.06 0.06 0.2 0.06 0.06 0.06 0.04 0.1 0.04 0.15 0.04 0.04 0.2 0.04 0.02 0.02 0.1 0.02 0.1 0.02 0.1 0.02 0.040.04 0.06 0.08 0.1 0.12 0.14 0 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50 0.1 0.2 0.3 0.4 0.5 0.60 0.1 0.2 0.3 0.4 0.5 0.6 0.70 ) GeV2/c 4 3π0 2M2(π0 ) GeV2/c 4 3π0 2M2 (π0 ) GeV2/c 4 3π0 2M2 (π0 ) GeV2/c 4 3π0 2M2(π0 ) GeV2/c 4 3π0 2M2(π0 =0.4-0.5 GeV/c2 pp MC Model MM =0.6-0.7 GeV/c2 pp MC Model MM =0.7-0.8 GeV/c2 pp MC Model MM =0.8-0.9 GeV/c2 pp MC Model MM =0.9-1.0 GeV/c2 pp MC Model MM 50000 0.14 45000 40000 0.12 35000 9000 8000 M2(π 01 π 02) GeV2/c4 0.5 2500 0.45 2000 0.4 0.35 1500 0.3 600 60 0.6 0.7 M2(π 0 π 0) GeV2/c4 12 π 0) GeV2/c4 2 M2(π 0 π 0) GeV2/c4 12 M2(π 0 π 0) GeV2/c4 12 0.35 0.6 500 50 0.50.3 7000 0.5 0.4 6000 400 300 40 0.25 0.4 M2(π 0 1 0.1 30000 25000 5000 30 0.2 0.3 0.25 4000 20000 1000 0.3 0.08 15000 0.15 3000 0.2 0.2 200 20 0.06 5000 10000 0.1 1000 2000 0.1 0.15 500 0.1 100 0.1 0.2 10 0.040.04 0.06 0.08 0.1 0.12 0.14 0 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50 0.1 0.2 0.3 0.4 0.5 0.60 0.1 0.2 0.3 0.4 0.5 0.6 0.70 ) GeV2/c 4 3π0 2M2(π0 ) GeV2/c 4 3π0 2M2 (π0 ) GeV2/c 4 3π0 2M2 (π0 ) GeV2/c 4 3π0 2M2(π0 ) GeV2/c 4 3π0 2M2(π0 Figure 95: Dalitz Plot 3π0 . M2(2π0)versus M2 (2π0). The rows from up to down correspond to: row 1: the experimentaldata,row2: theTotalEﬃciencyfunction(Eq.91),row3: thecorrected experimentaldata,row4: thestatistical errorof thecorrecteddata,row5:theMonte-Carlodeveloped model(Eq.47).Thecolumnsfromlefttoright correspond to thefollowing missing mass of twoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9−1.0 GeV/c2 . The plots are symmetrized against tree pions -each event is ﬁlled six times. The model is normalized to the same number of events as in the experimental data. It is seen that the Monte-Carlo developed model (Eq.47)row5describesthedata signiﬁcantlybetterthanthephase space model(homogeneousDalitzplot). Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. Figure 96: Dalitz Plot 3π0 projection to the M2(2π0) axis. The rows from up to down correspond to: row 1: the experimental data, row 2: the Total Eﬃciency function (Eq. 91), row 3: the corrected experimental data (black),theMonte-Carlodeveloped model(Eq.47)(red),theMonte-Carlophasespace(blue). Thecolumnsfrom left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5 GeV/c2 , column 2 MMpp =0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2 , column 5 MMpp =0.9−1.0 GeV/c2 . The plots are symmetrized against tree pions -each event is ﬁlled six times. The models are normalized to the same number of events asinthe experimentaldata. Itis seen that theMonte-Carlo developed model(Eq.47)row3describes thedata signiﬁcantlybetter than thephase space model. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Raw Data MM =0.4-0.5 GeV/c2 Raw Data MM =0.6-0.7 GeV/c2 Raw Data MM =0.7-0.8 GeV/c2 Raw Data MM =0.8-0.9 GeV/c2 Raw Data MM =0.9-1.0 GeV/c2 pppppp pppp 1.7 1.6 1.7 π 0) GeV/c2 π 0) GeV/c2 1.6 1000 1.6 1400 1.5 1200 18001800 1200 1.51.6 1600 1.5 1600 1400 8001000 1 1 1 1 1.5 1.5 1400 1 1 1 1 1 1.4 M(p M(p 1.4 1000 12001200 800 1.4 600 1.4 1.4 1000 800 1000 1.3 1.3 600 1.3 1.3 1.3 800 800 400600 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 136 BenedyktR.Jany 1π 10) GeV/c2 M(p π 0) GeV/c2 M(p π 0) GeV/c2 1 1 π 0) GeV/c2 M(p π 0) GeV/c2 π 0) GeV/c2 M(p π 0) GeV/c2 1.2 600 600 400 1.2 1.2 1.2 1.2 400 400 400 200 1.1 200 1.1 1.1 1.1 200 1.1 200 200 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.5 1.7 1.4 1.6 1.8 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 232323 2323 222 22 Total Efficiency MM =0.4-0.5 GeV/c2 Total Efficiency MM =0.6-0.7 GeV/c2 Total Efficiency MM =0.7-0.8 GeV/c2 Total Efficiency MM =0.8-0.9 GeV/c2 Total Efficiency MM =0.9-1.0 GeV/c2 pppppp pppp 0.05 0.045 0.05 0.05 0.05 0.045 0.05 0.045 1.5 0.04 1.7 1.7 π 0) GeV/c2 π 0) GeV/c2 1.6 1.6 0.045 0.04 0.045 0.04 1.6 1.6 1.5 0.04 0.035 0.04 0.035 1.5 1 1 1 1 1.5 1.5 0.035 0.035 0.035 1.4 1 1 1 1 1 M(p M(p M(p M(p 1.4 1.4 0.03 0.03 0.03 0.03 0.03 1.4 1.4 0.025 0.025 0.025 0.025 1.3 0.025 1.3 1.3 1.3 1.3 0.02 0.02 0.02 0.02 0.02 1.2 1.2 0.015 1.2 0.015 1.2 0.015 0.015 0.015 1.2 0.01 0.01 0.01 0.01 0.01 1.1 1.1 1.1 1.1 1.1 0.005 0.005 0.005 0.005 0.005 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 232323 2323 222 22 Corrected Data MM =0.4-0.5 GeV/c2 Corrected Data MM =0.6-0.7 GeV/c2 Corrected Data MM =0.7-0.8 GeV/c2 Corrected Data MM =0.8-0.9 GeV/c2 Corrected Data MM =0.9-1.0 GeV/c2 pppp pppppp 1000 1.6 900 800 1.5 700 1.7 3500 1.6 3000 1.5 2500 3000 1600 350 1.7 π 0) GeV/c2 1 π 0) GeV/c2 1 π 0) GeV/c2 1 π 0) GeV/c2 1.6 1.5 1400 1.51.6 1.5 3002500 1200 1 1 1 1 250 1 2000 1.4 M(p M(p M(p M(p M(p 1.4 1000 1.4 600 1.4 2000 200 1.4 500 1500 800 1.3 1.3 1.3 1.3 1500 1.3 150400 600 1000 1.2 1.2 300 1.2 1000 1.2 100 1.2 400 200 500 1.1 1.1 500 1.1 50200 1.1 1.1 100 01 01 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0 π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 232323 2323 222 22 Corrected Data Errors MM =0.4-0.5 GeV/c2 Corrected Data Errors MM =0.6-0.7 GeV/c2 Corrected Data Errors MM =0.7-0.8 GeV/c2 Corrected Data Errors MM =0.8-0.9 GeV/c2 Corrected Data Errors MM =0.9-1.0 GeV/c2 pp pppppppp 0.2 0.18 0.2 0.2 0.2 0.18 0.2 0.18 1.5 0.16 1.7 1.7 π 0) GeV/c2 1 π 0) GeV/c2 π 0) GeV/c2 π 0) GeV/c2 π 0) GeV/c2 1.6 1.6 0.18 0.16 0.18 0.16 1.6 1.6 1.5 0.16 0.14 0.16 0.14 1.5 1 1 1 1 1.5 1.5 0.14 0.14 0.14 1.4 1 1 1 1 1 M(p M(p M(p M(p M(p 1.4 1.4 0.12 0.12 0.12 0.12 0.12 1.4 1.4 0.1 0.1 0.1 1.3 0.1 1.3 0.1 1.3 0.08 1.3 0.08 1.3 0.08 0.08 0.08 1.2 0.06 1.2 0.06 1.2 0.06 1.2 0.06 1.2 0.06 0.04 0.04 0.04 0.04 0.04 1.1 0.02 1.1 0.02 1.1 0.02 1.1 0.02 1.1 0.02 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.11 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.11 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p =0.4-0.5 GeV/c2 pp MC Model MM =0.6-0.7 GeV/c2 pp MC Model MM =0.7-0.8 GeV/c2 pp MC Model MM =0.8-0.9 GeV/c2 pp MC Model MM =0.9-1.0 GeV/c2 pp MC Model MM 1000 1.6 900 800 1.5 700 1.7 3500 1.6 3000 1.5 2500 3000 1600 1400 350 1.7 π 0) GeV/c2 1 π 0) GeV/c2 π 0) GeV/c2 π 0) GeV/c2 π 0) GeV/c2 1.6 1.51.6 1.5 3002500 1.5 1200 1 1 1 1 250 1 1 1 1 1 2000 1.4 M(p M(p M(p M(p M(p 1.4 1000 1.4 600 1.4 2000 200 1.4 500 1500 1.3 800 1.3 1.2 1.3 300 400 1.2 1.3 1000 1500 1.2 1.3 1000 1.2 400 600 1.2 100 150 1.1 100 200 1.1 500 1.1 500 1.1 200 1.1 50 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.11 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.11 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p ) GeV/c2 3π0 2π0 2M(p Figure 97: Nyborg Plot. M2(p1π0)versus M(p2π0π0). The rows from up to down correspond to: row 1: the experi 1 23 mentaldata,row2: theTotalEﬃciencyfunction(Eq.91),row3: thecorrected experimentaldata,row4: thestatistical error of the correcteddata, row5: theMonte-Carlodeveloped model(Eq.47). The columnsfromleftto right correspond to thefollowing missing mass of twoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2,column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9 − 1.0 GeV/c2 . The plots are symmetrized against tree pions and two protons -each event is ﬁlled six times. The model is normalized to the same number of events as in the experimental data. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. Raw Data MM =0.4-0.5 GeV/c2 Raw Data MM =0.6-0.7 GeV/c2 Raw Data MM =0.7-0.8 GeV/c2 Raw Data MM =0.8-0.9 GeV/c2 Raw Data MM =0.9-1.0 GeV/c2 pppppppp pp 14000 40000 8000 12000 2500040000 35000 700035000 20000 3000010000 600030000 25000 5000250008000 15000 20000 400020000 6000 1000015000 300015000 4000 10000 200010000 5000 2000 5000 5000 1000 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 23 2323 2323 22222Total Efficiency MM =0.4-0.5 GeV/c2 Total Efficiency MM =0.6-0.7 GeV/c2 Total Efficiency MM =0.7-0.8 GeV/c2 Total Efficiency MM =0.8-0.9 GeV/c2 Total Efficiency MM =0.9-1.0 GeV/c2 pp pppppppp 0.04 0.04 0.045 0.05 0.035 0.03 0.035 0.04 0.03 0.035 0.04 0.03 0.025 0.03 0.025 0.025 0.02 0.03 0.025 0.02 0.02 0.015 0.02 0.015 0.02 0.015 0.015 0.01 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 23 2323 2323 22222Corrected Data MM =0.4-0.5 GeV/c2 Corrected Data MM =0.6-0.7 GeV/c2 Corrected Data MM =0.7-0.8 GeV/c2 Corrected Data MM =0.8-0.9 GeV/c2 Corrected Data MM =0.9-1.0 GeV/c2 pp pppppppp 600 300 dσ/dM2(p π 0 π 0) µb/GeV/c2 23 2 40 35 30 25 20 dσ/dM2(p π 0 π 0) µb/GeV/c2 23 2 150 dσ/dM2(p π 0 π 0) µb/GeV/c2 23 2 500 dσ/dM2(p π 0 π 0) µb/GeV/c2 23 dσ/dM2(p π 0 π 0) µb/GeV/c2 23 700 500 250200 600 400 200 2 2 400 300 150 100 300 15 200 100 200 10 50 100 50 100 5 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 M(p π0π0) GeV/c2 23 2323 2323 22222 Figure 98: Nyborg Plot projection to the M(p2π20π30) axis. The rows from up to down correspond to: row 1: the experimental data, row 2: the Total Eﬃciency function (Eq. 91), row 3: the corrected experimental data (black),theMonte-Carlodeveloped model(Eq.47)(red),theMonte-Carlophasespace(blue). Thecolumnsfrom left to right correspond to the following missing mass of two protons bins, column 1 MMpp =0.4 − 0.5 GeV/c2 , column 2 MMpp =0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2 , column 5 MMpp =0.9−1.0 GeV/c2 . The plots are symmetrized against tree pions and two protons -each event is ﬁlled six times. The models are normalized to the same number of events as in the experimental data. It is seen thattheMonte-Carlodeveloped model(Eq.47)row3describesthedata signiﬁcantlybetterthanthephase space model. Fully expandableand colored versionof the ﬁgureisavailableintheattached electronicversionof thethesis. 5 RESULTS AND ERROR DISCUSSION 5.1 The pp →pp3π 0 reaction Jagiellonian University 137 BenedyktR.Jany 5.1 The pp →pp3π 0 reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 138 BenedyktR.Jany Raw Data MM =0.4-0.5 GeV/c2 Raw Data MM =0.6-0.7 GeV/c2 Raw Data MM =0.7-0.8 GeV/c2 Raw Data MM =0.8-0.9 GeV/c2 Raw Data MM =0.9-1.0 GeV/c2 pppppppp pp 50000 30000 1000018000 50000 16000 25000 40000 8000 40000 20000 14000 12000 30000 600030000 10000 15000 8000 20000 400020000 100006000 4000 10000 200010000 5000 2000 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.1 1.2 1.3 1.4 1.5 1.6 1.1 1.2 1.3 1.4 1.5 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 111 11 11111Total Efficiency MM =0.4-0.5 GeV/c2 Total Efficiency MM =0.6-0.7 GeV/c2 Total Efficiency MM =0.7-0.8 GeV/c2 Total Efficiency MM =0.8-0.9 GeV/c2 Total Efficiency MM =0.9-1.0 GeV/c2 pp pppppppp 0.03 0.12 0.035 0.05 0.05 0.03 0.025 0.1 0.04 0.04 0.025 0.02 0.08 0.02 0.03 0.03 0.015 0.06 0.015 0.02 0.02 0.04 0.01 0.01 0.01 0.01 0.005 0.02 0.005 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.1 1.2 1.3 1.4 1.5 1.6 1.1 1.2 1.3 1.4 1.5 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 111 11 11111Corrected Data MM =0.4-0.5 GeV/c2 Corrected Data MM =0.6-0.7 GeV/c2 Corrected Data MM =0.7-0.8 GeV/c2 Corrected Data MM =0.8-0.9 GeV/c2 Corrected Data MM =0.9-1.0 GeV/c2 pp pppppppp 400 dσ/dM(p π 0) µb/GeV/c2 1 1 70 60 50 40 dσ/dM(p π 0) µb/GeV/c2 1 1000 dσ/dM(p π 0) µb/GeV/c2 1 1 dσ/dM(p π 0) µb/GeV/c2 1 300 dσ/dM(p π 0) µb/GeV/c2 1 700300 350 600800 250 1 1 1 500 250200 600 400 200 150 30 400 300 150 100 20 200 100 200 50 10 100 50 0 0 0 0 0 1.1 1.2 1.3 1.4 1.5 1.6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.1 1.2 1.3 1.4 1.5 1.6 1.1 1.2 1.3 1.4 1.5 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 M(p π0) GeV/c2 111 11 11111 Figure 99: Nyborg Plot projection to the M2(p1π10) axis. The rows from up to down correspond to: row 1: the experimentaldata, row2: theTotalEﬃciency function(Eq.91), row3: thecorrected experimentaldata(black),the Monte-Carlodeveloped model(Eq.47)(red),theMonte-Carlophase space(blue). The columnsfromleftto right correspondtothefollowing missing mass oftwoprotonsbins, column1 MMpp =0.4−0.5 GeV/c2, column2 MMpp = 0.6−0.7 GeV/c2, column 3 MMpp =0.7−0.8 GeV/c2, column 4 MMpp =0.8−0.9 GeV/c2, column 5 MMpp = 0.9−1.0 GeV/c2 .Theplotsaresymmetrized againsttreepionsand twoprotons -each eventisﬁlled sixtimes. The models are normalized to the same number of events as in the experimental data. It is seen that the Monte-Carlo developed model(Eq.47)row3describes thedata signiﬁcantlybetter than thephase space model. Fully expandable and colored version of the ﬁgure is available in the attached electronic version of the thesis. 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction 5.2 The pp→ ppη(3π0) reaction The pp → ppη reaction at an incident proton momentum 3.35 GeV/c (T = 2.541 GeV), which corresponds to the excess energy Q = 455 MeV, was also measured via η meson decay into three neutral pions. All ﬁnal state particles were detected, the signatures of the two protons were registered in theForwardDetector of theWASA, while the threepions were reconstructed from the decay into six photons in the Electromagnetic Calorimeter (see Section 4) the same way as in the case of the pp → pp3π0 reaction (see Section 5.1). Many previous high statistics experimental studies of the reaction dynamicsin the threshold region[79–81,85],forbeamkinetic energiesless than 2 GeV, show an important role of the N∗(1535)baryon resonance. The near threshold data were interpreted mostly in the framework of the one-boson exchange models[30–37](by exchange of variouslight mesonslike π,η,ρ,ω) and a dominant role of the resonance N∗(1535)S11. In the threshold region alsotheFinalStateInteractionplays animportant role[16,17]. For higher energies above 2 GeV beam kinetic energy, there are only few studies [86, 87] which consider two dominant production mechanisms: the resonantproduction(via excitation of N∗(1535))and the non resonant production. Available phase space and the observables After the background subtraction around 200k events of the pp→ ppη are available(Fig.100). Thedata analysis wasperformed asfollows. First the experimental accessibility of the phase space area was studied. Dalitzplot ppη for the Monte-Carlo simulation based on a homogeneous and isotropicphase spacepopulation andaproduction via excitation of N∗(1535), were prepared. One ﬁnds that the acceptance for this reaction is limited (Fig. 101). Since one wantsto study thedynamics of thisprocessitis convenientto introducethefollowing variables, accordinglytothe[86]. The momentum of the η meson in the Center of Mass qηCM ; the angle between the beam direction and the η meson in the Center of Mass θηCM ; the momentum of the protoninproton-proton restframepPP p ,the anglebetweenthebeamdirection and the proton direction in the proton-proton rest frame θpPP . Looking into the acceptance in four new deﬁned variables, by comparing the Monte-Carlo simulationfortrue events with the reconstructed ones(Figs.102,103,104), one concludes that only high η momenta could be measured. For the further studies the following region was selected: Jagiellonian University 141 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION FD MM vs Prob 3pi0 20000 Experimental Data 18000 Monte-Carlo PhSp 16000 Monte-Carlo The Model 14000 Entries 1106922 12000 Mean 0.6508 10000 RMS 0.1155 8000 6000 4000 2000 0.40 0.5 0.6 0.7 0.8 0.9 1 [GeV/c2]ppMM (a) Experimental Data(black), Monte-Carlo Phase Space(blue), the Model(Eq. 47)(red). FD MM vs Prob 3pi0 16000 Mean 0.546 14000 RMS 0.01489 Integral 2.126e+05 12000 DATA 10000 Monte-Carlo 8000 6000 4000 2000 0 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 MMpp [GeV/c2] (b) Subtractedpromptbackground. ExperimentalData(black), Monte-Carlo simulation(blue). Figure 100: Missing Mass of the two protons. Jagiellonian University 142 Benedykt R. Jany Dalitz Plot pp Eta Dalitz Plot pp Eta 4 14000 12000 3.5 4 η) GeV2/c4 η) GeV2/c4 1400 3.5 1200 2 2 M1(p M1(p 10000 1000 5 RESULTS AND ERROR DISCUSSION 5.2 The pp →ppη(3π 0 ) reaction Jagiellonian University 143 BenedyktR.Jany 3 8000 3 800 2.5 6000 2.5 600 4000 400 2 2000 2 200 2 2.5 3 η) GeV2/c4 2 M2(p 3.5 4 0 2 2.5 3 η) GeV2/c4 2 M2(p 3.5 4 0 (a) N∗ (1535)true events (b) Phase Space true events Dalitz Plot pp Eta Dalitz Plot pp Eta 4 140 120 3.5 4 70 60 η) GeV2/c4 η) GeV2/c4 3.5 50 2 2 M1(p M1(p 100 3 80 3 40 60 30 2.5 2.5 40 20 2 20 2 10 2 2.5 3 η) GeV2/c4 2 M2(p 3.5 4 0 2 2.5 3 η) GeV2/c4 2 M2(p 3.5 4 0 (c) N∗ (1535)reconstructed events (d)[PhaseSpacereconstructedevents Figure101: Monte-Carlo,DalitzPlot ppη -IM2(p1η)versus IM2(p2η).Theplotissymmetrized againsttwoprotons -each event is ﬁlled two times. 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION MomentumEtaCMvsThetaEtaCM MomentumEtaCM vs ThetaEtaCM Entries 2000000 . C.M . C.M qη [GeV/c] qη [GeV/c] 0.8 0.9 1 Mean x -0.00272 Mean y 0.3392 RMS x 0.5666 RMS y 0.1208 0.7 102 0.6 0.5 0.4 10 0.3 0.2 -1.50 0.1 -1 -0.5 0 0.5 1 CM)Cos(θη 1.5 1 (a) true events MomentumEtaCMvsThetaEtaCM MomentumEtaCM vs ThetaEtaCM Entries 17305 1 Mean x -0.7429 Mean y 0.4694 0.9 RMS x 0.194 RMS y 0.0888 0.8 10 0.7 0.6 0.5 1 0.4 0.3 0.2 0.1 10-1 0 -1.5 -1 -0.5 0 0.5 1 1.5 CM) Cos(θη (b) reconstructed events Figure 102: Monte-Carlo simulation, η momentum versus cos(θηCM ) in the Center of Mass system. Jagiellonian University 144 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction MomentumProtPP vs ThetaPPP MomentumProtPPvsThetaPPP [GeV/c] PP rest PP rest p [GeV/c] p 0.8 0.9 1 Entries Mean x Mean y RMS x RMS y 4000000 -5.034e-19 0.5658 0.6088 0.0884 103 0.7 0.5 0.6 102 0.4 0.3 10 0.2 0.1 -1.50 -1 -0.5 0 0.5 PP rest)pCos(θ 1 1.5 1 (a) true events p MomentumProtPPvsThetaPPP MomentumProtPP vs ThetaPPP Entries 34610 1 Mean x -1.269e-16 Mean y 0.4654 0.9 RMS x 0.7031 RMS y 0.09343 0.8 p 0.7 10 0.6 0.5 0.4 1 0.3 0.2 0.1 10-1 -1.50 -1 -0.5 0 0.5 PP rest)pCos(θ 1 1.5 (b) reconstructed events Figure 103: Monte-Carlo simulation, proton momentum versus cos(θppp) in the proton-proton rest frame. Jagiellonian University 145 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION Figure 104: Monte-Carlo Simulation Phase Space coverage, η momentum in theCM system,true events(blue), reconstructed events(red) qηCM =0.45−0.7 GeV/c cos(θηCM )=−1.0−0.0 (94) PP Theproton momentuminproton-proton restframepp is correlatedwith the η momentum in CM system qηCM (Fig. 105). The selection of the range CM PP in qimplies automatically the selection of the range in p. ηp As the initial state is symmetric i.e. two protons, the resulting angular distributions shouldbe symmetric around 90 deg. Toget thefullinformation one needs to measure only the half of the distribution, the other part is a reﬂection. For this studies the range from (−1,0) in the cos(θηCM )and the cos(θppp)were selected. Also the resolution for the cos(θηCM ) and the cos(θppp) was studied, to properly select width of the bins in those variables. Using Monte-Carlo simulation one compares the σ parameter of the ﬁtted Gaussian of the true minus reconstructed distribution as a function of the reconstructed value (Figs. 106, 107). The maximal bin size was chosen. Jagiellonian University 146 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction Figure 105: Monte-Carlo simulation, reconstructed events. Correlation between η momentum in CM system qηCM and proton momentum in proton-proton restframepPP p . Thebroadeningoftheline correspondstothedetector resolution eﬀect. Jagiellonian University 147 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION (a) True minus reconstructed value as a function of the reconstructed value of the cos(θηCM ). (b) The σ parameterof the ﬁttedGaussianpeak asafunctionof the reconstructed cos(θηCM ). Figure 106: Monte-Carlo simulation, resolution studies of the cos(θηCM ). Jagiellonian University 148 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction (a) True minus reconstructed value as a function of the reconstructed value of the cos(θppp). (b) The σ parameterof the ﬁttedGaussianpeak asafunctionof the reconstructed cos(θppp). Figure 107: Monte-Carlo simulation, resolution studies of the cos(θppp). Jagiellonian University 149 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION Figure108: Experimental Data,MissingMass of twoprotons versus M2(pη). The signal from the η meson as well as the background is seen. The production mechanism Before studying the angular distributions, the M2(pη)(invariant mass of the proton eta system) distribution was checked. To obtain the background free M2(pη) the missing mass of the two protons was plotted against the M2(pη) (Fig. 108). Then the background was ﬁtted outside the η meson peak by the second order polynomial and subtracted from the data, this was done for every M2 (pη)bin(Fig.109). Nextthebackground subtracted experimental data were compared with the Monte-Carlo simulation for two assumed production mechanisms [86], the phase space production and the production via excitation of the N∗(1532)(Fig. 110)(see Appendix C). It is seen that non of the Monte-Carlo modelsdescribes the experimental data. If one assumes that only those two mechanisms can contribute to the η production,one canperforma ﬁt of thetwoproductionmodelstotheexperimental data. Such a ﬁt was performed by using the χ2 method. The χ2 function was minimized: [Data−(bModel1 +(1 −b)Model2)]2 χ2 = (95) σ2 + b2σ2 +(1 −b)2σ2 Data Model1 Model2 where Model1 -is the η meson production via N∗(1532), Model2 -is the η meson phase space production. The σData is the error of the point for experimentaldata, σModel1 -istheerrorofthepointfor Model1 and σModel2 is the error of the point for Model2. The b parameter is the fraction of the production via N∗(1532)to the sum of phase space production and via Jagiellonian University 150 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp →ppη(3π 0 ) reaction Jagiellonian University 151 BenedyktR.Jany (a)(b) (c)(d) Figure 109: Examples of bin by bin background subtraction from the experimental M2(pη) distribution. The background was approximatedby the second orderpolynomial(greenline). 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION Figure 110: M2(pη)distribution. The experimental data are black line, the Monte-Carlo phase space blue line, the Monte-Carlo simulation assuming excitation of N∗(1535)redline. Theplotis symmetrized against twoprotons -each event is ﬁlled two times. The simulations are normalized to the same number of events as the experimental data. Jagiellonian University 152 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction N∗(1532). For the numerical purpose of doing the ﬁt algorithm, the χ2 function(Eq.95)was redeﬁnedto: χ2 [Data−(cModel1 ∗ +(1 −c)Model2 ∗)]2 = (96) σ2 + c2σ2 +(1 −c)2σ2 Data Model∗ Model∗ 12 The same as in the Section 5.1.1 on page 71. The χ2 function(Eq.96) was minimizein respect toparameter c. The (Fig. 111) shows the χ2 versus the searched parameter c, the function has one minimum. The estimated value of the c parameteris the one whichgives the smallest χ2, the error of the parameter is half of the distance for which the χ2 function changes by 1, this gives: χ2 195.3337 min = =4.54±0.22 (97) NDF 43 where χ2 value at minimum, and NDF is the Number ofDegrees min is the χ2 of Freedom. c = 0.6388±0.0075 (98) and corresponds to the: Model1 Model2 = 0.767±0.011 (99) giving the b = 0.4342±0.0084 (100) To check how the Monte-Carlo simulation based on the sum of two models, with the ﬁtted parameter, describes the experimental data, the comparison was done showing the models sum (Fig. 112). It it seen that sum of the models describes very good the event populations on the invariant mass spectrum. The same procedure was repeated for the background subtraction using polynomial of the ﬁrst order, after the ﬁt the results this gives: bpol1 =0.4141±0.0084 (101) The diﬀerence between the two background subtraction methods was used to estimate the systematic error: Δbsys. = |b−bpol1|=0.0201 (102) The ﬁnal value with the systematic error is Jagiellonian University 153 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION χ2 χ 2 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 0.2 0.4 0.6 0.8 1 c (a) 196.6 Δc=0.015 196.4 196.2 196 Δχ2=1 195.8 195.6 195.4 cmin=0.6388 195.2 0.628 0.63 0.632 0.634 0.636 0.638 0.64 0.642 0.644 0.646 0.648 c (b) Figure 111: χ2 versus the searched parameter c for the sum of phase space and N∗(1535)ﬁt. Jagiellonian University 154 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction Figure 112: M2(pη) distribution. The experimental data are black line, the Monte-Carlo phase space blue line, the Monte-Carlo simulation assuming excitation of N∗(1535) red line, the results of the ﬁt -sum of phase space (56.58%) and N∗(1535)(43.4%) production green line. The plot is symmetrized against two protons -each event is ﬁlled two times. The result of the ﬁt was normalized to the same amount of events as the experimental data. Jagiellonian University 155 Benedykt R. Jany 5.2 The pp→ ppη(3π0)reaction 5 RESULTS AND ERROR DISCUSSION b =0.434±0.008(stat.)±0.020(sys.) (103) As a ﬁnal results one gets that the production mechanism via N∗(1535) is the 43.4% of the total η production, atthis energy(T =2.54GeV)for the coveredpart of thephase space(Eq.94). The angular distributions Havingnowdescribedtheproduction mechanism(phase space and N∗(1535) excitation), onecanstarttolookintotheangulardistributionsforaccessible viaWasa-at-Cosypart of thephase space(Eq.94)(the conditionaldistributions). First the cos(θηCM )was studied for the four diﬀerent regions of the qηCM . To obtain the background free cos(θηCM ) the missing mass of the two protons was plotted against the cos(θηCM ). Then the background was ﬁtted outside the η meson peak by the second order polynomial and subtracted from the data, this was done bin by bin. The background subtracted experimentaldata were compared with aMonte-Carlo simulation(phase space and N∗(1535)excitation) without any assumed η angle anisotropy(Fig.113). It isseenthatforfourdiﬀerent rangesinthe η momentum the shape ofMonte-Carlo and experimental data changes diﬀerently. One sees the momentum dependence of theangulardistribution. Toextractthiseﬀect and correctfor acceptance and eﬃciency bias, the experimental data were divided by this Monte-Carlo simulation(Fig.114). Thedistributionwas compared with a Monte-Carlo simulation with η angle anisotropy model in PLUTO++ [71] (Integratedover all η momentum), whichisbased on[86]. One seesthe systematic changes of the angulardistribution when the η momentumincreases. The extracted anisotropy(experimental datadividedby theMonte-Carlo simulation ofthe ηmesonproduction(Fig.112)without any ηangle anisotropy) distributions for four diﬀerent region of the η meson momentum in the CM system were compared between each other and with the Monte-Carlo simulation with a anisotropyparametrizationbased onPluto++(Fig.115). The distributions were normalized so that for the cos(θηCM )= −0.67 all have the value 1. If one looks into that angular distributions, one sees that for the small momentum the data are consistent with this Monte-Carlo model. When the momentum increases the discrepancy between the model and the data is getting bigger. The model is almost ﬂat, where the data are more and more curved. One sees that the shape of the cos(θηCM )changes with the η meson momentum in the CM. Jagiellonian University 156 Benedykt R. Jany 5 RESULTS AND ERROR DISCUSSION 5.2 The pp →ppη(3π 0 ) reaction Jagiellonian University 157 BenedyktR.Jany CM CM (a) q=0.45−0.475 GeV/c (b) q=0.475−0.5 GeV/c ηη CM CM (c) q=0.5−0.55 GeV/c (d) q=0.55−0.7 GeV/c ηη Figure 113: cos(θηCM ) distribution for diﬀerent momenta of the η in CM system qηCM . Experimentaldata(black marker), Monte-Carlo simulation without any η angle anisotropy(N∗(1535)and Phase Space)(green line) 5.2 The pp →ppη(3π 0 ) reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 158 BenedyktR.Jany CM CM (a) q=0.45−0.475 GeV/c (b) q=0.475−0.5 GeV/c ηη CM CM (c) q=0.5−0.55 GeV/c (d) q=0.55−0.7 GeV/c ηη Figure 114: The extracted anisotropy (experimental data divided by the Monte-Carlo simulation of the η meson production(Fig.112)without any η angle anisotropy) versus cos(θηCM )fordiﬀerent momenta of the η inCM system qηCM .Experimentaldata(black marker),Monte-Carlosimulationwith anisotropy(arbitrary normalization -toguide theeye) -PLUTO++parametrization(redline) 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction Figure 115: The extracted anisotropy (experimental data divided by the Monte-Carlo simulation of the η mesonproduction(Fig.112) without any η angle anisotropy) as a function of cos(θηCM ) for four diﬀerent regions of the η momentum in the CM system, compared with the Monte-Carlo simulation with a η angle anisotropyparametrizationfromPluto++[71]. The shapes of the experimentaldistributions changes with the η momentum. The normalization of the histograms is described in text. Jagiellonian University 159 Benedykt R. Jany 5.2 The pp →ppη(3π 0 ) reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 160 BenedyktR.Jany CM CM (a) q=0.45−0.475 GeV/c (b) q=0.475−0.5 GeV/c ηη CM CM (c) q=0.5−0.55 GeV/c (d) q=0.55−0.7 GeV/c ηη Figure 116: cos(θppp) distribution for diﬀerent momenta of the η in CM system qηCM . Experimental data (black marker),Monte-Carlo simulation without anyproton angle anisotropy(greenline) 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction Next also the cos(θppp) was studied for the same four diﬀerent regions of the qηCM . To obtain the background free cos(θppp) the missing mass of the two protons was plotted against the cos(θppp). Then the background was ﬁtted outside the η meson peak by the second order polynomial and subtracted from the data, this was done bin by bin. The experimental data were compared with aMonte-Carlo simulation(phase space and N∗(1535) excitation) without any assumedproton angle anisotropy(Fig.116).Itis seen that for four diﬀerent ranges in the η momentum the shape of Monte-Carlo and experimental data changes diﬀerently. One see here also the momentum dependence of theangulardistribution. Toextractthiseﬀect and correctfor acceptance and eﬃciency bias, the experimental data were divided by the Monte-Carlo simulation(Fig.117). Thedistribution wascompared with a Monte-Carlo simulation with proton angle anisotropy model in PLUTO++ [71], whichisbased on[86]. The extracted anisotropy(experimental datadividedby theMonte-Carlo simulation of the η mesonproduction(Fig.112) without anyproton angle anisotropy) distributions for four diﬀerent region of the η meson momentum in the CM system were compared between each other and with the Monte-Carlo simulation with a proton angle anisotropy parametrization based on Pluto++[71](Fig.118). Thedistributions were normalized sothatforthe cos(θppp)= −0.44 all have the value 1. The eﬀect is not so prominent as in case of the cos(θηCM ). But still one sees a systematic change of the angular distribution with the η momentum. Concluding, one observes the momentum dependence of the angular distribution, for energy T =2.54GeV, which was measured for the ﬁrst time, for the availablepart of thephase space(Eq.94). The strongest eﬀectis seen in the cos(θηCM ) distribution, which is diﬀerent from the available η angle anisotropy model usedbyPLUTO++[71](Integrated over all η momentum), which isbased on[86](Fig.115). Since thedistribution changes from almost ﬂat to the curved one, when the momentum of the η increases, this indicates thatthediﬀerentpartial waves contribute[1,27]. The angular distributions of the η meson are available as tables of num-bers in Appendix G. Jagiellonian University 161 Benedykt R. Jany 5.2 The pp →ppη(3π 0 ) reaction 5 RESULTS AND ERROR DISCUSSION Jagiellonian University 162 BenedyktR.Jany CM CM (a) q=0.45−0.475 GeV/c (b) q=0.475−0.5 GeV/c ηη CM CM (c) q=0.5−0.55 GeV/c (d) q=0.55−0.7 GeV/c ηη Figure 117: The extracted anisotropy (experimental data divided by the Monte-Carlo simulation of the η meson production(Fig.112)without anyproton angle anisotropy) versus cos(θppp)for diﬀerent momenta of the η in CM system qηCM .Experimentaldata(black marker),Monte-Carlosimulationwith anisotropy(arbitrary normalization toguidetheeye) -PLUTO++parametrization(redline) 5 RESULTS AND ERROR DISCUSSION 5.2 The pp→ ppη(3π0)reaction Figure 118: The extracted anisotropy (experimental data divided by the Monte-Carlo simulation of the η mesonproduction(Fig.112) without any proton angle anisotropy) as a function of cos(θppp)for four diﬀerent regions of the η momentum in the CM system, compared with the Monte-Carlo simulation with a proton angle anisotropy parametrization from Pluto++. The normalization of the histograms is described in text. Jagiellonian University 163 Benedykt R. Jany 6 Summary and Conclusions For the ﬁrst time the prompt pp→ pp3π0 reaction channel was measured at incidentproton momentum of3.35 GeV/c withtheWASA-at-COSYdetector setup(seeSection2 onpage5). All the ﬁnal stateparticlesi.e. twoprotons and three pions, were identiﬁed and their four momenta reconstructed from thesignalsinthedetectors(seeSection4 onpage31)–around onemillion of the pp→ pp3π0 clean events were obtained. First, it was observed that the experimental data could not be described by the Monte-Carlo pp → pp3π0 assuming homogeneously and isotropically populatedphase space(seeFig.49 onpage71 andFig.50 onpage72) Later, the dynamics of the reaction was studied by the missing mass of the two protons MMpp dependent Dalitz and Nyborg plot analysis (see Section 5.1 on page 71) together with the proposed kinematic calculations by Monte-Carlo model based on simultaneous excitation of two baryon resonances Δ(1232) and N∗(1440) and their decays leading to the pp3π0 ﬁnal state (Eq. 26 on page 84). The model assumes two decay branches of the N∗(1440), the direct decay N∗(1440)→ Nππ and the sequential decay N∗(1440)→ Δ(1232)π → Nππ anditisbased onthekinematic calculations by PLUTO++ eventgenerator[71](seeAppendixC onpage187). This Work (Eq. 46on page 87) PDG [1] CELSIUS-WASA [88] Bonn-Gatchina (atmN∗
(1440)
= 1436 MeV/c2)[89] 0.039±0.011(stat.)±0.008(sys.) 0.166−0.5 1.0±0.1 1.20±0.11 Table 9: Comparison of the ratio R = Γ(N∗(1440)→ Nππ)/Γ(N∗(1440)→ Δ(1232)π → Nππ) of the partial decay widths for the decay of the Roper resonance N∗(1440).The ratioispresentedfortheRoper resonance mass of 1440 MeV/c2 . The fraction of those two decays was extracted from the experimental data by comparing event populations on Dalitz and Nyborg plots using the chi-square method (see Section 5.1.1 on page 80). The resulting value of the ratio R = Γ(N∗(1440) → Nππ)/Γ(N∗(1440) → Δ(1232)π → Nππ)= 0.039 ± 0.011(stat.)± 0.008(sys.) (Eq. 46 on page 87) together with comparison to the existing data is presented in Table 9. The obtained value diﬀers from the other experimental measurements [1, 88, 89] signiﬁcantly. The closestisthe estimationfromthePDG[1] R =0.166−0.5 (∼ 7 standard deviations diﬀerence), which is based on the Partial Wave Analysis [90]of many data sets. TheBonn-Gatchina value[89](based onthePartial WaveAnalysis[90]) as well astheCELSIUS-WASA[88](based onthe am 165 6 SUMMARY AND CONCLUSIONS plitude considerations of the pp→ pp2π0 reaction[25,91]) give value R ≈ 1 (∼ 96 standard deviations diﬀerence and ∼ 9 standard deviations including errors of Bonn-Gatchina and CELSIUS-WASA). The diﬀerence between the results might be also interpreted as complex multi-quark structure of the N∗(1440)resonance – “breathing modeofthenucleon” [92]. Thiscomplex structure mayleadto adiﬀerentbehaviordepending ontheproduction mechanism. Nevertheless, for the ﬁrst time one measures this ratio R in a direct way – by comparing experimental data with the Monte-Carlo simulations; notby the extractionfromthePartialWaveAnalysis[1,89] or asaresult of theamplitude analysis[88], whichisanindirect method. Itis alsoseenfrom the studies presented in this work that the N∗(1440)→ Δ(1232)π → Nππ sequential decay is a leading mode of the 3π0 production(constitutes ∼ 95% Eq. 47). To describe the event population in MMpp (proton-proton missingmass) themissing masspopulationfunctionwasintroduced, extractedfromtheexperimentaldata(Eq.47 onpage92). It was shownthatthe modeldescribes the data signiﬁcantly better than the homogeneously and isotropically populated phase space. Studying the kinematics of the reaction, two possible explanations of the origin of f(MMpp) (Eq.47) – the missing mass population function were considered. • The possibility of Δ(1232)N∗(1440) interaction via One Boson Exchange(OBE)(seeSection5.1.1 onpage97), which was excludeddue to completely diﬀerent behavior of the f(MMpp) as a function of the averagediﬀerence of thefour momentumthaninOBE[30–37]; • Next, it was shown that the MMpp is very sensitive to the structure of the spectral line shape of the N∗(1440) (see Section 5.1.1 on page 100). The proposed modiﬁcation of the spectral line mainly of the N∗(1440) → π0Δ(1232) substitutes the explicitly added f(MMpp); since the main eﬀectinﬂuencing thedescription ofthedataisdue tothe modiﬁcation of the N∗(1440)→ π0Δ(1232) line shape(itis theleading mode of 3π0 production ∼ 95%; see Eq. 47). In particular the proposed modiﬁcation of this spectral line was found to be very similar to the Breit-Wigner distribution. This might indicate that the proposed byPLUTO++[71] modiﬁcation ofthe N∗(1440)spectral line caused by decay to Δ(1232) (unstable hadron) is not so strong as proposed and seems to be not necessary in case of the reaction considered in this work. Nevertheless more detailed studies of this eﬀect are needed. For instance, it remains inconclusive whether the spectral line shape Jagiellonian University 166 Benedykt R. Jany 6 SUMMARY AND CONCLUSIONS of N∗(1440)remains the same in case of N∗(1440)→ pπ0π0 since the contribution of this branch consists of only ∼ 5% (see Eq. 47). Itwouldbe veryinteresting taskto seehowtheproposedinthis work spectral line of the N∗(1440)→ π0Δ(1232) works for the case of other reactions by high statistics experiments. Thepossibility of molecule orbound state creationof Δ(1232)N∗(1440)system as well as the excitations of the quark-gluon degrees of freedom was not excluded. It would require to consider the dynamical microscopic model ofthe reaction whichis missing(anyway such models are alwaysparameter dependent). Thedetailed validation of theMonte-CarlodevelopedModel(Eq.47)was performed(seeSection5.1.2 onpage5.1.2). To check the consistency ofthe model with experimentaldata, the model was compared with the experimental data for all considered spectra i.e. Dalitz and Nyborg plots for diﬀerent missing mass ranges – the statistical analysis wasperformed(see Section 5.1.2 onpage107). It was shown that the Monte Carlo developed model describes ∼ 80% of the experimental data within the statistical errors of one standard deviation. It was also shown that around 100% of the experimental data points is described by the proposed model withing the statistical error of ∼ 2.5 standard deviations. Concluding the developed model fully describes the data within the statistical precision of data and model. Five body pp → pp3π0 reaction is fully described by an initially two bodyprocess. Veriﬁcation ofapossibility of otherprocesses contributiontotheMonteCarlo developed model (Eq. 47) was performed using Monte-Carlo simulation of pp → pp3π0 reaction(based onhomogeneously andisotropicallypopulated phase space) to mimic otherprocesses(seeSection5.1.2 onpage113). The chi-square method was used for this task. Concluding, the contribution of other processes estimated here falls to value ∼ 2%. The cross section for the pp→ pp3π0 reactionchannel, usingknowncross section of pp → ppη(3π0) reaction as a normalization, was extracted (see Section 5.1.3 on page 117): σpp→pp3π0 =123±1(stat.)±8(sys.)±19(norm.)µb . The result was compared withthe availabledata[13] and modelsfor pp → pp3π0 cross sections [16–18], (Fig. 86 on page 123). The data conﬁrm the cross section scaling modelbased onDelofFinalStateInteraction[18]. One can predict also the cross section for the pp → ppπ+π−π0 reaction assuming that the reaction follows also via simultaneous excitation of the Jagiellonian University 167 Benedykt R. Jany 6 SUMMARY AND CONCLUSIONS Δ(1232) and N∗(1440)baryonresonances, seeTable3 onpage21. Thecross section for the reaction is predicted to be σpp→ppπ+π−π0 =861±147µb. Using the developed model (Eq. 47) the acceptance and eﬃciency correction of the missing mass of two protons dependent Dalitz and Nyborg plots was done (see Section 5.1.4 on page 124). The acceptance and eﬃciency corrected Dalitz and Nyborg plots are available as tables of numbers in Appendix G. The pp → ppη reaction was measured parallely via η meson decay into three neutral pions. Around 200k events after background subtraction was available(seeSection5.2 onpage141). First,the experimental accessibility of thephase space was studied. One ﬁnds that the acceptance for this reaction was limited. It could be expressed in the η meson momentum in the Center of Mass frame qηCM and cosine of the scattering angle of the η meson in the CM frame cos(θηCM )as follows: qηCM =0.45−0.7 GeV/c cos(θηCM )=−1.0−0.0 . Next, the production mechanism was investigated (see Section 5.2 on page 150). Two dominant production mechanisms were considered: the resonantproduction(via excitationof N∗(1535))and the nonresonantproduction. Thetwoscenariosweresimulatedby theMonte-Carloand ﬁtted tothe background subtracted experimental data distribution of invariant mass of proton-η system(seeFig.112 onpage155). The contribution oftheN∗(1535) resonancein theproduction mechanism was obtained and compared with existing experimentaldata,seeTable10. One ﬁndsthat whenthebeamkinetic energy T increases the N∗(1535)resonance contribution decreases and from T =2.54 GeV stabilizes at value ∼ 43%. One can try to compare the result obtained in this work with the closest value at T =2.5 GeV [86, 93]. Between these two measurements the T changes by 40 MeV and excess energy Q by 13 MeV in this range the value changes by ∼ 4 standard deviations, which is a signiﬁcant diﬀerence. Later, the angular distributions of the η meson in the CM system were studied as well astheproton angulardistributionsintheproton-proton rest frame. Both the distributions were studied for four ranges of the η meson momentum in the Center of Mass frame qηCM (see Section 5.2on page 156). For the ﬁrst time, one observes the momentum dependence of the angular distributions. The strongesteﬀectis seeninthe cos(θηCM )distribution, which Jagiellonian University 168 Benedykt R. Jany 6 SUMMARY AND CONCLUSIONS Beam Kinetic Energy T Excess Energy Q 2.15 GeV 325 MeV DISTO [86, 93] 2.5 GeV 442 MeV DISTO [86, 93] 2.54 GeV 455 MeV This Work (Eq. 103)on page 156 2.85 GeV 554 MeV DISTO [86, 93] 3.5 GeV 752 MeV HADES [87] N∗ (1535)Contribution 59% 51% 43.4±0.8(stat.)±2.0(sys.)% 43% 41% Table 10: Contribution of the N∗(1535)in the pp→ ppη production mechanism. isdiﬀerentfromtheη angle anisotropypredictedbyPLUTO++[71]based on [86](Fig.115onpage159). Sincethedistributionchangesfromalmost ﬂat to the curved one, when the momentum of the η increases, thisindicates that thediﬀerentpartial waves can contribute[1,27]. It seems that todescribe the angular distributions for the three highest ranges of qηCM atleastp wave is important. The appropriate model would be needed as well as the high statistics experimental measurements for diﬀerent beam energy covering full phase space are necessary to study this eﬀect in details. The angular distributions of the η meson are available as tables of numbers in Appendix G. Concluding, the multipion reactions in nucleon-nucleon collisions can be used as a precision tool to directly access the properties of the baryon resonancesby usingproposedin this work methods(one would nameit multipion spectroscopy): • thebaryon resonances areidentiﬁedby their uniquedecays topology on the missing mass of two protons MMpp dependent Dalitz and Nyborg plots – the strength of the diﬀerent resonances decays and branching ratios couldbeprecisely extracted(seeSection5.1.1 onpage71), • the missing mass of the two protons MMpp distribution is sensitive to thebaryon resonance spectralline shape andtotheinteractionbetween thebaryons(seeSection5.1.1 onpage97). Here also variable other than MMpp might be considered depending on the reaction details. It would be very valuable to investigate the properties of prompt pp → pp3π0 reaction for many diﬀerent energy regions to see the inﬂuence of the diﬀerent baryon resonances contributions as well as the pp → ppπ+π−π0 reaction to conﬁrm the predictions for the cross section and to study the dynamics of this reaction. Also never measured prompt pp → pp4π and pp → pp5π reactions both for charged and neutral pions in ﬁnal state could Jagiellonian University 169 Benedykt R. Jany 6 SUMMARY AND CONCLUSIONS beveryinteresting objectforstudiestounderstand theseprocesses; sincee.g. the pp → pp4π could proceed via simultaneous excitation of two N∗(1440) resonances and in the pp→ pp5π case both the N∗(1440)and higher baryon resonance might be involved. Besides,there exist nodynamical microscopic modelforthepromptpion productions in nucleon-nucleon collisions like 3π,4π,5π productions in contrast to the NN → NNππ reactions where complete dynamical microscopic model based on the excitations and decays of various baryon resonances ex-ists[25]. The resultspresentedinthis work(available astables of numbers in Appendix G) could be used as an input for testing such a model in future. Due to the high energy needed to excite 3π,4π,5π, it might be more plausible to use for the future model the microscopic approaches based on the QuantumChromoDynamics(QCD)(whichtakesinto account excitations ofquark-gluondegrees offreedom)[39–42] ratherthen common existing effective microscopic models[30–37]( which mimicinteractionby exchange of various light mesons like π,η,ρ,ω and are more applicable for lower energies [38])and which might be very diﬃcult theoretically. FullPartialWaveAnalysis[90]ofthe resultingdata can alsobeperformed in order to get the more details of the reaction mechanism, again it forms a formidable task. Other approaches to the analysis and visualization of the multidimensional data phase space of prompt multipion productions and model comparisons could be also considered. One may think about the generalization of theDalitzPlotto ﬁve-particleDalitzPlot(ﬁvedimensionalpentahedron representation) and to even more-particle Dalitz Plot in the way how it was proposed and successfully used tovisualize atomicbreak-upprocessesby usingFour-particleDalitzPlot(fourdimensionaltetrahedron representation) which visualizethe multidimensionaldataphase space[94,95]. AnotherideawouldbetousetheSOM –Self-OrganizingMap[96] orthe GTM – The Generative Topographic Mapping [97] techniques which visualizethe multidimensionaldataphase space as atwo-dimensionalplot[98]. TheAndrewsCurves[99]andtheir extensions[100] which represent a multidimensional data points as an orthogonal curves could be also taken into account. The methods like discussed above are mathematically complicated and may require time-demanding supercomputing. The multipion reactions seems to be promising ﬁeld of the future scientiﬁc exploitationsparticularly inthe area of thebaryon resonances spectroscopy. Jagiellonian University 170 Benedykt R. Jany AppendixA Kinematicsof ﬁveparticlephasespace One wouldliketo expressthe eventdistribution(Eq.12 onpage28)inthe following variables: −2 2 →→ 22 M2 =(E1 + E2)+(−p1 + p2 )=(E1 + E2)+ p (A.1) 12 12 √ (√ ) M2 25 = s −2 s(E3 + E4)−2E1 s + E2 −E4+ (A.2) − (E2 −E4)(E2 +2E3 + E4)+M2 123 √ M2 M2 = −s +2 s(E3 + E4) (A.3) 34 125 √ M2 2 = s −2 sE4 + m (A.4) 1235 4 M2 22 = (E1 + E2 + E3)+ p (A.5) 123 123 M2 (√ )22 = s −E1 −E2 −E3+ p (A.6) 45 123 and the inverse relations are E1 = s −M2 34 + M2 125 2 √ s + (A.7) − � 4sM2 25 + (s −M2 45)2 −2M2 123 (s + M2 45)+M4 123 2 √ s E2 = � 4sM2 25 + (s −M2 123)2 −2M2 45 (s + M2 123)+M4 45 2 √ s + (A.8) + m2 4 −M2 45 + M2 123 −M2 1235 2 √ s E3 = M2 34 −m2 4 −M2 125 + M2 1235 2 √ s (A.9) E4 = s + m2 4 −M2 1235 2 √ s (A.10) p12 p123 = + = 1 2 √ s � 2m 2 4 (s −M2 34 −M2 45 + M2 123 + M2 125 −M2 1235) + (s −M2 34 −M2 45 + M2 123 + M2 125 −M2 1235)2 −4sM2 12 + m 4 4 �1/2 � −2M2 45 (s + M2 123)+ (s −M2 123)2 + M4 45 2 √ s (A.11) (A.12) For the deﬁnitions of the symbols see Section 3.2 on page 27. The Jacobian 175 APPENDIX A KINEMATICS OF FIVE PARTICLE PHASE SPACE ∂(E1E2E3E4p12p123)1 Jac == √√√ (A.13) ∂(M2 12M2 34M2 123M2 8 C + E 25M2 1235 M2 45) BD where ()2 () B =4sM2 s −M2 −2M2 s + M2 + M4 (A.14) 25 + 45123 45123 ()()2 C = −2M2 s + M2 + s −M2 + M4 (A.15) 45 12312345 2 () D =2ms −M2 45 + M2 125 −M2 (A.16) 4 34 −M2 123 + M2 1235()24 E = s −M2 45 + M2 125 −M12352 −4sM2 m4 (A.17) 34 −M2 123 + M2 12 + one can rewrite(Eq.A.13): √ [()J−1 Jac = 32ss −2E1 s −M2 125p12p123 34 + M2 (A.18) with E1,p12,p123 given by(Eq. A.7, A.11, A.12). Now one can now rewritethe eventdistribution(Eq.12) intheinvariant masses π4dM2 dM2 dM2 dM2 dM2 dM22 122534123512345 d6P = √ |M|(A.19) 32s(s −2E1 s −M2 125)p12p123 34 + M2 Theboundaries of thephysical regionin E1,E2,E3,E4,p12,p123 variables are [√ J22 2 s −(E1 + E2 + E3)−E4 = m5 +(p123 ±p4)(A.20) p 2 = 2 (A.21) 12 (p1 ±p2)p 2 = 2 (A.22) 123 (p12 ±p3) Which could be transformed into new variables by the relations(Eq.A.7,A.8,A.9,A.10,A.11,A.12). In orderto obtain anytwodimensional eventdistributions onehastointegrate(Eq.A.19)overfour variables withtheboundary conditions(Eq.A.20,A.21,A.22). Jagiellonian University 176 Benedykt R. Jany AppendixB WASA-at-COSYDetectorCalibration The Calibration of the detectors is a conversion from electronic channels of ADC(Analog To Digital Converter) or TDC(Time To Digital Converter) i.e. arbitrary units of energy and time, which are measured by the detector electronics,tothephysical unitslikeGeV and ns. The calibrationprocedure is diﬀerent for diﬀerent detector type, since the principle of measurements are based on other physics phenomena. The WASA at Cosy detector setup section2.2isbuiltfromthreediﬀerentdetectortypesi.e. PlasticScintillators, Straw Tube Detectors and Electromagnetic Calorimeter. The appropriate calibration procedure for them is described below. Plastic Scintillators To obtain calibration of the plastic scintillators several eﬀects have to be taken into account which are causing the non linear conversion. One uses fastprotonsfromtheproton-proton elastic scatteringreaction which are close to the minimum ionization having speciﬁc constant energy loss in detector elements. FigureB.1:Thelight output nonuniformity check. ADC signal times cos(θ) as a function of the scattering angle θ of one element of the FTH detector for the elastic scattered protons. Red curve shows the ﬁt of the polynomial of the third degree[101]. First the possible non uniformity of light collection eﬃciency by scintillatorhastobe checked, whichdepends ongeometrical shape of scintillators. 179 APPENDIX B WASA-AT-COSY DETECTOR CALIBRATION The ADC signal times cos(θ)asafunction of thescattering angle θ in detector elementis checked(Fig.B.1). Thedeviationsfrom straightlineindicate thenonuniformity,the ﬁttothedataisperformed tocorrectforthiseﬀect. Figure B.2: Left Figure, ADC signal corrected for non uniformity of light collection eﬃciencyfor the two subsequent layers of theFRHdetectorfor the elastic scattered protons, characteristic points marked. On the right Figure, energydepositforthecharacteristicpoints, asmarked onleftFigure,forthe Monte-CarlosimulationinGeV versusADC value. The ﬁt(solidline) tothe correlation describes the calibration function. The linear function marked as a dotted line[102]. After the corrections for non uniformity of light collection eﬃciency for eachdetector elementthe correctedADC signal(energy loss) fortwo subsequentdetectorlayersisplotted(dE-Eplot)(B.2left)and compared with the same dependence for the Monte-Carlo simulation for the indicated points in the plot which correspond to: (0) Zero Point (1) Minimum Ionizing Point (3) Punch-through Point -the kinetic energy of the particle is larger then the stopping power of the detector (2),(4) EquilibriumPoints-theenergylossoftheparticlepunching through the currentlayer(2)is aslarge asthe energyloss oftheparticle stopped inthe currentlayer(4) (5) Maximum Deposit Point Jagiellonian University 180 Benedykt R. Jany APPENDIX B WASA-AT-COSY DETECTOR CALIBRATION Laterfor theindicatedpoints the energydeposit in onelayer is compared to the energy deposit from the Monte-Carlo simulation, resulting as a correlationplot(B.2 right). Nextthe ﬁttothecorrelationplotisperformed toget the conversion from theADC energydeposit(light output) to energydeposit in energy units. As it is seen the relation is not linear. The nonlinearities can rise from the nonlinearities of the photomultiplier tubes as well as the quenching eﬀect in scintillator. Straw Tube Detectors The Calibration of theStraw TubeDetectorslike MDC and FPC is essential to achieve high spatial resolution with these detectors. In addition to the positions ofthe anode wiresthedrifttimeis measuredwhichisthen converted using calibration function to the drift distance which is understood as a closest approach of the particle trajectory to the nearest anode wire of the strawtube. The time todistance relation(the calibration) depends onthe magnetic ﬁeld in which the detectors are, the gas mixture in the detectors andthe voltage applied tothe anode wires. Ithastobefoundfor eachchange of these conditions. To derive the calibration the following assumptions has be fulﬁlled: • the signals in the detectors are consequence of physical particle tracks -no noise • the straw tubes are 100% eﬃcient • the straw tubes are homogeneously irradiated dn NTot == const. (B.1) dr RTube where n -number of events, r -distance measured from anode wire, NTot -total number of events registered by straw tube, RTube -the radius of the straw tube. Now one can write the drift velocity v(t): dr dn dr RTube dn v(t)== = (B.2) dt dt dn NTot dt Togetthetimetodistance relation(the calibration)oneintegrates(Eq.B.2) and gets: � t � t n(t)dt n(t)dt T0 T0 D(t)= RTube = RTube (B.3) � Tmax T0 n(t)dt NTot Jagiellonian University 181 Benedykt R. Jany APPENDIX B WASA-AT-COSY DETECTOR CALIBRATION where T0 -starting point of the drift time measurement, Tmax -maximal drift time It is essential to precisely determine the T0 which is also the time reference oftheindividualTDCs. To eliminatethetriggertime andjitter one uses the relative time between the straws and nearestplastic scintillator. For the FPC the FTH detector is used and for the MDC the PSB detector respectively. Theprocess of theMDC calibrationis shown(Fig.B.3),thegas mixture used in the straw tubes Ar(80%)+ C2H6(20%) causes the linearity of the time to distance relation -the calibration. ×103 Events per 2 ns 3500 3000 140 120 100 R R max 2500 80 2000 60 1500 40 1000 20 500 0 0 Drift Time in ns Drift Time in ns Drift Time in ns FigureB.3:TheCalibration of theMDC,thegas mixture used(Ar(80%)+ C2H6(20%)). Left: the drift time spectra, Center: the integrated drift time distribution(red verticallinesindicateintegration range, afteridentiﬁcation the maximal drift range -blue horizontal lines), Right: the time to distance relation used as a calibration[101] Jagiellonian University 182 Benedykt R. Jany APPENDIX B WASA-AT-COSY DETECTOR CALIBRATION Electromagnetic Calorimeter The ﬁrst step in the calibration procedure of the electromagnetic Calorimeter wasto obtainthepreliminary calibration constantsby measuring the response of each individual crystal to photons coming from radioactive source [9]. Using these calibration as a ﬁrst step, for each data taking period the two photon decays of the mesons π0,η are used to obtain the actual set of calibration constants. For the experimental data the events with the two neutral tracks in CentralDetector were selected(seeAppendixD), regarded asphotons, andinvariant mass of them was computed: −γ →γ → IMγ1 γ2 =(E1 γ + E2 γ)2 −(−p1 + p2 )2 (B.4) =2k1k2E1 γE2 γ(1−cos(θ1γ,2)) γγ γ →γ →− where E1 ,E2 -energies of the photons,−p1 ,p2 -momenta vectors of the photons, cos(θ1γ,2) -opening angle between the photons, k1,k2 -new calibration factor for the photons. For each central crystal element of the two neutral clustersin calorimeter, deﬁnedasthe element withthehighest energydepositin cluster, theinvariant mass of thetwophotons IMγ1 γ2 was assigned. It is assumed that the central crystal element of the cluster has the highest impact on the invariant mass and the invariant mass shift is associated with it. To correct for the shift in invariant mass, the deviation from the nominal meson mass of π0 or η isused todeterminethenew calibrationfactorforthe photon energies for each crystal: M2 (γ )γπ0 ,η γ E= k1,2E= E(B.5) 1,21,21,2 New IM2 γ1 γ2 () Eγ where -individually calibrated energiesof thephotons, Mπ0 = 1,2 New 134.978 MeV/c2 , Mη =547.8 MeV/c2 from PDG[1].

Thisprocedureis repeatediterativelyfor each crystal until the correct meson mass position is reached and the result remains stable. After the iterations also the additional correction is used to improve the result and avoid overcompensation. The individually calibrated energies of (γ ) thephotons(E)are correctedforthe averageposition oftheinvariant 1,2 New mass: ()Avg IM2 ()kAvg ()γ1 γ2 () Eγ Eγ Eγ == (B.6) 1,21,21,21,2 Avg New New IM2 γ1 γ2 Jagiellonian University 183 Benedykt R. Jany APPENDIX B WASA-AT-COSY DETECTOR CALIBRATION ()()Avg where Eγ -average corrected energies of thephotons and IM2 1,2γ1 γ2 Avg invariant mass average over all crystals. The following way of the calibration implicitly corrects for the border crystal eﬀects and shower leakages. Jagiellonian University 184 Benedykt R. Jany AppendixC TheWASA-at-COSYMonte-Carlo Simulation It is a standard approach in the nuclear and high energy physics to have a full Monte-Carlo simulation of the experimental setup. The reasons for that are as follows: • one needs to cross check the reconstruction procedure • it is necessary to get the error parametrization for the Kinematic Fit • one needs to estimate the total reconstruction eﬃciency(geometrical acceptance, reconstruction eﬃciency) for later data correction • it could be also used to determine the detector resolution for the indirectly and directly measured observables To do all of the above tasks a Monte-Carlo simulation should fulﬁll two very important conditions: • The virtual detector in Monte-Carlo simulation should be a reﬂection of the physical one, as close as possible. It has to mimic the performances and status of all the components as it was present during the experiment • Thekinematics ofthe simulatedeventshasto reﬂectthe eventkinematicsduringthe experiment, onehastoknowthetruephysicalmechanism of the reaction or at least onehas to mimic it by the model as accurate as possible. One needs excellent tools to fulﬁll these conditions. The GEANT3 (Geometry and Tracking)program from CERN[103] is used for simulation of the physical processes of particles interactions with detector medium in which the whole geometry of WASA-at-COSY detector is virtualized. Thatincludesthe active materials asparticledetectorsthemselves as well as passive one e.g. ﬂanches, supports, air around detector etc. As anoutputfromGEANT onegetsthedetector responseforthekinematic conﬁguration(4-vectors of theparticles). For thegeneration of thekinematic conﬁguration of the reaction one uses Pluto++ Monte-Carlo event generator [71] version 5.31. In Pluto++ in addition to generation of homogeneously and isotropically populated phase space many realistic models of the reaction mechanism are implemented e.g. the production of η meson via N∗(1535)with angular anisotropy or decays 187 APPENDIX C THE WASA-AT-COSY MONTE-CARLO SIMULATION of baryon resonances like Δ(1232),N∗(1440). In the hadronic interactions atlow energiesforthe unstablehadrons(likebaryon resonances) onehasto take into account the deviations from the ﬁxed-width Breit-Wigner distribution. For realistic simulations this is done in Pluto++ by introducing the mass dependent width ΓTot(m)being afunction of thepartial widths Γk(m). The mass of the resonance m is sampled from the relativistic Breit-Wigner distribution h(m)with an appropriate staticpole mass MR Table C.1 of the resonance and the mass dependent width ΓTot(m)(Eq. C.1, C.2). 2ΓTot.(m) m h(m)= A (C.1) 2)22 (ΓTot.(m))2 (MR 2 −m+ m N ΓTot(m) = Γk(m) (C.2) k with N number of decay modes of the resonance. The constant A is chosen that the integral of the h(m)is equal to 1. Baryon Resonance name Pole Mass MR [MeV/c2 ] Static Width [MeV/c2 ] Δ(1232)P33 1232 120 N∗(1440)P11 1440 350 N∗(1535)S11 1535 150 Table C.1: Pole Mass and Static Width the selected baryon resonances used in Pluto++ event generator. Also the decays of the resonances to stable and unstable particles are considered by Pluto++. In case of the decays of the resonances to the unstable particles spectral line shape of the resonance is modiﬁed by the unstableparticle spectralline shape. In Pluto++ such a cases are threated explicitly giving a possibility to calculate the realistic spectral functions. All in all these eﬀects changes the spectral line shape of the resonances from the relativistic Breit-Wigner distribution shape. Realistic eﬀective spectral line shapes g(m) for Δ(1232), N∗(1440) and N∗(1535)inthe reactionspp→ pN∗(1535)→ ppη(3π0)and pp→ Δ(1232)N∗(1440) at incident proton momentum of 3.350 GeV/c2 calculated by Pluto++ are presentedin(Fig.C.1 andFig.C.2). The Δ(1232) decays into pπ0 (stable particles), theN∗(1440)decaysinto pπ0π0 (stableparticles)orinto pΔ(1232) Jagiellonian University 188 Benedykt R. Jany APPENDIX C THE WASA-AT-COSY MONTE-CARLO SIMULATION (unstable particle), when laterΔ(1232) decays into pπ0 . The strong diﬀerence between the N∗(1440)spectral line shape is seen in case of decays into stable(Fig.2(b))and unstableparticles(Fig.2(d)). The realistic eﬀective spectral line shape of the resonances dependents on the internal properties of the resonances as well as on their decay products. Thedetailinformations canbefound[71]. 3.350 GeV/c2 . The threshold for N∗(1535)→ pη is seen on the left. Calculations by Pluto++ (via Monte-Carlo method). Jagiellonian University 189 Benedykt R. Jany ×10-3 ×10-3 g(m) a.u. 0.5 g(m) a.u. 0.3 APPENDIX C THE WASA-AT-COSY MONTE-CARLO SIMULATION Jagiellonian University 190 BenedyktR.Jany 0.4 0.25 0.3 0.2 0.15 0.2 0.1 0.1 0.05 10 1.1 1.2 1.3 1.4 1.5 1.6 10 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 m 2][GeV/c m 2][GeV/c (a) Δ(1232) line shape, where N∗ (1440)→ pπ0π0 (b) N∗ (1440)line shape, where N∗ (1440)→ pπ0π0 (stable particles) (stable particles) ×10-3 g(m) a.u. 0.0012 g(m) a.u. 0.7 0.6 0.001 0.5 0.0008 0.4 0.0006 0.3 0.2 0.0004 0.1 0.0002 0 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 m [GeV/c2] m [GeV/c2] (c) Δ(1232) line shape, where N∗ (1440)→ π0Δ(1232) (d) N∗ (1440)line shape, where N∗ (1440)→ π0Δ(1232) (unstable particle) (unstable particle) Figure C.2: Realistic eﬀective spectral line shape g(m)of Δ(1232) and N∗(1440)in the pp→ Δ(1232)N∗(1440)reaction atincidentproton momentum of 3.350 GeV/c2 . The Δ(1232) decaysinto pπ0 (stableparticles), theN∗(1440) decays into pπ0π0 (stableparticles)orinto π0Δ(1232) (unstableparticle), whenlaterΔ(1232) decays into pπ0 . The strongdiﬀerencebetween the N∗(1440)spectralline shapein case ofdecaysinto stable and unstableparticles visible. In case of N∗(1440)the pπ0π0 threshold is seen on the left. Calculations by Pluto++ (via Monte-Carlo method). AppendixD TrackReconstructioninWASA-at-COSY The track reconstruction i.e. the reconstruction of particle trajectory from thedetectorinformationisthemainstep of theanalysis. Whenparticle ﬂies through the detector medium it interacts with single detector elements, it “hits the detector” and one is talking about the hit. Then one can combine the hits related to the one particle in one detector to a group which we call the cluster. Later the clusters from diﬀerent detectors related to one particle are combine to the particle trajectory called the track, which is the goal of the track reconstruction. These assignments are done by several diﬀerent computer algorithms optimizedforthegivendetectortype. Dueto the structure of theWASA-at-COSYdetector setup(Section2.2 onpage8) we have two types of tracks, described below. The Forward Detector Tracks These are the tracks in Forward Detector of the WASA-at-COSY identiﬁed as chargedparticles(usuallyprotons). The creation oftracksbegins withthe creation ofthe clustersfromthehits in the layers of Forward Detector components, here only time coincidence is used -the time of the cluster is calculated as an average time of contributing hits. Now the creation of the track from clusters begins. First the FTH clusters in all three layers of FTH detector are checked for time coincidence andgeometrical overlap forming uniquely deﬁnedpixelinFTH.Thetime of the track is set to the average time of the clusters forming FTH pixel. The position of thepixel setsthe coordinate of thetrack(polar θ and azimuthal φ angle) in coordinate system of theWASA-at-COSY with origin atbeamtarget overlappoint. Later thisinformationis reﬁned byprojecting theFTH pixel to the FPC planes, in each layer of the four FPC the straw tube is selected which is the closest one to the FTH pixel and then the crossing point of these tubes is calculated which gives the corrected new coordinate of the track. By using this method with FPC detector, the so called binary mode, the angular resolution improves by factor two giving FWHMθ ∼ 0.2 [deg] and FWHMφ ∼ 3 [deg] (mainlydue to the high granularity of the FPC). In the last step the information from clusters from the FWC, FTH and FVHdetectors(TheFRIis not used) isincorporatedintothetrackby comparing time coincidence and azimuthal angle overlap between the track and those clusters. 193 APPENDIX D TRACK RECONSTRUCTION IN WASA-AT-COSY The Central Detector Tracks These are the tracks in Central Detector of the WASA-at-COSY identiﬁed as neutral or chargedparticles(i.e. photons, electrons,pions,protons). The Central Detector consists of three completely diﬀerent detector components MDC, PSB, SEC which can contribute to the track creation and track type. The same as for the FD tracks procedure begins with a creation of the clusters fromhitsin these threedetector components, if a contribution of the detectorintrack wasjustiﬁed. Thetracks withMDC and/orPSB cluster are charged tracks identiﬁed with a charged particle. The absence of MDC and/orPSB cluster(the veto condition)determines the neutralparticleidentiﬁed with a photon. To distinguish between the charged and neutral track one needs only onefromtwodetectorinformation(MDC orPSB). ThePSBis aplastic scintillator anditgives very fastbinary information if there was a charged particle. If one is interested in more details like particle momentum one needs to to use theMDCin thetrackbuilding(MDC is a drift chamber). Since in this work the goal is to detect the photons from the pion decays it is enough to use the contributions of PSB and SEC detectors in track building process for distinguishing between neutral and charged tracks. Also from the point of view of computing time the PSB decision information is around one order of magnitude faster then the MDC information. For pp → pp3π0 → pp6γ reaction only tracks with PSB and SEC detector contribution were considered. The details about the MDC reconstruction couldbefoundin[104]. First the clusterinPSBisbuildfrom overlapping elementsif the elements are in time coincidence, then the azimuthal angle of the cluster is calculated as an average of the elements. The polar angle is ﬁxed for diﬀerent parts of thePSBdetector,forforward 30[deg], central 90[deg]andbackward 140[deg]. The time of the cluster is calculated as an average of the time for elements and the energy deposit as a maximum from the elements. Next the cluster is SEC detector is found. Since the photons hitting SEC detector produce electromagnetic showers, their transversal development depends on the Moliere Radius which exceeds the size of the crystal so one photon hitting the crystal can develop to other crystals and the cluster building has to combine it. The cluster building works iteratively, the neighboring elements are checked for time coincidence(maximum 50 ns)and minimum energy deposit of 2 MeV and combined to the cluster. The center of the cluster is taken as a contributing crystal with highest energy deposit. The time is taken from the cluster center. The energy deposit of the cluster − → is the sum of all contributing elements. The position of the cluster R is Jagiellonian University 194 Benedykt R. Jany APPENDIX D TRACK RECONSTRUCTION IN WASA-AT-COSY calculated asa weighted average of thepositions of thecontributing crystals −→ r i −→ R = i wi −→ r i i wi (D.1) where wi = MAX 0;w0 + ln � Edepi i Edepi � (D.2) here w0 =5 and Edepi -energy deposit of the i−th crystal contributing to the cluster. Now the PSB cluster and SEC cluster information is combined to the track by checking the angular overlapping between the clusters and time coincidence. The neutraltracks(identiﬁed withphotons) arethose without assignedPSB cluster(the veto condition). Jagiellonian University 195 Benedykt R. Jany Appendix E The Kinematic Fit The idea of the kinematic ﬁt The experimentalist deals with the measurements mi which are always biased by the measurements speciﬁc uncertainly σi (i.e. accuracy of the m detectors, reconstruction accuracy etc.). All this informations has to be taken into account in the data analysis, so that in the ﬁnal results all those eﬀects are compensated. One way ofhandling thatproblemisthekinematic ﬁtting[105, 106]. The kinematic ﬁtting procedure is a data transformation technique which takes into account the information about the errors σi of m the measured quantities mi to compute the most probable value fi of the true unknown value ti, the estimator of the true value fi = T({mj,σj }) (E.1) m Considering reaction → 1+2+3+4+... + k (E.2) Pin Pout where Pin = PA + PB -4-momentum vector of the input channel Pout = P1 + P2 + P3 + P4 + ... -4-momentum vector of the output channel from the conservation of energy and momentum the relation has to be fulﬁlled C = Pin −Pout = O (E.3) and one knows that the number of independent variables to describe this reaction is 3k−4 (E.4) where k -numberof ﬁnal stateparticles[63]. Usuallytheinitialstate Pin iswellknownandis ﬁxed(itisnotapurposeof the ﬁt). Whatismeasured,arethethe ﬁnal stateparticles i =1,2,3,...,k. The following parameters of the ﬁnal state particles are measured: their Energy(Ei),polar angle(θi)anazimuthal angle(φi)which deﬁne clearly the particle. Lets call such a parameters for one given particle mi, of course each measurement has its uncertainly σi , where i denotes the ﬁnal state m particles. Nowdeﬁne a vector ofall measurements M =[mi]and a covariance matrix for the measurements V =[cov(p,l)](where p,l =1,2,3,...,k) on the diagonal of this matrix are (σi )2, in case of no correlations between the m measurements the matrix is diagonal. What we want from this information istheset of newvariableswhich will ”replace“themeasurements. Theyhave 197 APPENDIX E THE KINEMATIC FIT to take into account all the uncertainties V.Such anewvector willbecalled F =[fi]. To realize the task to ﬁnd the estimator F (Eq. E.1)one can use the Least Square Method(The Chi-square Method). So ﬁrst lets build the the Chi-square functional: H(F)=(F −M)V−1 (F −M)T . (E.5) Toincorporatethekinematic condition(Eq. E.3) tothefunctional one uses the Lagrange Multipliers, now the functional has the form: H(F)=(F −M)V−1 (F −M)T + λC (E.6) here λ denotes the Lagrange multiplier vector. In general in addition tothekinematic condition(Eq.E.3) one can request other condition asfor example: that two ﬁnal state particles come from a intermediate resonance, lets call these extra conditions D and rewriting functional as H(F)=(F −M)V−1 (F −M)T + λCC + λDD (E.7) The solution for F onegetsfor H(F)having minimum. This corresponds to well known mathematical problem of minimization of the functional with extra boundary conditions. The minimalH(F)= χ2 ,in case ofGaussian V,isdistributed as a χ2 FitNDF distribution with Number Degrees of Freedom NDF =4+ND [107], ND number of extra constraints D. Ingeneral not all ofthree variables(Ei,θi,φi) foreach ﬁnal stateparticleshavetobemeasured toidentiﬁed completely the reaction (Eq. E.2), (Eq. E.4) tells us about it. The unmeasured variables couldbe retrievedfromthe availableinformation,by solving(Eq.E.7). Denoting Nu -number of unmeasured variables, the NDF of χ2 distribution Fit changes now to NDF =4+ND −Nu. Knowing theProbability Distribution Function of χ2 one can test thedeviationfrom theoretical χ2 ,itis easily Fit NDF doneby introducing theProbability of theFit(Complementary Cumulative Distribution Function or survival function): � ∞ ProbFit(χ2 )= χ2 (k)dk =1−F(χ2 ,NDF) (E.8) FitNDFFit χ2 Fit where F(χ2 ,NDF)is the Cumulative Distribution Function(CDF). Since Fitthe ProbFit isatypeofCumulativeFunctionitshouldhavea ﬂatdistribution on range 0−1. Oneusesnumerical methodsto ﬁnd asolutionof(Eq.E.7)forgiven M,V [108–110] Jagiellonian University 198 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT Example of the kinematic ﬁt As an Example of kinematic ﬁt lets consider the following reaction: pp→ ppπ0π0π0 → ppγγγγγγ (E.9) One has 8 particles in the ﬁnal state so the number of independent variables to describe this reaction is 3∗ 8−4 =20. If we would measure for each of the particles Energy(E),polar angle(θ) and azimuthal angle(φ) then we would have 8∗ 3 =24 variables, the information would be redundant. Lets consider that we measure E,θ,φ for photons and only θ,φ for the protons, then we have still 6∗ 8+2∗ 2 =22 independent variables, that is still enough to fully describe our reaction. Having the following situation we can now do the kinematic ﬁtting, one has number of unmeasured variables Nu =2 (E for two protons -will come as a result of the ﬁt) and additional number constraints ND =3 (the mass constraints for the photons which should give three π0). The Number Degrees of Freedom NDF for the χ2 Fit ﬁt is NDF =4+ND −Nu =4+3−2 =5 and if the errors are distributed as Gaussian, the χ2 should be distributed as a χ2 distribution with Fit NDF NDF =5. Fit reaction pp → pp3π0 → pp6γ for the Gaussian distributions of the errors. Theblackhistogram corresponds to theMonte-Carlo simulation, the redline denotes theoretical χ2 . Number of events is shown on vertical axis. In the(Fig. E.1) is presented the χ2 distribution for the Kinematic Fit Fit for the above case for the Monte-Carlo simulation assuming homogeneous Jagiellonian University 199 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT and isotropic populated Phase Space with Gaussian smearing used for the variables. One sees that the χ2 agrees with a theoretical χ2 distribution Fit NDF with NDF =5. Also the ProbFit distributionispresentedin(Fig.E.2). It is ﬂat, it conﬁrms ones more that χ2 agrees with a theoretical χ2 and Fit NDF that the errors of the variables are Gaussian. When the errors are not Gaussian distributed, or the reaction hypothesis is not true(e.g. background reaction contamination) the χ2 does not agrees Fit with a theoretical χ2 distribution see(Fig.E.3). Thisis also reﬂected on NDF the ProbFit whichis nothomogeneouslydistributed anymore, thepopulation of smallprobability valuesishigher(Fig.E.4)in case of nonGaussian errors. The ProbFit is slightly non homogeneous with very prominent increase at smallprobability(Fig.E.5) in case the reactionhypothesisis not true. To dealwith thisproblem one selects eventsfor which theprobability ProbFit for thesignal reactionisingood agreement withﬂatdistribution,inthisexample that would be ProbFit > 0.2. Of course the bigger the ProbFit threshold for selection the ”cleaner“ will be the data sample but the statistics will reduce -one has to ﬁnd always the compromise. In addition to the probability function checks one can construct the ”Residual spectra” for ﬁt: residuali = mi −fi (E.10) in case ofGaussian errors andno correlationsbetween the measured variables Jagiellonian University 200 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT Fit the reaction pp → pp3π0 → pp6γ for the Non Gaussian distributions of the errors. The black histogram corresponds to the Monte-Carlo simulation, the red line denotes theoretical χ2 . Number of events is shown on vertical axis. Jagiellonian University 201 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT reaction. The same amount of each reaction type was simulated. (i.e. the V matrix diagonal) the residuali variable should be distributed as Gaussian distribution with mean 0 and standard deviation σi given by residual the relation[111]: σi =(σi )2 −(σi )2 (E.11) residual mf which may still ”puzzle many users“ [112]. One can also deﬁne the ”Pulls” distribution: residuali pulli = (E.12) σi residual the pulli variable should be distributed as standard Gaussian distribution with mean 0 and sigma 1. Jagiellonian University 202 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT Illustration of the kinematic ﬁt -simple example To illustrate the idea of the kinematic ﬁt and how it works look at some simple example. Lets assume that we have measured x =0.5 with the error σx =1 and y =0.5 with the error σy =1 and we know that the relation xy =1 (E.13) shouldholdforthe variables(asconstraint; it correspondstothe(Eq.E.3)). Now we want to combine this information together, as in the kinematic ﬁt. One constructs the function to minimize: where x correspond tothe mostprobable value which we arelooking h(x ∗ ,y ∗ ) = � x ∗ − x �2 + � y ∗ −y �2 (E.14) σx σy with the constraint: x ∗ y ∗ = 1 (E.15) ∗ ∗ ,y for. Onehasto ﬁnd theminimumof(Eq.E.14)with condition(Eq.E.15). To solve thisproblem one uses theLagrange multipliers method. One constructs the functional to minimize: ∗∗∗ H(x ,y ∗ )= h(x ,y ∗ )+λ(xy ∗ −1) (E.16) Jagiellonian University 203 Benedykt R. Jany APPENDIX E THE KINEMATIC FIT here λ is Lagrange multiplier. One gets the equations: ∂H(x ∗ ,y ∗) =0 (E.17) ∂x∗ ∂H(x ∗ ,y ∗) =0 ∂y∗ x ∗ y ∗ −1 =0 There aretwo candidatesfor solution of these equations(Fig.E.6): 1. x ∗ =1, y ∗ =1, λ = −1(P1) 2. x ∗ = −1, y ∗ = −1, λ = −3(P2) Since H(P1)
The MLE methodis the mostgeneral one with the optimalperformances of the estimators soit was chosen. The estimator of thekinetic energy of the particle T(Ekin)is the one which maximizes the likelihood function L which corresponds to minimization of l = −log(L). The likelihood for the dE-E telescope is deﬁned as the conditional probability f that the energy of the particleE is equaltothetruekinetic energyEkin underthe conditionthat one hasthe energydepositsinlayers ofFRH telescope E1 ,E2 ,E3 ,E4 ,E5 depdepdepdepdep i.e. () L(E)= fE = Ekin/Edep1 dep,E3 dep,E5 ,E2 dep,E4 dep(F.1) Now using Bayestheorem[115]we can write 207 APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION () L(E)= fE = Ekin/E1 dep,E3 dep,E5 (F.2) dep,E2 dep,E4 dep () fE1 ,E2 ,E3 ,E4 ,E5 /E= depdepdepdepdepEkin = fE1 ,E2 ,E3 ,E4 ,E5 /E= EkindEkin depdepdepdepdep The only thing which one has to derive is the probability function () fE1 ,E2 ,E3 ,E4 ,E5 /E= whichis ﬁvedimensionalprobability depdepdepdepdepEkin functionin energydeposits conditionedby energy(sixdimensionalfunction), to simplify the problem one may assume the factorization: () L(E)= f dep,E2 dep,E4 dep/E= Ekin(F.3) E1 dep,E3 dep,E5 = f1 ∗ f2 ∗ f3 ∗ f4 ∗ f5 where fi are independent and deﬁned as () E1 f1 = f dep/E1 = Ekin(F.4) () E2 f2 = f dep/E2 = Ekin −E1 (F.5) dep () f3 = fE3 /E3 = Ekin −E1 −E2 (F.6) depdep dep () f4 = f dep/E4 = dep −E2 dep(F.7) E4 Ekin −E1 dep −E3 () E5 −E2 −E3 −E4 f5 = f dep/E5 = Ekin −E1 dep dep dep(F.8) dep The fi functions were derived using Monte-Carlo GEANT3 simulation for the single particle tracks in FRH telescope for the given kinetic energy. Example of thederivedprobability functionforprotonsis on(Fig.F.1). Jagiellonian University 208 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION f3 Edep3 0.18 0.2 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E3 = Ek-Edep1-Edep2 Figure F.1: f3 probability function derived using Monte-Carlo GEANT3 simulated single protons tracks in FRH telescope. On vertical axis energy loss in third layer of FRH in GeV (here denoted as Edep3). On horizontal axis the diﬀerence between the kinetic energy and energy loss in ﬁrst and second layer of FRH in GeV (here denotedE3) The reconstruction properties Usingabovedeﬁnedlikelihoodfunction L(E)(Eq.F.4)one may now reconstruct the kinetic energy of the particles. The reconstruction of the protons kinetic energy is in the most common interest of WASA-at-COSY collaboration, due to interest of studying the decays of the η ′ meson[116, 117]. Using Monte-Carlo GEANT3 simulation of the WASA-at-COSY detector, the single proton tracks illuminating FRH detector are generated with the kinetic energy from 50 MeV to 2 GeV. The kinetic energy was reconstructed usingproposed method. Onthe(Fig.F.2) the relative resolution for the reconstructed kinetic energy (Etrue −Erec)/Etrue as a function of the minimized likelihood l = −log(L)ispresented, as onecan see whenthelikelihood l is getting larger(the degree of belief2
is getting smaller) the relative resolution becomes worse. Cutting on the likelihood value rejects the events with the bad resolution i.e. events witch have low degree of belief. Since the likelihood function was derived using Monte-Carlo with some ﬁnal statistical sample of the data, to control this systematical eﬀect, in addition to the likelihood L one may calculate the relative error of the minimized likelihood Lerr =ΔL/L, using standard errorpropagationmethodfor 2 term taken from[115] Jagiellonian University 209 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION (Etrue-E5)/Etrue (Eq. F.4). The relative resolution for the reconstructed kinetic energy as a function of the error of the minimized likelihood Lerr isshown(Fig.F.3). Onesees that the energy resolution of the values above 0.25 of the Lerr is very bad. Performing a cut on the Lerr can suppress this systematical eﬀect on L. 1 0.5 0 -0.5 -1 1234567 Llog5 FigureF.2: Relative resolutionforthe reconstructedkinetic energyversusthe minimizedLikelihoodfunction l (heredenotedas Llog5)for theMonte-Carlo single proton tracks. When oneplotstruekinetic energy versustheLikelihood l (Fig. F.4)one immediately sees that these variables are directly correlated, so performing a cut on the Likelihood l one can select a certain range of the true kinetic energy of the particle. Having now good established Likelihood reconstruction, one may think of doing particle identiﬁcation i.e. one wants to test the hypothesis of the particle type. Lets assume that we would like to distinguish protons from charged pions knowing the energy deposits in FRH. The straightforward way is to compare the probability function f conditioned by the particle hypothesis: (EProton ) fE = ,Proton/E1 ,E2 ,E3 ,E4 ,E5 kin depdepdepdepdep = (F.9) RPID (EPion ) fE = ,Pion/E1 ,E2 ,E3 ,E4 ,E5 kin depdepdepdepdepL(EProton ,Proton) = L(EPion ,Pion) Jagiellonian University 210 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION (Etrue-E5)/Etrue 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 LErr5 Figure F.3: Relative resolution for the reconstructed kinetic energy versus the error of the minimized likelihood function Lerr (here denotedas LErr5) for the Monte-Carlo single proton tracks. Etrue 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 Llog5 FigureF.4:TruekineticenergyinGeV(i.e. assumedintheMonte-Carlo simulations, here denoted as Etrue) versus the likelihood l for the Monte-Carlo single proton tracks. Jagiellonian University 211 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION which is exactly equal, from the deﬁnition, to the ratio of the likelihood functionforproton andpion(PID staysforParticleIdentiﬁcation). Using the(Eq.F.10)thePID(F.5a,b). Restricting ourself to theparticular range of theLikelihood andits error(F.5c) one canget agood separation between protons, and pions up to ∼ 700MeV. Itisessential toknow, asmentioned above,howthereconstructionworks for the protons coming from the pp → ppη ′ production at incident proton energy T =2.54 GeV. These protons have the kinetic energy range of 300− 800 MeV. Once more the single proton tracks are generated using Monte-Carlo GEANT3 simulation with energy range up to 2 GeV later the cut on reconstructed kinetic energy was performed Erec =300−800 MeV. The quality of the reconstruction for both reconstruction methods is shown (Figs. F.6, F.7). The beneﬁts of the likelihood method are seen on (Fig. F.8): the wrongly reconstructed particles are cut out without loosing eﬃciency. For the conditions(l< 4.4 and Lerr < 0.25) the reconstruction eﬃciency of 0.57 was reached. Jagiellonian University 212 Benedykt R. Jany 80 80 a) b) 70 70 APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION Jagiellonian University 213 BenedyktR.Jany LProton/Lpion 60 Lproton/Lpion 60 50 Protons 50 Pions 40 40 30 30 20 20 10 10 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Eproton Eproton [GeV] LProton/LPion 80 70 Protons Pions c) 60 50 40 30 20 10 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 EProton [GeV] FigureF.5: ParticleIdentiﬁcation using likelihood function. RPID ration(Eq. F.10) versus the reconstructed kinetic energy in GeV assuming proton as a particle for the single Monte-Carlo tracks of a) protons, b) charged pions, c) protons(blue) and pions(red) with lproton < 4.4 and Lproton < 0.25 condition. err APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION Etrue2 a) 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0.30 0.4 0.5 0.6 0.7 0.8 Erec Etrue b) 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.3 0.4 0.5 0.6 0.7 0.8 ErecBRJ FigureF.6:TruekineticenergyinGeV(i.e. assumedintheMonte-Carlo simulations, here denoted as Etrue) versus reconstructed kinetic energy in GeV using:a) standard reconstruction b) Likelihood based for l< 4.4 and Lerr < 0.25; for the Monte-Carlo single proton tracks with Erec = 300 − 800 MeV. Jagiellonian University 214 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION Jagiellonian University 215 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION Veriﬁcation of the method UsingMonte-Carlo simulation thepp→ ppη ′ production atincidentproton energy T =2.54 GeV was generated, and the missing mass of the two ﬁnal state protons was reconstructed using standard and likelihood one method ofkinetic energy reconstruction(Fig.F.9). UsingtheLikelihoodbased reconstruction one gets sharper η ′ peak. The pp→ ppη WASA-at-COSY data from April 2007 at incident proton energy T =1.4 GeV were analyzed. The comparison of the missing mass of thetwoprotonsforstandard andLikelihood methodis shown(Fig.F.10). For the Likelihood method the η signal is much more pronounced and the multipion background is suppressed. One concludes that the developed Bayesian likelihood kinetic energy reconstruction method had a better performances in all tests then the standard reconstruction method since it uses the full multidimensional information simultaneously. The method could be easily generalized for the full FD telescope, since it is based on the likelihood, also future detectors could be treated the same way like planned TOF and DIRC detector. Nevertheless further developments are required to fully adopt and optimize the method for the experimental data, like derivation of the likelihood Jagiellonian University 216 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION function using real experimental detector response. Jagiellonian University 217 Benedykt R. Jany APPENDIX F BAYESIAN LIKELIHOOD ENERGY RECONSTRUCTION Jagiellonian University 218 Benedykt R. Jany Appendix G Data Tables – Results Dalitz Plot ppX: MA{2)(p {1)p {2)) versus MA{2)(p {1,2)3pi0) - Each event is filled two times Acceptance and Efficiency Corrected for MM {pp)=0.4-0.5 GeV/cA{2) Error of MA{2)(p {1,2)3pi0): 0.0575 GeVA{2)/cA{4) Error of MA2(p {1)p {2)): 0.0775 GeVA{2)/cA{4) The data are in the following format: {{MA{2)(p {1,2)3pi0) [GeVA{2)/cA{4)], MA{2)(p {1)p {2)) [GeVA{2)/cA{4)], sigma [mub], Error of sigma [mub] ),...) {{3.1375,3.4775,0.0174797,0.018807),{3.2525,3.4775,0.261445,0.0701279),{3.3675,3.4775,0.491976,0.131835),{3.4825,3.4775,0.337561,0.0996555), {3.5975,3.4775,0.00502498,0.00592007),{2.9075,3.6325,1.02536,0.184544),{3.0225,3.6325,2.12291,0.231854),{3.1375,3.6325,3.63787,0.35924), {3.2525,3.6325,3.68257,0.358165),{3.3675,3.6325,3.2539,0.294329),{3.4825,3.6325,4.1901,0.420464),{3.5975,3.6325,2.69199,0.266603),{3.7125,3.6325,0.764874,0.159824), {2.6775,3.7875,1.41741,0.282268),{2.7925,3.7875,3.73763,0.325932),{2.9075,3.7875,4.7208,0.364872),{3.0225,3.7875,4.88511,0.407791),{3.1375,3.7875,3.69048,0.284186), {3.2525,3.7875,3.88397,0.308961),{3.3675,3.7875,3.88398,0.298999),{3.4825,3.7875,5.00449,0.413006),{3.5975,3.7875,4.70849,0.354093),{3.7125,3.7875,4.2155,0.391736), {2.4475,3.9425,0.0328254,0.0224952),{2.5625,3.9425,3.0355,0.315005),{2.6775,3.9425,6.59154,0.448956),{2.7925,3.9425,6.09635,0.430104), {2.9075,3.9425,5.66769,0.449077),{3.0225,3.9425,4.22978,0.325428),{3.1375,3.9425,4.4695,0.358734),{3.2525,3.9425,4.08232,0.317733),{3.3675,3.9425,5.00882,0.391812), {3.4825,3.9425,5.37028,0.383899),{3.5975,3.9425,6.97521,0.489063),{3.7125,3.9425,6.07407,0.432689),{2.3325,4.0975,0.100675,0.0482124), {2.4475,4.0975,4.6543,0.385888),{2.5625,4.0975,8.16989,0.523706),{2.6775,4.0975,7.11624,0.47761),{2.7925,4.0975,6.61185,0.539786),{2.9075,4.0975,4.94218,0.423436), {3.0225,4.0975,5.51691,0.512892),{3.1375,4.0975,5.5405,0.51213),{3.2525,4.0975,4.84894,0.425921),{3.3675,4.0975,6.21782,0.517028),{3.4825,4.0975,7.62919,0.528102), {3.5975,4.0975,7.43044,0.459133),{3.7125,4.0975,6.07615,0.458178),{2.2175,4.2525,0.014781,0.00742517),{2.3325,4.2525,3.67241,0.289163), {2.4475,4.2525,8.9962,0.528426),{2.5625,4.2525,7.71055,0.534302),{2.6775,4.2525,6.87624,0.670021),{2.7925,4.2525,6.48654,0.708842),{2.9075,4.2525,5.087,0.568579), {3.0225,4.2525,6.90866,0.842781),{3.1375,4.2525,5.46198,0.649803),{3.2525,4.2525,5.84001,0.610077),{3.3675,4.2525,6.52555,0.594335),{3.4825,4.2525,8.27967,0.537363), {3.5975,4.2525,8.79431,0.50209),{3.7125,4.2525,2.10085,0.222115),{2.2175,4.4075,2.23241,0.244879),{2.3325,4.4075,8.39249,0.479131),{2.4475,4.4075,7.19706,0.531163), {2.5625,4.4075,7.18008,0.931462),{2.6775,4.4075,7.114,1.1608),{2.7925,4.4075,6.20508,0.998695),{2.9075,4.4075,8.51726,1.61476),{3.0225,4.4075,7.86981,1.38567), {3.1375,4.4075,5.45658,0.866037),{3.2525,4.4075,6.87668,1.02521),{3.3675,4.4075,7.66655,0.828966),{3.4825,4.4075,8.19602,0.474229),{3.5975,4.4075,5.76453,0.387057), {2.1025,4.5625,0.437278,0.094854),{2.2175,4.5625,5.86262,0.380862),{2.3325,4.5625,8.88623,0.863359),{2.4475,4.5625,6.00173,1.15924),{2.5625,4.5625,7.08138,2.5577), {2.6775,4.5625,2.6916,1.00795),{2.7925,4.5625,20.5465,16.8269),{2.9075,4.5625,4.72404,1.91132),{3.0225,4.5625,3.70918,1.7851),{3.1375,4.5625,3.60967,1.17574), {3.2525,4.5625,6.2935,1.39343),{3.3675,4.5625,8.25147,0.949508),{3.4825,4.5625,6.81706,0.415901),{3.5975,4.5625,0.635083,0.117729), {1.9875,4.7175,0.0765214,0.0837539),{2.1025,4.7175,3.42536,0.368101),{2.2175,4.7175,8.42509,0.796023),{2.3325,4.7175,7.59134,1.50078), {2.4475,4.7175,4.10324,1.47949),{2.5625,4.7175,9.29245,8.65981),{2.6775,4.7175,28.3254,31.8561),{2.7925,4.7175,9.53683,13.2541),{3.0225,4.7175,5.30418,3.79407), {3.1375,4.7175,4.78785,1.55479),{3.2525,4.7175,9.05299,1.61104),{3.3675,4.7175,8.42633,0.750061),{3.4825,4.7175,1.49538,0.188709),{1.9875,4.8725,0.33857,0.0908039), {2.1025,4.8725,6.3783,0.662743),{2.2175,4.8725,8.67542,1.38354),{2.3325,4.8725,6.52215,1.91261),{2.4475,4.8725,6.2972,3.61201),{2.5625,4.8725,3.42587,3.85261), {2.7925,4.8725,1.0444,1.38457),{2.9075,4.8725,4.52859,3.48024),{3.0225,4.8725,5.65446,2.03865),{3.1375,4.8725,7.08134,1.43643),{3.2525,4.8725,9.11126,1.04574), {3.3675,4.8725,1.95546,0.274247),{1.9875,5.0275,1.7606,0.357606),{2.1025,5.0275,8.29153,1.0781),{2.2175,5.0275,12.1079,2.72267),{2.3325,5.0275,8.40841,3.89482), {2.4475,5.0275,2.51329,1.51365),{2.6775,5.0275,1.40782,1.8404),{2.7925,5.0275,4.16865,2.97703),{2.9075,5.0275,8.43391,4.11218),{3.0225,5.0275,9.61317,2.35597), {3.1375,5.0275,8.34345,1.14221),{3.2525,5.0275,2.27993,0.401442),{1.8725,5.1825,0.178989,0.175937),{1.9875,5.1825,2.11612,0.392844),{2.1025,5.1825,7.9617,1.29348), {2.2175,5.1825,7.65753,2.42021),{2.3325,5.1825,23.3057,9.22882),{2.4475,5.1825,7.11902,7.45962),{2.6775,5.1825,7.31083,6.91889),{2.7925,5.1825,12.313,6.06399), {2.9075,5.1825,7.91734,1.91426),{3.0225,5.1825,5.61624,0.90616),{3.1375,5.1825,1.79766,0.339612),{1.8725,5.3375,0.271656,0.155461),{1.9875,5.3375,3.00048,0.642847), {2.1025,5.3375,6.70178,1.52439),{2.2175,5.3375,7.95728,2.79232),{2.3325,5.3375,9.40674,5.55021),{2.4475,5.3375,10.6207,8.53026),{2.5625,5.3375,62.0422,46.1063), {2.6775,5.3375,7.29463,3.16797),{2.7925,5.3375,7.79153,1.98928),{2.9075,5.3375,5.09323,1.10407),{3.0225,5.3375,0.570924,0.182446),{1.8725,5.4925,0.265949,0.201073), {1.9875,5.4925,3.60774,1.18242),{2.1025,5.4925,4.91299,1.6903),{2.2175,5.4925,56.4728,34.6586),{2.3325,5.4925,43.6614,46.4431),{2.4475,5.4925,26.4614,29.7595), {2.5625,5.4925,170.782,169.419),{2.6775,5.4925,8.58401,2.55137),{2.7925,5.4925,2.41678,0.737233),{2.9075,5.4925,0.474332,0.323947),{1.8725,5.6475,0.414746,0.325622), {1.9875,5.6475,1.47571,1.03575),{2.2175,5.6475,5.03593,5.05213),{2.4475,5.6475,4.01481,3.59535),{2.5625,5.6475,55.9362,56.5585),{2.6775,5.6475,0.907701,0.588852)) Table G.1: Acceptance and eﬃciency corrected Dalitz Plot ppX, for MMpp =0.4 − 0.5 GeV/c2 (Fig. 92). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of theabsolutenormalization of 19% have to beincluded. Fully expandable version of thetableis availablein the attached electronic version of the thesis. 221 APPENDIX G DATA TABLES – RESULTS Dalitz Plot ppX: MA{2)(p {1)p {2)) versus MA{2)(p {1,2)3pi0) -Each event is filled two times Acceptance and Efficiency Corrected for MM {pp)=0.6-0.7 GeV/cA{2) Error of MA{2)(p {1,2)3pi0): 0.023 GeVA{2)/cA{4) Error of MA2(p {1)p {2)): 0.031 GeVA{2)/cA{4) The data are in the following format: {{MA{2)(p {1,2)3pi0) [GeVA{2)/cA{4)], MA{2)(p {1)p {2)) [GeVA{2)/cA{4)], sigma [mub], Error of sigma [mub] ),...) {{3.425,3.493,0.144261,0.172171),{3.471,3.493,0.638871,0.522185),{3.517,3.493,0.253877,0.303098),{3.195,3.555,0.717029,0.233791),{3.241,3.555,3.83872,0.476267),{3.287,3.555,9.46322,0.754229), {3.333,3.555,15.2864,1.00203),{3.379,3.555,20.0221,1.23572),{3.425,3.555,24.4798,1.49261),{3.471,3.555,26.61,1.69081),{3.517,3.555,23.2614,1.40015),{3.563,3.555,20.566,1.25609), {3.609,3.555,13.077,0.886915),{3.655,3.555,5.38522,0.581806),{3.701,3.555,0.195404,0.130282),{3.057,3.617,0.149706,0.0721678),{3.103,3.617,3.00061,0.440607),{3.149,3.617,13.0436,1.06606), {3.195,3.617,23.6314,1.41962),{3.241,3.617,29.8858,1.60859),{3.287,3.617,26.8697,1.36303),{3.333,3.617,26.559,1.32986),{3.379,3.617,26.415,1.3126),{3.425,3.617,25.0646,1.26055), {3.471,3.617,24.8321,1.23632),{3.517,3.617,25.1389,1.24424),{3.563,3.617,29.5594,1.52274),{3.609,3.617,28.1906,1.4553),{3.655,3.617,29.8835,1.61796),{3.701,3.617,15.9822,1.16582), {3.747,3.617,0.644988,0.358961),{2.965,3.679,0.288225,0.12942),{3.011,3.679,6.4197,0.981585),{3.057,3.679,17.2574,1.36347),{3.103,3.679,30.7316,1.72462),{3.149,3.679,34.3012,1.69064), {3.195,3.679,35.0059,1.70844),{3.241,3.679,31.4577,1.50473),{3.287,3.679,28.4133,1.31107),{3.333,3.679,27.9521,1.31875),{3.379,3.679,29.6771,1.40924),{3.425,3.679,28.4411,1.35058), {3.471,3.679,27.1365,1.2417),{3.517,3.679,31.7607,1.51368),{3.563,3.679,31.5902,1.5051),{3.609,3.679,33.4146,1.61966),{3.655,3.679,35.363,1.70522),{3.701,3.679,37.3202,1.97896), {3.747,3.679,14.3286,1.50794),{2.873,3.741,0.0143844,0.0172694),{2.919,3.741,3.87257,0.733035),{2.965,3.741,17.859,1.5641),{3.011,3.741,34.5995,1.98575),{3.057,3.741,40.1951,1.88501), {3.103,3.741,38.2233,1.73174),{3.149,3.741,38.7862,1.78108),{3.195,3.741,33.2322,1.49098),{3.241,3.741,33.1528,1.50957),{3.287,3.741,31.2907,1.4849),{3.333,3.741,32.1185,1.54781), {3.379,3.741,31.2274,1.50362),{3.425,3.741,30.2092,1.42707),{3.471,3.741,34.9165,1.65465),{3.517,3.741,31.4878,1.4015),{3.563,3.741,36.226,1.66019),{3.609,3.741,38.8549,1.76344), {3.655,3.741,41.6888,1.90005),{3.701,3.741,42.948,1.93472),{3.747,3.741,32.4979,2.22148),{2.827,3.803,2.1658,1.04709),{2.873,3.803,10.6503,1.2397),{2.919,3.803,27.6651,1.8263), {2.965,3.803,44.2222,2.09437),{3.011,3.803,45.3381,1.95219),{3.057,3.803,41.8986,1.76933),{3.103,3.803,39.6302,1.71343),{3.149,3.803,34.5192,1.46707),{3.195,3.803,32.3433,1.44908), {3.241,3.803,32.1632,1.46806),{3.287,3.803,30.8247,1.44642),{3.333,3.803,33.8229,1.70295),{3.379,3.803,32.0661,1.5403),{3.425,3.803,32.0895,1.48761),{3.471,3.803,33.713,1.49602), {3.517,3.803,33.2292,1.42591),{3.563,3.803,39.9125,1.73875),{3.609,3.803,40.1507,1.70665),{3.655,3.803,45.373,1.90635),{3.701,3.803,52.1246,2.23425),{3.747,3.803,43.1802,2.48887), {2.781,3.865,3.40495,0.861508),{2.827,3.865,19.4547,1.67642),{2.873,3.865,36.1683,1.9497),{2.919,3.865,52.2152,2.22498),{2.965,3.865,50.9528,2.103),{3.011,3.865,49.3339,2.14991), {3.057,3.865,38.9265,1.60366),{3.103,3.865,38.1441,1.67092),{3.149,3.865,33.1037,1.45733),{3.195,3.865,31.5594,1.46627),{3.241,3.865,33.7099,1.69539),{3.287,3.865,34.0871,1.67815), {3.333,3.865,32.7975,1.60359),{3.379,3.865,31.1836,1.52926),{3.425,3.865,33.7499,1.58717),{3.471,3.865,33.8605,1.48204),{3.517,3.865,37.3275,1.60237),{3.563,3.865,40.6225,1.70671), {3.609,3.865,49.2943,2.08146),{3.655,3.865,50.2841,2.05291),{3.701,3.865,58.4505,2.35322),{3.747,3.865,48.8172,2.73593),{2.735,3.927,7.68382,1.51098),{2.781,3.927,19.8714,1.41204), {2.827,3.927,38.7701,1.89632),{2.873,3.927,55.4878,2.2747),{2.919,3.927,52.9977,2.10571),{2.965,3.927,51.8445,2.14225),{3.011,3.927,46.562,1.94641),{3.057,3.927,42.2038,1.83451), {3.103,3.927,36.3797,1.61101),{3.149,3.927,36.7138,1.83261),{3.195,3.927,34.5673,1.75746),{3.241,3.927,32.0003,1.68171),{3.287,3.927,33.1107,1.73357),{3.333,3.927,33.52,1.7569), {3.379,3.927,35.9518,1.83147),{3.425,3.927,34.4101,1.60538),{3.471,3.927,38.836,1.70814),{3.517,3.927,45.2493,1.8976),{3.563,3.927,50.2887,2.10006),{3.609,3.927,50.5588,2.02899), {3.655,3.927,58.2437,2.29508),{3.701,3.927,61.1107,2.38403),{3.747,3.927,34.6982,2.23201),{2.643,3.989,0.0531627,0.0564928),{2.689,3.989,6.5707,0.9202),{2.735,3.989,21.0642,1.36645), {2.781,3.989,43.4424,2.10188),{2.827,3.989,62.7041,2.54167),{2.873,3.989,61.5103,2.43077),{2.919,3.989,53.9182,2.16059),{2.965,3.989,50.6911,2.09508),{3.011,3.989,43.7194,1.87368), {3.057,3.989,40.0628,1.85487),{3.103,3.989,34.1084,1.69998),{3.149,3.989,36.4598,1.99341),{3.195,3.989,35.4993,1.98808),{3.241,3.989,33.8443,1.95982),{3.287,3.989,31.8205,1.80214), {3.333,3.989,37.2815,2.03491),{3.379,3.989,36.364,1.90325),{3.425,3.989,39.3679,1.89527),{3.471,3.989,42.0368,1.76679),{3.517,3.989,50.9886,2.11547),{3.563,3.989,53.4128,2.12016), {3.609,3.989,60.4144,2.4051),{3.655,3.989,67.5091,2.58451),{3.701,3.989,65.0478,2.54022),{3.747,3.989,12.6589,1.32259),{2.597,4.051,0.027313,0.0341311),{2.643,4.051,7.57316,1.15879), {2.689,4.051,22.8623,1.56678),{2.735,4.051,39.017,1.88111),{2.781,4.051,55.4656,2.19686),{2.827,4.051,62.2322,2.40422),{2.873,4.051,57.6914,2.24606),{2.919,4.051,53.4548,2.1529), {2.965,4.051,48.5409,2.15705),{3.011,4.051,43.4623,2.10545),{3.057,4.051,42.068,2.32784),{3.103,4.051,37.3865,2.22251),{3.149,4.051,35.5012,2.16803),{3.195,4.051,31.3188,1.87673), {3.241,4.051,32.0951,1.89777),{3.287,4.051,37.5537,2.31741),{3.333,4.051,39.1608,2.34492),{3.379,4.051,38.548,2.13666),{3.425,4.051,42.7995,2.09257),{3.471,4.051,49.3859,2.03334), {3.517,4.051,52.502,2.07887),{3.563,4.051,57.4665,2.20477),{3.609,4.051,66.0408,2.53283),{3.655,4.051,69.5935,2.61622),{3.701,4.051,46.9997,2.26238),{3.747,4.051,0.99606,0.558987), {2.597,4.113,4.71585,0.842909),{2.643,4.113,19.8313,1.42338),{2.689,4.113,39.1664,1.97831),{2.735,4.113,56.486,2.36803),{2.781,4.113,63.0916,2.34244),{2.827,4.113,58.3352,2.18325), {2.873,4.113,55.799,2.27764),{2.919,4.113,49.0367,2.18374),{2.965,4.113,42.0463,2.16554),{3.011,4.113,39.8805,2.42539),{3.057,4.113,42.1758,2.82593),{3.103,4.113,43.3186,2.96548), {3.149,4.113,35.1955,2.50588),{3.195,4.113,38.8833,2.75193),{3.241,4.113,38.2756,2.69546),{3.287,4.113,40.3469,2.75788),{3.333,4.113,37.4419,2.39645),{3.379,4.113,40.1755,2.33869), {3.425,4.113,46.4974,2.31982),{3.471,4.113,51.8368,2.07773),{3.517,4.113,56.0568,2.1271),{3.563,4.113,64.6047,2.42611),{3.609,4.113,69.4012,2.53003),{3.655,4.113,69.8826,2.76126), {3.701,4.113,15.4626,1.46443),{2.551,4.175,4.25382,1.3963),{2.597,4.175,18.5564,1.59063),{2.643,4.175,35.2896,1.99804),{2.689,4.175,51.6835,2.32702),{2.735,4.175,66.1687,2.55854), {2.781,4.175,63.4544,2.41241),{2.827,4.175,60.3965,2.4999),{2.873,4.175,57.9965,2.87242),{2.919,4.175,50.5191,3.0099),{2.965,4.175,44.9843,3.15789),{3.011,4.175,42.1336,3.17901), {3.057,4.175,44.2907,3.59051),{3.103,4.175,38.967,3.06642),{3.149,4.175,39.6153,3.28028),{3.195,4.175,39.1764,3.26179),{3.241,4.175,40.6149,3.3274),{3.287,4.175,46.2392,3.60612), {3.333,4.175,45.237,3.246),{3.379,4.175,45.8973,2.91685),{3.425,4.175,51.4414,2.64798),{3.471,4.175,60.639,2.42905),{3.517,4.175,68.8115,2.66167),{3.563,4.175,72.4284,2.69462), {3.609,4.175,73.5339,2.79023),{3.655,4.175,38.3783,2.25999),{3.701,4.175,1.08112,0.661606),{2.505,4.237,0.402073,0.223792),{2.551,4.237,11.8316,1.26736),{2.597,4.237,24.5324,1.4951), {2.643,4.237,42.748,2.01764),{2.689,4.237,55.3675,2.22101),{2.735,4.237,68.8567,2.64515),{2.781,4.237,59.2493,2.51246),{2.827,4.237,57.9358,3.10789),{2.873,4.237,52.4683,3.59872), {2.919,4.237,47.7223,3.89709),{2.965,4.237,40.3251,3.56093),{3.011,4.237,40.4581,3.91241),{3.057,4.237,47.4946,4.90564),{3.103,4.237,46.24,4.87079),{3.149,4.237,45.7184,4.79325), {3.195,4.237,40.9939,4.02245),{3.241,4.237,48.0861,4.72831),{3.287,4.237,40.9423,3.65294),{3.333,4.237,47.0065,3.83271),{3.379,4.237,50.9387,3.44258),{3.425,4.237,55.1873,2.90565), {3.471,4.237,64.0341,2.48021),{3.517,4.237,71.4964,2.7224),{3.563,4.237,68.9222,2.47684),{3.609,4.237,46.0148,2.18838),{3.655,4.237,5.00599,0.792282),{2.505,4.299,4.46097,0.675549), {2.551,4.299,19.8465,1.44936),{2.597,4.299,35.5658,1.89779),{2.643,4.299,48.1858,2.14341),{2.689,4.299,61.1955,2.54002),{2.735,4.299,72.6967,3.4844),{2.781,4.299,64.7497,3.96514), {2.827,4.299,49.6674,3.91378),{2.873,4.299,46.1367,4.57907),{2.919,4.299,52.1594,6.15658),{2.965,4.299,47.6052,6.04418),{3.011,4.299,41.6091,5.53629),{3.057,4.299,31.6337,4.09214), {3.103,4.299,50.7322,7.33156),{3.149,4.299,43.4793,5.83978),{3.195,4.299,38.0384,4.62714),{3.241,4.299,53.1742,6.91472),{3.287,4.299,49.6991,5.59634),{3.333,4.299,44.6475,4.05995), {3.379,4.299,56.1519,4.2504),{3.425,4.299,57.2986,3.0796),{3.471,4.299,72.475,2.86059),{3.517,4.299,75.7439,2.86783),{3.563,4.299,51.0204,2.29531),{3.609,4.299,10.6896,1.0375), {2.459,4.361,0.601168,0.1865),{2.505,4.361,10.642,0.978843),{2.551,4.361,22.8818,1.34773),{2.597,4.361,43.1364,2.23558),{2.643,4.361,52.0761,2.50468),{2.689,4.361,64.3925,3.30985), {2.735,4.361,61.1183,4.14111),{2.781,4.361,50.4032,4.75403),{2.827,4.361,40.5252,4.96765),{2.873,4.361,40.04,5.96831),{2.919,4.361,49.5764,9.37772),{2.965,4.361,53.4931,10.5415), {3.011,4.361,30.1589,6.06161),{3.057,4.361,48.3943,9.98658),{3.103,4.361,36.9757,7.78824),{3.149,4.361,38.1644,7.22504),{3.195,4.361,39.1454,7.0769),{3.241,4.361,38.4315,6.36286), {3.287,4.361,41.2377,5.44834),{3.333,4.361,50.609,5.02549),{3.379,4.361,60.8605,4.81604),{3.425,4.361,63.7239,3.37104),{3.471,4.361,66.2881,2.53834),{3.517,4.361,52.2727,2.35753), {3.563,4.361,15.7113,1.2887),{3.609,4.361,0.309411,0.31468),{2.459,4.423,4.07119,0.596565),{2.505,4.423,16.5633,1.23075),{2.551,4.423,29.1227,1.77748),{2.597,4.423,43.9937,2.70432), {2.643,4.423,51.2921,3.27995),{2.689,4.423,63.8634,4.86576),{2.735,4.423,56.4557,5.62711),{2.781,4.423,38.6193,5.50876),{2.827,4.423,30.3689,5.35817),{2.873,4.423,31.2735,7.04847), {2.919,4.423,28.0368,7.62295),{2.965,4.423,26.4236,9.69854),{3.011,4.423,14.6128,4.52117),{3.057,4.423,22.8037,8.54623),{3.103,4.423,22.6749,7.58352),{3.149,4.423,39.2671,10.9438), {3.195,4.423,27.8679,6.53348),{3.241,4.423,28.5345,4.95333),{3.287,4.423,42.0843,5.50446),{3.333,4.423,56.2813,5.91429),{3.379,4.423,56.548,4.11414),{3.425,4.423,69.2695,3.66999), {3.471,4.423,45.0144,2.10419),{3.517,4.423,18.1297,1.3963),{3.563,4.423,1.30844,0.63738),{2.413,4.485,0.463358,0.199692),{2.459,4.485,7.97979,0.839153),{2.505,4.485,21.4915,1.69457), {2.551,4.485,31.8167,2.48306),{2.597,4.485,42.558,3.40464),{2.643,4.485,58.8061,5.19329),{2.689,4.485,55.9491,5.7302),{2.735,4.485,43.5742,5.76107),{2.781,4.485,33.4507,5.72106), {2.827,4.485,41.6297,9.51024),{2.873,4.485,29.3798,8.26221),{2.919,4.485,38.0493,13.3758),{2.965,4.485,27.9417,10.8859),{3.011,4.485,23.6044,10.176),{3.057,4.485,14.4564,5.72774), {3.103,4.485,43.9394,13.9957),{3.149,4.485,32.7033,8.25429),{3.195,4.485,36.4494,7.45209),{3.241,4.485,31.3777,4.80991),{3.287,4.485,50.5284,6.41199),{3.333,4.485,61.0926,5.50282), {3.379,4.485,61.4932,4.20196),{3.425,4.485,44.0103,2.70878),{3.471,4.485,16.0161,1.24978),{3.517,4.485,1.61627,0.585959),{2.413,4.547,2.4363,0.568296),{2.459,4.547,18.4005,2.28067), {2.505,4.547,22.9379,2.2597),{2.551,4.547,40.8317,4.37353),{2.597,4.547,36.9286,3.94591),{2.643,4.547,39.2364,4.46694),{2.689,4.547,45.8631,5.84876),{2.735,4.547,33.6438,5.17781), {2.781,4.547,39.1077,7.83832),{2.827,4.547,37.9885,10.45),{2.873,4.547,23.2044,7.36817),{2.919,4.547,15.1104,5.36088),{2.965,4.547,18.3443,7.82452),{3.011,4.547,39.6167,14.5645), {3.057,4.547,21.687,7.40696),{3.103,4.547,20.2717,5.99748),{3.149,4.547,48.7341,10.1322),{3.195,4.547,31.9706,5.18294),{3.241,4.547,41.4402,5.54246),{3.287,4.547,43.626,4.47743), {3.333,4.547,55.791,5.13602),{3.379,4.547,40.2302,3.31349),{3.425,4.547,19.4098,2.04582),{3.471,4.547,1.90457,0.720502),{2.413,4.609,6.30301,1.36675),{2.459,4.609,15.2214,2.04923), {2.505,4.609,21.7626,2.67584),{2.551,4.609,31.7661,4.12463),{2.597,4.609,38.7882,5.28434),{2.643,4.609,52.4904,7.97358),{2.689,4.609,33.6357,5.44371),{2.735,4.609,35.3312,6.5847), {2.781,4.609,37.973,8.83723),{2.827,4.609,29.9238,8.78161),{2.873,4.609,12.0821,4.28929),{2.919,4.609,67.6724,27.379),{2.965,4.609,75.1691,35.9211),{3.011,4.609,24.9811,7.49905), {3.057,4.609,19.4684,5.41544),{3.103,4.609,42.1141,9.51937),{3.149,4.609,49.5744,9.06433),{3.195,4.609,37.1929,5.40555),{3.241,4.609,50.406,6.6041),{3.287,4.609,39.2706,4.34676), {3.333,4.609,31.9268,3.36458),{3.379,4.609,15.1112,2.1041),{3.425,4.609,1.12973,0.461008),{2.367,4.671,0.633611,0.381235),{2.413,4.671,7.14765,1.56599),{2.459,4.671,13.9177,2.38597), {2.505,4.671,17.006,2.65554),{2.551,4.671,22.8918,3.7697),{2.597,4.671,20.7486,3.35375),{2.643,4.671,43.2814,7.85502),{2.689,4.671,26.3332,4.78274),{2.735,4.671,36.7716,7.77473), {2.781,4.671,36.1986,8.82383),{2.827,4.671,40.6846,12.2846),{2.873,4.671,25.0683,8.11145),{2.919,4.671,30.6655,8.63852),{2.965,4.671,28.9526,7.98011),{3.011,4.671,33.2095,9.45153), {3.057,4.671,33.9936,8.05818),{3.103,4.671,34.8928,6.27734),{3.149,4.671,40.538,6.60997),{3.195,4.671,26.0896,3.73444),{3.241,4.671,29.4438,4.31377),{3.287,4.671,19.196,2.56429), {3.333,4.671,11.8388,2.32734),{3.379,4.671,0.877673,0.640201),{2.367,4.733,1.4021,0.715946),{2.413,4.733,4.8321,1.11405),{2.459,4.733,7.62263,1.40086),{2.505,4.733,16.1741,3.02994), {2.551,4.733,17.0752,3.37875),{2.597,4.733,37.4981,7.90221),{2.643,4.733,23.0773,4.9682),{2.689,4.733,34.7287,8.5636),{2.735,4.733,26.9653,6.87024),{2.781,4.733,30.5296,9.15828), {2.827,4.733,42.3338,14.5582),{2.873,4.733,53.957,19.1325),{2.919,4.733,47.3451,16.1284),{2.965,4.733,37.6077,11.6539),{3.011,4.733,29.5043,7.24798),{3.057,4.733,33.9121,7.19882), {3.103,4.733,36.0949,7.01971),{3.149,4.733,23.3199,4.22773),{3.195,4.733,20.9588,3.64159),{3.241,4.733,10.9724,1.85421),{3.287,4.733,4.59442,1.08266),{3.333,4.733,0.477141,0.429059), {2.367,4.795,0.231953,0.18841),{2.413,4.795,4.01954,1.2731),{2.459,4.795,14.5235,3.89933),{2.505,4.795,14.2696,3.40708),{2.551,4.795,21.3805,5.50705),{2.597,4.795,18.3249,4.43829), {2.643,4.795,30.5313,8.2078),{2.689,4.795,32.5454,9.53823),{2.735,4.795,28.0636,9.89345),{2.781,4.795,35.1749,11.2322),{2.827,4.795,50.6965,21.9356),{2.873,4.795,32.9233,10.4416), {2.919,4.795,33.5074,11.8218),{2.965,4.795,34.7746,10.8712),{3.011,4.795,25.0079,6.29517),{3.057,4.795,33.7762,8.02497),{3.103,4.795,18.6091,4.11651),{3.149,4.795,21.4859,4.77062), {3.195,4.795,8.48632,2.15625),{3.241,4.795,3.08051,1.60066),{2.367,4.857,0.959558,0.72585),{2.413,4.857,7.31385,3.2277),{2.459,4.857,7.17975,2.17674),{2.505,4.857,13.0281,4.40499), {2.551,4.857,21.0534,6.70446),{2.597,4.857,28.9242,9.71405),{2.643,4.857,24.3098,8.42981),{2.689,4.857,15.3401,5.69077),{2.735,4.857,54.3945,25.2527),{2.781,4.857,19.1526,9.11692), {2.827,4.857,49.0496,24.1411),{2.873,4.857,17.7386,8.18861),{2.919,4.857,30.363,9.90057),{2.965,4.857,29.1988,9.2546),{3.011,4.857,32.8848,10.2226),{3.057,4.857,18.3278,5.83254), {3.103,4.857,13.4319,3.82878),{3.149,4.857,5.14593,1.89952),{3.195,4.857,1.49941,0.958287),{2.413,4.919,2.52784,1.33743),{2.459,4.919,8.60308,3.64227),{2.505,4.919,8.48279,2.86586), {2.551,4.919,11.7693,4.59508),{2.597,4.919,13.8249,5.14464),{2.643,4.919,7.62002,3.58178),{2.689,4.919,17.5545,8.32096),{2.735,4.919,22.8095,11.0562),{2.781,4.919,22.6861,10.1746), {2.827,4.919,13.3277,5.9915),{2.873,4.919,8.67166,3.81856),{2.919,4.919,14.8063,5.9107),{2.965,4.919,15.343,5.71295),{3.011,4.919,8.9813,2.93801),{3.057,4.919,12.6894,5.35604), {3.103,4.919,0.80873,0.446559),{2.413,4.981,1.77224,1.24784),{2.459,4.981,7.24842,3.8566),{2.505,4.981,9.2968,4.96044),{2.551,4.981,6.43142,3.19105),{2.597,4.981,40.4776,22.1099), {2.643,4.981,16.876,8.46348),{2.689,4.981,191.205,124.36),{2.735,4.981,260.624,168.304),{2.781,4.981,17.849,10.4009),{2.827,4.981,49.994,32.5627),{2.873,4.981,20.8781,9.78121), {2.919,4.981,6.85202,3.46595),{2.965,4.981,7.28471,3.73134),{3.011,4.981,5.07957,2.75352),{3.057,4.981,4.05249,3.43353),{2.413,5.043,0.352789,0.490017),{2.459,5.043,19.7416,15.4054), {2.505,5.043,3.95804,2.72192),{2.551,5.043,14.5895,10.2331),{2.597,5.043,14.2865,13.433),{2.643,5.043,25.3673,21.9232),{2.689,5.043,6.84269,6.14103),{2.735,5.043,5.22176,4.14115), {2.781,5.043,14.7335,13.6318),{2.827,5.043,9.68665,7.84269),{2.873,5.043,16.4074,14.398),{2.459,5.105,0.891613,0.989223),{2.505,5.105,1.98153,2.36992),{2.551,5.105,1.76259,2.35199), {2.597,5.105,14.3157,12.902),{2.643,5.105,13.253,16.0458),{2.735,5.105,1.73547,2.15932),{2.781,5.105,1.36469,1.83189),{2.827,5.105,16.1499,19.325),{2.873,5.105,0.342169,0.475249)) Table G.2: Acceptance and eﬃciency corrected Dalitz Plot ppX, for MMpp =0.6 − 0.7 GeV/c2 (Fig. 92). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of the absolutenormalization of 19% have to beincluded. Fullyexpandable version ofthe tableis availablein the attached electronic version of the thesis. Jagiellonian University 222 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Dalitz Plot ppX: MA{2)(p {1)p {2)) versus MA{2)(p {1,2)3pi0) -Each event is filled two times Acceptance and Efficiency Corrected for MM {pp)=0.7-0.8 GeV/cA{2) Error of MA{2)(p {1,2)3pi0): 0.0115 GeVA{2)/cA{4) Error of MA2(p {1)p {2)): 0.0155 GeVA{2)/cA{4) The data are in the following format: {{MA{2)(p {1,2)3pi0) [GeVA{2)/cA{4)], MA{2)(p {1)p {2)) [GeVA{2)/cA{4)], sigma [mub], Error of sigma [mub] ),...) {{3.5285,3.5085,0.252593,0.209134),{3.5745,3.5085,1.03778,1.15619),{3.5975,3.5085,0.126218,0.152197),{3.3445,3.5395,0.0411685,0.0237064),{3.3675,3.5395,1.32124,0.421478),{3.3905,3.5395,4.6884,1.04784),{3.4135,3.5395,9.61662,1.45562),{3.4365,3.5395,15.487,1.94094),{3.4595,3.5395,20.914,2.29714),{3.4825,3.5395,27.3881,3.09326), {3.5055,3.5395,30.8862,3.31281),{3.5285,3.5395,29.8593,3.14235),{3.5515,3.5395,35.902,3.95484),{3.5745,3.5395,30.9428,3.32454),{3.5975,3.5395,31.2906,3.3542),{3.6205,3.5395,23.3751,2.64219),{3.6435,3.5395,13.7311,1.7436),{3.6665,3.5395,5.70949,1.02307),{3.6895,3.5395,0.570463,0.239677),{3.2755,3.5705,0.451741,0.203402), {3.2985,3.5705,3.12184,0.652584),{3.3215,3.5705,6.403,0.946969),{3.3445,3.5705,13.5222,1.56144),{3.3675,3.5705,22.2067,2.0906),{3.3905,3.5705,28.7629,2.52667),{3.4135,3.5705,41.0806,3.33766),{3.4365,3.5705,41.3634,3.31759),{3.4595,3.5705,35.2164,2.82513),{3.4825,3.5705,38.2975,3.086),{3.5055,3.5705,40.7017,3.37456), {3.5285,3.5705,32.8401,2.58412),{3.5515,3.5705,40.8257,3.43324),{3.5745,3.5705,36.7673,2.91038),{3.5975,3.5705,39.2874,3.14648),{3.6205,3.5705,38.2196,3.08324),{3.6435,3.5705,35.3623,2.72492),{3.6665,3.5705,32.7048,2.64759),{3.6895,3.5705,21.8321,2.21628),{3.7125,3.5705,6.64431,1.21258),{3.7355,3.5705,0.580575,0.648683), {3.2065,3.6015,0.352032,0.37726),{3.2295,3.6015,2.35454,0.795028),{3.2525,3.6015,7.01694,1.20946),{3.2755,3.6015,20.3787,2.88698),{3.2985,3.6015,23.3532,2.53914),{3.3215,3.6015,34.4877,3.05981),{3.3445,3.6015,40.5282,3.28829),{3.3675,3.6015,41.3123,3.12279),{3.3905,3.6015,39.4156,2.97725),{3.4135,3.6015,36.5623,2.57115), {3.4365,3.6015,39.4856,2.91866),{3.4595,3.6015,38.8017,2.82227),{3.4825,3.6015,36.4486,2.60171),{3.5055,3.6015,36.9471,2.70797),{3.5285,3.6015,40.6068,2.98972),{3.5515,3.6015,41.5934,3.01397),{3.5745,3.6015,38.0231,2.73457),{3.5975,3.6015,38.2001,2.80531),{3.6205,3.6015,41.1762,3.09379),{3.6435,3.6015,46.673,3.60667), {3.6665,3.6015,46.222,3.60351),{3.6895,3.6015,40.3973,3.13532),{3.7125,3.6015,30.9024,2.72952),{3.7355,3.6015,9.75826,1.95323),{3.1605,3.6325,0.316342,0.248371),{3.1835,3.6325,4.37959,1.36282),{3.2065,3.6325,10.629,1.96052),{3.2295,3.6325,19.1693,2.51245),{3.2525,3.6325,24.1943,2.33631),{3.2755,3.6325,37.5547,3.25016), {3.2985,3.6325,39.3475,2.98401),{3.3215,3.6325,45.9985,3.42726),{3.3445,3.6325,44.0944,3.12316),{3.3675,3.6325,50.2333,3.71839),{3.3905,3.6325,46.1593,3.33445),{3.4135,3.6325,44.2074,3.05637),{3.4365,3.6325,40.4906,2.79033),{3.4595,3.6325,37.0869,2.5401),{3.4825,3.6325,44.2224,3.10327),{3.5055,3.6325,38.4898,2.64481), {3.5285,3.6325,44.3514,3.16759),{3.5515,3.6325,44.3778,3.06472),{3.5745,3.6325,41.6881,2.91977),{3.5975,3.6325,43.9785,3.12045),{3.6205,3.6325,44.6936,3.09373),{3.6435,3.6325,48.6665,3.49511),{3.6665,3.6325,48.6672,3.52618),{3.6895,3.6325,49.2213,3.61125),{3.7125,3.6325,41.8196,3.24893),{3.7355,3.6325,33.6519,3.46324), {3.7585,3.6325,2.45041,1.46215),{3.1375,3.6635,5.08424,1.99491),{3.1605,3.6635,12.5635,2.42748),{3.1835,3.6635,17.069,2.11122),{3.2065,3.6635,25.0282,2.5179),{3.2295,3.6635,40.7561,3.71369),{3.2525,3.6635,45.8184,3.66003),{3.2755,3.6635,58.5776,4.36018),{3.2985,3.6635,60.4469,4.401),{3.3215,3.6635,52.0032,3.58502), {3.3445,3.6635,44.7957,2.8891),{3.3675,3.6635,49.8981,3.41555),{3.3905,3.6635,41.5248,2.72488),{3.4135,3.6635,36.0502,2.34783),{3.4365,3.6635,43.8139,2.94716),{3.4595,3.6635,50.1832,3.54956),{3.4825,3.6635,43.1411,2.90652),{3.5055,3.6635,45.9575,3.11671),{3.5285,3.6635,40.3768,2.71492),{3.5515,3.6635,50.8437,3.3935), {3.5745,3.6635,45.5437,3.03363),{3.5975,3.6635,48.6942,3.30774),{3.6205,3.6635,52.7119,3.66168),{3.6435,3.6635,48.7462,3.3082),{3.6665,3.6635,52.3901,3.60615),{3.6895,3.6635,60.2405,4.30632),{3.7125,3.6635,54.887,4.01539),{3.7355,3.6635,45.7099,3.73328),{3.7585,3.6635,16.9489,5.29946),{3.0915,3.6945,5.27111,3.09325), {3.1145,3.6945,11.2831,2.80184),{3.1375,3.6945,20.3957,2.89762),{3.1605,3.6945,29.613,3.45974),{3.1835,3.6945,37.0971,3.34776),{3.2065,3.6945,53.9237,4.353),{3.2295,3.6945,51.843,3.59312),{3.2525,3.6945,60.2731,4.11492),{3.2755,3.6945,61.0564,4.14775),{3.2985,3.6945,53.291,3.46658),{3.3215,3.6945,46.6272,2.95202), {3.3445,3.6945,55.7152,3.73222),{3.3675,3.6945,55.6428,3.92175),{3.3905,3.6945,48.2982,3.2351),{3.4135,3.6945,47.2027,2.9892),{3.4365,3.6945,50.753,3.37454),{3.4595,3.6945,43.6394,2.75471),{3.4825,3.6945,46.566,3.06414),{3.5055,3.6945,51.144,3.44527),{3.5285,3.6945,46.608,3.03713),{3.5515,3.6945,51.9476,3.38804), {3.5745,3.6945,56.6867,3.78362),{3.5975,3.6945,58.6367,3.90482),{3.6205,3.6945,51.5289,3.34693),{3.6435,3.6945,51.3203,3.34978),{3.6665,3.6945,63.5641,4.3806),{3.6895,3.6945,58.1513,3.8327),{3.7125,3.6945,62.96,4.23883),{3.7355,3.6945,55.2691,4.03874),{3.7585,3.6945,59.6855,14.8524),{3.0685,3.7255,5.18304,1.63521), {3.0915,3.7255,13.8994,2.32203),{3.1145,3.7255,25.0548,2.96928),{3.1375,3.7255,34.0083,3.16964),{3.1605,3.7255,48.676,4.0266),{3.1835,3.7255,50.0663,3.56177),{3.2065,3.7255,66.1183,4.54787),{3.2295,3.7255,66.9216,4.35722),{3.2525,3.7255,65.1915,4.30134),{3.2755,3.7255,58.2359,3.71568),{3.2985,3.7255,55.2947,3.53856), {3.3215,3.7255,55.6423,3.49289),{3.3445,3.7255,52.2671,3.33051),{3.3675,3.7255,53.0028,3.51538),{3.3905,3.7255,46.9334,3.03508),{3.4135,3.7255,46.6471,2.94528),{3.4365,3.7255,49.1834,3.34934),{3.4595,3.7255,49.1123,3.21702),{3.4825,3.7255,44.3058,2.81752),{3.5055,3.7255,59.4091,3.92393),{3.5285,3.7255,48.3599,3.06873), {3.5515,3.7255,55.0875,3.60446),{3.5745,3.7255,55.8087,3.67601),{3.5975,3.7255,54.702,3.47772),{3.6205,3.7255,58.4731,3.84841),{3.6435,3.7255,64.9028,4.22133),{3.6665,3.7255,65.9064,4.18026),{3.6895,3.7255,60.0649,3.77238),{3.7125,3.7255,70.2574,4.49796),{3.7355,3.7255,59.8912,4.0058),{3.7585,3.7255,50.5751,10.5693), {3.0225,3.7565,6.70857,6.88315),{3.0455,3.7565,16.5657,4.49346),{3.0685,3.7565,20.5424,2.73416),{3.0915,3.7565,32.945,3.36828),{3.1145,3.7565,50.8764,4.69719),{3.1375,3.7565,60.5955,4.79037),{3.1605,3.7565,56.8225,3.73317),{3.1835,3.7565,71.1842,4.72615),{3.2065,3.7565,65.9633,4.16678),{3.2295,3.7565,67.5486,4.28495), {3.2525,3.7565,61.7699,3.95857),{3.2755,3.7565,51.6403,3.29254),{3.2985,3.7565,52.1013,3.28921),{3.3215,3.7565,48.4211,3.09168),{3.3445,3.7565,52.8458,3.4007),{3.3675,3.7565,55.4652,3.56517),{3.3905,3.7565,56.409,3.7269),{3.4135,3.7565,51.523,3.28696),{3.4365,3.7565,55.456,3.68818),{3.4595,3.7565,49.9743,3.14805), {3.4825,3.7565,56.1027,3.72041),{3.5055,3.7565,53.8251,3.38497),{3.5285,3.7565,51.7264,3.32015),{3.5515,3.7565,52.2169,3.43401),{3.5745,3.7565,59.9044,3.92131),{3.5975,3.7565,55.8378,3.45739),{3.6205,3.7565,66.7732,4.24674),{3.6435,3.7565,57.7903,3.54939),{3.6665,3.7565,70.0676,4.3874),{3.6895,3.7565,72.04,4.5954), {3.7125,3.7565,72.2166,4.52011),{3.7355,3.7565,75.3128,4.95042),{3.7585,3.7565,67.5965,12.8425),{2.9995,3.7875,6.20359,2.54419),{3.0225,3.7875,20.7079,3.88414),{3.0455,3.7875,30.4956,3.69455),{3.0685,3.7875,40.9625,4.02857),{3.0915,3.7875,45.7357,3.65426),{3.1145,3.7875,52.9385,3.76985),{3.1375,3.7875,59.3697,3.83206), {3.1605,3.7875,68.3407,4.3058),{3.1835,3.7875,67.3475,4.19058),{3.2065,3.7875,68.9886,4.33673),{3.2295,3.7875,64.2604,3.98831),{3.2525,3.7875,56.9523,3.66225),{3.2755,3.7875,56.7425,3.46647),{3.2985,3.7875,56.1055,3.45083),{3.3215,3.7875,57.5323,3.7857),{3.3445,3.7875,49.9303,3.23004),{3.3675,3.7875,49.8101,3.20714), {3.3905,3.7875,57.3654,3.86861),{3.4135,3.7875,49.2083,3.18668),{3.4365,3.7875,46.3923,3.02468),{3.4595,3.7875,47.7178,2.998),{3.4825,3.7875,48.7212,3.08359),{3.5055,3.7875,53.7233,3.42775),{3.5285,3.7875,59.7213,3.88576),{3.5515,3.7875,54.4394,3.36517),{3.5745,3.7875,56.1264,3.43189),{3.5975,3.7875,63.5505,3.90736), {3.6205,3.7875,67.3765,4.1349),{3.6435,3.7875,70.3545,4.47216),{3.6665,3.7875,65.6534,4.03128),{3.6895,3.7875,71.1476,4.37738),{3.7125,3.7875,76.7658,4.6511),{3.7355,3.7875,82.1366,5.32789),{3.7585,3.7875,58.9708,12.2579),{2.9535,3.8185,0.959846,0.748619),{2.9765,3.8185,7.71691,2.08614),{2.9995,3.8185,25.2087,3.82955), {3.0225,3.8185,33.0327,3.55538),{3.0455,3.8185,35.7361,3.0624),{3.0685,3.8185,45.8504,3.48668),{3.0915,3.8185,58.0947,3.97419),{3.1145,3.8185,70.2696,4.5394),{3.1375,3.8185,72.0263,4.46324),{3.1605,3.8185,72.6667,4.43539),{3.1835,3.8185,72.2716,4.61035),{3.2065,3.8185,69.2064,4.37645),{3.2295,3.8185,58.2662,3.54004), {3.2525,3.8185,60.3911,3.83317),{3.2755,3.8185,59.6402,3.8446),{3.2985,3.8185,56.8237,3.50534),{3.3215,3.8185,51.093,3.21137),{3.3445,3.8185,57.5979,3.73697),{3.3675,3.8185,52.038,3.28772),{3.3905,3.8185,50.2245,3.34356),{3.4135,3.8185,55.3139,3.73931),{3.4365,3.8185,53.3442,3.53807),{3.4595,3.8185,51.7,3.31732), {3.4825,3.8185,52.4728,3.14355),{3.5055,3.8185,60.2827,3.80608),{3.5285,3.8185,58.9752,3.68436),{3.5515,3.8185,67.6247,4.27237),{3.5745,3.8185,62.03,3.79615),{3.5975,3.8185,66.0173,4.11467),{3.6205,3.8185,69.8961,4.29838),{3.6435,3.8185,71.266,4.43969),{3.6665,3.8185,68.4056,4.06229),{3.6895,3.8185,76.0467,4.60517), {3.7125,3.8185,82.7392,4.9578),{3.7355,3.8185,87.8181,5.81884),{3.7585,3.8185,26.5526,5.35866),{2.9305,3.8495,4.98853,4.53759),{2.9535,3.8495,10.0959,2.31437),{2.9765,3.8495,24.2591,3.4081),{2.9995,3.8495,26.3094,2.619),{3.0225,3.8495,38.1409,3.25808),{3.0455,3.8495,57.0622,4.39167),{3.0685,3.8495,56.3637,3.8595), {3.0915,3.8495,63.1103,3.90532),{3.1145,3.8495,68.7279,4.06208),{3.1375,3.8495,76.227,4.67347),{3.1605,3.8495,65.6703,3.96483),{3.1835,3.8495,76.2626,5.00246),{3.2065,3.8495,64.1573,4.02435),{3.2295,3.8495,61.2416,3.86525),{3.2525,3.8495,60.9125,3.90203),{3.2755,3.8495,53.8203,3.43713),{3.2985,3.8495,55.9722,3.65683), {3.3215,3.8495,53.9701,3.53305),{3.3445,3.8495,47.5123,3.11646),{3.3675,3.8495,57.1527,3.88801),{3.3905,3.8495,47.9416,3.18968),{3.4135,3.8495,48.3426,3.21672),{3.4365,3.8495,46.6271,2.99501),{3.4595,3.8495,52.9691,3.2629),{3.4825,3.8495,66.2097,4.32035),{3.5055,3.8495,51.9984,3.22857),{3.5285,3.8495,67.1313,4.37613), 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{3.3215,4.4075,22.8043,4.78329),{3.3445,4.4075,43.3836,8.58222),{3.3675,4.4075,26.7597,4.93041),{3.3905,4.4075,27.4266,5.23048),{3.4135,4.4075,15.024,3.08781),{3.4365,4.4075,5.84887,2.02746),{2.7005,4.4385,2.41812,0.96913),{2.7235,4.4385,12.6025,3.77338),{2.7465,4.4385,14.9368,3.46314),{2.7695,4.4385,20.9234,4.96487), {2.7925,4.4385,26.0854,7.02046),{2.8155,4.4385,19.6616,4.56798),{2.8385,4.4385,29.6859,7.7462),{2.8615,4.4385,19.7972,5.80332),{2.8845,4.4385,27.0841,8.14585),{2.9075,4.4385,57.7773,21.7316),{2.9305,4.4385,22.2891,7.24943),{2.9535,4.4385,13.6088,5.13531),{2.9765,4.4385,11.2083,4.49922),{2.9995,4.4385,22.1838,8.73765), {3.0225,4.4385,22.6564,8.90612),{3.0455,4.4385,9.14883,4.26533),{3.0685,4.4385,16.8539,9.58254),{3.0915,4.4385,14.9579,5.61304),{3.1145,4.4385,15.8899,7.47748),{3.1375,4.4385,11.1659,4.63984),{3.1605,4.4385,17.676,7.13163),{3.1835,4.4385,32.5182,10.3598),{3.2065,4.4385,15.7649,6.27912),{3.2295,4.4385,24.6613,7.84096), {3.2525,4.4385,41.5004,12.5363),{3.2755,4.4385,24.554,6.35573),{3.2985,4.4385,24.8996,5.65369),{3.3215,4.4385,35.7318,7.98902),{3.3445,4.4385,42.1277,9.39245),{3.3675,4.4385,16.3102,3.34549),{3.3905,4.4385,9.45303,2.38478),{3.4135,4.4385,2.84885,1.82238),{2.6775,4.4695,0.0244971,0.0328611),{2.7005,4.4695,1.60955,0.623549), {2.7235,4.4695,7.00517,2.3093),{2.7465,4.4695,9.47948,2.8663),{2.7695,4.4695,13.2982,4.13414),{2.7925,4.4695,16.2672,4.24612),{2.8155,4.4695,18.2522,5.74312),{2.8385,4.4695,27.8156,8.56083),{2.8615,4.4695,14.4434,5.12802),{2.8845,4.4695,26.004,9.22048),{2.9075,4.4695,25.61,9.34063),{2.9305,4.4695,19.1905,7.65073), {2.9535,4.4695,20.1081,8.62418),{2.9765,4.4695,14.1279,7.30578),{2.9995,4.4695,16.029,7.71538),{3.0225,4.4695,4.6791,5.09202),{3.0455,4.4695,40.4512,20.4054),{3.0685,4.4695,17.9007,9.55587),{3.0915,4.4695,1.91524,1.98601),{3.1145,4.4695,14.9465,8.349),{3.1375,4.4695,34.2863,15.2344),{3.1605,4.4695,20.9056,8.1638), {3.1835,4.4695,27.6216,9.69343),{3.2065,4.4695,21.1872,8.01627),{3.2295,4.4695,15.0443,5.24877),{3.2525,4.4695,40.5882,12.4947),{3.2755,4.4695,23.3639,6.40745),{3.2985,4.4695,21.9675,6.14532),{3.3215,4.4695,15.3618,3.91777),{3.3445,4.4695,10.4694,2.69331),{3.3675,4.4695,4.90452,1.71771),{3.3905,4.4695,0.373861,0.288549), {2.7005,4.5005,2.92764,1.44828),{2.7235,4.5005,6.28112,2.2751),{2.7465,4.5005,13.4268,5.6958),{2.7695,4.5005,7.36715,2.61055),{2.7925,4.5005,10.1834,3.42491),{2.8155,4.5005,12.9968,5.03605),{2.8385,4.5005,13.5572,5.48834),{2.8615,4.5005,24.8571,9.25589),{2.8845,4.5005,16.028,5.75445),{2.9075,4.5005,39.927,16.6527), {2.9305,4.5005,15.8718,9.97874),{2.9535,4.5005,23.5947,13.9596),{2.9765,4.5005,7.07701,4.67294),{2.9995,4.5005,38.4279,20.8554),{3.0225,4.5005,29.0107,13.6813),{3.0455,4.5005,61.4232,35.1873),{3.0685,4.5005,26.7139,15.8796),{3.0915,4.5005,3.59324,3.84791),{3.1145,4.5005,27.4861,13.937),{3.1375,4.5005,21.9624,11.985), {3.1605,4.5005,37.7095,16.2269),{3.1835,4.5005,24.4066,8.81265),{3.2065,4.5005,19.5213,6.14527),{3.2295,4.5005,16.5635,6.20474),{3.2525,4.5005,15.6211,5.46904),{3.2755,4.5005,14.9146,4.61565),{3.2985,4.5005,12.5574,4.27639),{3.3215,4.5005,6.82749,2.27963),{3.3445,4.5005,4.23746,2.43937),{2.7005,4.5315,4.87018,5.03641), {2.7235,4.5315,4.29711,2.04125),{2.7465,4.5315,5.63609,2.4013),{2.7695,4.5315,5.24036,2.54337),{2.7925,4.5315,37.165,17.7566),{2.8155,4.5315,10.6446,5.09725),{2.8385,4.5315,17.4747,8.6531),{2.8615,4.5315,27.3173,12.5648),{2.8845,4.5315,22.8244,14.8386),{2.9075,4.5315,20.0014,8.19497),{2.9305,4.5315,20.8164,12.3465), {2.9535,4.5315,15.8975,9.41982),{2.9765,4.5315,25.2944,19.1192),{2.9995,4.5315,4.46531,3.49183),{3.0225,4.5315,6.47306,7.36256),{3.0455,4.5315,22.507,18.9574),{3.0685,4.5315,17.2556,9.49744),{3.0915,4.5315,14.1138,9.05897),{3.1145,4.5315,17.1535,8.93333),{3.1375,4.5315,17.3958,7.41199),{3.1605,4.5315,40.694,20.4497), {3.1835,4.5315,20.516,8.90539),{3.2065,4.5315,25.132,12.87),{3.2295,4.5315,25.0896,11.6015),{3.2525,4.5315,11.7114,4.51948),{3.2755,4.5315,4.90943,1.87396),{3.2985,4.5315,8.55661,5.12817),{2.7005,4.5625,0.714966,0.579084),{2.7235,4.5625,5.9248,3.84565),{2.7465,4.5625,7.27802,4.34633),{2.7695,4.5625,12.1609,7.0527), {2.7925,4.5625,5.93838,2.93276),{2.8155,4.5625,5.33385,2.94142),{2.8385,4.5625,3.80111,1.99654),{2.8615,4.5625,48.2898,35.9621),{2.8845,4.5625,8.99791,5.18115),{2.9075,4.5625,23.4524,13.5315),{2.9305,4.5625,10.6949,5.23856),{2.9535,4.5625,7.16189,4.96506),{2.9765,4.5625,3.45088,3.77064),{2.9995,4.5625,26.6719,20.428), {3.0225,4.5625,6.14402,4.99685),{3.0455,4.5625,15.1352,7.84356),{3.0685,4.5625,16.5153,10.4077),{3.0915,4.5625,11.8775,6.37248),{3.1145,4.5625,9.50349,4.5788),{3.1375,4.5625,29.76,16.912),{3.1605,4.5625,6.10132,2.84231),{3.1835,4.5625,5.76674,3.19099),{3.2065,4.5625,8.41547,3.73382),{3.2295,4.5625,10.0178,5.31473), {3.2525,4.5625,21.1036,18.0297),{3.2755,4.5625,1.96175,1.35328),{2.7005,4.5935,0.767939,1.07268),{2.7235,4.5935,6.06901,5.20404),{2.7465,4.5935,4.06202,2.99778),{2.7695,4.5935,5.46024,3.93031),{2.7925,4.5935,8.40456,5.24129),{2.8155,4.5935,5.88343,3.39854),{2.8385,4.5935,24.7732,20.5306),{2.8615,4.5935,11.69,6.69719), {2.8845,4.5935,11.7583,7.42288),{2.9075,4.5935,17.6922,15.5862),{2.9305,4.5935,20.1331,20.8873),{2.9535,4.5935,5.51026,4.65436),{2.9765,4.5935,2.6608,2.91903),{2.9995,4.5935,5.03454,5.82767),{3.0225,4.5935,42.8772,42.7074),{3.0455,4.5935,6.30216,5.26363),{3.0685,4.5935,3.33352,3.73057),{3.0915,4.5935,29.9575,21.1037), {3.1145,4.5935,26.644,15.5511),{3.1375,4.5935,5.4808,3.89481),{3.1605,4.5935,5.05576,2.7986),{3.1835,4.5935,5.89832,3.9673),{3.2065,4.5935,8.56513,8.95124),{3.2295,4.5935,10.4529,7.88071),{3.2525,4.5935,1.35206,1.88437),{2.7235,4.6245,1.0294,1.22875),{2.7465,4.6245,8.38945,8.13946),{2.7695,4.6245,1.93663,1.68447), {2.7925,4.6245,2.23174,1.86074),{2.8155,4.6245,7.03073,5.16623),{2.8385,4.6245,32.4076,33.9495),{2.8615,4.6245,11.6605,9.02757),{2.8845,4.6245,8.62129,5.86396),{2.9075,4.6245,8.97515,6.95946),{2.9305,4.6245,11.2248,9.4565),{2.9995,4.6245,14.7153,11.3263),{3.0225,4.6245,11.7878,9.4118),{3.0455,4.6245,5.24552,4.50767), {3.0685,4.6245,13.5841,9.01595),{3.0915,4.6245,9.7302,6.06872),{3.1145,4.6245,3.95476,3.51116),{3.1375,4.6245,5.38912,4.37042),{3.1605,4.6245,2.58336,1.91989),{3.1835,4.6245,59.5019,63.3231),{3.2065,4.6245,0.315396,0.437101),{2.7695,4.6555,5.05877,5.80446),{2.7925,4.6555,2.97026,2.85306),{2.8155,4.6555,7.75419,8.65822), {2.8385,4.6555,7.72243,6.67371),{2.8615,4.6555,8.70859,9.92836),{2.9075,4.6555,50.4862,70.1687),{2.9305,4.6555,27.1678,32.9366),{2.9535,4.6555,87.418,78.34),{2.9765,4.6555,51.9313,72.177),{3.0225,4.6555,3.38004,4.16807),{3.0455,4.6555,52.7498,51.9216),{3.0685,4.6555,10.7657,9.10936),{3.0915,4.6555,1.72522,2.02837), {3.1145,4.6555,4.70317,5.22278),{2.7925,4.6865,1.7133,2.38),{2.8155,4.6865,0.768124,0.857873),{2.8845,4.6865,4.03888,5.61188),{2.9075,4.6865,37.9591,52.7602),{2.9535,4.6865,3.73804,4.96495),{2.9995,4.6865,3.70655,5.14997),{3.0225,4.6865,1.45575,1.6875),{3.0685,4.6865,1.69299,2.35173)) Table G.3: Acceptance and eﬃciency corrected Dalitz Plot ppX, for MMpp =0.7 − 0.8 GeV/c2 (Fig. 92). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of theabsolutenormalization of 19% have to beincluded. Fully expandable version of thetableis availablein the attached electronic version of the thesis. Jagiellonian University 223 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Dalitz Plot ppX: MA{2)(p {1)p {2)) versus MA{2)(p {1,2)3pi0) -Each event is filled two times Acceptance and Efficiency Corrected for MM {pp)=0.8-0.9 GeV/cA{2) Error of MA{2)(p {1,2)3pi0): 0.0092 GeVA{2)/cA{4) Error of MA2(p {1)p {2)): 0.0124 GeVA{2)/cA{4) The data are in the following format: {{MA{2)(p {1,2)3pi0) [GeVA{2)/cA{4)], MA{2)(p {1)p {2)) [GeVA{2)/cA{4)], sigma [mub], Error of sigma [mub] ),...) {{3.6044,3.5116,0.541648,0.655714),{3.6412,3.5116,0.596555,0.683178),{3.678,3.5116,0.281725,0.339916),{3.4572,3.5364,0.145345,0.140382),{3.4756,3.5364,1.0515,0.398442),{3.494,3.5364,3.11392,0.826368), {3.5124,3.5364,10.7203,2.21179),{3.5308,3.5364,12.3399,2.15593),{3.5492,3.5364,24.8505,3.72135),{3.5676,3.5364,27.6237,3.81123),{3.586,3.5364,38.3653,5.04571),{3.6044,3.5364,38.2992,5.04538), {3.6228,3.5364,42.2633,5.30799),{3.6412,3.5364,46.5501,6.08295),{3.6596,3.5364,31.6086,3.94031),{3.678,3.5364,29.125,3.82414),{3.6964,3.5364,22.541,3.41809),{3.7148,3.5364,6.98936,1.63808), {3.7332,3.5364,0.713549,0.577758),{3.402,3.5612,0.179433,0.12123),{3.4204,3.5612,2.60262,0.949159),{3.4388,3.5612,5.95235,1.19979),{3.4572,3.5612,15.4347,2.58025),{3.4756,3.5612,16.0202,2.02926), {3.494,3.5612,26.3014,2.73451),{3.5124,3.5612,41.6138,4.27907),{3.5308,3.5612,43.697,4.0533),{3.5492,3.5612,47.6182,4.18813),{3.5676,3.5612,45.9377,3.95962),{3.586,3.5612,49.7589,4.37177),{3.6044,3.5612,48.0736,4.13747), {3.6228,3.5612,49.8813,4.38438),{3.6412,3.5612,60.7416,5.41131),{3.6596,3.5612,49.9747,4.27317),{3.678,3.5612,47.2564,3.91117),{3.6964,3.5612,54.2294,4.70592),{3.7148,3.5612,34.6426,3.29021),{3.7332,3.5612,19.118,2.61881), 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{3.0892,4.1564,13.7151,2.71517),{3.1076,4.1564,14.6308,3.26162),{3.126,4.1564,18.2105,4.12295),{3.1444,4.1564,19.2336,4.21815),{3.1628,4.1564,18.2778,4.72072),{3.1812,4.1564,25.2151,7.19435),{3.1996,4.1564,23.265,7.28325), {3.218,4.1564,16.1006,5.01785),{3.2364,4.1564,14.2855,4.46278),{3.2548,4.1564,25.808,7.81172),{3.2732,4.1564,31.2789,11.6688),{3.2916,4.1564,20.7115,7.41865),{3.31,4.1564,24.7443,7.3544),{3.3284,4.1564,12.2214,4.09904), {3.3468,4.1564,39.5594,12.0329),{3.3652,4.1564,25.565,6.69811),{3.3836,4.1564,13.7442,3.95946),{3.402,4.1564,17.7125,3.83353),{3.4204,4.1564,28.6056,5.9236),{3.4388,4.1564,17.3026,3.69543),{3.4572,4.1564,17.2206,2.99953), {3.4756,4.1564,12.1266,2.34581),{3.494,4.1564,8.86166,2.25885),{3.5124,4.1564,1.69684,1.37674),{3.034,4.1812,1.21504,0.640146),{3.0524,4.1812,5.54975,1.78701),{3.0708,4.1812,7.35031,1.92793), {3.0892,4.1812,15.5914,4.53038),{3.1076,4.1812,13.3691,3.67478),{3.126,4.1812,38.946,12.2376),{3.1444,4.1812,18.4208,5.38207),{3.1628,4.1812,22.1486,6.91151),{3.1812,4.1812,15.6299,5.79175),{3.1996,4.1812,37.8517,18.2315), {3.218,4.1812,20.792,7.83949),{3.2364,4.1812,12.7001,4.29214),{3.2548,4.1812,11.4643,4.44256),{3.2732,4.1812,18.2489,6.54639),{3.2916,4.1812,16.1179,5.49495),{3.31,4.1812,19.6857,8.84526),{3.3284,4.1812,20.3001,7.25277), {3.3468,4.1812,22.6281,7.95405),{3.3652,4.1812,32.4795,10.6271),{3.3836,4.1812,24.2847,6.22168),{3.402,4.1812,26.985,7.79272),{3.4204,4.1812,22.657,5.59509),{3.4388,4.1812,7.91989,1.96937),{3.4572,4.1812,10.6854,3.05768), {3.4756,4.1812,2.71919,1.2359),{3.034,4.206,0.816142,0.616354),{3.0524,4.206,2.06616,0.91632),{3.0708,4.206,5.23513,1.95759),{3.0892,4.206,8.26661,2.76115),{3.1076,4.206,14.9823,5.81256),{3.126,4.206,9.68809,3.43061), {3.1444,4.206,13.165,5.5148),{3.1628,4.206,14.7655,5.77858),{3.1812,4.206,14.2607,6.46316),{3.1996,4.206,9.66212,4.55877),{3.218,4.206,14.2166,6.26859),{3.2364,4.206,32.8381,17.0796),{3.2548,4.206,7.87469,4.11545), {3.2732,4.206,14.028,6.39904),{3.2916,4.206,14.3712,6.40101),{3.31,4.206,18.1137,7.68866),{3.3284,4.206,6.89773,3.63304),{3.3468,4.206,18.8365,6.9313),{3.3652,4.206,12.0486,4.23701),{3.3836,4.206,20.7724,7.56222), {3.402,4.206,11.0418,3.48477),{3.4204,4.206,7.47736,2.55613),{3.4388,4.206,2.02392,0.944024),{3.4572,4.206,0.364887,0.349027),{3.0524,4.2308,2.26344,1.48567),{3.0708,4.2308,9.3843,6.56842),{3.0892,4.2308,9.0859,3.98671), {3.1076,4.2308,10.9767,4.59996),{3.126,4.2308,11.6408,6.02872),{3.1444,4.2308,20.3358,9.13652),{3.1628,4.2308,8.27375,5.35175),{3.1812,4.2308,4.91122,2.93674),{3.1996,4.2308,17.9302,10.8106),{3.218,4.2308,5.09078,2.99451), {3.2364,4.2308,15.345,10.4242),{3.2548,4.2308,2.72816,2.11165),{3.2732,4.2308,23.5394,15.3062),{3.2916,4.2308,9.53481,4.79664),{3.31,4.2308,8.4258,4.81307),{3.3284,4.2308,11.3536,4.97114),{3.3468,4.2308,28.4976,14.4999), {3.3652,4.2308,11.8869,4.24955),{3.3836,4.2308,7.09535,3.74228),{3.4204,4.2308,0.959876,0.779587),{3.0708,4.2556,5.50331,5.97248),{3.0892,4.2556,1.488,1.03081),{3.1076,4.2556,5.52501,3.17668), {3.126,4.2556,3.45685,1.91136),{3.1444,4.2556,2.50333,2.02746),{3.1628,4.2556,5.27362,3.08059),{3.1812,4.2556,10.6615,5.98882),{3.1996,4.2556,14.3533,9.60369),{3.218,4.2556,2.93211,1.93199),{3.2364,4.2556,2.52741,1.99616), {3.2548,4.2556,12.0129,6.66796),{3.2732,4.2556,7.12498,4.3569),{3.2916,4.2556,5.39021,3.09412),{3.31,4.2556,4.49606,2.83741),{3.3284,4.2556,2.06684,1.35792),{3.3468,4.2556,4.33655,2.335),{3.3652,4.2556,5.57506,6.263), {3.126,4.2804,3.36102,4.6692),{3.1444,4.2804,28.1174,21.208),{3.1628,4.2804,4.57747,3.50608),{3.1812,4.2804,1.30385,1.43217),{3.1996,4.2804,7.76386,8.82662),{3.218,4.2804,14.8058,15.2554),{3.2364,4.2804,0.991845,1.06937), {3.2548,4.2804,4.45939,3.98615),{3.2732,4.2804,11.7601,12.4348),{3.31,4.2804,0.712409,0.808439),{3.126,4.3052,0.794005,1.10194),{3.1812,4.3052,2.15931,2.69578),{3.1996,4.3052,17.5949,24.4622), {3.218,4.3052,4.3079,4.71797)) Table G.4: Acceptance and eﬃciency corrected Dalitz Plot ppX, for MMpp =0.8 − 0.9 GeV/c2 (Fig. 92). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of the absolutenormalization of 19% have to beincluded. Fullyexpandable version ofthe tableis availablein the attached electronic version of the thesis. Jagiellonian University 224 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Dalitz Plot ppX: MA{2)(p {1)p {2)) versus MA{2)(p {1,2)3pi0) -Each event is filled two times Acceptance and Efficiency Corrected for MM {pp)=0.9-1.0 GeV/cA{2) Error of MA{2)(p {1,2)3pi0): 0.0092 GeVA{2)/cA{4) Error of MA2(p {1)p {2)): 0.0124 GeVA{2)/cA{4) The data are in the following format: {{MA{2)(p {1,2)3pi0) [GeVA{2)/cA{4)], MA{2)(p {1)p {2)) [GeVA{2)/cA{4)], sigma [mub], Error of sigma [mub] ),...) {{3.6964,3.5116,1.0775,1.28044),{3.7148,3.5116,0.425199,0.46287),{3.586,3.5364,1.088,0.581925),{3.6044,3.5364,4.10825,1.31608),{3.6228,3.5364,10.1694,2.42972), {3.6412,3.5364,13.8889,2.66161),{3.6596,3.5364,28.3338,4.75683),{3.678,3.5364,25.7707,4.10794),{3.6964,3.5364,44.0459,6.2),{3.7148,3.5364,39.2999,5.49888), {3.7332,3.5364,48.9846,7.02437),{3.7516,3.5364,11.1061,3.60382),{3.5492,3.5612,4.09424,2.95001),{3.5676,3.5612,8.52081,2.54972),{3.586,3.5612,18.7441,3.45899), {3.6044,3.5612,18.0005,2.55105),{3.6228,3.5612,22.9513,2.95998),{3.6412,3.5612,37.8487,4.12865),{3.6596,3.5612,45.1759,4.59595),{3.678,3.5612,52.5015,4.93813), {3.6964,3.5612,54.5681,4.7755),{3.7148,3.5612,54.5423,4.78838),{3.7332,3.5612,57.2934,5.30767),{3.7516,3.5612,44.4538,7.18247),{3.5124,3.586,2.17021,1.89152), {3.5308,3.586,8.56278,2.35432),{3.5492,3.586,12.9857,2.44847),{3.5676,3.586,20.5402,2.83451),{3.586,3.586,29.842,3.41664),{3.6044,3.586,39.1989,4.01342), {3.6228,3.586,48.0789,4.35772),{3.6412,3.586,47.3077,4.06359),{3.6596,3.586,47.2198,3.77053),{3.678,3.586,53.3684,4.01019),{3.6964,3.586,60.6264,4.48387), {3.7148,3.586,62.7895,4.59452),{3.7332,3.586,53.0228,4.19947),{3.7516,3.586,55.3409,7.04509),{3.494,3.6108,3.67921,2.94916),{3.5124,3.6108,11.5252,2.44566), {3.5308,3.6108,19.8939,3.17339),{3.5492,3.6108,29.3319,3.58173),{3.5676,3.6108,36.8485,3.85317),{3.586,3.6108,40.3689,3.51075),{3.6044,3.6108,45.8728,3.77083), {3.6228,3.6108,48.8176,3.80095),{3.6412,3.6108,49.096,3.65526),{3.6596,3.6108,53.5801,3.78733),{3.678,3.6108,64.8683,4.59833),{3.6964,3.6108,66.3661,4.58204), {3.7148,3.6108,66.9233,4.63416),{3.7332,3.6108,52.61,4.08817),{3.7516,3.6108,63.3909,9.48434),{3.4572,3.6356,21.8167,24.5824),{3.4756,3.6356,8.019,2.64738), {3.494,3.6356,12.3298,2.02029),{3.5124,3.6356,17.9398,2.29273),{3.5308,3.6356,28.7945,2.95722),{3.5492,3.6356,32.5416,2.99692),{3.5676,3.6356,40.1561,3.3603), {3.586,3.6356,46.5011,3.61828),{3.6044,3.6356,47.3817,3.5049),{3.6228,3.6356,59.9733,4.30498),{3.6412,3.6356,60.8496,4.16414),{3.6596,3.6356,56.762,3.9139), {3.678,3.6356,58.256,3.89411),{3.6964,3.6356,58.7354,3.93614),{3.7148,3.6356,60.8809,4.20389),{3.7332,3.6356,42.9737,3.48417),{3.7516,3.6356,38.0801,5.91269), {3.4388,3.6604,1.56585,1.33792),{3.4572,3.6604,6.54396,1.70104),{3.4756,3.6604,15.3001,2.43577),{3.494,3.6604,21.0138,2.45588),{3.5124,3.6604,26.7118,2.69277), {3.5308,3.6604,29.7678,2.70924),{3.5492,3.6604,31.8194,2.70862),{3.5676,3.6604,41.7181,3.43892),{3.586,3.6604,46.0584,3.43123),{3.6044,3.6604,56.0223,4.13854), {3.6228,3.6604,48.254,3.42812),{3.6412,3.6604,56.2067,3.85777),{3.6596,3.6604,59.2985,4.10662),{3.678,3.6604,55.3039,3.84596),{3.6964,3.6604,55.0392,4.03299), {3.7148,3.6604,44.0797,3.42391),{3.7332,3.6604,41.2827,3.78728),{3.7516,3.6604,15.0369,3.02156),{3.4204,3.6852,0.894607,0.491971),{3.4388,3.6852,4.55871,1.02497), {3.4572,3.6852,10.142,1.62238),{3.4756,3.6852,21.2014,2.42236),{3.494,3.6852,31.5513,3.34651),{3.5124,3.6852,32.7149,3.23813),{3.5308,3.6852,28.8195,2.55733), {3.5492,3.6852,33.7302,2.74491),{3.5676,3.6852,42.7144,3.19144),{3.586,3.6852,41.8897,3.15508),{3.6044,3.6852,45.4207,3.38755),{3.6228,3.6852,48.1656,3.43189), {3.6412,3.6852,51.1071,3.57869),{3.6596,3.6852,45.4037,3.35283),{3.678,3.6852,49.799,3.60203),{3.6964,3.6852,45.837,3.57627),{3.7148,3.6852,39.8847,3.53836), {3.7332,3.6852,22.3566,2.61245),{3.7516,3.6852,3.31321,1.179),{3.402,3.71,0.160139,0.166911),{3.4204,3.71,3.81797,0.863741),{3.4388,3.71,8.63425,1.35253), {3.4572,3.71,16.9862,2.223),{3.4756,3.71,21.1661,2.19113),{3.494,3.71,22.3635,2.21029),{3.5124,3.71,35.5557,3.26246),{3.5308,3.71,31.906,2.74521), {3.5492,3.71,39.5898,3.22534),{3.5676,3.71,41.5412,3.29632),{3.586,3.71,54,4.26425),{3.6044,3.71,48.4893,3.56644),{3.6228,3.71,49.4734,3.73524), {3.6412,3.71,44.8552,3.42158),{3.6596,3.71,41.7322,3.30434),{3.678,3.71,38.8343,3.09577),{3.6964,3.71,33.1182,2.93646),{3.7148,3.71,21.8157,2.22913), {3.7332,3.71,9.88459,2.09701),{3.402,3.7348,3.09035,1.44484),{3.4204,3.7348,6.6052,1.22141),{3.4388,3.7348,11.8983,1.59969),{3.4572,3.7348,22.6036,2.78528), {3.4756,3.7348,24.5573,2.60234),{3.494,3.7348,30.3698,2.9251),{3.5124,3.7348,32.7103,2.99359),{3.5308,3.7348,37.0517,3.0659),{3.5492,3.7348,37.9202,3.1523), {3.5676,3.7348,37.4015,2.99926),{3.586,3.7348,40.782,3.12576),{3.6044,3.7348,40.2155,3.31869),{3.6228,3.7348,39.7779,3.17161),{3.6412,3.7348,46.0783,3.67756), {3.6596,3.7348,38.2724,3.25016),{3.678,3.7348,33.2902,3.12109),{3.6964,3.7348,24.3341,2.68516),{3.7148,3.7348,11.7111,2.03705),{3.7332,3.7348,0.853918,0.590315), {3.3836,3.7596,0.161222,0.113296),{3.402,3.7596,5.39269,1.41308),{3.4204,3.7596,8.40068,1.55422),{3.4388,3.7596,15.453,2.10565),{3.4572,3.7596,23.2474,2.80325), {3.4756,3.7596,22.156,2.24853),{3.494,3.7596,27.7043,2.79803),{3.5124,3.7596,29.3648,2.71917),{3.5308,3.7596,27.1587,2.39816),{3.5492,3.7596,32.4354,2.72203), {3.5676,3.7596,33.4371,2.86698),{3.586,3.7596,37.191,3.12856),{3.6044,3.7596,36.025,3.05069),{3.6228,3.7596,32.2491,2.78948),{3.6412,3.7596,30.6365,2.82116), {3.6596,3.7596,28.1558,2.80294),{3.678,3.7596,21.9471,2.72436),{3.6964,3.7596,13.5639,2.44125),{3.7148,3.7596,5.1907,5.08971),{3.3836,3.7844,1.00224,0.49848), {3.402,3.7844,6.49195,1.53528),{3.4204,3.7844,10.2974,1.67567),{3.4388,3.7844,20.4549,2.93153),{3.4572,3.7844,24.3366,2.88273),{3.4756,3.7844,24.4117,2.61193), {3.494,3.7844,27.6985,2.82562),{3.5124,3.7844,26.7612,2.4979),{3.5308,3.7844,29.4337,2.78451),{3.5492,3.7844,30.9272,2.88552),{3.5676,3.7844,34.9994,3.1298), {3.586,3.7844,29.3023,2.78373),{3.6044,3.7844,28.7854,2.65615),{3.6228,3.7844,31.9494,3.29711),{3.6412,3.7844,34.4153,3.87953),{3.6596,3.7844,17.986,2.42686), {3.678,3.7844,10.4044,2.08388),{3.6964,3.7844,2.03006,1.41357),{3.3836,3.8092,0.806857,0.354865),{3.402,3.8092,5.15213,1.14316),{3.4204,3.8092,9.84351,1.80258), {3.4388,3.8092,12.8243,1.85638),{3.4572,3.8092,11.2312,1.37833),{3.4756,3.8092,23.3284,2.70778),{3.494,3.8092,19.7567,2.12815),{3.5124,3.8092,24.0041,2.44644), {3.5308,3.8092,24.4478,2.52008),{3.5492,3.8092,30.0236,2.97719),{3.5676,3.8092,24.0834,2.51753),{3.586,3.8092,21.4622,2.30775),{3.6044,3.8092,18.4112,2.05086), {3.6228,3.8092,15.6093,2.08309),{3.6412,3.8092,11.1373,1.78823),{3.6596,3.8092,6.47241,1.54199),{3.678,3.8092,11.0496,12.4655),{3.3836,3.834,0.973906,0.583844), {3.402,3.834,7.22493,1.79797),{3.4204,3.834,6.87385,1.43348),{3.4388,3.834,11.9947,1.77737),{3.4572,3.834,13.4002,1.76851),{3.4756,3.834,17.5411,2.21993), {3.494,3.834,16.0445,1.94794),{3.5124,3.834,18.1786,1.98188),{3.5308,3.834,17.5137,2.01631),{3.5492,3.834,18.2053,2.12584),{3.5676,3.834,17.8764,2.13284), {3.586,3.834,16.4922,2.195),{3.6044,3.834,11.3769,1.7231),{3.6228,3.834,11.0186,2.36772),{3.6412,3.834,3.75586,1.86658),{3.3836,3.8588,0.0579833,0.0692122), {3.402,3.8588,2.56175,0.834902),{3.4204,3.8588,5.33095,1.33137),{3.4388,3.8588,8.92011,1.55534),{3.4572,3.8588,8.38127,1.23217),{3.4756,3.8588,15.0434,2.09252), {3.494,3.8588,17.0784,2.29198),{3.5124,3.8588,15.6666,2.25888),{3.5308,3.8588,13.9856,1.88049),{3.5492,3.8588,12.8281,1.77336),{3.5676,3.8588,12.1413,2.05577), {3.586,3.8588,4.91756,1.13806),{3.6044,3.8588,3.72473,1.31289),{3.6228,3.8588,0.442436,0.613198),{3.402,3.8836,0.431228,0.388931),{3.4204,3.8836,7.34865,2.4861), {3.4388,3.8836,4.09088,1.04183),{3.4572,3.8836,5.86519,1.1768),{3.4756,3.8836,4.54023,0.889984),{3.494,3.8836,7.87931,1.42473),{3.5124,3.8836,5.65284,1.01553), {3.5308,3.8836,5.54509,1.22505),{3.5492,3.8836,4.79992,1.25107),{3.5676,3.8836,7.20018,3.4687),{3.4388,3.9084,0.809838,0.470609),{3.4572,3.9084,1.03597,0.430993), {3.4756,3.9084,1.15242,0.518849),{3.494,3.9084,1.04034,0.397789),{3.5124,3.9084,1.21577,0.620887)) Table G.5: Acceptance and eﬃciency corrected Dalitz Plot ppX, for MMpp =0.9 − 1.0 GeV/c2 (Fig. 92). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of theabsolutenormalization of 19% have to beincluded. Fully expandable version of thetableis availablein the attached electronic version of the thesis. Jagiellonian University 225 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Da tzMot 3p0 i'{0}_{1}pi'{0}_{2}) i'{0}_{2}pi'{0}_{3}) - Eac. is l ed sx imes i iM'{2}(pversis M'{2}(peventiit Acceptance and Ellicienc Corrected l4-0. or MM_{pp}00.5 Ge1/c'{2} Error ol M'{2}(pi'{0}_{2}pi'{0}_{3}) 0.00152 Ge1'{2}/c'{4} Error ol M'{2}(pi'{0}_{1}pi'{0}_{2}) 0.00152 Ge1'{2}/c'{4} T.e data are in t lo o�ng l {{M'{2}(pi'{0}_{2}p'{0}_{3}) [Ge1'{2}/c'{4}] M'{2}(pi'{0}_{1}p'{0}_{2}) [Ge1'{2}/c'{4}], sgma [mil] Error o sgma [mil] }... .eiormat i,ii,li,} {{0.07496,0.07192,2538.01,1793.71},{0.078,0.07192,1419.37,461.649},{0.08104,0.07192,3334.68,1235.94},{0.08408,0.07192,1924.84,598.086},{0.08712,0.07192,1404.78,394.034},{0.09016,0.07192,3380.76,1203.18},{0.0932,0.07192,3913.58,1303.82}, {0.,0.07192,3592.63,1266.19},{0.,2207.08630.63},{0.10232,0.07192,2474.93,640.239},{0.10536,0.07192,279921,740.962},{0.1084,0.07192,2128.36,539.397},{0.11144,0.07192,2385.13,712.151},{0.11448,0.07192,4902.7,2810.81}, 0962409928,0.07192,.{0.11752,0.07192,2265.88,1816.41},{0.07192,0.07496,2538.01,1793.71},{0.07496,0.07496,21280.3,2488.76},{0.078,0.07496,28604.2,2881.52},{0.08104,0.07496,28583.9,2433.64},{0.08408,0.07496,31021.9,2594.86},{0.08712,0.07496,30800.1,2514.06}, {0.09016,0.07496,30274.6,2389.32},{0.0932,0.07496,35813,3028.89},{0.09624,0.07496,29562.9,2316.16},{0.09928,0.07496,33850.6,2664.14},{0.10232,0.07496,28609,2172.54},{0.10536,0.07496,30215.1,2529.08},{0.1084,0.07496,28114.3,2523.19}, {0.11144,0.07496,19829.4,1889.25},{0.11448,0.07496,21276.6,2591.24},{0.11752,0.07496,15433.2,2192.83},{0.12056,0.07496,6525.41,963.023},{0.1236,0.07496,3337.06,772.815},{0.12664,0.07496,810.584,334.516},{0.07192,0.078,1419.37,461.649}, {0.07496,0.078,28604.2,2881.52},{0.078,0.078,40868.4,3395.18},{0.08104,0.078,41139.9,2976.78},{0.08408,0.078,47782.1,3437.36},{0.08712,0.078,49295.2,3415.36},{0.09016,0.078,43140.3,2807.55},{0.0932,0.078,44007.8,2823.18}, {0.09624,0.078,44955.3,2859.78},{0.09928,0.078,40392.4,2699.6},{0.10232,0.078,35845.1,2529.94},{0.10536,0.078,30587.9,2287.11},{0.1084,0.078,30873.7,2664.7},{0.11144,0.078,20417.2,1743.17},{0.11448,0.078,18980.7,1889.2}, {0.11752,0.078,13999.4,1464.99},{0.12056,0.078,9163.9,1048.6},{0.1236,0.078,11527.3,2025.61},{0.12664,0.078,3046.15,526.386},{0.12968,0.078,1258.71,490.212},{0.07192,0.08104,3334.68,1235.94},{0.07496,0.08104,28583.9,2433.64}, {0.078,0.08104,41139.9,2976.78},{0.08104,0.08104,44614.3,2902.28},{0.08408,0.08104,52330.7,3342.9},{0.08712,0.08104,60882.2,3911.26},{0.09016,0.08104,51509.2,3110.43},{0.0932,0.08104,50454.1,3090.58},{0.09624,0.08104,40717.6,2545.59}, {0.09928,0.08104,42743.7,3025.45},{0.10232,0.08104,35756.6,2624.12},{0.10536,0.08104,33058.2,2592.71},{0.1084,0.08104,26596.4,2214.08},{0.11144,0.08104,22096,1911.06},{0.11448,0.08104,17355.1,1615.19},{0.11752,0.08104,17941.5,1960.17}, {0.12056,0.08104,14323.6,1810.15},{0.1236,0.08104,7341.57,954.096},{0.12664,0.08104,4442.43,677.872},{0.12968,0.08104,1873.55,405.916},{0.13272,0.08104,417.675,425.303},{0.07192,0.08408,1924.84,598.086},{0.07496,0.08408,31021.9,2594.86}, {0.078,0.08408,47782.1,3437.36},{0.08104,0.08408,52330.7,3342.9},{0.08408,0.08408,62280.5,4085.57},{0.08712,0.08408,59184.5,3557.77},{0.09016,0.08408,54444.5,3356.24},{0.0932,0.08408,52414.2,3465.73},{0.09624,0.08408,41135,2732.47}, {0.09928,0.08408,43030.7,3169.64},{0.10232,0.08408,31955,2343.16},{0.10536,0.08408,31763.3,2541.46},{0.1084,0.08408,29077,2567.87},{0.11144,0.08408,25068.3,2297.13},{0.11448,0.08408,17482.4,1718.25},{0.11752,0.08408,15289.4,1608.92}, {0.12056,0.08408,13642,1673.61},{0.1236,0.08408,14883.2,2367.16},{0.12664,0.08408,7972.96,1413.88},{0.12968,0.08408,3405.88,735.116},{0.13272,0.08408,1057.94,627.235},{0.07192,0.08712,1404.78,394.034},{0.07496,0.08712,30800.1,2514.06}, {0.078,0.08712,49295.2,3415.36},{0.08104,0.08712,60882.2,3911.26},{0.08408,0.08712,59184.5,3557.77},{0.08712,0.08712,49129.6,2946.77},{0.09016,0.08712,45237.7,2826.57},{0.0932,0.08712,45501.7,3024.24},{0.09624,0.08712,37033.9,2475.23}, {0.09928,0.08712,34151.8,2398.3},{0.10232,0.08712,29587.7,2149.13},{0.10536,0.08712,27653.3,2235.21},{0.1084,0.08712,22826.8,1883.48},{0.11144,0.08712,21367.8,1983.05},{0.11448,0.08712,18528.3,1848.77},{0.11752,0.08712,14855.8,1610.02}, {0.12056,0.08712,14077.1,1844.82},{0.1236,0.08712,10624.1,1553.14},{0.12664,0.08712,5562.6,932.138},{0.12968,0.08712,3641.58,805.157},{0.13272,0.08712,1316.61,883.037},{0.07192,0.09016,3380.76,1203.18},{0.07496,0.09016,30274.6,2389.32}, {0.078,0.09016,43140.3,2807.55},{0.08104,0.09016,51509.2,3110.43},{0.08408,0.09016,54444.5,3356.25},{0.08712,0.09016,45237.7,2826.57},{0.09016,0.09016,42089.3,2759.44},{0.0932,0.09016,37912.6,2545.3},{0.09624,0.09016,35946.1,2617.82}, {0.09928,0.09016,30790.3,2308.9},{0.10232,0.09016,22454.2,1626.98},{0.10536,0.09016,26031.3,2178.62},{0.1084,.7,1972.3},,0.09016,21785.2,2134.39},{0.11448,0.09016,18883.9,2077.23},{0.11752,0.09016,12041.1,1.04}, 0.09016,22224{0.11144268{0.12056,0.09016,10975.6,1393.68},{0.1236,0.09016,7518.78,1038.76},{0.12664,0.09016,5874.3,1073.87},{0.12968,0.09016,1723.28,374.349},{0.13272,0.09016,699.444,543.604},{0.07192,0.0932,3913.58,1303.82},{0.07496,0.0932,35813.1,3028.89}, {0.078,0.0932,44007.8,2823.18},{0.08104,0.0932,50454.1,3090.58},{0.08408,0.0932,52414.2,3465.73},{0.08712,0.0932,45501.7,3024.24},{0.09016,0.0932,37912.6,2545.3},{0.0932,0.0932,37839.1,2696.59},{0.09624,0.0932,35570.2,2728.53}, {0.09928,0.0932,25741.7,1867.91},{0.10232,0.0932,25677,2071.84},{0.10536,0.0932,22567.8,1895.18},{0.1084,0.0932,24015.7,2375.42},{0.11144,0.0932,20983,2181.12},{0.11448,0.0932,15118.6,1585.43},{0.11752,0.0932,12618.9,1538.29}, {0.12056,0.0932,8756.85,1150.21},{0.1236,0.0932,6325.38,987.484},{0.12664,0.0932,3673.45,643.928},{0.12968,0.0932,2009.66,610.964},{0.07192,0.09624,3592.63,1266.19},{0.07496,0.09624,29562.9,2316.16},{0.078,0.09624,44955.3,2859.78}, {0.08104,0.09624,40717.5,2545.59},{0.08408,0.09624,41135,2732.47},{0.08712,0.09624,37033.9,2475.23},{0.09016,0.09624,35946.1,2617.82},{0.0932,0.09624,35570.2,2728.53},{0.09624,0.09624,27077.6,2050.47},{0.09928,0.09624,25326.2,1974.29}, {0.10232,0.09624,23637.2,2010.43},{0.10536,0.09624,19406.5,1691.81},{0.1084,0.09624,18985.3,1827.42},{0.11144,0.09624,16654.7,1799.04},{0.11448,0.09624,17025.5,2104.79},{0.11752,0.09624,15474.9,2099.14},{0.12056,0.09624,9828.83,1418.25}, {0.1236,0.09624,7525.84,1395.18},{0.12664,0.09624,2949.11,607.889},{0.12968,0.09624,745.511,292.016},{0.07192,0.09928,2207.08,630.63},{0.07496,0.09928,33850.6,2664.14},{0.078,0.09928,40392.4,2699.6},{0.08104,0.09928,42743.7,3025.45}, {0.08408,0.09928,43030.7,3169.64},{0.08712,0.09928,34151.8,2398.3},{0.09016,0.09928,30790.3,2308.9},{0.0932,0.09928,25741.7,1867.91},{0.09624,0.09928,25326.2,1974.29},{0.09928,0.09928,30030.1,2754.31},{0.10232,0.09928,22001.4,1996.48}, {0.10536,0.09928,21019.8,2032.48},{0.1084,0.09928,21088,2315.85},{0.11144,0.09928,16456.3,1956.57},{0.11448,0.09928,18251,2627.35},{0.11752,0.09928,9122.12,1255.9},{0.12056,0.09928,7522.05,1182.75},{0.1236,0.09928,4649.55,915.851}, {0.12664,0.09928,1201.02,289.554},{0.12968,0.09928,210.022,194.221},{0.07192,0.10232,2474.93,640.239},{0.07496,0.10232,28609,2172.54},{0.078,0.10232,35845.1,2529.94},{0.08104,0.10232,35756.6,2624.12},{0.08408,0.10232,31955,2343.16}, {0.08712,0.10232,29587.7,2149.13},{0.09016,0.10232,22454.2,1626.98},{0.0932,0.10232,25677,2071.84},{0.09624,0.10232,23637.2,2010.43},{0.09928,0.10232,22001.4,1996.48},{0.10232,0.10232,16415.2,1470.11},{0.10536,0.10232,16439.7,1612.66}, {0.1084,0.10232,16269.8,.48},{0.11144,0.10232,15091.9,1934.51},{0.11448,0.10232,10180.3,1318.86},{0.11752,0.10232,670966,937.837},{0.12056,0.,4687.09,803.866},{0.1236,0.10232,3685.97,985.736},{0.1266499.113}, 1841.10232,0.10232,834.361,3{0.07192,0.10536,2799.21,740.962},{0.07496,0.10536,30215.1,2529.08},{0.078,0.10536,30587.9,2287.11},{0.08104,0.10536,33058.2,2592.71},{0.08408,0.10536,31763.3,2541.46},{0.08712,0.10536,27653.3,2235.21},{0.09016,0.10536,26031.3,2178.62}, {0.0932,0.10536,22567.8,1895.18},{0.09624,0.10536,19.81},{0.09928,0.10536,21019.8,2032.48},{0.10232,0.10536,16439.7,1612.66},{0.10536,010536,20517.6,2415.87},{0.1084,0.10536,12663.2,1488.27},{0.11144,0.10536,8779.54,1049.7}, 406.5,1691.{0.11448,0.10536,10140.1,1607.5},{0.11752,0.10536,6616.65,1187.33},{0.12056,0.10536,2938.83,563.058},{0.1236,0.10536,1780.79,727.786},{0.07192,0.1084,2128.36,539.397},{0.07496,0.1084,28114.3,2523.19},{0.078,0.1084,30873.7,2664.7}, {0.08104,0.1084,26596.4,2214.08},{0.08408,0.1084,29077,2567.87},{0.08712,0.1084,22826.8,1883.48},{0.09016,0.1084,22224.7,1972.3},{0.0932,0.1084,24015.7,2375.42},{0.09624,0.1084,18985.3,1827.42},{0.09928,0.1084,21088,2315.85}, {0.10232,0.1084,16269.8,1841.48},{0.10536,0.1084,12663.2,1488.27},{0.1084,0.1084,12311.7,1647.29},{0.11144,0.1084,9009.3,1278.66},{0.11448,0.1084,9431.62,1818.31},{0.11752,0.1084,6813.62,1637.26},{0.12056,0.1084,2160.68,657.701}, {0.07192,0.11144,2385.13,712.151},{0.07496,0.11144,19829.4,1889.25},{0.078,0.11144,20417.2,1743.17},{0.08104,0.11144,22096,1911.06},{0.08408,0.11144,25068.3,2297.13},{0.08712,0.11144,21367.8,1983.05},{0.09016,0.11144,21785.2,2134.39}, {0.0932,0.11144,20983,2181.12},{0.09624,0.11144,16654.7,1799.04},{0.09928,0.11144,16456.3,1956.57},{0.10232,0.11144,15091.9,1934.51},{0.10536,0.11144,8779.54,1049.7},{0.1084,0.11144,9009.3,1278.66},{0.11144,0.11144,6123.94,923.801}, {0.11448,0.11144,4457.57,902.223},{0.11752,0.11144,5088.75,1951.21},{0.12056,0.11144,438.96,374.22},{0.07192,0.11448,4902.7,2810.81},{0.07496,0.11448,21276.6,2591.24},{0.078,0.11448,18980.7,1889.2},{0.08104,0.11448,17355.1,1615.19}, {0.08408,0.11448,17482.4,1718.25},{0.08712,0.11448,18528.3,1848.77},{0.09016,0.11448,18883.9,2077.23},{0.0932,0.11448,15118.6,1585.43},{0.09624,0.11448,17025.5,2104.79},{0.09928,0.11448,18251,2627.35},{0.10232,0.11448,10180.3,1318.86}, {0.10536,0.11448,10140.1,1607.5},{0.1084,0.11448,9431.62,1818.31},{0.11144,0.11448,4457.57,902.223},{0.11448,0.11448,3344.34,1020.28},{0.11752,0.11448,4028.88,2944.23},{0.07192,0.11752,2265.88,1816.41},{0.07496,0.11752,15433.2,2192.83}, {0.078,0.11752,13999.4,1464.99},{0.08104,0.11752,17941.5,1960.17},{0.08408,0.11752,15289.4,1608.92},{0.08712,0.11752,14855.8,1610.02},{0.09016,0.11752,12041.1,1268.04},{0.0932,0.11752,12618.9,1538.29},{0.09624,0.11752,15474.9,2099.14}, {0.09928,0.11752,9122.12,1255.9},{0.10232,0.11752,6709.66,937.837},{0.10536,0.11752,6616.65,1187.33},{0.1084,0.11752,6813.62,1637.26},{0.11144,0.11752,5088.75,1951.21},{0.11448,0.11752,4028.88,2944.23},{0.07496,0.12056,6525.41 ,963.023},{0.078,0.12056,9163.9,1048.6},{0.08104,0.12056,14323.6,1810.15},{0.08408,0.12056,13642,1673.61},{0.08712,0.12056,14077.1,1844.82},{0.09016,0.12056,10975.6,1393.68},{0.0932,0.12056,8756.85,1150.21},{0.09624,0.12056,9828.83,1418.25}, {0.09928,0.12056,7522.05,1182.75},{0.10232,0.12056,4687.09,803.866},{0.10536,0.12056,2938.83,563.058},{0.1084,0.12056,2160.68,657.701},{0.11144,0.12056,438.96,374.22},{0.07496,0.1236,3337.06,772.815},{0.078,0.1236,11527.3,2025.61}, {0.081040.1236,7341.57,954.096},{0.08408,0.1236,14.16},{0.08712,0.1236,10624.1,1553.14},{0.09016,0.1236,7518.78,1038.76},{0.0932,0.1236,6325.38,987.484},{0.09624,0.1236,7525.84,1395.18},{0.09928,0.1236,4649.55,915.851}, ,883.2,2367{0.10232,0.1236,3685.97,985.737},{0.10536,0.1236,1780.79,727.786},{0.07496,0.12664,810.584,334.516},{0.078,0.12664,3046.15,526.386},{0.08104,0.12664,4442.43,677.872},{0.08408,0.12664,7972.96,1413.88},{0.08712,0.12664,5562.6,932.138}, {0.09016,0.12664,5874.3,1073.87},{0.0932,0.12664,3673.45,643.928},{0.09624,0.12664,2949.11,607.889},{0.09928,0.12664,1201.02,289.554},{0.10232,0.12664,834.361,399.113},{0.078,0.12968,1258.71,490.212},{0.08104,0.12968,1873.55,405.916}, {0.08408,0.12968,3405.88,735.116},{0.08712,0.12968,3641.58,805.157},{0.09016,0.12968,1723.28,374.349},{0.0932,0.12968,2009.66,610.964},{0.09624,0.12968,745.511,292.016},{0.09928,0.12968,210.022,194.221},{0.08104,0.13272,417.675,425.303}, {0.08408,0.13272,1057.94,627.235},{0.08712,0.13272,1316.61,883.037},{0.09016,0.13272,699.444,543.604}} Table G.6: Acceptance and eﬃciency corrected Dalitz Plot 3π0 , for MMpp =0.4 − 0.5 GeV/c2 (Fig. 95). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of the absolutenormalization of 19% have to beincluded. Fullyexpandable version ofthe tableis availablein the attached electronic version of the thesis. Jagiellonian University 226 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Da itz M ot 3pi0 M'{2}(pi'{0} _{1}pi'{0}_{2}) versis M'{2}(pi'{0}_{2}pi'{0}_{3}) -Eac. event is li ed six times Acceptance and Ellicienc Corrected lor MM_{pp}00.6-0.7 Ge1/c'{2} Error ol M'{2}(pi'{0}_{2}pi'{0}_{3}) 0.0019 Ge1'{2}/c'{4} Error ol M'{2}(pi'{0}_{1}pi'{0}_{2}) 0.0019 Ge1'{2}/c'{4} T.e data are in t.e lo o ing lormat {{M'{2}(pi'{0} _{2}pi'{0}_{3}) [Ge1'{2}/c'{4}], M'{2}(pi'{0}_{1}pi'{0}_{2}) [Ge1'{2}/c'{4}], sigma [mil], Error ol sigma [mil] },...} {{0.155 9,0.0723,575.3 68,433.976},{0.1597,0.0723,224.43 1,101.824},{0.1635,0.0723,487.254,151. 819},{0.1673,0.0723,1108.95,287.212},{0.1711,0.0723, 1589. 99,339.919},{0.1749,0.0723,2265.5,443. 843},{0.1787,0.0723,2400.33,40 7.69},{0.1825,0.0723,2060.54,333.58},{0.1863,0.0723,2335.55,388.299},{0. 1901,0.072 3,2677.24,445.444},{0.1939,0.0723,2049.48,339.886},{0.1977,0.0723,2292.53,389.558}, {0.2015,0.0723,1700.7 1,275.516}, {0.2053,0.0723,1747.83,274. 912},{0. 2091,0.0723,1730.59,28 6.828},{0.2129,0.072 3,1600.25,274.719},{0.2167,0.0723,1507.51,260.855},{ 0.2205,0.0723, 2036.93,426.967},{0.2243,0.0723,1619.58,33 7.091},{0.2281, 0.0723,1084.2,220.17},{0.2319,0.0723,1751.5,5 01.443},{0.2 357,0.0723,813.184,208.007},{0.2395,0.0723,520.464,141.779},{0.2433,0.0723,437.665,151.98}, {0.2471,0.0723,259.12 5,108.337}, {0.2547,0.0723,30.8331,34.34},{0.1407,0.0761,755.536,423.267},{0.1445,0.0761,633.441,142.803},{0.1483,0.0 761,124 7.52, 214.876},{0.1521,0.0761,23 85.12,3 40.18},{0.1559,0.0761,3761.05,437.963},{0.1597,0.0761,550 8.76,574.432},{0.1635,0.0761,5679.48,5 30.807},{0.1 673,0.0761,7405.01,663.96},{0.1711,0.0761,668 3. 65,526.374},{ 0.1749,0.0761,6408.28,467.309}, {0.1787,0.0761,9463.7 2,724},{0.1825,0.0761,8736.04,628.535},{0.1863,0.0761,10558.1,745.49 2}, {0.1901,0.0761,10632,736.459}, {0.1939,0.0761,9665.32,636. 285},{0.1977,0.0761,1 0812.7,772.7 83},{0.20 15,0.076 1,9073.07, 618.70 3},{0.2053,0.0761,8601 .96,613.016},{0.2091,0.0761,7797.34,587.594},{0.2129,0.0761, 6370.73,490.2 98},{0.21 67, 0.0761,66 81.27,554.103},{0.220 5,0.0761,5555.73,469.042}, {0.2243,0.0761,5749.9 1,555.405}, {0.2281,0.0761,4479.8,419.489},{0.2319,0.0761,4105.67,410.659},{0.2357,0.0761,2918.96,289.39},{0.2395,0.0 761,317 3.03, 354.047},{0.2433,0.0761,3211.52,3 99.089},{0.2471,0.0761,2713.56,353.392}, {0.2509,0.0761,1723.63,244.165},{0.2547,0.0761,1381.76,210.829},{0. 2585,0.076 1,936.285,159.51 6},{0.2623,0.0761,961.795,216.858},{0.2661,0.0761,621.345,154.259}, {0.2699,0.0761,1862.04,989.714}, {0.2737,0.0761,138.222,63.2077},{0. 2775,0.0761,227.45,173.195},{0.1293,0.0799,356.997,150.626}, {0.1331,0.0799,947.286,207. 433},{0. 1369,0.0799,2651.11,488.272},{0.1407,0.079 9,3035.31,374.34}, {0.1445,0.0799,4351.19,489.261},{0.1483,0.0799,4624.44,433.113},{0. 1521,0.079 9,5293.11,471.05 1},{0.1559,0.0799,6947,575.379},{0.1597,0.0799, 6810.82,536.757}, {0.1635,0.0799,7909.18,595.755}, {0.1673,0.0799,8417.24,591. 184},{0. 1711,0.0799,8564.01,59 0.897},{0.1749,0.079 9,10555.8,756.484},{0.1787,0.0799,8666.57,566.244},{ 0.1825,0.0799, 9080.12,610.725},{0.1863,0.0799,8901.42,57 1.338},{0.1901, 0.0799,10050.3,672.666},{0.1939,0.0799,10745.1,732.556},{0.1977,0.0799,10661.7,738. 453},{0.2015,0.0799,9577.75,64 2. 247},{0.2053,0. 0799,9210.33,630.214}, {0.2091,0.0799,9319.51,667.229}, {0.2129,0.0799,7839.78,562. 451},{0. 2167,0.0799,7846.86,58 3.493},{0.2205,0.079 9,7115.24,560.359},{0.2243,0.0799,5721.89,452.529},{ 0.2281,0.0799, 5661.77,486.642},{0.2319,0.0799,5395.48,48 5.463},{0.2357, 0.0799,5225.65,507.296},{0.2395,0.0799,4234.7 8,394.633},{0.2433,0.0799,3844.26,400. 148},{0.2471,0.0799,3041.48,31 0. 946},{0.2509,0. 0799,2647.76,303.108}, {0.2547,0.0799,2575.41,315.074}, {0.2585,0.0799,2252.89,289. 82},{0.2623,0.0799,1619.03,221.846},{0.2661,0.0799,1945.68,319.266}, {0.2699,0.0799,961.799,156. 82},{0.2737,0.0799,930. 648,1 72.667},{0.2775,0.0799,840.436,219.293}, {0.2813,0.0799,311.733,78.2426},{0.2851,0.0799,170.885,61.7406 },{0. 2889,0.079 9,347.118,260.71 2},{0.1217,0.0837,459. 134,156.957},{0.1255,0.0837,1485.01,278.741}, {0.1293,0.0837,2846.92,415.52},{0.1331,0.0 837,4036.53,466.087},{0.1369,0.0837,3977.96,383.421},{0.1407,0.0837,5460.6,495.662},{0.1445,0.0 837,630 2.59, 533.458},{0.1483,0.0837,5324.72,4 07.314},{0.1521,0.0837,6824.58,502.175}, {0.1559,0.0837,6735.41,475.042},{0.1597,0.0 837,8218.02,577.788 },{0. 1635,0.083 7,8446.25,587.86 5},{0.1673,0.0837,8831.12,598.335},{0.1711,0.0837,9445.37,649.399}, {0.1749,0.0837,8585.02,591.453}, {0.1787,0.0837,94 34.44,657. 114},{0. 1825,0.0837,9210.59,62 3.733},{0.1863,0.083 7,10688.2,761.96}, {0.1901,0.0837,9150.74,628. 532},{0. 1939,0.0837,9244.32,643.436},{0.1977,0.083 7,8756.51,586.346},{0.2015,0.0837,9659.18,671.212},{0.2053,0.0837,8471.17,567.9},{0.2 091,0.0837,9241.27,627.142},{0.2129,0.0837,8876. 84,626.127},{0.2167,0.0837,7979.73,569.564 }, {0.2205,0.0837,7182.96,538.569}, {0.2243,0.0837,66 33.73,509. 677},{0. 2281,0.0837,6684.25,53 8.858},{0.2319,0.083 7,5972.28,505.511},{0.2357,0.0837,4842.91,409.337},{ 0.2395,0.0837, 4985.81,464.801},{0.2433,0.0837, 4639.12,43 8.124},{0.2471, 0.0837,3664.56,364.074},{0.2509,0.0837,2971.9 8,301.693},{0.2547,0.0837,3008.18,331. 01},{0.2585, 0.0837,2231.94,255.969},{0.2623,0.0837,2628.46,338.965}, {0.2661,0.0837,1747.02,229.632}, {0.2699,0.0837,13 98.46,185. 404},{0. 2737,0.0837,1596.88,25 4.625},{0.2775,0.083 7,1526.12,271.563},{0.2813,0.0837,738.172,141.685},{ 0.2851,0.0837, 908.056,214.448},{0.2889,0.0837, 624.181,17 2.35},{0.2927,0.0837,303.769,104.239},{0.2965,0.0837,121.547,64.185 9},{0.3003,0.0837, 185.84,209.977},{0.1141,0.0875,201. 864,77.7923},{0.1179,0.0875,1578.8,314.103 }, {0.1217,0.0875,2344.5,327.657},{0.1255,0.0 875,326 9.41,355.058},{0.1293,0.0875,4185.59,405.7},{0.133 1, 0.0875,6057.14,553.039},{0.1369,0.08 75, 6308.7,54 4.802},{0.14 07,0.0875,6545.18,519.0 59},{0.1445, 0.0875,6679.77,493.923},{ 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{0.1217,0.2775,1439.59,181.032}, {0.12 55,0.2775,17 44.33,244.914},{0. 129 3,0.2 775,1823. 48,278.27},{ 0.1331,0.2775,1753.48,263.053}, {0.1369,0.2775,1631.89,260.143},{0. 140 7,0.2 775,117 0. 44,176.226},{0.1445,0.2775,1175.23,180.441},{ 0.1483,0.2775,1109.1,168.8 03},{0.1521,0.2 775,1057.83,174.422 },{0. 1559,0.2775,1038.43,1 86.414},{ 0.1597,0.277 5,840. 236,153. 347},{0.1 635,0.2775,836.938,179.265}, {0.1673,0.2775,594.37,128.302},{0.171 1, 0.2 775,637.9 67, 170.64},{0.1749,0.277 5,483.771,135. 675},{0.1787,0.2775,770.455,263.0 12},{0.1825,0.2 775,308.25,124.333},{0.1 863,0. 277 5,317.971,18 6. 111},{0.1901,0.2775,1 6.9121,16.12 4},{0.0799,0.2 813,311.733,78.2426}, {0.0 837,0.281 3,738.172,14 1. 685},{0.0875,0.2813,1 501.77, 260.3 35},{0.0913, 0.2813,139 6. 02,209.05 7},{0.0951,0.2 813,1760.78,278.176}, {0.0989,0.2813,1687.86,255.193}, {0.10 27,0.2813,13 72.28,188.512},{0. 106 5,0.2 813,1645. 5,229.085},{ 0.1103,0.2813,1268.51,161.966}, {0.1141,0.2813,1210.32,160.412},{0. 117 9,0.2 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7},{0.1255,0.2 851,1193.43,183.172}, {0.1293,0.2851,942.416,151.286}, {0.13 31,0.2851,88 2.316,138.464},{0. 136 9,0.2 851,1057. 6,174.832},{ 0.1407,0.2851,922.564,152.454}, {0.1445,0.2851,997.897,185.258},{0. 148 3,0.2 851,727 .2 08,135.118},{0.1521,0.2851,1009.75,245.658},{ 0.1559,0.2851,822.737,196. 3},{0.1597,0.28 51,895.391,2 49.924},{ 0.1635,0.2851, 569.091,15 1.562},{0.1673,0.2851,306.378,92.39 97},{0.17 11,0.2851,519.522,233.198 }, {0.1749,0.2851,151.893,63.0906}, {0.17 87,0.2851,44.9 438,26.4463},{0. 079 9,0.2 889,347.1 18,260.712},{0.0837,0.288 9,624.181,172.35}, {0.0875,0.2889,589.785,114.631},{0. 091 3,0.2 889,533 .4 13,95.6061},{0.0951,0.2889,951.746,169.152},{ 0.0989,0.2889,1092.2,181.0 34},{0.1027,0.2 889,1017.86,158.308 },{0. 1065,0.2889,1336,219. 525},{0.1 103,0.2889,1 269.34,21 1.485},{0.11 41,0.2889, 904.943,136.203}, {0.1179,0.2889,1328.63,228.59},{0.121 7, 0.2 889,835.4 73, 134.543},{0.1255,0.28 89,940.55 7,160.631},{ 0.1293,0.2889,887.574,149.815}, {0.1331,0.2889,930.034,169.835},{0. 136 9,0.2 889,666 .7 26,130.386},{0.1407,0.2889,810.4,158.2 29},{0.1445, 0.2889,599.418,109.95 },{0.1 483,0.288 9,633.615,13 5. 149},{0.1521,0.2889,3 96.719, 90.22 35},{0.1559, 0.2889,395.1 77,101.95 6},{0.1597,0.2 889,242.828,61.2312}, {0.1635,0.2889,438.513,164.208}, {0.16 73,0.2889,36 1.033,138.161},{0. 171 1,0.2 889,248.8 61,160.727},{0.1749,0.288 9,222.614,199.759},{0.0837,0.2927,303.769,104.239},{0.08 75,0.2927,57 6. 454,141.339},{0.0913, 0.2927, 712.588,13 5.553}, {0.0951,0.2927,837.892,154.622},{0.0989,0.2927,742.927,129.228},{0.1027,0.2927,10 31.58,181.059},{0.1065,0.2 927,920.3 16,159. 55},{ 0. 1103,0.2927,1171.88,216.926}, {0.1141,0.2927,1052.89,178.681}, {0.11 79,0.2927,62 3.852,105.272},{0. 121 7,0.2 927,1047. 88,193.19},{ 0.1255,0.2927,1032.12,199.444}, {0.1293,0.2927,692.64, 126.976},{0.1331,0.29 27,683.17 6,1 31.455},{ 0.1369,0.2927, 1047.04,253.187},{0.1407,0.2927,658.322,132.7 34},{0.1445,0.2 927,414.63,8 3. 2764},{ 0.1483,0.2927, 471.998,11 0.193},{0.1521,0.2927,422.913,106.6 63},{0.15 59,0.2927,218.341,60.7433 }, {0.1597,0.2927,647.867,272.848}, {0.16 35,0.2927,24 9.499,109.063},{0. 167 3,0.2 927,123.9 35,73.4697},{0.0837,0.296 5,121.547,64.1859},{0.0875,0.2965,284.066,82.1287},{0.09 13,0.2965,54 2. 601,122.488},{0.0951, 0.2965, 495.329,96.4858}, {0.0989,0.2965,519.946,96.6128},{0.1027,0.2965,711.318,140.932},{0.1065,0.2965,48 5.978,86.3202},{0.1103,0.2 965,612.0 72,119. 016},{0.1141,0. 2965,517.451,89.3151}, {0.1179,0.2965,627.053,121.085}, {0.12 17,0.2965,54 4.562,110.432},{0. 125 5,0.2 965,569.9 98,109.778},{0.1293,0.296 5,488.534,99.1689},{0.1331,0.2965,342.57,67.4906},{0. 136 9,0.2 965,472 .3 39,109.754},{0.1407,0.2965,542.37,130.37},{0.1445, 0.2965,317.668,80.527 9},{0.1483,0.29 65,302.959,8 3. 1247},{ 0.1521,0.2965, 316.674,10 4.379},{0.1 559,0.2965,120.272,41.90 33},{0.15 97,0.2965,385.774,307.048 }, {0.1635,0.2965,183.055,152.962}, {0.08 37,0.3003,18 5.84, 209.977},{0.0875,0.30 03,160.29 2,55. 2487},{ 0.0913,0.3003,591.705,181.02},{0.0951,0.3 003,544.317, 147.675},{0.0989,0 .30 03,728.01 1,1 99.728},{ 0.1027,0.3003, 845.045,196.806},{0.1065,0.3003,683.622,148.5 9},{0.1103,0.30 03,605.34,12 2. 205},{0.1141,0.3003,6 08.047, 119.9 87},{0.11 79, 0.3003,590.2 93,122.42 6},{0.1217,0.3 003,683.503,143.841}, {0.1255,0.3003,508.031,104.24},{0.129 3, 0.3 003,868.7 44, 244.271},{0.1331,0.30 03,652.55,181. 954},{0.1369,0.3003,471.707,129.2 16},{0.1407,0.3 003,549.78,189.403},{0.1 445,0. 300 3,341.283,11 5. 16},{0.1483, 0.3003,19 4.618,71.387 7},{0.1521,0.3 003,177.024,81.5752}, {0.1 559,0.300 3,84.9016,47 .1 042},{0.1597,0.3003,1 96.234, 235.4 06},{0.08 75, 0.3041,69.37 86,45.67},{ 0.0913,0.3041,3 09.828,15 2.592}, {0.0951,0.3041,724.922,311.312}, {0.09 89,0.3041,48 8.581,131.359},{0. 102 7,0.3 041,387.3 05,94.3996},{0.1065,0.304 1,551.545,154.103},{0.1103,0.3041,476.959,118.936},{0.11 41,0.3041,42 1. 281,116.617},{0.1179, 0.3041, 414.709,108.924}, {0.1217,0.3041,358.373,100.779},{0.1255,0.3041,322.665,84.9144},{0.1293,0.3041,58 5.677,175.842},{0.1331,0.3 041,329.1 17,89.4295},{0.1369,0.3041,671.776,307.13}, {0.1407,0.3041,349.564,163.714}, {0.14 45,0.3041,13 4.992,73.3726},{0. 148 3,0.3 041,104.5 86,57.6547},{0.1521,0.304 1,108.598,113.328},{0.0913,0.3079,66.3763,36.6491},{0.09 51,0.3079,24 5. 096,101.353},{0.0989, 0.3079, 340.869,131.331}, {0.1027,0.3079,291.338,87.3016},{0.1065,0.3079,351.578,105.276},{0.1103,0.3079,26 4.701,75.9226},{0.1141,0.3 079,254.4 64,70.2101},{0.1179,0.3079,343.589,107.365}, {0.1217,0.3079,327.431,111.919}, {0.12 55,0.3079,15 0.471,43.4708},{0. 129 3,0.3 079,351.9 68,144.3},{0.1331,0.3079,241.638,102.0 66},{0.1369,0.3 079,155.204, 64.936},{0.1 407,0. 307 9,157.074,81.6 152},{0.1445,0.3079,3 4.979,27.664 9},{0.0951,0.3 117,48.36 42,36.4876}, {0.0989,0.311 7,160.047,70 .8 213},{0.1027,0.3117,3 02.07,148.72 8}, {0.106 5,0.3 117,242.51 1,97.7311 },{0.1103,0.31 17, 342.051,1 32.08}, {0.1141,0.3117,180.048,51.209},{0.117 9, 0.3 117,227.6 75, 87.6486},{0.1217,0.31 17,253.23 5,103.49},{0.1255,0.3117,303.105,155.9 95},{0.1293,0.3 117,249.925, 136.06},{0.1 331,0. 311 7,65.3558,39.0 687},{0.1369,0.3117,4 7.9452,37.80 54},{0.0989, 0.3155,372. 189,397.37 3},{0.1027,0.31 55,110.632,8 0. 3667},{ 0.1065,0.3155, 36.3678,23.5 719},{0.1 103,0.3155,63.9 721,30.57 22},{0.11 41,0.3155,62.2029,28.3764 }, {0.1179,0.3155,73.2448,49.4029}, {0.12 17,0.3155,82.9 168,62.1799},{0. 125 5,0.3 155,63.43 91,48.4107},{0.1293,0.315 5,417.637,445.739},{0.1141,0.3193,22.2383,27.0536}} Table G.7: Acceptance and eﬃciency corrected Dalitz Plot 3π0 , for MMpp =0.6 − 0.7 GeV/c2 (Fig. 95). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of theabsolutenormalization of 19% have to beincluded. Fully expandable version of thetableis availablein the attached electronic version of the thesis. Jagiellonian University 227 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Da itz M ot 3pi0 M'{2}(pi'{0}_{1}pi'{0}_{2}) versis M'{2}(pi'{0}_{2}pi'{0}_{3}) -Eac. event is li ed six times Acceptance and Ellicienc Corrected lor MM_{pp}00.7-0.8 Ge1/c'{2} Error ol M'{2}(pi'{0}_{2}pi'{0}_{3}) 0.0038 Ge1'{2}/c'{4} Error ol M'{2}(pi'{0}_{1}pi'{0}_{2}) 0.0038 Ge1'{2}/c'{4} T.e data are in t.e lo o ing lormat {{M'{2}(pi'{0}_{2}pi'{0}_{3}) [Ge1'{2}/c'{4}], M'{2}(pi'{0}_{1}pi'{0}_{2}) [Ge1'{2}/c'{4}], sigma [mil], Error ol sigma [mil] },...} {{0.1882,0.0742,21.5862,11.217},{0.1958,0.0742,187.466,43.5553},{0.2034,0.0742,473.956,79.5377},{0.211,0.0742,861.758,103.907},{0.2186,0.0742,941.111,88.9312},{0.2262,0.0742,1435.05,118.383},{0.2338,0.0742,1575.75,112.073},{0.2414,0.0742,1881.01,123.749},{0.249,0.0742,2054.49,128.898},{0.2566,0.0742,2220.1,137.693}, {0.2642,0.0742,2212.84,138.773},{0.2718,0.0742,1951.44,124.618},{0.2794,0.0742,1921.77,127.753},{0.287,0.0742,1470.46,102.35},{0.2946,0.0742,1131.26,86.1821},{0.3022,0.0742,1022.81,83.5233},{0.3098,0.0742,869.855,80.7469},{0.3174,0.0742,698.91,77.8231},{0.325,0.0742,486.847,57.859},{0.3326,0.0742,352.103,52.5768}, {0.3402,0.0742,280.518,51.6069},{0.3478,0.0742,126.437,24.2839},{0.3554,0.0742,76.3822,22.4097},{0.363,0.0742,49.2372,19.0688},{0.3706,0.0742,7.47951,5.11123},{0.1578,0.0818,115.858,123.699},{0.1654,0.0818,165.364,38.1155},{0.173,0.0818,369.195,44.5642},{0.1806,0.0818,902.971,86.2472},{0.1882,0.0818,1435.91,109.632}, {0.1958,0.0818,1806.08,115.805},{0.2034,0.0818,2190.73,128.54},{0.211,0.0818,2289.58,121.696},{0.2186,0.0818,2720.13,142.08},{0.2262,0.0818,2859.03,143.821},{0.2338,0.0818,3059.78,155.979},{0.2414,0.0818,3360.72,167.065},{0.249,0.0818,3254.17,160.199},{0.2566,0.0818,3294.76,160.891},{0.2642,0.0818,2932.8,143.336}, {0.2718,0.0818,3319.69,163.83},{0.2794,0.0818,3089.51,151.95},{0.287,0.0818,2991.17,147.809},{0.2946,0.0818,3008.49,155.227},{0.3022,0.0818,2447.53,134.461},{0.3098,0.0818,2100.74,122.91},{0.3174,0.0818,1649.69,100.14},{0.325,0.0818,1574.62,106.593},{0.3326,0.0818,1205.75,87.1604},{0.3402,0.0818,1002.13,79.6507}, {0.3478,0.0818,942.305,87.1523},{0.3554,0.0818,638.935,66.8321},{0.363,0.0818,438.391,51.1929},{0.3706,0.0818,299.352,42.7863},{0.3782,0.0818,200.104,36.7135},{0.3858,0.0818,93.6766,22.5144},{0.3934,0.0818,29.7539,11.0765},{0.401,0.0818,7.06652,6.28563},{0.1426,0.0894,58.8327,27.0785},{0.1502,0.0894,343.169,55.4934}, {0.1578,0.0894,730.212,68.6895},{0.1654,0.0894,1170.1,89.0934},{0.173,0.0894,1648,107.479},{0.1806,0.0894,2006.18,119.288},{0.1882,0.0894,2491.3,146.282},{0.1958,0.0894,2527.95,136.401},{0.2034,0.0894,2678.22,139.341},{0.211,0.0894,2671.04,136.084},{0.2186,0.0894,2507.13,123.964},{0.2262,0.0894,3063.41,157.725}, {0.2338,0.0894,3079.31,156.745},{0.2414,0.0894,2903.44,143.401},{0.249,0.0894,2878.31,141.268},{0.2566,0.0894,2788.96,133.004},{0.2642,0.0894,2801.34,137.079},{0.2718,0.0894,2792.46,137.292},{0.2794,0.0894,3036.77,151.402},{0.287,0.0894,2709.39,135.108},{0.2946,0.0894,2786.19,142.485},{0.3022,0.0894,2704.3,141.024}, {0.3098,0.0894,2468.57,131.449},{0.3174,0.0894,2284.57,129.077},{0.325,0.0894,1823.38,108.206},{0.3326,0.0894,1652.62,106.362},{0.3402,0.0894,1264.65,85.689},{0.3478,0.0894,1106.14,82.7562},{0.3554,0.0894,931.205,78.8035},{0.363,0.0894,731.982,67.2289},{0.3706,0.0894,633.975,65.1623},{0.3782,0.0894,463.161,54.9294}, {0.3858,0.0894,336.435,45.9563},{0.3934,0.0894,253.749,40.9123},{0.401,0.0894,100.714,21.7858},{0.4086,0.0894,87.7671,27.5485},{0.4162,0.0894,6.42152,3.74763},{0.1274,0.097,14.6497,10.8822},{0.135,0.097,236.668,37.9439},{0.1426,0.097,631.994,57.5046},{0.1502,0.097,1135.21,85.1612},{0.1578,0.097,1392.01,88.3054}, {0.1654,0.097,1633.45,91.8196},{0.173,0.097,1831.5,99.7128},{0.1806,0.097,2107.11,111.189},{0.1882,0.097,2223.66,114.994},{0.1958,0.097,2211.45,111.686},{0.2034,0.097,2786.15,144.704},{0.211,0.097,2263.21,111.076},{0.2186,0.097,2684.85,132.136},{0.2262,0.097,2662.72,130.855},{0.2338,0.097,2547.43,123.811}, {0.2414,0.097,2729.3,132.222},{0.249,0.097,2438.56,117.085},{0.2566,0.097,2643.49,128.027},{0.2642,0.097,2642.51,128.79},{0.2718,0.097,2581.69,129.493},{0.2794,0.097,2275.9,107.647},{0.287,0.097,2787.2,139.368},{0.2946,0.097,2461.62,121.993},{0.3022,0.097,2367.92,118.228},{0.3098,0.097,2183.14,112.035}, {0.3174,0.097,2278.39,120.424},{0.325,0.097,1885.54,105.094},{0.3326,0.097,1521.4,90.8244},{0.3402,0.097,1327,81.7352},{0.3478,0.097,1170.43,78.6175},{0.3554,0.097,919.689,67.4885},{0.363,0.097,768.599,64.1094},{0.3706,0.097,645.442,60.0347},{0.3782,0.097,459.604,43.3752},{0.3858,0.097,446.312,49.3932},{0.3934,0.097,272.95,33.4076}, {0.401,0.097,170.925,24.6781},{0.4086,0.097,156.779,29.6289},{0.4162,0.097,70.3786,17.1528},{0.4238,0.097,69.4424,37.7365},{0.1198,0.1046,122.803,54.7261},{0.1274,0.1046,412.77,48.4041},{0.135,0.1046,900.638,70.5444},{0.1426,0.1046,1204.46,78.343},{0.1502,0.1046,1542.37,91.9522},{0.1578,0.1046,1850.78,102.404}, {0.1654,0.1046,2090.46,115.947},{0.173,0.1046,1997.64,106.175},{0.1806,0.1046,2241.11,119.406},{0.1882,0.1046,2101.15,107.95},{0.1958,0.1046,2365.72,119.304},{0.2034,0.1046,2430.53,120.107},{0.211,0.1046,2567.53,131.21},{0.2186,0.1046,2780.15,136.848},{0.2262,0.1046,2641.42,131.141},{0.2338,0.1046,2598.8,125.77}, {0.2414,0.1046,2840.32,140.115},{0.249,0.1046,2700.51,132.291},{0.2566,0.1046,2760.03,133.896},{0.2642,0.1046,2565.06,122.649},{0.2718,0.1046,2659.73,129.021},{0.2794,0.1046,2385.51,114.857},{0.287,0.1046,2470.97,121.882},{0.2946,0.1046,2143.48,105.783},{0.3022,0.1046,2320,116.089},{0.3098,0.1046,2418.55,123.632}, {0.3174,0.1046,2182.52,116.283},{0.325,0.1046,1719.97,90.4329},{0.3326,0.1046,1706.35,101.05},{0.3402,0.1046,1388.95,86.0681},{0.3478,0.1046,1209.66,80.7702},{0.3554,0.1046,945.439,67.2619},{0.363,0.1046,839.045,65.9951},{0.3706,0.1046,698.06,60.5077},{0.3782,0.1046,499.625,44.8762},{0.3858,0.1046,436.069,44.7397}, {0.3934,0.1046,358.675,43.3165},{0.401,0.1046,258.845,32.7632},{0.4086,0.1046,159.341,25.0929},{0.4162,0.1046,122.247,24.3663},{0.4238,0.1046,44.7217,11.2495},{0.4314,0.1046,56.2404,28.0247},{0.1122,0.1122,71.6941,19.5364},{0.1198,0.1122,585.214,60.3176},{0.1274,0.1122,1120,85.2193},{0.135,0.1122,1377.99,85.2253}, {0.1426,0.1122,1769.19,103.178},{0.1502,0.1122,1762.23,95.9832},{0.1578,0.1122,1845.59,98.6398},{0.1654,0.1122,2125.98,115.618},{0.173,0.1122,1982.21,100.698},{0.1806,0.1122,2382.69,123.341},{0.1882,0.1122,2440.16,124.672},{0.1958,0.1122,2229.05,111.7},{0.2034,0.1122,2389.59,117.035},{0.211,0.1122,2418.01,119.194}, {0.2186,0.1122,2448.68,116.779},{0.2262,0.1122,2545.69,122.779},{0.2338,0.1122,2667.08,128.686},{0.2414,0.1122,2602.55,125.311},{0.249,0.1122,2342.16,112.023},{0.2566,0.1122,2450.05,117.636},{0.2642,0.1122,2459.64,118.155},{0.2718,0.1122,2539.4,122.716},{0.2794,0.1122,2455.75,120.655},{0.287,0.1122,2439.17,121.387}, {0.2946,0.1122,2199.96,107.33},{0.3022,0.1122,2186.7,107.647},{0.3098,0.1122,2262.7,112.21},{0.3174,0.1122,2279.94,118.485},{0.325,0.1122,1930.01,103.178},{0.3326,0.1122,1529.32,86.8105},{0.3402,0.1122,1429.2,88.9704},{0.3478,0.1122,1327.09,89.0671},{0.3554,0.1122,1003.95,71.3127},{0.363,0.1122,880.836,67.7372}, {0.3706,0.1122,799.018,67.9731},{0.3782,0.1122,579.027,50.7853},{0.3858,0.1122,409.877,40.6948},{0.3934,0.1122,387.938,42.9504},{0.401,0.1122,311.059,39.1201},{0.4086,0.1122,184.908,28.2357},{0.4162,0.1122,141.284,25.512},{0.4238,0.1122,117.531,25.3576},{0.4314,0.1122,70.5542,22.7903},{0.439,0.1122,1.8653,1.56294}, {0.1046,0.1198,122.803,54.7261},{0.1122,0.1198,585.214,60.3176},{0.1198,0.1198,958.42,68.0274},{0.1274,0.1198,1296.81,80.2426},{0.135,0.1198,1652.82,91.9812},{0.1426,0.1198,1813.6,100.874},{0.1502,0.1198,1624.83,83.7027},{0.1578,0.1198,2060.87,110.195},{0.1654,0.1198,2265.44,118.363},{0.173,0.1198,1966.61,98.7203}, {0.1806,0.1198,2313.74,117.548},{0.1882,0.1198,2268.28,111.085},{0.1958,0.1198,2427.17,119.87},{0.2034,0.1198,2412.37,117.567},{0.211,0.1198,2408.39,116.468},{0.2186,0.1198,2612.09,127.049},{0.2262,0.1198,2509.47,118.988},{0.2338,0.1198,2510.66,119.662},{0.2414,0.1198,2540.13,117.859},{0.249,0.1198,2494.82,117.11}, {0.2566,0.1198,2486.53,116.668},{0.2642,0.1198,2473.6,117.574},{0.2718,0.1198,2197.66,104.296},{0.2794,0.1198,2255.48,107.354},{0.287,0.1198,2281.92,108.633},{0.2946,0.1198,2271.44,111.353},{0.3022,0.1198,2206.64,108.532},{0.3098,0.1198,1922.84,96.0542},{0.3174,0.1198,2102.69,109.349},{0.325,0.1198,1764.43,92.6606}, {0.3326,0.1198,1823.11,107.718},{0.3402,0.1198,1279.41,76.5526},{0.3478,0.1198,1171.02,75.407},{0.3554,0.1198,986.123,67.1215},{0.363,0.1198,855.532,65.2758},{0.3706,0.1198,641.782,53.0888},{0.3782,0.1198,538.943,46.8992},{0.3858,0.1198,410.064,39.005},{0.3934,0.1198,371.821,42.2745},{0.401,0.1198,316.628,40.9401}, {0.4086,0.1198,188.948,26.3843},{0.4162,0.1198,155.578,27.6432},{0.4238,0.1198,93.6411,19.7519},{0.4314,0.1198,77.0973,23.1112},{0.439,0.1198,8.33795,6.31021},{0.097,0.1274,14.6497,10.8822},{0.1046,0.1274,412.77,48.4041},{0.1122,0.1274,1120,85.2193},{0.1198,0.1274,1296.81,80.2426},{0.1274,0.1274,1649.96,90.8916}, {0.135,0.1274,1768.11,94.7219},{0.1426,0.1274,1936.19,104.053},{0.1502,0.1274,1830.98,94.3663},{0.1578,0.1274,2120.4,109.05},{0.1654,0.1274,2131.07,105.564},{0.173,0.1274,2083.58,102.187},{0.1806,0.1274,2192.43,106.942},{0.1882,0.1274,2284.21,108.681},{0.1958,0.1274,2396.77,114.21},{0.2034,0.1274,2551.47,122.822}, {0.211,0.1274,2457.17,114.637},{0.2186,0.1274,2453.9,116.598},{0.2262,0.1274,2771,134.565},{0.2338,0.1274,2871.65,138.068},{0.2414,0.1274,2448.43,113.016},{0.249,0.1274,2506.86,117.74},{0.2566,0.1274,2357.19,108.109},{0.2642,0.1274,2668.99,129.805},{0.2718,0.1274,2495.97,119.158},{0.2794,0.1274,2566.92,127.017}, {0.287,0.1274,2623.68,129.935},{0.2946,0.1274,2324.22,113.105},{0.3022,0.1274,2033.76,99.3636},{0.3098,0.1274,1952.16,95.0077},{0.3174,0.1274,2078.2,107.12},{0.325,0.1274,1759.28,92.0503},{0.3326,0.1274,1665.94,96.2963},{0.3402,0.1274,1403.76,86.969},{0.3478,0.1274,1181.41,75.532},{0.3554,0.1274,910.175,62.4778}, {0.363,0.1274,900.64,69.5731},{0.3706,0.1274,688.456,56.0249},{0.3782,0.1274,499.52,42.9235},{0.3858,0.1274,424.055,40.8838},{0.3934,0.1274,320.162,34.6909},{0.401,0.1274,255.88,32.3747},{0.4086,0.1274,186.476,26.5374},{0.4162,0.1274,135.998,21.8566},{0.4238,0.1274,108.331,21.9528},{0.4314,0.1274,33.6486,9.57991}, {0.439,0.1274,5.91762,2.66459},{0.097,0.135,236.668,37.9439},{0.1046,0.135,900.638,70.5444},{0.1122,0.135,1377.99,85.2253},{0.1198,0.135,1652.82,91.9812},{0.1274,0.135,1768.11,94.7219},{0.135,0.135,1815.62,95.2792},{0.1426,0.135,1720.85,87.1729},{0.1502,0.135,1886.29,94.7125},{0.1578,0.135,2151.73,106.36}, {0.1654,0.135,2143.51,105.286},{0.173,0.135,2384.4,119.916},{0.1806,0.135,2205.42,106.929},{0.1882,0.135,2499.35,121.909},{0.1958,0.135,2327.99,110.174},{0.2034,0.135,2507.09,119.6},{0.211,0.135,2346.87,109.824},{0.2186,0.135,2530.02,120.493},{0.2262,0.135,2305.92,108.267},{0.2338,0.135,2364.93,107.641}, {0.2414,0.135,2568.33,122.45},{0.249,0.135,2521.48,120.318},{0.2566,0.135,2545.73,121.904},{0.2642,0.135,2506.47,119.728},{0.2718,0.135,2281.94,107.736},{0.2794,0.135,2374.25,115.509},{0.287,0.135,2232.29,106.938},{0.2946,0.135,2205.07,106.103},{0.3022,0.135,2093.06,100.943},{0.3098,0.135,2135.54,105.897}, 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{0.1274,0.363,900.64,69.5731},{0.135,0.363,909.579,72.5207},{0.1426,0.363,747.581,57.9245},{0.1502,0.363,609.339,47.243},{0.1578,0.363,825.815,69.7862},{0.1654,0.363,718.957,63.1671},{0.173,0.363,619.356,56.8775},{0.1806,0.363,427.654,40.5032},{0.1882,0.363,505.482,52.574},{0.1958,0.363,458.356,47.1954}, {0.2034,0.363,456.607,54.0606},{0.211,0.363,329.646,41.8489},{0.2186,0.363,293.115,39.4797},{0.2262,0.363,218.357,33.7418},{0.2338,0.363,215.156,39.2899},{0.2414,0.363,155.767,32.8611},{0.249,0.363,123.576,45.1739},{0.2566,0.363,10.9726,8.04218},{0.0742,0.3706,7.47951,5.11123},{0.0818,0.3706,299.352,42.7863}, {0.0894,0.3706,633.975,65.1623},{0.097,0.3706,645.442,60.0347},{0.1046,0.3706,698.06,60.5077},{0.1122,0.3706,799.018,67.9731},{0.1198,0.3706,641.782,53.0888},{0.1274,0.3706,688.456,56.0249},{0.135,0.3706,631.769,50.8996},{0.1426,0.3706,672.493,55.731},{0.1502,0.3706,581.124,48.7565},{0.1578,0.3706,556.871,49.9559}, {0.1654,0.3706,503.856,46.6351},{0.173,0.3706,488.38,47.0866},{0.1806,0.3706,451.481,46.98},{0.1882,0.3706,428.4,47.0606},{0.1958,0.3706,372.135,46.8718},{0.2034,0.3706,407.965,55.4129},{0.211,0.3706,235.46,33.5288},{0.2186,0.3706,244.646,39.8193},{0.2262,0.3706,186.191,34.7254},{0.2338,0.3706,133.974,36.2504}, {0.2414,0.3706,110.652,46.1025},{0.249,0.3706,4.90484,3.45881},{0.0818,0.3782,200.104,36.7135},{0.0894,0.3782,463.161,54.9294},{0.097,0.3782,459.604,43.3752},{0.1046,0.3782,499.625,44.8762},{0.1122,0.3782,579.027,50.7853},{0.1198,0.3782,538.943,46.8992},{0.1274,0.3782,499.52,42.9235},{0.135,0.3782,464.693,39.741}, {0.1426,0.3782,502.58,47.6731},{0.1502,0.3782,533.643,50.7725},{0.1578,0.3782,442.08,42.0049},{0.1654,0.3782,382.141,38.6686},{0.173,0.3782,359.47,36.6025},{0.1806,0.3782,330.607,37.139},{0.1882,0.3782,260.117,30.8426},{0.1958,0.3782,261.098,36.1799},{0.2034,0.3782,230.54,33.4532},{0.211,0.3782,204.02,34.8175}, {0.2186,0.3782,135.973,27.0699},{0.2262,0.3782,56.1965,14.9719},{0.2338,0.3782,73.7942,34.9339},{0.0818,0.3858,93.6766,22.5144},{0.0894,0.3858,336.435,45.9563},{0.097,0.3858,446.312,49.3932},{0.1046,0.3858,436.069,44.7397},{0.1122,0.3858,409.877,40.6948},{0.1198,0.3858,410.064,39.005},{0.1274,0.3858,424.055,40.8838}, {0.135,0.3858,491.69,47.9068},{0.1426,0.3858,395.725,40.0251},{0.1502,0.3858,263.994,27.1664},{0.1578,0.3858,375.303,39.4153},{0.1654,0.3858,296.634,35.3453},{0.173,0.3858,248.811,28.3347},{0.1806,0.3858,315.643,43.2269},{0.1882,0.3858,246.182,33.9548},{0.1958,0.3858,216.147,34.9026},{0.2034,0.3858,141.354,24.5043}, {0.211,0.3858,99.9317,21.5535},{0.2186,0.3858,76.6263,22.05},{0.2262,0.3858,18.5386,10.3082},{0.0818,0.3934,29.7539,11.0765},{0.0894,0.3934,253.749,40.9123},{0.097,0.3934,272.95,33.4076},{0.1046,0.3934,358.675,43.3165},{0.1122,0.3934,387.938,42.9504},{0.1198,0.3934,371.821,42.2745},{0.1274,0.3934,320.162,34.6909}, {0.135,0.3934,287.136,28.8762},{0.1426,0.3934,354.541,39.7653},{0.1502,0.3934,279.62,31.4042},{0.1578,0.3934,273.869,34.8753},{0.1654,0.3934,246.178,32.0659},{0.173,0.3934,302.333,47.4259},{0.1806,0.3934,198.156,30.1207},{0.1882,0.3934,149.053,25.3867},{0.1958,0.3934,105.157,20.4525},{0.2034,0.3934,146.998,39.6391}, {0.211,0.3934,55.6588,17.3197},{0.2186,0.3934,33.3093,27.6147},{0.0818,0.401,7.06652,6.28563},{0.0894,0.401,100.714,21.7858},{0.097,0.401,170.925,24.6781},{0.1046,0.401,258.845,32.7632},{0.1122,0.401,311.059,39.1201},{0.1198,0.401,316.628,40.9401},{0.1274,0.401,255.88,32.3747},{0.135,0.401,261.195,32.6463}, {0.1426,0.401,270.544,34.3547},{0.1502,0.401,247.173,32.964},{0.1578,0.401,184.085,25.7698},{0.1654,0.401,233.973,37.7601},{0.173,0.401,135.852,21.8179},{0.1806,0.401,95.052,17.5678},{0.1882,0.401,70.6506,16.2833},{0.1958,0.401,104.732,33.0615},{0.2034,0.401,26.5984,10.1283},{0.211,0.401,2.3816,2.1026}, {0.0894,0.4086,87.7671,27.5485},{0.097,0.4086,156.779,29.6289},{0.1046,0.4086,159.341,25.0929},{0.1122,0.4086,184.908,28.2357},{0.1198,0.4086,188.948,26.3843},{0.1274,0.4086,186.476,26.5374},{0.135,0.4086,195.949,27.5871},{0.1426,0.4086,196.863,29.0933},{0.1502,0.4086,128.934,19.2099},{0.1578,0.4086,187.651,34.7021}, {0.1654,0.4086,112.141,21.5852},{0.173,0.4086,99.7201,21.5681},{0.1806,0.4086,86.0825,21.6372},{0.1882,0.4086,120.404,44.6872},{0.1958,0.4086,31.2799,16.1845},{0.0894,0.4162,6.42152,3.74763},{0.097,0.4162,70.3786,17.1528},{0.1046,0.4162,122.247,24.3663},{0.1122,0.4162,141.284,25.512},{0.1198,0.4162,155.578,27.6432}, {0.1274,0.4162,135.998,21.8566},{0.135,0.4162,127.141,23.0946},{0.1426,0.4162,113.674,21.1265},{0.1502,0.4162,146.872,29.9235},{0.1578,0.4162,92.2553,20.849},{0.1654,0.4162,81.3937,18.8963},{0.173,0.4162,61.4856,18.6917},{0.1806,0.4162,21.7324,7.88759},{0.1882,0.4162,62.1706,46.75},{0.097,0.4238,69.4424,37.7365}, {0.1046,0.4238,44.7217,11.2495},{0.1122,0.4238,117.531,25.3576},{0.1198,0.4238,93.6411,19.7519},{0.1274,0.4238,108.331,21.9528},{0.135,0.4238,91.2193,18.8638},{0.1426,0.4238,90.0727,19.3713},{0.1502,0.4238,77.4007,19.4417},{0.1578,0.4238,32.5991,9.55119},{0.1654,0.4238,41.2141,16.0497},{0.173,0.4238,30.0512,21.3808}, {0.1046,0.4314,56.2404,28.0247},{0.1122,0.4314,70.5542,22.7903},{0.1198,0.4314,77.0973,23.1112},{0.1274,0.4314,33.6486,9.57991},{0.135,0.4314,56.009,17.0844},{0.1426,0.4314,53.1886,16.1534},{0.1502,0.4314,44.6754,18.4967},{0.1578,0.4314,28.8984,18.0197},{0.1654,0.4314,7.61761,7.47231},{0.1122,0.439,1.8653,1.56294}, {0.1198,0.439,8.33795,6.31021},{0.1274,0.439,5.91762,2.66459},{0.135,0.439,5.62522,4.12154},{0.1426,0.439,3.71108,3.35619}} Table G.8: Acceptance and eﬃciency corrected Dalitz Plot 3π0 , for MMpp =0.7 − 0.8 GeV/c2 (Fig. 95). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of the absolutenormalization of 19% have to beincluded. Fullyexpandable version ofthe tableis availablein the attached electronic version of the thesis. Jagiellonian University 228 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Da itz M ot 3pi0 M'{2}(pi'{0}_{1}pi'{0}_{2}) versis M'{2}(pi'{0}_{2}pi'{0}_{3}) -Eac. event is li ed six times Acceptance and Ellicienc Corrected lor MM_{pp}00.8-0.9 Ge1/c'{2} Error ol M'{2}(pi'{0}_{2}pi'{0}_{3}) 0.0076 Ge1'{2}/c'{4} Error ol M'{2}(pi'{0}_{1}pi'{0}_{2}) 0.0076 Ge1'{2}/c'{4} T.e data are in t.e lo o ing lormat {{M'{2}(pi'{0}_{2}pi'{0}_{3}) [Ge1'{2}/c'{4}], M'{2}(pi'{0}_{1}pi'{0}_{2}) [Ge1'{2}/c'{4}], sigma [mil], Error ol sigma [mil] },...} {{0.1996,0.078,1.11463,1.56555},{0.2148,0.078,29.7945,6.89685},{0.23,0.078,91.1146,10.8706},{0.2452,0.078,206.896,17.3053},{0.2604,0.078,319.49,21.7803},{0.2756,0.078,437.862,26.1315},{0.2908,0.078,484.918,25.4043},{0.306,0.078,550.259,27.8981},{0.3212,0.078,672.058,32.4128},{0.3364,0.078,675.745,32.2104}, {0.3516,0.078,733.434,36.2814},{0.3668,0.078,613.758,31.0891},{0.382,0.078,573.977,30.5733},{0.3972,0.078,433.795,26.0353},{0.4124,0.078,275.181,18.8589},{0.4276,0.078,182.123,14.8278},{0.4428,0.078,123.652,12.5835},{0.458,0.078,61.008,7.77846},{0.4732,0.078,37.1574,6.27166},{0.4884,0.078,23.1172,6.98904}, {0.5036,0.078,5.21031,2.44401},{0.5188,0.078,0.0945187,0.103864},{0.1692,0.0932,10.3075,2.30187},{0.1844,0.0932,95.861,9.2912},{0.1996,0.0932,244.818,16.5765},{0.2148,0.0932,355.102,19.7768},{0.23,0.0932,455.732,22.6882},{0.2452,0.0932,483.173,21.9438},{0.2604,0.0932,615.046,27.6051},{0.2756,0.0932,590.613,25.1192}, {0.2908,0.0932,639.934,27.4212},{0.306,0.0932,730.181,31.1472},{0.3212,0.0932,707.028,29.9057},{0.3364,0.0932,680.164,27.5597},{0.3516,0.0932,708.372,29.2262},{0.3668,0.0932,661.263,28.2457},{0.382,0.0932,612.032,25.9119},{0.3972,0.0932,616.338,27.2738},{0.4124,0.0932,570.383,26.5166},{0.4276,0.0932,455.227,22.5115}, {0.4428,0.0932,290.36,16.4775},{0.458,0.0932,244.421,15.8421},{0.4732,0.0932,142.531,11.4461},{0.4884,0.0932,99.8162,9.6208},{0.5036,0.0932,64.0351,7.67582},{0.5188,0.0932,36.063,6.27109},{0.534,0.0932,14.2948,3.37259},{0.5492,0.0932,2.12469,1.12511},{0.1388,0.1084,5.85016,2.48385},{0.154,0.1084,76.8611,8.11118}, {0.1692,0.1084,248.152,15.5971},{0.1844,0.1084,325.145,17.165},{0.1996,0.1084,369.995,18.1293},{0.2148,0.1084,467.719,22.0876},{0.23,0.1084,460.628,20.4835},{0.2452,0.1084,560.44,24.3349},{0.2604,0.1084,545.632,22.7299},{0.2756,0.1084,598.159,24.4714},{0.2908,0.1084,661.434,27.028},{0.306,0.1084,646.586,26.3828}, {0.3212,0.1084,643.265,25.3738},{0.3364,0.1084,582.68,23.0161},{0.3516,0.1084,643.752,26.1633},{0.3668,0.1084,655.253,26.8797},{0.382,0.1084,589.116,24.3474},{0.3972,0.1084,547.162,23.8072},{0.4124,0.1084,495.54,21.8105},{0.4276,0.1084,433.375,19.6533},{0.4428,0.1084,371.084,18.5575},{0.458,0.1084,280.54,15.8155}, {0.4732,0.1084,181.644,11.6947},{0.4884,0.1084,135.724,10.2672},{0.5036,0.1084,99.8336,9.01057},{0.5188,0.1084,59.0028,6.58913},{0.534,0.1084,35.9109,5.21457},{0.5492,0.1084,13.8742,2.76189},{0.5644,0.1084,6.29255,2.2155},{0.1236,0.1236,13.7261,3.64761},{0.1388,0.1236,148.053,11.3703},{0.154,0.1236,254.789,14.299}, {0.1692,0.1236,309.447,15.7924},{0.1844,0.1236,324.068,15.5491},{0.1996,0.1236,409.632,18.5165},{0.2148,0.1236,475.909,20.5021},{0.23,0.1236,520.459,21.5182},{0.2452,0.1236,549.836,22.3401},{0.2604,0.1236,576.607,22.9915},{0.2756,0.1236,572.715,22.5208},{0.2908,0.1236,570.336,22.149},{0.306,0.1236,595.04,23.0369}, {0.3212,0.1236,614.246,23.0931},{0.3364,0.1236,598.839,23.4405},{0.3516,0.1236,617.456,24.1865},{0.3668,0.1236,561.959,22.0482},{0.382,0.1236,550.217,21.9908},{0.3972,0.1236,516.31,21.167},{0.4124,0.1236,454.41,19.1641},{0.4276,0.1236,402.752,17.7906},{0.4428,0.1236,408.417,19.4688},{0.458,0.1236,269.295,14.1509}, 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{0.3668,0.3364,198.01,13.1592},{0.382,0.3364,159.691,12.4506},{0.3972,0.3364,120.295,11.3665},{0.4124,0.3364,73.3065,8.89433},{0.4276,0.3364,44.2265,6.80649},{0.4428,0.3364,23.3695,7.02071},{0.458,0.3364,0.732625,0.9325},{0.078,0.3516,733.434,36.2814},{0.0932,0.3516,708.372,29.2262},{0.1084,0.3516,643.751,26.1632}, {0.1236,0.3516,617.456,24.1865},{0.1388,0.3516,561.569,21.3871},{0.154,0.3516,529.39,19.9772},{0.1692,0.3516,508.712,19.5506},{0.1844,0.3516,556.604,21.3046},{0.1996,0.3516,543.426,20.4017},{0.2148,0.3516,510.922,19.5171},{0.23,0.3516,511.857,19.0267},{0.2452,0.3516,578.511,22.3895},{0.2604,0.3516,590.989,22.4313}, {0.2756,0.3516,575.678,22.6942},{0.2908,0.3516,506.209,22.2},{0.306,0.3516,421.171,20.0436},{0.3212,0.3516,346.977,18.1853},{0.3364,0.3516,275.52,16.3985},{0.3516,0.3516,216.199,14.4663},{0.3668,0.3516,156.294,11.9344},{0.382,0.3516,112.923,10.5604},{0.3972,0.3516,73.7929,8.52808},{0.4124,0.3516,51.9417,7.94228}, {0.4276,0.3516,17.0887,4.40456},{0.4428,0.3516,1.04591,1.0235},{0.078,0.3668,613.758,31.0891},{0.0932,0.3668,661.263,28.2457},{0.1084,0.3668,655.253,26.8797},{0.1236,0.3668,561.959,22.0482},{0.1388,0.3668,514.862,19.576},{0.154,0.3668,544.264,21.2848},{0.1692,0.3668,511.224,19.4691},{0.1844,0.3668,516.335,19.9139}, {0.1996,0.3668,496.387,19.0528},{0.2148,0.3668,542.52,20.7892},{0.23,0.3668,563.748,21.5616},{0.2452,0.3668,565.229,22.1919},{0.2604,0.3668,491.417,20.0899},{0.2756,0.3668,410.489,17.6235},{0.2908,0.3668,415.928,20.2426},{0.306,0.3668,314.64,16.7364},{0.3212,0.3668,275.411,16.18},{0.3364,0.3668,198.01,13.1592}, {0.3516,0.3668,156.294,11.9344},{0.3668,0.3668,151.883,15.3978},{0.382,0.3668,88.496,10.4649},{0.3972,0.3668,42.6024,6.77605},{0.4124,0.3668,21.8751,5.72653},{0.4276,0.3668,5.05644,4.56619},{0.078,0.382,573.977,30.5733},{0.0932,0.382,612.032,25.9119},{0.1084,0.382,589.116,24.3474},{0.1236,0.382,550.217,21.9908}, {0.1388,0.382,523.168,20.7944},{0.154,0.382,506.332,19.9752},{0.1692,0.382,480.471,18.6776},{0.1844,0.382,480.198,18.6842},{0.1996,0.382,485.218,19.1639},{0.2148,0.382,570.729,22.5809},{0.23,0.382,514.658,20.6243},{0.2452,0.382,486.748,20.2291},{0.2604,0.382,458.284,20.8966},{0.2756,0.382,346.107,16.8742}, {0.2908,0.382,316.662,17.0147},{0.306,0.382,252.462,15.3018},{0.3212,0.382,189.772,12.6666},{0.3364,0.382,159.691,12.4506},{0.3516,0.382,112.923,10.5604},{0.3668,0.382,88.496,10.4649},{0.382,0.382,61.5721,9.14579},{0.3972,0.382,24.6857,6.14187},{0.4124,0.382,2.01419,1.50021},{0.078,0.3972,433.795,26.0353}, {0.0932,0.3972,616.338,27.2738},{0.1084,0.3972,547.162,23.8072},{0.1236,0.3972,516.31,21.167},{0.1388,0.3972,450.764,18.1287},{0.154,0.3972,484.751,20.0944},{0.1692,0.3972,474.048,18.7545},{0.1844,0.3972,459.518,18.4683},{0.1996,0.3972,515.064,20.7184},{0.2148,0.3972,476.152,19.7598},{0.23,0.3972,423.535,18.0122}, {0.2452,0.3972,399.103,18.5481},{0.2604,0.3972,328.452,16.6049},{0.2756,0.3972,268.977,14.8725},{0.2908,0.3972,217.814,13.2853},{0.306,0.3972,186.587,13.0521},{0.3212,0.3972,150.11,12.3615},{0.3364,0.3972,120.295,11.3665},{0.3516,0.3972,73.7929,8.52808},{0.3668,0.3972,42.6024,6.77605},{0.382,0.3972,24.6857,6.14187}, {0.078,0.4124,275.181,18.8589},{0.0932,0.4124,570.383,26.5166},{0.1084,0.4124,495.54,21.8105},{0.1236,0.4124,454.41,19.1641},{0.1388,0.4124,425.604,17.8391},{0.154,0.4124,395.253,16.376},{0.1692,0.4124,414.585,17.3034},{0.1844,0.4124,418.922,17.6301},{0.1996,0.4124,430.087,18.298},{0.2148,0.4124,382.172,17.2323}, {0.23,0.4124,326.478,15.9281},{0.2452,0.4124,281.114,14.5143},{0.2604,0.4124,235.683,13.4971},{0.2756,0.4124,206.161,13.1372},{0.2908,0.4124,173.533,12.1398},{0.306,0.4124,135.245,11.0044},{0.3212,0.4124,101.538,9.37982},{0.3364,0.4124,73.3065,8.89433},{0.3516,0.4124,51.9417,7.94228},{0.3668,0.4124,21.8751,5.72653}, {0.382,0.4124,2.01419,1.50021},{0.078,0.4276,182.123,14.8278},{0.0932,0.4276,455.227,22.5115},{0.1084,0.4276,433.375,19.6533},{0.1236,0.4276,402.752,17.7906},{0.1388,0.4276,410.861,17.9989},{0.154,0.4276,364.608,15.7223},{0.1692,0.4276,378.074,16.5711},{0.1844,0.4276,347.896,15.7035},{0.1996,0.4276,330.726,16.2458}, {0.2148,0.4276,292.265,15.1822},{0.23,0.4276,234.652,12.6293},{0.2452,0.4276,223.509,13.1864},{0.2604,0.4276,187.727,12.3713},{0.2756,0.4276,169.584,12.3996},{0.2908,0.4276,118.599,9.92273},{0.306,0.4276,109.403,11.0783},{0.3212,0.4276,61.6029,7.42243},{0.3364,0.4276,44.2265,6.80649},{0.3516,0.4276,17.0887,4.40456}, {0.3668,0.4276,5.05644,4.56619},{0.078,0.4428,123.652,12.5835},{0.0932,0.4428,290.36,16.4775},{0.1084,0.4428,371.084,18.5575},{0.1236,0.4428,408.417,19.4688},{0.1388,0.4428,363.229,16.6481},{0.154,0.4428,354.438,17.0612},{0.1692,0.4428,314.354,15.1885},{0.1844,0.4428,253.2,12.9554},{0.1996,0.4428,276.305,14.9629}, {0.2148,0.4428,209.212,12.5712},{0.23,0.4428,200.139,12.7713},{0.2452,0.4428,161.531,11.3805},{0.2604,0.4428,127.73,9.39854},{0.2756,0.4428,132.395,11.9168},{0.2908,0.4428,87.8163,9.23651},{0.306,0.4428,55.3502,7.1837},{0.3212,0.4428,37.0586,6.32321},{0.3364,0.4428,23.3695,7.02071},{0.3516,0.4428,1.04591,1.0235}, {0.078,0.458,61.008,7.77846},{0.0932,0.458,244.421,15.8421},{0.1084,0.458,280.54,15.8155},{0.1236,0.458,269.295,14.1509},{0.1388,0.458,257.171,13.6065},{0.154,0.458,241.256,13.4636},{0.1692,0.458,240.376,13.7078},{0.1844,0.458,182.779,10.8038},{0.1996,0.458,200.055,12.4275},{0.2148,0.458,152.193,10.2451}, {0.23,0.458,145.917,10.3841},{0.2452,0.458,128.378,10.2822},{0.2604,0.458,98.451,8.71627},{0.2756,0.458,62.9612,6.7566},{0.2908,0.458,47.5112,6.3063},{0.306,0.458,29.4378,5.80966},{0.3212,0.458,11.2693,3.19972},{0.3364,0.458,0.732625,0.9325},{0.078,0.4732,37.1574,6.27166},{0.0932,0.4732,142.531,11.4461}, {0.1084,0.4732,181.644,11.6947},{0.1236,0.4732,209.986,12.9067},{0.1388,0.4732,179.879,10.7042},{0.154,0.4732,162.612,9.97856},{0.1692,0.4732,160.054,10.5852},{0.1844,0.4732,149.289,10.2388},{0.1996,0.4732,131.806,9.76234},{0.2148,0.4732,111.38,8.94705},{0.23,0.4732,107.081,9.74696},{0.2452,0.4732,83.8534,8.1542}, {0.2604,0.4732,61.126,7.15099},{0.2756,0.4732,35.7154,5.46181},{0.2908,0.4732,30.848,5.88024},{0.306,0.4732,6.88974,2.15445},{0.078,0.4884,23.1172,6.98904},{0.0932,0.4884,99.8162,9.62079},{0.1084,0.4884,135.724,10.2672},{0.1236,0.4884,139.394,9.79079},{0.1388,0.4884,142.443,9.73912},{0.154,0.4884,126.901,9.12214}, {0.1692,0.4884,114.727,8.47104},{0.1844,0.4884,112.917,9.0277},{0.1996,0.4884,85.9994,7.66096},{0.2148,0.4884,75.0009,7.06632},{0.23,0.4884,87.4159,9.57525},{0.2452,0.4884,48.9255,5.99768},{0.2604,0.4884,42.5671,6.96857},{0.2756,0.4884,23.6255,5.87618},{0.2908,0.4884,6.34425,2.68719},{0.078,0.5036,5.21031,2.44401}, {0.0932,0.5036,64.0351,7.67582},{0.1084,0.5036,99.8336,9.01057},{0.1236,0.5036,83.0313,6.95703},{0.1388,0.5036,97.3468,7.72241},{0.154,0.5036,90.3812,7.6127},{0.1692,0.5036,85.3067,7.36646},{0.1844,0.5036,73.4334,7.08561},{0.1996,0.5036,70.982,7.57071},{0.2148,0.5036,43.6189,5.031},{0.23,0.5036,35.109,4.79955}, {0.2452,0.5036,24.4714,4.37976},{0.2604,0.5036,26.0647,6.37618},{0.2756,0.5036,2.72321,1.13922},{0.078,0.5188,0.0945187,0.103864},{0.0932,0.5188,36.063,6.27109},{0.1084,0.5188,59.0028,6.58913},{0.1236,0.5188,72.716,7.29625},{0.1388,0.5188,62.0301,6.13131},{0.154,0.5188,65.2164,6.54917},{0.1692,0.5188,52.0761,5.35445}, {0.1844,0.5188,50.0954,5.75913},{0.1996,0.5188,46.9338,6.40338},{0.2148,0.5188,26.834,4.3914},{0.23,0.5188,16.9176,3.78487},{0.2452,0.5188,11.7808,3.31591},{0.2604,0.5188,2.41563,1.22034},{0.0932,0.534,14.2948,3.37259},{0.1084,0.534,35.9109,5.21457},{0.1236,0.534,32.9009,4.21369},{0.1388,0.534,44.2847,5.49628}, {0.154,0.534,38.5855,4.98875},{0.1692,0.534,32.9675,4.40264},{0.1844,0.534,29.0599,4.78967},{0.1996,0.534,12.8218,2.18219},{0.2148,0.534,19.5218,4.61073},{0.23,0.534,7.87705,2.76856},{0.2452,0.534,1.40784,1.04981},{0.0932,0.5492,2.12469,1.12511},{0.1084,0.5492,13.8742,2.76189},{0.1236,0.5492,23.8922,3.77977}, {0.1388,0.5492,24.8603,3.96066},{0.154,0.5492,24.1278,3.68267},{0.1692,0.5492,26.0451,4.64906},{0.1844,0.5492,14.4514,3.4049},{0.1996,0.5492,6.55786,1.64513},{0.2148,0.5492,2.83706,1.28224},{0.1084,0.5644,6.29255,2.2155},{0.1236,0.5644,6.87072,2.09481},{0.1388,0.5644,10.1397,2.29526},{0.154,0.5644,7.88754,2.04463}, {0.1692,0.5644,6.53783,2.19514},{0.1844,0.5644,4.71495,1.8719},{0.1996,0.5644,1.32437,0.798851},{0.1236,0.5796,1.1685,0.719288},{0.1388,0.5796,5.06709,2.07976},{0.154,0.5796,2.82324,1.44086},{0.1692,0.5796,1.11737,1.13421}} Table G.9: Acceptance and eﬃciency corrected Dalitz Plot 3π0 , for MMpp =0.8 − 0.9 GeV/c2 (Fig. 95). The errors of invariant masses are determined by selected bin sizes and chosen as a half of the bin size. Additionalglobal uncertainties of theabsolutenormalization of 19% have to beincluded. Fully expandable version of thetableis availablein the attached electronic version of the thesis. Jagiellonian University 229 Benedykt R. Jany APPENDIX G DATA TABLES – RESULTS Da itz M ot 3pi0 M'{2}(pi'{0}_{1}pi'{0}_{2}) versis M'{2}(pi'{0}_{2}pi'{0}_{3}) -Eac. event is li ed six times Acceptance and Ellicienc Corrected lor MM_{pp}00.9-1.0 Ge1/c'{2} Error ol M'{2}(pi'{0}_{2}pi'{0}_{3}) 0.0095 Ge1'{2}/c'{4} Error ol M'{2}(pi'{0}_{1}pi'{0}_{2}) 0.0095 Ge1'{2}/c'{4} T.e data are in t.e lo o ing lormat {{M'{2}(pi'{0}_{2}pi'{0}_{3}) [Ge1'{2}/c'{4}], M'{2}(pi'{0}_{1}pi'{0}_{2}) [Ge1'{2}/c'{4}], sigma [mil], Error ol sigma [mil] },...} {{0.3155,0.0685,3.97196,3.17704},{0.3345,0.0685,3.11625,1.16597},{0.3535,0.0685,11.0104,2.85879},{0.3725,0.0685,12.6238,3.15231},{0.3915,0.0685,20.3641,4.46382},{0.4105,0.0685,23.6206,5.17257},{0.4295,0.0685,22.9771,4.87041},{0.4485,0.0685,24.9041,5.92463},{0.4675,0.0685,13.4976,4.14261},{0.4865,0.0685,9.16804,3.08867}, 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{0.3345,0.4675,52.6223,5.93617},{0.3535,0.4675,25.371,3.58279},{0.3725,0.4675,25.0652,4.34013},{0.3915,0.4675,14.4946,3.2893},{0.4105,0.4675,16.9468,5.76736},{0.4295,0.4675,4.85503,2.06573},{0.4485,0.4675,9.05805,9.61934},{0.4675,0.4675,0.689202,0.634858},{0.0685,0.4865,9.16804,3.08867},{0.0875,0.4865,62.9978,7.73089}, {0.1065,0.4865,51.3271,5.40147},{0.1255,0.4865,49.5775,4.65713},{0.1445,0.4865,42.5685,3.96548},{0.1635,0.4865,45.75,4.23623},{0.1825,0.4865,42.4659,4.03904},{0.2015,0.4865,51.7767,4.78348},{0.2205,0.4865,50.8278,4.54367},{0.2395,0.4865,43.3495,3.99624},{0.2585,0.4865,44.4951,4.12173},{0.2775,0.4865,46.8065,4.49029}, {0.2965,0.4865,47.8196,5.04798},{0.3155,0.4865,42.1452,4.91878},{0.3345,0.4865,31.4954,4.44731},{0.3535,0.4865,21.5933,4.00369},{0.3725,0.4865,12.4137,2.91113},{0.3915,0.4865,8.57888,2.50074},{0.4105,0.4865,6.5802,4.3546},{0.4295,0.4865,10.6419,9.93439},{0.0685,0.5055,3.11593,1.61466},{0.0875,0.5055,38.317,4.53889}, {0.1065,0.5055,42.466,4.5346},{0.1255,0.5055,41.5386,4.09707},{0.1445,0.5055,36.4374,3.51207},{0.1635,0.5055,36.7354,3.73007},{0.1825,0.5055,36.7906,3.56154},{0.2015,0.5055,41.265,4.14047},{0.2205,0.5055,40.0487,3.85742},{0.2395,0.5055,39.9475,3.83309},{0.2585,0.5055,32.3131,3.14304},{0.2775,0.5055,37.9929,3.87726}, {0.2965,0.5055,33.7773,4.42523},{0.3155,0.5055,20.9533,3.05106},{0.3345,0.5055,12.8353,2.49834},{0.3535,0.5055,13.7911,3.57729},{0.3725,0.5055,6.29409,1.86656},{0.3915,0.5055,8.79837,4.93488},{0.4105,0.5055,1.66468,1.31072},{0.0685,0.5245,18.7201,16.0054},{0.0875,0.5245,53.5248,7.3026},{0.1065,0.5245,39.2646,4.60549}, {0.1255,0.5245,40.7495,4.48593},{0.1445,0.5245,29.685,3.29943},{0.1635,0.5245,31.7476,3.53904},{0.1825,0.5245,32.5155,3.56129},{0.2015,0.5245,32.4464,3.55861},{0.2205,0.5245,36.64,4.05136},{0.2395,0.5245,35.0632,3.74227},{0.2585,0.5245,34.6477,3.94509},{0.2775,0.5245,33.2849,4.70461},{0.2965,0.5245,19.4361,3.21547}, 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{0.2395,0.5625,16.2594,2.86893},{0.2585,0.5625,7.0335,1.3756},{0.2775,0.5625,9.44074,2.6054},{0.2965,0.5625,3.14873,1.05957},{0.3155,0.5625,3.38682,1.38636},{0.3345,0.5625,3.5955,2.50816},{0.3535,0.5625,4.22558,3.50343},{0.0875,0.5815,6.7379,2.35075},{0.1065,0.5815,20.7582,3.2207},{0.1255,0.5815,16.2812,2.33075}, {0.1445,0.5815,19.9983,2.68218},{0.1635,0.5815,20.6327,2.7522},{0.1825,0.5815,17.7264,2.52971},{0.2015,0.5815,13.2101,2.11258},{0.2205,0.5815,15.5719,2.81415},{0.2395,0.5815,8.01947,1.81122},{0.2585,0.5815,8.02287,2.35195},{0.2775,0.5815,5.06431,2.07099},{0.2965,0.5815,3.78826,1.6233},{0.3155,0.5815,1.34401,0.999205}, {0.0875,0.6005,1.4239,0.785351},{0.1065,0.6005,6.63276,1.50157},{0.1255,0.6005,13.3396,2.25221},{0.1445,0.6005,7.13794,1.25564},{0.1635,0.6005,7.3382,1.33131},{0.1825,0.6005,10.1139,1.96871},{0.2015,0.6005,6.225,1.32059},{0.2205,0.6005,4.79651,1.20365},{0.2395,0.6005,8.57177,3.09542},{0.2585,0.6005,2.34191,0.987549}, {0.2775,0.6005,2.0145,0.948254},{0.2965,0.6005,1.01986,0.921736},{0.0875,0.6195,0.896828,0.704717},{0.1065,0.6195,5.21076,1.67351},{0.1255,0.6195,6.53643,1.56298},{0.1445,0.6195,5.90938,1.29983},{0.1635,0.6195,7.77485,1.81502},{0.1825,0.6195,5.05688,1.36253},{0.2015,0.6195,5.23207,1.55929},{0.2205,0.6195,3.13266,1.16276}, {0.2395,0.6195,2.80524,1.25995},{0.2585,0.6195,0.64709,0.523234},{0.2775,0.6195,0.466253,0.393135},{0.2965,0.6195,1.14837,1.36904},{0.1065,0.6385,3.04008,1.54642},{0.1255,0.6385,2.43984,0.850332},{0.1445,0.6385,5.49462,1.57809},{0.1635,0.6385,6.28299,2.26355},{0.1825,0.6385,2.2207,0.780873},{0.2015,0.6385,2.79979,1.13026}, {0.2205,0.6385,1.39932,0.895534},{0.2395,0.6385,2.17456,1.49124},{0.2775,0.6385,0.479937,0.665312},{0.1065,0.6575,0.304439,0.342498},{0.1255,0.6575,3.05421,1.69894},{0.1445,0.6575,2.4132,1.09994},{0.1635,0.6575,1.11023,0.476283},{0.1825,0.6575,1.93575,0.976311},{0.2015,0.6575,1.27863,0.871074},{0.2205,0.6575,1.51767,1.70034}, {0.1065,0.6765,0.160787,0.194473},{0.1255,0.6765,0.53072,0.37809},{0.1445,0.6765,0.188002,0.20369},{0.1825,0.6765,0.647467,0.500329},{0.2015,0.6765,0.162418,0.183953},{0.2205,0.6765,2.66589,3.71589},{0.1445,0.6955,0.275476,0.356917},{0.1635,0.6955,0.0453464,0.0485538}} Table G.10: Acceptance and eﬃciency corrected Dalitz Plot 3π0 , for MMpp =0.9 − 1.0 GeV/c2 (Fig. 95). 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First of allI wouldliketo express my enormousgratitudeto my supervisorProf. ZbigniewRudy for ﬁrstintroductiontothe secrets ofdata analysis,for the time spent on manydiscussions, for his guidance, encouragement, support and patience. I am also very grateful to Dr. Volker Hejny from FZ-J¨ulich, for showing me the aspects of the advanced data analysis and for the guidance through the analysis steps. I am also verygratefultoProf. BugusławKamysfor allowing me to prepare this dissertation in the Nuclear Physics Departmentof theJagiellonianUniversity andforhisadvices, support and discussions. I would like also to thank Prof. Hans Str¨oher for possibility of stay in Forschungszentrum-J¨ulich and for giving me the opportunity to work with WASA at COSY collaboration. I want to express my appreciation to Prof. Lucjan Jarczyk for many interesting ideas and fruitful discussions. I also thank all colleagues from WASA at COSY collaboration, specially: Dr. AndrzejKupsc,Dr. hab. SusanSchadmand, Dr. MagnusWolke,Dr. hab. FrankGoldenbaum,Dr. Christian Pauly and Dr. Christoph Redmer. I thank all my colleagues from the IKP ForschungszentrumJ¨ulich and from the Nuclear Physics Department of the Jagiellonian University for the pleasant atmosphere of daily work. I also want to express my gratitude to my beloved wife for sharing daily life and fascination to physics with me. 255