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U-Spin Symmetry Test of the Σ∗+ Electromagnetic Decay
A dissertation presented to
the faculty of
the College of Arts and Sciences of Ohio University
In partial fulfillment
of the requirements for the degree
Doctor of Philosophy
Dustin M. Keller
November 2010
© 2010 Dustin M. Keller. All Rights Reserved.
2
This dissertation titled
U-Spin Symmetry Test of the Σ∗+ Electromagnetic Decay
by
DUSTIN M. KELLER
has been approved for
the Department of Physics and Astronomy
and the College of Arts and Sciences by
Kenneth H. Hicks
Professor of Physics and Astronomy
Benjamin M. Ogles
Interim Dean, College of Arts And Sciences
3
ABSTRACT
KELLER, DUSTIN M., Ph.D., November 2010, Physics
U-Spin Symmetry Test of the Σ∗+ Electromagnetic Decay (288 pp.)
Director of Dissertation: Kenneth H. Hicks
This dissertation presents analysis for electromagnetic decay of the Σ0(1385) from
the reaction γ p→ K+Σ∗0. Also presented is the first ever measurement of the
electromagnetic decay of the Σ+(1385) from the reaction γ p→ K0Σ∗+. Both results are
extracted from the g11a data set taken using the CLAS detector at Thomas Jefferson
National Accelerator Facility. A real photon beam with a maximum energy of 3.8 GeV
was incident on a liquid hydrogen target during the experiment resulting in the
photoproduction of the kaon and Σ∗ hyperons. Kinematic fitting is used to separate signal
from background in each case. For the first time, a method to kinematically fit the neutron
in the Electromagnetic Calorimeter (EC) of CLAS was performed, leading to a high
statistics study of the neutron resolutions in the EC. New techniques in neutron resolution
matching for Monte Carlo simulation using dynamic variable smearing are also
developed. The results from the Σ0(1385) electromagnetic decay lead to smaller statistical
and systematic uncertainties than the previous measurement by Taylor et al. A U-spin
symmetry test using the U-spin SU(3) multiplet representation gave a prediction for the
Σ∗+→ Σ+γ partial width and the Σ∗0→ Λγ partial width. The latter agrees, within the
experimental uncertainties, with the prediction from U-spin symmetry, but the former
reaction is much smaller than its prediction.
Approved:
Kenneth H. Hicks
Professor of Physics and Astronomy
4
TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.1 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 The Constituent Quark Model . . . . . . . . . . . . . . . . . . . . . . . . 221.4 Electromagnetic decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.5 The CLAS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6 Previous Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 281.7 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Theoretical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.1 The NRQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.1.2 The RCQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1.3 The χCQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.4 The MIT Bag Model . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.5 The Soliton Model . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.6 The Skyrme Model . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1.7 The Algebraic Model . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 U-Spin Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2.2 U-Spin prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Meson Cloud effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Jefferson Lab, CEBAF and CLAS detector . . . . . . . . . . . . . . . . . . . . . 533.1 Continuous Electron Beam Accelerator Facility . . . . . . . . . . . . . . . 543.2 The Bremsstrahlung Photon Tagger . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 The Radiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 The Magnetic Spectrometer . . . . . . . . . . . . . . . . . . . . . 603.2.3 The Hodoscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4 The Tagger Readout . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 The CLAS Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.1 The g11a Cryotarget . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.2 The Start Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.3 The Superconducting Toroidal Magnet . . . . . . . . . . . . . . . 663.3.4 The Drift Chambers . . . . . . . . . . . . . . . . . . . . . . . . . 68
5
3.3.5 The Time-of-Flight Detector . . . . . . . . . . . . . . . . . . . . . 703.3.6 The Forward Electromagnetic Calorimeter . . . . . . . . . . . . . . 72
3.4 The Beamline Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.5 The g11a Trigger and Data Acquisition . . . . . . . . . . . . . . . . . . . 753.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
I Electromagnetic decay of the Σ∗0 78
4 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.1 Run Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Energy Loss Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 Tagger Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Momentum Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 Effectiveness of Corrections . . . . . . . . . . . . . . . . . . . . . . . . . 844.6 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.7 Vertex Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.8 Beam Photon Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.9 Detector Performance Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6 Kinematic fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.1 Monte Carlo Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.2 Monte Carlo Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3 Monte Carlo Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.4 Trigger Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.5 Matching Data and Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 Extraction Methods and Constraints . . . . . . . . . . . . . . . . . . . . . . . . 1398.1 Method-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.1.1 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2 Method-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Method-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9 Systematic Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6
II Electromagnetic decay of the Σ∗+ 187
10 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18810.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11 Neutron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19411.1 Neutron Detection Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19511.2 Neutron Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19611.3 Neutron Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19611.4 Neutron Momentum Correction . . . . . . . . . . . . . . . . . . . . . . . 19711.5 Neutron Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20111.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
12 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
13 Neutron Kinematic Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22313.1 Neutron EC Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 223
13.1.1 Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22413.1.2 Off-Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . 225
13.2 Neutron Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
14 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
15 Extraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24115.1 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24115.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24815.3 Systematic Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25015.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
16 Overall Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Appendix A: Ratio Calculation Details . . . . . . . . . . . . . . . . . . . . . . . . . 273
Appendix B: BOS CLAS Data Structure . . . . . . . . . . . . . . . . . . . . . . . . 278
7
LIST OF TABLES
1.1 Theoretical predictions of the decay widths for the model shown of electro-magnetic decays of various baryons (all in units of keV). . . . . . . . . . . . . 27
4.1 Bad Time-of-flight scintillators. . . . . . . . . . . . . . . . . . . . . . . . . . 984.2 The list of run numbers that have problems with the forward part of the TOF. . 99
7.1 The set of Monte Carlo channel generated for acceptance studies. . . . . . . . . 130
8.1 Counts for nΛ found though the different methods; the raw counts rejectedfrom the π0 hypothesis, and the estimated number of nΛ from Monte Carlo.The uncertainties are fit uncertainties combined with statistical uncertainties. . . 153
8.2 Counts for nK∗ found though different methods. Uncertainties are fituncertainties combined with statistical uncertainties. . . . . . . . . . . . . . . 156
8.3 Acceptances (in units of 10−3) for the channels used in the calculation of thebranching ratios. Here there is a 10% confidence level used as upper and lowerP(χ2) cuts; the Pxy cut was 0.03 GeV. The uncertainties are statistical only.The three columns contain the acceptance for each hypothesis Aγ , Aπ , and thecounts that made all other cuts but did not satisfy either γ or π0 hypothesisdenoted as Aγπ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.4 Breakdown of statistics for each term in Eq. 15.5 for the Λ(γ) and Λ(π0)hypothesis. Each listed channel is subtracted from the raw counts directly fromthe kinematic fit to obtain the final ratio. The uncertainties listed are statisticalonly. The K∗+ counts are not used in method-1 in order to follow Taylor [10]. . 159
8.5 Dependence of the corrected branching ratio on the confidence level cuts. . . . 1618.6 Optimization points for each Pa
π0 and Pbγ from Ref. [71]. . . . . . . . . . . . . . 168
8.7 Acceptances (in units of 10−3) for the channels used in the calculation of thebranching ratios. Here Pb
γ (χ2) > 10% and Paπ (χ2) < 1% while Pb
π (χ2) > 10%and Pa
γ (χ2) < 1% and the Pxy cut was 0.03 GeV. The uncertainties are statisticalonly. The three columns contain the acceptance for each hypothesis Aγ , Aπ ,and the counts that made all other cuts but did not satisfy either the γ or π0
hypothesis denoted as Aγπ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.8 Counts for nΛ found through the different methods; the raw counts rejected
from the π0 hypothesis, and the estimated number of nΛ from Monte Carlo.The uncertainties are fit uncertainties combined with statistical uncertainties. . . 173
8.9 Counts for nK∗ found through the different methods. Uncertainties are fituncertainties combined with statistical uncertainties. . . . . . . . . . . . . . . 173
9.1 Branching ratio for excluded channels in (%). . . . . . . . . . . . . . . . . . . 1829.2 Dependence of the corrected branching ratio on the confidence level cuts for
the selected systematic range. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8
9.3 Ranges of systematic variation in resulting ratio in (%) showing L(Low)-contribution and H(High)-contribution and rang in each case. . . . . . . . . . . 185
14.1 The set of Monte Carlo channels and amount of events generated for theacceptance studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
15.1 The cuts used to extract the final radiative and π0 counts. (See text for details.) 24415.2 Acceptances (in units of 10−3) for the channels used in the calculation of the
branching ratios. All the cuts used to obtain the acceptance values are listed inTable 15.1. The uncertainties are statistical only. The two columns contain theacceptance for each hypothesis Aγ , Aπ . In some cases the values are roundedup to 0.0001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
15.3 Breakdown of statistics for each term in Eq. 15.4 for the Σ(γ) and Σ(π0)hypothesis. Each counts for each hypothesis is subtracted accordingly asshown in Eq. 15.4. Th raw count are taken directly from the kinematic fitto use in the final ratio calculation. The uncertainties listed are statistical only. . 249
15.4 The resulting contribution to the ratio for a f value of 10%. Some values arerounded up. Uncertainty is statistical only. . . . . . . . . . . . . . . . . . . . . 254
15.5 Ranges of systematic variation in resulting ratio in (%) showing L(Low)-contribution and H(High)-contribution and rang in each case. . . . . . . . . . . 262
16.1 Comparison of theorectical model predictions for the radiative decay widthswith the present results, all in keV. . . . . . . . . . . . . . . . . . . . . . . . . 265
9
LIST OF FIGURES
1.1 The missing mass of the γ p→ K+X reaction in the photon energy range (a)1.5 < Eγ < 2.0 GeV and (b) 2.0 < Eγ < 2.4. The data points are shownin closed circles. The strengths of the Σ∗0 production were obtained fromthe Λπ0 decay mode and the other reactions were obtained by fitting. Thesolid fit line, open circle, dashed lines, and dot-dashed indicate the spectrafrom K+Λ(1405), K+Σ(1385), nonresonant K+Σπ and K∗0Σ+ production,respectively. Source [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.2 DESY Bubble Chamber Group total cross section [48] in with CLAS totalcross section with parameterization fit for γ p→ K+Σ(1385). Source [49]. . . . 30
2.1 Baryon decuplet, hypercharge (Y ) versus isospin (I3). . . . . . . . . . . . . . . 452.2 Baryon decuplet, plotted for U-spin multiplets with charge (Q) versus u-spin
(U3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3 The diagrams for the dressed γN → ∆ vertex. The meson cloud diagrams are
in brackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4 The magnetic dipole transition form factor G∗(Q2) for γ∗∆→ ∆(1232). The
experimental points (empty circles) are for the inclusive data from pre-1990experiments at DESY and SLAC [42], and exclusive data (filled squares) arefrom BATES [43], MAMI [44], and JLAB [45]. The Solid curve indicates thedressed calculation, while the dotted line is without the meson cloud effect. . . 52
3.1 The aerial view of CEBAF at Thomas Jefferson National Labs. The“racetrack”-shaped area indicates where the accelerator ring lies underground.Each experimental hall is underneath the three grassy sectors near the bottomof the picture. Hall B is the middle hall. Image source:[35]. . . . . . . . . . . . 53
3.2 A schematic diagram of the Continuous Electron Beam Accelerator Facility.The linear red tubes represent the LINACs made of superconducting RFcavities grouped into 20 cryomodules. A magnification of the recirculationarcs is shown in the top right. The experimental halls are shown on the bottomleft. Image source:[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 A picture of a pair of superconducting niobium Radio Frequency (RF) cavities.CEBAF uses 338 uperconduction cavities, like the RF cavity shown. Imagesource:[36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 A schematic diagram of the superconducting RF cavity in operation. Theacceleration gradient is provided by establishing a standing wave, leading toa continuous positive electric force on the electron. The phase of the wavesadvances the position of the electron bunch in the cavity creating the gradient.Image source:[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
10
3.5 A schematic of the photon tagging system. The tagger is setup to allow indirectmeasurements of the photon beam energy. The recoil electron is directed intothe tagger spectrometer to that is energy can be measured to deduce the photonenergy produced. Image source: [56]. . . . . . . . . . . . . . . . . . . . . . . 59
3.6 A schematic of the tagger magnet and hodoscope used in the taggingsystem. The trajectories of the recoil electrons are depicted by the dashedlines. Electrons from various trajectories in the spectrometer correspond tobremsstrahlung photons of a given energy. Image source: [56]. . . . . . . . . . 61
3.7 A schematic of the tagger logic setup. The T-counter hits are also used to setthe event trigger. The common stop to the E-counter TDC array is controlledby the the CLAS Level 1 trigger. Image source: [56]. . . . . . . . . . . . . . . 62
3.8 A photograph of the CLAS detector from inside Hall B. The Time-of-flightscintillators are pulled away from the drift chambers showing inside. Imagesource: [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 A schematic diagram of the full CLAS detector. The drift chambers areshown in violet, the toroidal magnet is shown in light blue, the Time-of-flightscintillators are shown in red, and the electromagnetic calorimeter is shown ingreen. Image source: [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 A diagram of the g11a Cryotarget. The dimensions of the target cell used were40 cm long and 4 cm in diameter. The cell was filled with liquid H2. Imagesource: [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 A diagram of the new g11a Start Counter. The detector is a six-sectorscintillation device with four mounted photomultiplier tubes for each sector.One sector is cut away in the diagram to show the target space. Image source:[58]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.12 A photograph of the coils of the CLAS toroidal magnet before installation ofthe rest of the detector subsystem. Image source: [59]. . . . . . . . . . . . . . 67
3.13 (a) The contours of constant absolute magnetic field of the CLAS toroid in themidplane between two of the coils. (b)The field vectors for the CLAS toroidtransverse to the beam. The field lines represent the field strength. The sixcoils are shown in the cross section. Image source: [52]. . . . . . . . . . . . . 68
3.14 A diagram look down showing the CLAS drift chamber region relative to theother subsystems. The dashed lines outline the location of the toroidal magnetcoils. Image source: [59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.15 (a) An isolated view of the time-of-flight paddles in one sector. The setdesignated for forward angle detection are on the right. (b) The schematic ofthe light guide used to connect the backward angle TOF paddles to the PMTs. . 71
3.16 A vertical slice of the EC light readout system. PMT - Photomultiplier TubeLG - Light Guide, FOBIN - Fiber Optic Bundle Inner, FOBOU - Fiber OpticBundle Outer, SC - Scintillators, Pb - 2.2 mm Lead sheets, IP - Inner Plate(closest to target or face of EC) Image source: [55]. . . . . . . . . . . . . . . . 73
11
3.17 (a) View of one of the six CLAS electromagnetic calorimeter triangularmodules showing the three projection planes. (b) The diagram of eventreconstruction in the EC. Energy deposition profile is shown along each stereoview. Image source: [55]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.18 A schematic layout of the beamline and flux monitoring devices. The beamlineenters from the left. Image source: [56]. . . . . . . . . . . . . . . . . . . . . . 76
4.1 ∆E with respect to momentum for left: proton, and right: K+ . . . . . . . . . . 814.2 Left: ∆Eγ/Ebeam vs. E-counter for the reaction (γ)p→ pπ+π− used to find the
tagger correction. Right: The extracted Gaussian mean from the ∆Eγ/Ebeam vs.E-counter fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Left: ∆p+ found for the K+ in the topology γ p→ K+Λ→ K+pπ− for Sector1, θ ∈ (20,25) and φ ∈ (−15,−10). Right: The Gaussian fit for ∆p+. . . . 84
4.4 Left: Invariant mass of (pπ−), Right: Missing mass off the K+. Beforecorrections is shown in black and after is shown in red. . . . . . . . . . . . . . 85
4.5 Left: Invariant mass of (pπ−) with Gaussian fit before corrections, Right:Invariant mass of (pπ−) with Gaussian fit after corrections. . . . . . . . . . . . 86
4.6 Left: Missing mass off the K+ with Gaussian fit before corrections, Right:Missing mass off the K+ with Gaussian fit after corrections. . . . . . . . . . . 86
4.7 ∆t giving the PID timing cut. Clusters of events at (±2 ns,±2 ns),(±4 ns,±4ns), etc. are from photons from a different beam bucket. The dashed linesrepresent the timing cut used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.8 Calculated mass based on the PART bank information with no cuts used.Visible accidentals result from other RF buckets that are present. . . . . . . . . 89
4.9 Left: K+ β distribution before any cuts, Right:K+ β distribution after timing cut. 904.10 ∆β for the K+ before any cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . 914.11 ∆β for the K+ after a ±1 ns timing cut. . . . . . . . . . . . . . . . . . . . . . 914.12 K+ calculated mass versus momentum after a ±1 ns timing cut. . . . . . . . . 924.13 π− calculated mass versus momentum after a ±1 ns timing cut. . . . . . . . . . 924.14 Proton calculated mass versus momentum after a ±1 ns timing cut. . . . . . . . 934.15 Left: Invariant mass of p,π− before DOCA cut. Right: Invariant mass of p,π−
after DOCA cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.16 Kaon vertex distribution. The dashed lines are the implemented cuts. . . . . . . 954.17 Distance of closest approach; the dashed line is the implemented cut. . . . . . . 954.18 Angular distribution of the kaons before fiducial cuts. . . . . . . . . . . . . . . 974.19 Angular distribution of the kaons after fiducial cuts. . . . . . . . . . . . . . . . 98
5.1 Pion contamination: Mass squared (M2X ) of any missing particle for the
γ p→ pπ+π−(X) reaction where the π+ was a potentially mis-identified kaon.Events with M2
x < 0.01 GeV2 were removed. . . . . . . . . . . . . . . . . . . 1015.2 Pion contamination: Mass squared (M2
X ) of any missing particle for the γ p→π+π−(X) reaction, where the π+ was a potentially mis-identified kaon. . . . . 101
5.3 Λ peak from the invariant mass of the π− and proton. . . . . . . . . . . . . . . 102
12
5.4 No kinematic cuts applied. Top left: Invariant mass of the p-π−. Top right:Missing mass squared of all detected particles. Bottom left: Missing mass offthe p-π−. Bottom right: Missing mass off the K+. All units are (GeV). . . . . . 104
5.5 Plots after Λ cut only. Top left: Missing energy of all particles detected. Topright: Missing mass squared of all detected particles. Bottom left: Missingmass off the Λ. Bottom right: Transverse missing momentum. All units are(GeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Missing mass off the K+ after the cut on the Λ. The fit to the Σ∗ uses arelativistic Breit-Wigner and quadratic background. The dashed line indicatesthe cut around the Σ∗0. The Λ(1520) is also clearly visible. . . . . . . . . . . . 106
5.7 Plots after Σ∗ cut only. Top left: Missing energy of all particles detected. Topright: Missing mass squared of all detected particles. Bottom left: Missingmass off the Λ. Bottom right: Transverse missing momentum. All units are(GeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 Plots after Σ∗ and Λ restrictions. Top left: Missing energy of all particlesdetected. Top right: Missing mass squared of all detected particles. BottomLeft: Missing mass off the Λ (counts below the dashed line are cut). Bottomright: Transverse missing momentum. All units are (GeV). . . . . . . . . . . . 108
5.9 Missing energy produced from simulations for the reaction γ p→ K+Λπ0. . . . 1095.10 Left: Transverse missing momentum Pxy after only one tagged photon is
selected. Right: P2xy after only one tagged photon is selected with cut
implemented, seen as the dotted line. . . . . . . . . . . . . . . . . . . . . . . . 1105.11 Simulations for the perpendicular momentum Pxy of missing mass candidates.
The blue distribution is for the simulated γ p → K+Λ(π0) reaction and theyellow is for the γ p→ K+Λ(γ) reaction. . . . . . . . . . . . . . . . . . . . . . 111
5.12 Magnification of the simulation of the perpendicular momentum Pxy of missingmass candidates and the implemented cut showing little effect on signals. . . . 112
5.13 Plots after all mentioned cuts. Top left: Missing energy of all particles detected.Top right: Missing mass squared of all detected particles. Bottom left: Missingmass off the Λ. Bottom right: Transverse missing momentum. All units are(GeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.1 Left: momentum resolution from data used to match Monte Carlo. Right: thematching of Monte Carlo to data by smearing out the measured resolution, redis Monte Carlo and blue is data. . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Comparison between data and Monte Carlo width of the missing mass squareddistribution. The left plot show the data with a Gaussian fit, while the middle isthe Monte Carlo before smearing and the right is the Monte Carlo after smearing.125
7.3 Decay time for the Λ from the Monte Carlo generated for γ p→K+Σ∗→K+Λπ0.1287.4 Monte Carlo distributions for the left: K∗, middle: Λ(1405), and right: Σ∗0. A
Breit-Wigner fit is used to demonstrate accurate width and mass in each case. . 129
13
7.5 Left: Photon energy distribution for data using unskimed data without anycuts. Right: Photon energy distribution taken from the Monte Carlo generatorshowing the bremsstrahlung distribution. . . . . . . . . . . . . . . . . . . . . . 130
7.6 Left: Cross section and fit function used in Monte Carlo generation for thereaction Σ∗ → Λπ0. Right: Comparison between data and generated crosssection after using correction to the photon energy distribution. . . . . . . . . . 131
7.7 Approximated differential cross section for γ p→ K+Σ∗0→ K+Λπ0 in cosθKin the center-of-mass frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.8 Final comparison between Monte Carlo (lines) and data (points with errors)for the reaction γ p→ K+Σ∗0 → K+Λπ0. Top left: cosθ of kaon in the labframe. Top right: kaon momentum distribution. Bottom left: pion momentumdistribution. Bottom right: proton momentum distribution. . . . . . . . . . . . 134
7.9 Monte Carlo for the γ p→K+Σ∗0→K+Λγ channel. Left: the K+pπ− missingmass squared. Right: the missing mass off the Λ. . . . . . . . . . . . . . . . . 135
7.10 Monte Carlo for the γ p → K+Σ∗0 → K+Λπ0 channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . . 1357.11 Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ0π0 channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . . 1357.12 Monte Carlo for the γ p→ K+Λ(1405)→ K+Λγ channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . 1367.13 Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ0γ channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . 1367.14 Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ+π− channel (Before Cuts).
Left: the K+pπ− missing mass squared. Right: the missing mass off the Λ. . . 1367.15 Monte Carlo for the γ p → K+Σ∗ → K+Σ+π− channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . 1377.16 Monte Carlo for the γ p→ K∗+Λ→ ΛK+γ channel. Left: the K+pπ− missing
mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . . . . . . 1377.17 Monte Carlo for the γ p→K∗+Λ→ΛK+π0 channel. Left: the K+pπ− missing
mass squared. Right: the missing mass off the Λ. . . . . . . . . . . . . . . . . 1377.18 Left: Monte Carlo missing energy for the γ p→ K∗+Λ→ ΛK+π0 channel.
Right: Monte Carlo missing energy for the γ p → K+Λ(1405) → K+Σ0π0
channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.1 Left: χ2 distribution for the π0 hypothesis with a fit using Eq. 8.2. Right:confidence level distribution for the π0 hypothesis. . . . . . . . . . . . . . . . 143
8.2 Left: χ2 distribution for the γ hypothesis with a fit using Eq. 8.2. Right:confidence level distribution for the γ hypothesis. . . . . . . . . . . . . . . . . 143
8.3 Left: χ2 distribution for γ hypothesis with a fit using Eq.8.2. Right confidencelevel distribution for γ hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . 145
14
8.4 Missing mass squared distribution for the events that are going into the secondstep of the kinematic fitting procedure. The kinematic fit to π0 satisfyingP(χ2) < 10% shows the radiative candidates (yellow), as well at the rejectedbackground from the Λ(1405) (green). The white region shows the π0
candidates from a P(χ2)≥ 10% cut. . . . . . . . . . . . . . . . . . . . . . . . 1468.5 Left: data (error bars) with Monte Carlo (line) from the Λπ0 channel and the
full spectrum of Λ(1405) filled to match the data. Right: data (error bars) withMonte Carlo (line) from the Λ(1405) only. . . . . . . . . . . . . . . . . . . . . 152
8.6 Left: fit with a Gaussian and quadratic background for the hyperon mass range1.34-1.43 GeV. Right: fit with polynomial to the point derived from Gaussianfits of various mass windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.7 Missing mass off of the Λ for the match of data and Monte Carlo to obtain thecounts from K∗. The lines is data and the points are from Monte Carlo afteradding in the K∗ and Λ(1405). . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.8 A study over a variation in confidence level; each cut corresponds to Pπ0(χ2)= Pγ(χ2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.9 Only the selected confidence level cuts with the best quality signal based on theP1 parameter in the χ2 fit (values seen in Table 8.5). Pπ0(χ2)/Pγ(χ2) is used onthe x-axis to obtain a distinguishable point for each ratio value. . . . . . . . . . 163
8.10 The fit to the χ2 distribution from Monte Carlo for various mixtures ofradiative signal and Λπ0 events for small statistics, after cutting events withPa
π (χ2) < 1%. The amount of π0 background still present in each case can bedetermined by looking at the number of events added from the pure singal casein the top left plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.11 Left: χ2 distribution and fit with two degrees of freedom for the π0 candidatesfrom the g11a data after the Pa
γ (χ2) < 1% cut and before the Pbπ (χ2) > 10%
cut; Right: the corresponding confidence level distribution for the π0 candidates.1708.12 Left: χ2 distribution and fit with two degrees of freedom for the γ candidates
from the g11a data before the Paπ (χ2) < 1% cut; Right: the corresponding
confidence level distribution for the γ candidates. . . . . . . . . . . . . . . . . 1708.13 Left: χ2 distribution and fit with two degrees of freedom for the γ candidates
from the g11a data after the Paπ (χ2) < 1% cut and before the Pb
γ (χ2) > 10%cut; Right: the corresponding confidence level distribution for the γ candidates. 171
8.14 Left: the nπ counts extracted using the confidence level cuts Paγ < 0.01 and
Pbπ > 0.1. Middle: the nγ counts extracted using the confidence level cuts
Paπ < 0.01 and Pb
γ > 0.1. Right: the counts nπ and nγ shown in the spectrumbefore any kinematic fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.15 The confidence level distribution with the additional constraint on the invariantmass of the Σ∗0 using a missing π0 hypothesis. . . . . . . . . . . . . . . . . . 176
8.16 The χ2 and confidence level distribution with the additional constraint on theinvariant mass of the Σ∗0 with the missing γ hypothesis. . . . . . . . . . . . . . 177
15
9.1 The variation in the DOCA cuts in centimeters with the acceptance correctedbranching ratio shown in (%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 The variation in the perpendicular momentum cuts with the acceptancecorrected branching ratio shown in (%). . . . . . . . . . . . . . . . . . . . . . 180
9.3 Left: Variation in the acceptance corrected branching ratio for various t-slopesetting in the generator for the π0 channel; right: for the radiative channel. . . . 181
10.1 The ∆β distribution for the π− and the cut applied to clean up identification. . . 18910.2 Invariant mass of the π+-π− combination for the two different π+ detected,
prior to any π+ organization to optimize the K0 cut. . . . . . . . . . . . . . . . 19010.3 Invariant mass of the π+-π− combination for the two different π+ after π+
organization to optimize the K0 peak. . . . . . . . . . . . . . . . . . . . . . . 19010.4 The confidence level distribution for the (2-C) kinematic fit with constraint on
π+ and π− to be the mass of the K0 with a missing mass off the K0 of the Σ∗+. 19110.5 (a) The invariant mass of the π+ π− after the best π+ is selected. (b) The
invariant mass of the π+ and π− after the P(χ2) > 0.1% confidence level cut. . 19210.6 (a) The missing mass off the K0 before the confidence level cut. (b) The
missing mass off the K0 after the P(χ2) > 0.1% confidence level cut. . . . . . . 193
11.1 (a) ∆p before the correction, showing EC counts for inner and outer blocklayers. (b) The pk f it vs pmeas for EC counts for inner and outer block layersbefore corrections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.2 (a) ∆p after correction showing EC counts for inner and outer block layerstogether. (b) The pk f it vs pmeas for EC counts for inner and outer block layersafter corrections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.3 Demonstration of cosθ vs φ for the neutron candidates. . . . . . . . . . . . . . 19911.4 Demonstration of cosθ vs φ for the detected neutrons. . . . . . . . . . . . . . . 20011.5 The g11a data set is used to obtain ∆p over a range of p (upper left), ∆p over a
range of φ (upper right), ∆p over a range of θ (lower left), ∆p over a range ofnuetron path length in (cm) (lower right). . . . . . . . . . . . . . . . . . . . . 202
11.6 Monte Carlo is used to obtain ∆p over a range of p (upper left), ∆p over a rangeof φ (upper right), ∆p over a range of θ (lower left), ∆p over a range of neutronpath length in (cm) (lower right). . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.7 The g11a study of σ(p). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required rangeof p. The lower right plot shows the χ2 from each Gaussian fit for every binused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
11.8 The g11a study of σ(θ). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required rangeof θ . The lower right plot shows the χ2 from each Gaussian fit for every binused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
16
11.9 The g11a study of σ(φ). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required rangeof φ . The lower right plot shows the χ2 from each Gaussian fit for every binused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
11.10The Monte Carlo study of σ(p). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussianfits. The lower left plot shows the values of σ from the Gaussian fits over therequired range of p. The lower right plot shows the χ2 from each Gaussian fitfor every bin used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
11.11The Monte Carlo study of σ(θ). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussianfits. The lower left plot shows the values of σ from the Gaussian fits over therequired range of θ . The lower right plot shows the χ2 from each Gaussian fitfor every bin used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.12The Monte Carlo study of σ(φ). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussianfits. The lower left plot shows the values of σ from the Gaussian fits over therequired range of φ . The lower right plot shows the χ2 from each Gaussian fitfor every bin used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.13The g11a data study of σ(θ). The upper left plot shows ∆θ over a range ofmomentum. The upper left shows the resolutions from the Gaussian fits of ∆θ
as a function of momentum. The middle right shows the ∆θ over a range ofφ . The middle left shows the resolutions from the Gaussian fits of ∆θ as afunction of φ . The bottom left plot shows ∆θ over a range of θ . The bottomleft shows the resolutions from the Gaussian fits of ∆θ as a function of θ . A cuton the cosine of the angle between the missing neutron vector and the detectedneutron vector of 3 is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.14The g11a data study of σ(φ). The upper left plot shows ∆φ over a range ofmomentum. The upper left shows the resolutions from the Gaussian fits of ∆φ
as a function of momentum. The middle right shows the ∆φ over a range ofφ . The middle left shows the resolutions from the Gaussian fits of ∆φ as afunction of φ . The bottom left plot shows ∆φ over a range of θ . The bottomleft shows the resolutions from the Gaussian fits of ∆φ as a function of θ . A cuton the cosine of the angle between the missing neutron vector and the detectedneutron vector of 3 is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17
11.15The Monte Carlo study of σ(θ). The upper left plot shows ∆θ over a rangeof momentum. The upper left shows the resolutions from the Gaussian fits of∆θ as a function of momentum. The middle right shows the ∆θ over a rangeof φ . The middle left shows the resolutions from the Gaussian fits of ∆θ as afunction of φ . The bottom left plot shows ∆θ over a range of θ . The bottomleft shows the resolutions from the Gaussian fits of ∆θ as a function of θ . A cuton the cosine of the angle between the missing neutron vector and the detectedneutron vector of 3 is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.16The Monte Carlo study of σ(φ). The upper left plot shows ∆φ over a rangeof momentum. The upper left shows the resolutions from the Gaussian fits of∆φ as a function of momentum. The middle right shows the ∆φ over a rangeof φ . The middle left shows the resolutions from the Gaussian fits of ∆φ as afunction of φ . The bottom left plot shows ∆φ over a range of θ . The bottomleft shows the resolutions from the Gaussian fits of ∆φ as a function of θ . A cuton the cosine of the angle between the missing neutron vector and the detectedneutron vector of 3 is used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.1 The invariant mass of the π+1 -π−, (upper left), missing mass off the π
+1 -
π− (upper right), the n-π+2 invariant mass (lower left), and the missing mass
squared of all the detected particles (lower right). All distributions are beforeany kinematic constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
12.2 The invariant mass of the n-π+2 , (upper left), missing mass off the K0 (upper
right), the missing mass off the n-π+2 combination (lower left), and the missing
mass squared of all the detected particles (lower right), after the±0.005 K0 peak.21912.3 The missing mass off the K0 (upper left), and the missing mass off the Σ+
(upper right), the missing mass squared of all detected particles (lower left),and a magnification of the missing mass squared with Gaussian fit to the π0
region (lower right) . These distribution are made after take a±0.02 cut aroundthe Σ+ peak as well as the prior cut of ±0.005 on the K0 peak. . . . . . . . . . 220
12.4 The comparison between the final missing mass squared distribution from theset of cuts discribed after a ±0.05 cut around the Σ∗+ mass (upper left) andfrom the P(χ2) > 0.1% confidence level cut used obtain the previous K0 andΣ∗+ candidates. The same ±0.02 cut around the Σ+ peak is used in both. . . . . 221
12.5 The missing mass off the Σ+ after all cuts described (no kinematic fit). Thedistribution is similar for the the kinematic fit (not shown). . . . . . . . . . . . 222
13.1 Confidence level distribution for a (4-C) kinematic fit π+π−π+n. . . . . . . . . 22713.2 Left:Showing the (4-C) kinematic fit to get the invariant mass of π+n, Right:
improvement in missing mass off the Σ+ (both in GeV). . . . . . . . . . . . . . 22813.3 Left:Showing the (5-C) kinematic fit to the invariant mass of π+n, Right:
improvement to the missing mass off the Σ+ (both in GeV). . . . . . . . . . . . 22813.4 Pull distributions for a (4-C) kinematic kit π+π−π+n. . . . . . . . . . . . . . . 229
18
14.1 The missing mass off the N∗ using a ±0.05 invariant mass cut on the π+ncombination at a mass of 1440 MeV (left) demonstrating a clear ω(782) peak;(right) at a mass of the Σ+ demonstrating that the ω(782) peak is still present. 231
14.2 The γ p→ K0Σ∗+→ K0Σ+γ Monte Carlo distibutions for the π+1 -π− invariant
mass, the missing mass off the π+1 -π− combination, the missing energy from
all detected particle, and the missing mass squared of all detected particles. . . 23514.3 The γ p→K0Σ∗+→K0Σ+π0 Monte Carlo distibutions for the π
+1 -π− invarinat
mass, the missing mass off the π+1 -π− combination, the missng energy from
all detected particle, and the missing mass squared of all detected particles. . . 23614.4 The ωN(1440) → π+π−π0nπ+ Monte Carlo distributions for the π
+1 -π−
invarinat mass, the missing mass off the π+1 -π− combination, the missing
energy from all detected particle, and the missing mass squared of all detectedparticles, no cuts yet applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
14.5 The ρN(1520) → π+π−π0π+n Monte Carlo distibutions for the π+1 -π−
invariant mass, the missing mass off the π+1 -π− combination, the missing
energy from all detected particle, and the missing mass squared of all detectedparticles, no cuts yet applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
14.6 The ηnπ+ → π+π−π0π0nπ+ Monte Carlo distibutions for the π+1 -π−
invarinat mass, the missing mass off the π+1 -π− combination, the missng
energy from all detected particle, and the missing mass squared of all detectedparticles, no cuts yet applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14.7 The K∗0Σ+ → K0Σ+π0 Monte Carlo distributions for the π+1 -π− invariant
mass, the missing mass off the π+1 -π− combination, the missing energy from
all detected particle, and the missing mass squared of all detected particles, nocuts yet applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
15.1 The variation in the ratio due to the confidence level cut from the kinematic fitof π
+1 -π− combination to the invariant mass of K0 with missing mass of Σ∗0. . 251
15.2 Variation in the acceptance corrected branching ratio for various t-slope settingin the generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
15.3 Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts are equal for eachhypothesis such that Pγ(χ2) = Pπ(χ2). . . . . . . . . . . . . . . . . . . . . . . 256
15.4 Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts for the π0 hypothesisis kept at 10% while the radiative hypothesis confidence level cut, Pγ(χ2), isvaried. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
15.5 Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts for the radiativehypothesis is kept at 10% while the π0 hypothesis confidence level cut,Pπ0(χ2), is varied.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
15.6 Variation in the ratio for a range of missing energy cuts on the final extractionof the radiative signal. Here the cuts are only applied to the radiative hypothesis. 260
19
15.7 The variation in ratio from the cut on the Σ+ invariant mass. . . . . . . . . . . 261
20
1 INTRODUCTION
The quark model tells us that the properties of hadrons that arise from the quantum
numbers are determined by their valence quarks. For example consider the proton, which
is made of two up quarks of charge +2/3 and one down quark of charge −1/3. Adding
these together make a total charge of +1. However, quarks also carry color charge,
hadrons have zero total color charge because of a phenomenon known as color
confinement. This means that hadrons are always colorless. To achieve a colorless
combinations consider a Baryon with three quarks, each with different colors, or a meson
with a quark of one color and an antiquark with the required anticolor.
The hadrons are held together by the strong interaction. The theory that encompasses
the physics that bind the quarks into hadrons is Quantum Chromodynamcs (QCD). QCD
dictates the behavior of the strong force and the interaction between the quarks and gluons.
It is a non-linear theory and there are at present no analytic solutions. This means that
only an approximate modeling of the nucleon spectrum is possible. There are a variety of
theories which attempt to incorporate empirical information into functional models.
There has been a great deal of attention given to excited states of the nucleon. The
transition form factors posses characteristic information on such things as the exited state
wave function and the spin structure of the transition. Precise measurements can test
baryon structure models and lead to a greater understanding of the strong interaction.
1.1 Quantum Chromodynamics
In the representation of the color SU(3) gauge group, quarks are the fermionic flavor
carriers. Gluons, with field four-vector Aaµ with a = 1, . . .8 are the exchange particles (or
gauge bosons) for the color force, and represent the gauge fields in the octet
representation. Flavor is a degree of freedom independent of color.
21
The QCD Lagrangian is expressed as
LQCD =−12
trGµνGµν + ∑f =u,...,b
q f (i /D−m)q f (1.1)
where the notation is defined by
Gµν = ∂µAν −∂νAµ − ig[Aµ ,Aν ],
Dµ = ∂ − igAµ
Aµ =8
∑a=1
λ a
2Aa
µ ,
where the λ a are the Gell-Mann matrices. For the strong interaction, the mass matrix m is
diagonal in flavor space. The quintessential characteristics of QCD are found in the
behavior of asymptotic freedom and infrared slavery. Asymptotic freedom is the property
that causes the interaction between quarks to become arbitrarily weak at very small length
scales (sub-femtometer) or, equivalently, large energy scales (tens of GeV). Infrared
slavery, or quark confinement, is the phenomenon that color quarks cannot be isolated
singularly, and ultimately cannot be directly observed. Both can be characterized by the
momentum dependent coupling constant from the theory of renormalization. The
coupling of each QCD vertex is therefore energy dependent and is on the order of unity at
low energies. At this energy, more complicated diagrams may contribute on the same
order as the leading-order diagrams. The opposite is true for high energy, making higher
order diagrams essentially negligible in most cases.
From experiments, it is seen that quark-antiquark pairs (mesons) can be found in
nature such as the π− (du) and K+ (us). The three-quark combinations (baryons) are
prominent such as the proton (uud), neutron (udd) or the Λ (uds). The masses of the up
and down quarks in the proton make up only ∼ 1% of the total proton mass. Clearly there
is additional phenomena and/or highly-relativistic quarks which are not contained in many
models. Ultimately, at the energy scale of nuclear physics, phenomenological tools and
models with degrees of freedom different than true QCD are the most successful.
22
1.2 Spectroscopy
The central focus of hadronic physics is to develop more detailed descriptions of the
transition between quark-gluon and hadronic degrees of freedom. Much effort in recent
years has taken place on improving the picture of the nucleon spectrum. Atomic
spectroscopy uses excitations of atoms to investigate the governing interactions. Studying
the nucleon excitations or resonances (N∗) provides essential information on the behavior
of quarks and the baryon structures. Quantum mechanics describes energy transitions in
discrete intervals leading to descriptions of excited baryons states that also exist in
discrete intervals. This corresponds to observation. Using the experimental results,
theories are able to derive highly successful models that are only approximations of full
QCD, but yield practical predictions.
1.3 The Constituent Quark Model
The Constituent Quark Model (CQM) has been reasonably successful in its
predictions. The CQM uses only the valence quarks as relevant degrees of freedom. A
simplified picture is employed such that excitations can be expressed as the result of
changes to the angular momentum of these quarks. This type of simplification makes
predictions such as the masses of baryon resonances possible to calculate. It is found that
the number of states predicted by the CQM calculation compared with those actually
observed is approximately 4:1 [60]. Because of the many successes of the CQM, these
missing states have been the central focus and motivation of many experiments and
analysis. The Missing Baryons Problem may be an indication that the model, though very
useful in some cases, does not reflect nature well enough to provide reliable predictions.
23
1.4 Electromagnetic decays
A nice demonstration of the CQM predictive capabilities is the calculation of the
magnetic moments of the low-mass baryons, using the SU(6) wave functions [1, 2]. These
calculations [3] are within ∼10% of the current measured values [60]. In the CQM,
quarks are treated as point-like Dirac dipoles. However, nucleons are known to be far
more structured. For example a third of the of the spin of the proton comes from the
valence quarks [4], with the rest of the spin coming from a combination of the gluon spins
and orbital motion of the quarks [5, 6]. Despite these complexities, the CQM captures the
essential degrees of freedom that are relevant to the magnetic moments. Each new high
precision measurement of a baryon magnetic moment leads to new model restrictions and
a deeper understanding of the nature of the baryon wave-functions.
Measurements of electromagnetic (EM) decays can be challenging as they are often
quite small and the EM transitions of decuplet-to-octet baryons can be overwhelmed by
the strong decays. For example the branching ratio of the Σ∗0→ Λγ has been shown [10]
to be on the order of 1% but the signal is buried by the stronger Σ∗0→ Λπ0 decay.
The resulting width from the previous measurement of the Σ∗0→ Λγ [10] is much
larger than most current theoretical predictions. This could be due to meson cloud effects,
which were not included in these calculations. There is a theoretical basis [9] that suggests
that pion cloud effects contribute on the order of ∼40% to the γ p→ ∆+ magnetic dipole
transition form factor, G∆M(Q2), for low Q2. The CQM [11] indicates that the value of
G∆M(0) is directly proportional to the proton magnetic moment [9], and measurements of
G∆M for low Q2 are rationalized in the frame work of the model if the experimental
magnetic moment is lowered by about 25%.
Hyperons produced through low-rate strangeness-conserving reactions can be used to
measure EM transitions to other decuplet baryons [10]. These small electromagnetic
24
decay branching ratios are, as stated above, difficult to measure directly. Ultimately these
measurements are needed to understanding the nature of the baryon wave-functions.
If the EM transition form factors for decuplet baryons with strangeness are also
sensitive to meson cloud effects, models attempting to make prediction of the decuplet
radiative decay widths will need revisions to incorporate this effect. Comparison of data
for the EM decay of decuplet hyperons, Σ∗, to the present predictions of quark models
provides a measure of the importance of meson cloud diagrams in the Σ∗→ Y γ transition.
Establishing experimental results for all three charged states has not yet been achieved.
With theoretical predictions for the degree at which the meson cloud effect plays a
role, it is then possible to test SU(3) flavor symmetry breaking (and the degree at which it
is broken) with a measurement of Σ∗→ Y γ decays. This can be best achieved by
measuring both Γ(Σ∗−→ Σ−γ) and Γ(Σ∗+→ Σ+γ) decay widths.
U-spin symmetry forbids radiative decay of specific decuplet baryons. U-spin is
analogous to isospin in the sense that it is a symmetry in the exchange of the d and s
quarks rather than the u and d quarks, see Section 2.2. A value U of U-spin can be
assigned to each baryon based on the quark composition. The Σ∗− and the Ξ∗− of the
baryon decuplet have U = 3/2 whereas the octet baryons Σ− and Ξ− have U = 1/2. Since
the photon has U = 0, this implies that
Σ∗−→ Σ
−γ and Ξ
∗−→ Ξ−
γ
have zero amplitude in the equal-mass limit due to U-spin symmetry. This can also be
understood in the context of the SU(6) wavefunctions for these baryons. The M1
transition operator is written between initial and final states :
〈Σ−SU(6)|∑q
2mqσq ·(
k× ε∗λ)|Σ∗−SU(6)〉= 0.
25
Here the sum is over all q constituent quarks, and mq is the mass of the q quark. One can
also show that doing the same for the Σ∗+ gives a non-zero amplitude. U-spin invariance
implies a large difference in the radiative decay widths of Σ∗− and Σ∗+.
The chiral symmetry for U-spin is strongly broken because the constituent mass of
the strange quark, ms, is approximately 1.5 times greater than the non-strange quarks, m.
The magnetic moment is inversely proportional to the mass, and so there is no
cancellation like in the equal-mass SU(6) case above. From Ref. [12], an estimate of the
correction givesΓ(Σ∗−→ Σ−γ)Γ(Σ∗+→ Σ+γ)
=19
(1− m
ms
)2
which is a ratio of about 1%, suggesting that U-spin symmetry breaking for radiative
decays is at the level of only a few percent. This implies that U-spin is an effective tool,
even considering the quark mass difference.
Detailed corrections have been carried out by several groups [13, 14], all of which
come up with corrections that are of a few percent. In lattice QCD, the quarks have much
different interactions with the photon than for the CQM, but these too have ratios (for the
above equation) within a few percent [15]. This makes a stronger case for the usefulness
of U-spin symmetry.
There has been much theoretical interest in radiative baryon decays. However, there
are only a few measurements. Recently, a measurement the radiative decay of the Σ∗−
been attempted by the SELEX collaboration [16], resulting in only an upper limit. The
90% confidence level upper bound of Γ = 9.5 keV was reported, however most models
predict a value of less than 4 keV. Ultimately this result is not very useful in constraining
theoretical estimates. More experimental measurements are necessary with better
experimental constraints.
A set of steps have been carried out to investigate the various Σ∗ electromagnetic
decay at CLAS. First, a check on the previous CLAS measurement of the EM decay of the
26
Σ∗0 was done. The previous measurement is quite high compared to nearly all available
calculations, and confirmation of the result is required to determine if the result is reliable
and hence if meson cloud effects are indeed contributing significantly. The next step was
to measure the Σ∗+ electromagnetic decay which has not been done before. The topology
of the reaction γ p→ K0Σ∗+→ K0Σ+γ in its final set of decay products is
γ p→ π+π−nπ+γ , compared to the Σ∗− case γ p→ K+Σ∗−→ K+nπ−γ . The advantages
of the Σ∗+ is that its rate is expected to be much larger, at nearly an order of magnitude
larger than the Σ∗− electromagnetic decay. Also, the CLAS data set g11a, available for the
positive channel, is quite large.
The first part of this thesis is dedicated to developing a method to extract the small
radiative signal out from under a large π0 background. The result from the first of the
present measurements is the EM decay width Σ∗0→ Λγ normalized to the strong decay
Σ∗0→ Λπ0. These results can be compared to previous measurements of the Σ∗0 EM
decay [10] that had a larger uncertainty (∼25% statistical and ∼15% systematic
uncertainty). The smaller uncertainties here are due to a larger data set (more than 10
times bigger) and subsequently a better control over systematic uncertainties. The reduced
uncertainty is important because, as mentioned above, meson cloud effects are predicted
to be on the order of ∼25-40%. In order to know quantitatively the effect of meson clouds
for baryons with non-zero strangeness, it is desirable to keep measurement uncertainties
below ∼10%.
There are many calculations of the EM decays of decuplet hyperons such as: the
non-relativistic quark model (NRQM) [17, 23], a relativistic constituent quark model
(RCQM) [24], a chiral constituent quark model (χCQM) [25], the MIT bag model[26],
the bound-state soliton model[27], a three-flavor generalization of the Skyrme model that
uses the collective approach[28, 29], an algebraic model of hadron structure[30], among
27
others. Table 1.1 summarizes the theoretical predictions and experimental branching
ratios for the EM transitions of interest.
Table 1.1: Theoretical predictions of the decay widths for the model shown ofelectromagnetic decays of various baryons (all in units of keV).
Model ∆→ Nγ Σ→ Λγ Σ∗0→ Λγ Σ∗+→ Σ+γ
NRQM[17, 23, 26] 360 8.6 273 104
RCQM[24] 4.1 267
χCQM[25] 350 265 105
MIT Bag[26] 4.6 152 117
Soliton[27] 243 91
Skyrme[28, 29] 309-326 157-209 47
Algebraic model[30] 341.5 8.6 221.3 140.7
Experiment[60] 660±47 9.1±0.9 470±160
1.5 The CLAS detector
A comprehensive investigation of electromagnetic strangeness production has been
carried out using the CLAS detector at Thomas Jefferson National Accelerator Facility.
Many data on ground-state hyperon photoproduction have already been published
[32, 33, 34] using data from the so-called g1 and g11 data sets. The g1 experiment had an
open trigger [32] and lower data acquisition speed, whereas the g11 experiment required
at least two particles to be detected [34], lowering the trigger rate, yet had a much higher
data acquisition speed. The g11 data had a higher average beam current, resulting in over
20 times more useful triggers than for the g1 data. The present results used the g11 data
set whereas the previous CLAS measurement used the g1 data set. Previous CLAS results
28
for other known reactions give confidence to the corresponding calibration of these data
sets [34].
As previously mentioned, the EM decay of the Σ∗0 is only about 1% of the total
decay width. To isolate this signal from the dominant strong decay Σ∗0→ Λπ0, the
missing mass of the detected particles, γ p→ K+Λ(X) is calculated. Because of its
proximity to the π0 peak in the mass spectrum from strong decay, the EM decay signal is
difficult to separate using simple peak-fitting methods. The strategy here is to understand
and eliminate as much background as possible using standard kinematic cuts, and then use
a kinematic fitting procedure for each channel. As described later, by varying the cut
points on the confidence levels of each kinematic fit, the systematic uncertainty associated
with the extracted branching ratio for EM decay can be quantitatively determined. The
increased statistics for the g11 data also helps greatly to reduce the systematic uncertainty.
The second half of this thesis is dedicated to development of a method that can be
used to extract the electromagnetic decay signal from the more difficult topology of the
Σ∗+. The Σ∗+→ Σ+γ is estimated to be event smaller than that of the Σ∗0→ Λγ but with
the same proximity to the π0 peak in the mass spectrum. The same strategy (to understand
and eliminate as much background as possible using standard kinematic cuts and then use
a kinematic fitting procedure) is used for each channel.
1.6 Previous Experimental Results
The total and differential cross section are analyzed in the present analysis to improve
the Monte Carlo simulations of the γ p→ K+Σ∗0 reaction. The differential cross section
has been measured in the photon energy range from 1.5 to 2.4 GeV and the angular range
of 0.8 < cosθK < 1.0 in the K+ center-of-mass angle at SPing-8 from the LEPS
collaboration [46].
29
In the LEPS experiment, forward going K+’s from the γ p→ K+X reaction were
detected in the spectrometer. A time projection chamber was used together with the
spectrometer to assist in the detection of the decay products. The main decay mode was
identified by detecting the Λ and the K+. The decay to Σπ is also measured. The missing
mass spectrum of the γ p→ K+X reaction for events with K+Σπ is shown if Figure 1.1.
The experimental results are shown as closed circles. The data was fit with a shape
determined by MC simulations. The solid histograms show the fit results. The strengths of
the Σ∗0 production were obtained from the Λπ0 decay mode and the other reaction
channels were obtained by fitting. The solid fit line, open circle, dashed lines, and
dot-dashed indicate the spectra of the K+Λ(1405), K+Σ(1385), nonresonant K+Σπ and
K∗0Σ+ production, respectively.
Figure 1.1: The missing mass of the γ p→ K+X reaction in the photon energy range (a)1.5 < Eγ < 2.0 GeV and (b) 2.0 < Eγ < 2.4. The data points are shown in closed circles.The strengths of the Σ∗0 production were obtained from the Λπ0 decay mode and the otherreactions were obtained by fitting. The solid fit line, open circle, dashed lines, and dot-dashed indicate the spectra from K+Λ(1405), K+Σ(1385), nonresonant K+Σπ and K∗0Σ+
production, respectively. Source [46].
30
CLAS has also looked at the production cross section for γ p→ K+Σ∗0 in a
preliminary analysis with the g11a data set [47]. The two competing processes
γ p→ K+Σ∗0 and γ p→ K∗+Λ are studied. Simulations assuming a t-channel process
where used to correct for the detector acceptance. The Σ(1385) yield was achieved using a
p-wave Breit-Wigner function with background shape generated using K∗+Λ simulations.
Only total cross sections are presented.
There has also been some earlier work by the DESY Bubble Chamber Group [48]
that measured a total cross section. The CLAS and DESY results are shown in Figure 1.2.
Figure 1.2: DESY Bubble Chamber Group total cross section [48] in with CLAS total crosssection with parameterization fit for γ p→ K+Σ(1385). Source [49].
There has been previous work on the branching ratio of Σ∗0→ Λγ to Σ∗0→ Λπ0 by
Taylor [10] using g1c data set at CLAS. The result with its statistical uncertainty is
31
1.53±0.39. The g11a data set has many more triggered events, ultimately leading to a
reasonable reduction of statistical uncertainty. Considerable effort is made to reduce
systematic uncertainty and to include systematic studies not performed in Taylor [10].
Here, other methods are used with better control over various systematic effects. The
essential differences in the steps taken by Taylor [10] include: (a) using a kinematic fit to
the Λ hypothesis prior to the final missing π0 or radiative hypothesis, (b) the method of
extracting background counts, and (c) using a Gaussian fit after the final kinematic fit to
obtain the counts and uncertainties. In computer simulations, Taylor used a model to
approximate the angular dependence of the γ p→ K+Σ∗0→ K+Λπ0 cross section
distribution. In the present analysis the angular distributions from the data are used.
Various degrees of improvement to the simulations and extraction of the radiative signal
are discussed in the analysis section of this thesis.
For completeness, some earlier work is mentioned that made the an attempt to
measure the radiative decay of the Σ∗0. Meisner [21] observed a single event consistent
with K0 p→ Σ(1385)π+→ Λγπ+. From this measurement a branching ratio of
0.17±0.17 was obtained. Colas et al. [22] determined the limits on the Σ(1385) radiative
decay of
Γ(Σ(1385)→ Λγ)Γ(Σ(1385)→ Λπ0)
< 0.06 (1.2)
Γ(Σ(1385)→ Σ0γ)Γ(Σ(1385)→ Λπ0)
< 0.05. (1.3)
The Σ∗+ electromagnetic decay has not been observed before and there are no
published cross sections. However, the CLAS collaboration is presently working on
differential cross section for the γ p→ K0Σ∗+ reaction.
32
1.7 The Experiment
Here, a brief description is given, with more details in chapters to follow. The present
experiment was carried out in Hall B of the Thomas Jefferson National Accelerator
Facility. The particle physics detector is described in detail in Chapter 2. Below, the main
points of the present measurement are given.
In this experiment, a proton at rest is hit with a photon of known energy. The reaction
γ p→ KY ∗ is detected, where Y ∗ denotes the excited state hyperon of interest. The trigger
and detector timing information is used to select events. Each channel of interest has its
own set of competing background channels that must be understood. For example, the
γ p→ KY ∗ reaction can have the same topology as the γ p→ K∗Y . Other vector mesons
can have similar contaminating effects such as the γ p→ ωN∗, which can have the same
final decay products as the K0Σ∗+→ π+π−π+nπ0. The K+Σ∗0 is very close in the mass
sepctrum to the K+Λ(1405), complicating the separation of the two without signal loss.
The photon from the electromagnetic decay is not directly measured in the detectors
but is detected through the missing mass of the other final decay products. For example
the reaction γ p→ K+Σ∗0 has a possible final state of K+Σ∗0→ K+pπ−γ for its
electromagnetic decay. All the charged particle can be detected in the drift chambers and
the γ can be detected using conservation of energy and momentum such that
Pµx = Pµ
tot−Pµ
K −Pµp −Pµ
π , where Pµ
tot is the total incoming four-momentum, and Pµx is the
missing four-momentum. More explicitly the missing energy and momentum are,
Ex = Eγ +Mp−EK−Ep−Eπ
~Px = ~Pγ −~PK−~Pp−~Pπ
leading to a missing mass of,
Mx =√
E2x −~P2
x .
33
This technique improves resolution and statistics compare to detecting all final decay
products. Additionally kinematic checks and constraints can be used by studying the
missing mass off a known decay product. For example in this case the missing mass off
the K+ will give the Σ∗0 mass.
For each γ p→ KΣ∗ reaction measured, both a radiative decay and a strong decay to a
π0 are possible. The work presented here is rooted in the development of the extraction
techniques of small radiative signals among other backgrounds. Its is only through
progress in event-weighting analysis that small signals are able to be observed in the
presence of overwhelming background.
By extracting these electromagnetic decay signals from the Σ∗0→ Λγ and
Σ∗+→ Σ+γ final states, a test for U-spin invariance is achieved by comparing the U-spin
SU(3) prediction for the Σ∗+→ Σ+γ partial width and the Σ∗0→ Λπ0 partial width to
same widths measured experimentally.
Through the results of this work it is possible to more accurately determine which
theoretical models provide reasonable predictions, and the approximate degree of
contribution from the meson cloud effects in the wave-functions of baryons.
34
2 THEORETICAL OVERVIEW
Theoretical investigations of electromagnetic decays provide a framework to test
against. This provides a language to describe the possible distribution of different quark
configurations in hadronic matter. Modeling electromagnetic decays can indicate where
the language has become too simple. The interaction of photons with electric charges of
the quark fields provides an invaluable probe into strong interactions and helps to
eliminate models. The radiative processes allow a more complete theoretical
interpretation than that of the purely hadronic interactions. Some highlights of each model
listed in Table 1.1 is now discussed.
2.1 Theoretical Background
2.1.1 The NRQM
In the non-relativistic quark model (NRQM) [3], the force binding quarks together is
approximated to be linear defined by their separation distance, resulting in simple
harmonic oscillator wave-function at zeroth-order. The quarks naturally carry spin and
interact with each under the hyperfine interaction
H i jhyp =
2αS
3mim j
[8π
3δ
3(~ri j)~si ·~s j +1r3
i j(3~si · ri j~s j · ri j−~si ·~s j)
](2.1)
where the mass of quark i ( j) is mi (m j), the spin is si (s j) and the distance between quarks
is~ri j, and αS is a free parameter. Spin-orbit coupling is ignored, as the effects are
estimated to be relatively small.
35
The wave-functions presented in Ref. [23] for the hyperons is∣∣∣Λ(12+)⟩
= 0.93∣∣Λ8
2SS⟩−0.30
∣∣Λ82S′S⟩−0.20
∣∣Λ82SM
⟩−0.05
∣∣Λ12SM
⟩, (2.2)∣∣∣Σ(1
2+)⟩
= 0.97∣∣Σ8
2SS⟩−0.18
∣∣Σ82S′S⟩−0.16
∣∣Σ82SM
⟩−0.02
∣∣Σ102SM
⟩, (2.3)∣∣∣Σ(3
2+)⟩
=∣∣Σ10
4SS⟩, (2.4)∣∣∣Λ(1
2−)⟩
= 0.90∣∣Λ1
2PM⟩+0.43
∣∣Λ82PM
⟩−0.06
∣∣Λ84PM
⟩, (2.5)∣∣∣Λ(3
2−)⟩
= 0.91∣∣Λ1
2PM⟩+0.40
∣∣Λ82PM
⟩+0.01
∣∣Λ84PM
⟩. (2.6)
The first term in the “ket” notation gives the baryon with a subscript of the SU(3)
multiplet, then a superscript 2J +1 where J is the total spin. The orbital angular
momentum is S or P with a subscript representing the SU(6) permutation symmetry. S′S
indicates the baryon excitation corresponding to the n = 2 harmonic oscillator level.
The Isgur and Karl model is used to calculate the radiative widths and amplitudes by
Darewych, Horbatsch, and Koniuk in Ref. [17]. In this calculation, the photoemission is
assumed to come from the de-excitation of a single quark. A nonrelativistic reduction of
the quark-photon interaction is put between the initial and final baryon states to obtain the
T-matrix element,
〈B′(p′,s′)γ(K,λ )|T |B(p,s)〉 =
−3ie(2π)3/2 〈B′(p′,s′)| e3
e
[~σ3 · (
~K×~ε∗)2m3
+ i~p′3·~ε
∗
m3
]× e−i~K·~r3 |B(p,s)〉 (2.7)
for the B(p,s)→ B′(p′,s′)γ process with momentum p and spin s dependence. Here,
ε(K,λ ) is the photon polarization vector and e3,~r, m3, and 1/2~σ3 are the third quark
charge, position, mass and spin. The momentum of the third quark in B′ is denoted as ~p3′.
Three possible independent helicity amplitudes can be obtained using all photon
polarizations and spin states. The radiative widths are calculated as
Γγ =1
(2J +1)MB′
MB
q2
2π4∑
Jz
|AJz(q)|2, (2.8)
where J is the angular momentum and MB is the mass of the hyperon.
36
2.1.2 The RCQM
The relativistic constituent quark model (RCQM) calculation in Table 1.1 comes
from the work of Warns, Pfeil, and Rollnik [24]. The model specifies using hyperons that
are built up of three massive point-like quarks in the RCQM framework. The amplitudes
for a process are related to the matrix elements
〈ΨY ′|HEM |ΨY 〉 , (2.9)
for the process Y ′→ Y + γ∗ where Y is the initial hyperon state and Y ′ is the final and the
photon γ∗ is either real or virtual. The RCQM uses electromagnetic transition amplitudes
that also assume a single quark transition hypothesis such that the incoming photon is
coupled to only one of the constituent quarks. After taking additional relativistic effects
into account and the center-of-mass motion of the three-quarks the electromagnetic
Hamiltonian H EM can be expressed as,
H EM = HnrSQ +H(2)
SQ +H(3)SQ +H(2)
I +H(3)I +H(2)
CM +H(3)CM +O
(1c4
), (2.10)
where HnrSQ is the non-relativistic Hamiltonian from the coupling between the photon field
and a single quark, and the second and third order relativistic corrections in 1/c are H(2)SQ
and H(3)SQ , similarly for the interaction potential between the quarks, H(2)
I and H(3)I . Finally
H(2)CM and H(3)
CM are the adjustments needed for the center-of-mass motion correction of the
three quark system.
The Isgur-Karl wave-functions (equations 2.2-2.6) were used with the u and d quark
masses at 350 MeV each and with the strange quark mass of 580 MeV. Numerical
calculations of the electromagnetic transition amplitudes with both transversly and
longitudinally polarized photons. Calculations of the form factors and electoexcitation
amplitudes were achieved in the equal velocity frame, valid only for a momentum transfer
of Q2 ≤ 3 GeV2. Only calculations for the neutral channel were done.
37
2.1.3 The χCQM
In the chiral constituent quark model, spontaneously broken chiral symmetry of
low-energy QCD leads to constituent quarks and the pseudoscalar mesons as pertinent
degrees of freedom. The Hamiltonian, using three different quark masses mi is,
H =3
∑i=1
(mi +
Pi2
2mi
)− P2
2M−ac
3
∑i< j
λC ·λ C(ri− rj)2 +
3
∑i<j
Vres(ri,rj) (2.11)
where λ C ·λ C is the color space scalar product of the Gell-Mann matrices, P and M are
the center of mass momentum and mass respectively, ac is the confinement strength and
ri j is the distance between quarks i and j. This expression uses a quadratic confinement
potential. The V res term is the residual interaction potential which contains the one-gluon
exchange contribution as well as the chiral interaction potential from the pseudoscalar
meson exchange. The electromagnetic currents are built from the nonrelativistic reduction
of the Feynman diagrams that consist of the impulse approximation, the pseudoscalar
meson-pair current, the pseudoscalar meson in-flight current, the gluon pair current and
the scalar-exchange current. The helicity amplitudes as a function of the square of the
photon momentum are denoted as A3/2(q2) and A1/2(q2) which enables one to write the
expression for the partial decay width,
ΓM1 =ω2
π
Moct
Mdec
22J +1
∣∣A3/2(q
2)+A1/2(q2)∣∣. (2.12)
Here the total spin J = 3/2 is from the excited state baryon and ω is the resonance
frequency in the center of mass of the decaying hyperon. The E1 transition follows from
the same expression using the corresponding helicity amplitudes. The decay widths are
primarily governed by M1 transitions, which are determined by the impulse
approximation. This is due to a predominant two-body current cancellation.
38
2.1.4 The MIT Bag Model
The concept of the bag model originated from the central idea of confining the colors
of quarks and gluons inside the hadron. The color electric fields can only exist within the
hadron interior where the vacuum is restored to the perturbative vacuum. The region
where the quarks and gluons are allowed is called the “bag.” The Lagrangian of the MIT
bag model is,
L =i2
ψ(←−/∂ −−→/∂ )ψθ(R− r)− 1
2ψψδ (r−R), (2.13)
where θ(R− r) is the Heaveside’s step function, ψ is the quark field, R is the bag radius,
with exterior mass M. The quark mass inside the bag is zero.
The starting point is to consider the Λ(12−) wave function and perform the SU(3)
symmetry breaking. Taking a non-zero strange quark mass allows for mixing of the
flavor-singlet∣∣Λ1
2⟩ and flavor-octet∣∣Λ8
2⟩. The basis states can be represented using the
SU(3) flavor multiplet, using again spin and parity of the excited quark. Using standard
MIT bag parameters, the first lowest-lying Λ(12−) state wavefunctions at 1364 MeV is,
|Λ〉1 = 0.39∣∣∣∣Λ1,
12
−⟩+0.42
∣∣∣∣Λ8,12
−⟩A+0.46
∣∣∣∣Λ8,12
−⟩B+0.67
∣∣∣∣Λ8,32
−⟩,(2.14)
where index A indicates that the state is totally symmetric while B indicate a mixed
symmetric wave-function in pseudospin for the octet states. The large (8, 32−
) component
is indicative of the low eigenfrequency of the p 32
mode in the bag. There is coupling of the
s 12
quarks to S = 1 and so it can couple to the Σ state in the radiative decay. The next state
predicted at 1446 MeV is,
|Λ〉2 = 0.93∣∣∣∣Λ1,
12
−⟩−0.11
∣∣∣∣Λ8,12
−⟩A−0.21
∣∣∣∣Λ8,12
−⟩B−0.26
∣∣∣∣Λ8,32
−⟩,(2.15)
which is closer to the Λ(12−) ground state in the SU(3) limit. An evaluation of the
radiative decay width is made for both Λ1 and Λ2 to gain a crude measure of the
sensitivity of the bag wave function.
39
The MIT bag model radiative width calculation is then determined as,
ΓJ f Ji = 2k 12Ji+1 ∑Mi,M f ∑λ=±1
∣∣∣⟨J f M f∣∣∫ d3r~ε ∗
λ(~k) ·~j(~r) e−i~k·~r |JiMi〉
∣∣∣2 , (2.16)
~j(~r) = e f ψ†(~r)~αψ(~r),~α =
0 ~σ
~σ 0
, (2.17)
where Ji, Mi are the initial state total angular momentum and the z component of the
angular momentum and J f , M f are for the final state. The ~σ are just the Pauli spin
matrices while the quark associated with the radiative transition is the Dirac spinor ψ(~r).
The momentum vector of the photon is represented by~k ≡ kz. The inital and final mass of
the baryon is mi and m f respectively. The flavor-dependent quark charge is e f . The photon
polarization vector is~ε ∗λ(~k). Numerical calculations for the radiative widths for U-spin
allowed transitions are done as well as the widths of the lower-lying neutral hyperons.
2.1.5 The Soliton Model
The bound state soliton model is used to obtain total widths and E2/M1 ratios
corresponding to decuplet-to-octet electromagnetic transitions in the paper by Schat,
Gobbi, and Scoccola [27]. The bound state soliton model uses the effective SU(3) chiral
action with a symmetry breaking term,
Γ =∫
d4x− f 2
π
4Tr(LµLµ)+
132e2 Tr[Lµ ,Lν ]
+ΓWZ +Γsb. (2.18)
Here, ΓWZ is the non-local Wess-Zumino action and Γsb is the symmetry breaking term.
The left current is Lµ = U†∂µU , where U is the chiral field. The Callan-Klebanov ansatz
is implemented and then an expansion up to second order is made in the kaon field. The
Lagrangian density is expressed as the sum of a pure SU(2) Lagrangian (dependent on the
chiral field) and the effective Lagrangian, which contains the interaction of the soliton and
the kaon fields. A solution set can be achieved by minimizing the classical SU(2) energy
40
for the equation,[− 1
r2ddr
(r2h
ddr
)+m2
k +V Λ,le f f − f ω
2Λ,l−2λωΛ,l
]kΛ,l(r) = 0 (2.19)
where the bound state energy for quantum numbers Λ = L+T and l is ωΛ,l . The quantum
numbers are expressed in a mode of decomposition of the kaon field where L is the
angular momentum operator and T is the isospin operator. The chiral angle produces the
terms h, f ,λ , and V Λ,le f f .
To calculate the radiative widths of the decuplet hyperons for the M1 transition a
multipole expansion of the electromagnetic field is implemented. The partial width is
expressed as,
ΓM1 = 18αq|⟨M3(q)
⟩|2. (2.20)
The matrix element is between the initial state octet hyperon and the final state deculpet
hyperon. The fine structure constant is α = 1/137 and q is the photon momentum. The
M3(q) operator is defined by
M3(q) =12
ε3i j
∫d3r
j1(qr)r
riJemj . (2.21)
Here j1(qr) is from the l = 1 spherical Bessel function and Jemj is simply the spatial
component of the electromagnetic current.
2.1.6 The Skyrme Model
The Skyrme model for the nucleon is based on an extended version of the non-linear
sigma model that unifies the description of mesons and nucleons at low energies. The
framework was developed by treating the nucleon as a collective excitation of a meson
field rather than a single particle state of quarks. From topological analysis the nucleon is
considered in 1+3 dimensions leading to a solution of a non-linear theory of the three
component pion. In the work by Abada, Weigel, and Reinhard [28], the hyperons are
41
considered as kaons bound in the background of the static soliton field. Strange degrees of
freedom are treated as SU(3) collective excitations of the non-strange soliton. This
collective approach starts with the non-linear representation of the pseudoscalar nonet.
The Skyrme model contains the Lagrangian component of the non-linear σ model as well
as the flavor symmetric fourth-order stabilizing term,
LS = Tr(− f 2
π
4αµα
µ +1
32e2 [αµ ,αν ][αµ ,αν ])
. (2.22)
Here the αµ = U†∂U for the field U = exp(iΦ) of the pseudoscalar nonet Φ. The physical
pion decay constant fπ = 93 MeV and f 2π = f 2
π +8β ′, whre β ′ =−26.4 MeV2. The SU(3)
symmetry breaking component to the Lagrangian is,
LSB = Tr(T + xS)[β′(Uαµα
µ +αµαµU†)+δ
′(U +U†−2)], (2.23)
where T projects onto the non-strange degrees of freedom and similarly S projects onto
strange degrees of freedom. The masses of the pion and kaon along with their decay
constants determine the needed parameters. To take into account the axial anomaly, the
Wess-Zumino term is used and written as,
ΓWZ =− iNc
240π2
∫d5xε
µνρσκTr(αµαναρασ ακ). (2.24)
To include the electromagnetic properties of baryons at finite momentum transfer a direct
derivative coupling to the photon field, Aµ , is used,
L9 = iL9(∂µAν −∂νAµ)Tr(
ξ†[
λ3 +1√3
λ8
]ξ [ξ †
αµ
ανξ +ξ α
µα
νξ
†])
. (2.25)
Here the Gell-Mann matrices are used with the square root of the chiral field such that
ξ = U1/2. The L9 is a dimensionless coefficient with value of 6.9×10−3. The total action
can then be expressed as,
Γ =∫
d4xLS +LSB +L9 +ΓWZ. (2.26)
42
To obtain information about the radiative decays of the 32+
baryons, the quadrupole and
monopole pieces of the electric and magnetic from factors are required. E(q) denotes the
electric quadrapole operator that excludes contributions with a total angular momentum of
zero. The magnetic monopole operator is denoted as M(q) for momentum q. The decay
width for the radiative transition from 32+
baryons to 12+
baryons can be calculated from
the matrix elements of E and M,
ΓE2 =6758
αh f q∣∣∣∣⟨B(
12
+)∣∣∣∣ E(q)
∣∣∣∣B′(32
+)⟩∣∣∣∣2 , (2.27)
ΓM1 = 18αh f q∣∣∣∣⟨B(
12
+)∣∣∣∣M(q)
∣∣∣∣B′(32
+)⟩∣∣∣∣2 , (2.28)
where q is the momentum of the photon in the rest frame of the 32+
baryon, and αh f is the
fine structure constant. The total decay widths in the Skyrme model calculation are found
to be strongly dominated by the M1 contribution leading to E2/M1 ratios of a few percent.
2.1.7 The Algebraic Model
The algebraic model of hadron structure, presented by Bijker, Iachello, and Leviatan
[30], introduces a spectrum generating algebra for the radial excitations. The algebra for
mesons is taken to be U(4), and for baryons, U(7). The method unifies the harmonic
oscillator quark model, U(4)⊃ U(3) for mesons and U(7)⊃ U(6) for baryons using a
formalism that treats the orbital excitations as rotations and vibrations of string structures.
The mass spectrum is presented along with the strong and electromagnetic decay widths.
This is done using a dynamical symmetry framework where the spin-flavor symmetry is
broken by the masses.
The Algebraic Model builds baryons using three constituent parts which have
internal and spatial degrees of freedom. The internal degrees of freedom are chosen to be
flavor-triplet, spin-doublet, and color triplet. The motion degrees of freedom of the three
43
quarks are given by the relative Jaccobi,
ρ =1√2(r1− r2), (2.29)
λ =1√
m21 +m2
2 +(m1 +m2)2(m1r1 +m2r2− (m1 +m2)r3), (2.30)
where mi and r are the mass and coordinates of the ith quark. As in the constituent quark
model, the electromagnetic interaction comes from the coupling of the constituent parts to
the electromagnetic field. The nonrelativistic portion of the transverse electromagnetic
coupling for the left-handed photon emission from B→ B′+ γ can be described as,
HEM = 2√
π
k0
3
∑j=1
µ je j
[ks je−ik·rj +
12g j
(p je−ik·rj + e−ik·rj p j)]
(2.31)
for the e−ik·rj wave with rj, pj, and sj for the coordinate, momentum, and spin for the jth
constituent, respectively. Also, k0 is the photon energy and k = kz is the momentum
carried by the emitted photon and g j is the strength coefficient. The transverse helicity
amplitude for the excited state of the baryon resonance is
Aν =∫
dβg(β )⟨Ψ′(0,S′,S′,ν−1)
∣∣HEM |Ψ(L,S,J,ν)〉 (2.32)
where ν gives the helicity. L,S,and J are the orbital angular momentum, the spin, and the
total angular momentum respectively. The g(β ) is the charge and magnetization
distribution. The radiative hyperon decay widths are calculated using the expression,
Γ(B→ B′+ γ) = 2πρ1
(2π)32
2J +1 ∑ν>0|Aν(q)|2 . (2.33)
The electromagnetic decay widths are calculated assuming SUs f (6) spin-flavor symmetry
while using the rest frame of the excited state baryon.
44
2.2 U-Spin Symmetry
It is possible to show that in SU(3) flavor space, the quark content of the non-strange
sector can be found by combining the (d,u) I-spin doublets. In the same way, other states
can be found such that the two (s,d) quarks are used for the U-spin doublets, and the (u,s)
quarks are used for the V-spin doublets. States of a given multiplet are transformed into
each other by means of the X-spin operators. Within an I-spin multiplet, the mass
differences are of the order of magnitude of 1 MeV, or about 1% of the particle mass.
Hence, isospin symmetry is weakly broken. I-spin respects chiral symmetry but is broken
by the electromagnetic interaction. The photon couples to the charge (and magnetic
moment) of the quarks, which are different for u and d quarks. U-spin is expected to be
strongly broken by chiral symmetry, however it respects charge symmetry. The other
subgroup, V-spin, is broken by both chiral and charge symmetry.
Baryons can be represented in symmetric spin multiples of I-spin versus hypercharge,
as shown Figure 2.1. Particles of nearly-equal mass are put in horizontal rows. The same
representation for U-spin versus charge can be shown as in Figure 2.2. In this case,
particles of equal charge form horizontal rows. This representation of the baryon decuplet
can be useful when considering electromagnetic transitions, such as radiative decays.
SU(3) f can be written in terms of its subgroups as SU(2)I × U(1)Y or equivalently as
SU(2)U × U(1)Q.
To look closer at how U-spin symmetry is useful, first consider two examples of
U-spin symmetry applied to baryon decays. It is possible to show that U-spin conservation
forbids some baryon radiative decays, and can in principle suggest an experimental test of
U-spin symmetry. A calculation of the corrections, based on the known mass difference of
the s and d quarks, gives a prediction of the degree of U-spin symmetry breaking expected
in the present measurements, as shown next.
45
Figure 2.1: Baryon decuplet, hyper-charge (Y ) versus isospin (I3).
Figure 2.2: Baryon decuplet, plottedfor U-spin multiplets with charge (Q)versus u-spin (U3).
2.2.1 Examples
To test for U-spin invariance, consider the example of comparing the decays of
∆−→ nπ− and Σ∗−→ Λπ−. Both the ∆− (U3 =−3/2) and the Σ− (U3 =−1/2) are
members of the U = 3/2 multiplet [37]. The mesons can also be arranged in U-spin
multiplets, with the π− having U = 1/2 and U3 =−1/2. The neutron has definite U-spin,
with U = 1 and U3 =−1. However the Λ mixes with the Σ0, so each one has both U = 1
and U = 0 components [38]. Using the U-spin unitary rotation in SU(2) leads to
|ΛU〉=√
32|U = 1〉+ 1
2|U = 0〉 .
As with I-spin, the usefulness of U-spin comes from its ability to predict transition
probabilities using just Clecsh-Gorden (CG) coefficients. A decuplet decay from an initial
state of U = 3/2 to a final state with one particle having U = 1/2, requires a ∆U = 1
transition. For the decay of the ∆−, the U-spin (CG) is unity. The amplitude is denoted as
M (∆−→ nπ−)≡M1 .
46
The Σ∗− decay has a CG of 〈32 −
12 |1 0 1
2 −12〉=
√3/2 where only the U = 1 component
of the Λ contributes. Under U-spin symmetry the amplitude is,
M (Σ∗−→ Λπ−) =
1√2M1.
Taking into account the phase space factors, which are proportional to
|p3|(E ′+M′)/(MRM′2), for a P-wave (J = 3/2→ J = 1/2) decay it is possible to
compare the decay widths. The p in the phase space factor is the center-of-mass
momentum of each decay particle, E ′ is the energy of the decay baryon, and MR the mass
of the initial baryon resonance. U-spin symmetry gives,
Γ(∆−→ nπ−)Γ(Σ∗−→ Λπ−)
= 1.769× 2|M1|2
|M1|2= 3.54 .
The experimental results lead to a ratio of,
Γ(∆−→ nπ−)Γ(Σ∗−→ Λπ−)
=118 MeV
0.87×39.4 MeV= 3.44±0.20
from Ref. [60], where 0.87±0.015 is the branching ratio and 39.4±2.0 MeV is the full
width of the Σ∗. Here U-spin is very much in agreement with the physical result.
Note that U-spin is not necessarily a symmetry of the strong interaction, since there is
a big difference in the masses of the d and s quarks, so this good agreement with
experiment is somewhat surprising. However, this is only one case, and more tests of
U-spin symmetry (cases which are not already constrained by I-spin, charge or other
symmetry laws) are needed before any conclusion can be reached.
2.2.2 U-Spin prediction
It is possible using the U-spin SU(3) multiplet representation to obtain a prediction
for the ratio of the ∆0→ nγ partial width to the Σ∗0→ Λγ partial width. This implies that
the experimental partial width of the ∆0→ nγ reaction can be used with the U-spin
Clebsh-Gordon coefficients along with the corresponding phase space factors to obtain the
47
expected partial width of the Σ∗0→ Λγ . In the strict limit of SU(3) symmetry, U-spin is
conserved for all processes. Only radiative transition between states with the same value
of U-spin can occur within this limit.
In U-spin space, the unitary rotation in the SU(2) symmetry space can be used to
describe the mixing between the neutral ground state SU(3) multiplet members Λ and Σ0
such that, ∣∣Σ0U⟩
=12|U = 1〉+
√3
2|U = 0〉
|ΛU〉 =12|U = 0〉−
√3
2|U = 1〉 (2.34)
The radiative decays B(32+)→ B′(1
2+)γ are M1 and E2 electromagnetic transitions with
the spin flip of one of the quarks in B. The E2 amplitudes are very small and are
negligible. The amplitude requires the Clebsch-Gordon coefficients, which can be found
by contraction of the initial excited state baryon with the final state baryon and the emitted
photon, where the photon is a U-spin scalar with U = 0, resulting in⟨∆
0|nγ⟩
= 〈1−1|1−1 0 0〉= 1⟨Σ∗0|Λγ
⟩= −
√3
2〈1 0|1 0 0 0〉=−
√3
2. (2.35)
The ratio strictly based on this rotation is then
|〈∆0|nγ〉|2
|〈Σ∗0|Λγ〉|2=
43. (2.36)
The phase space factors required are presented in Ref. [17] and use a radiative width
Γγ ∝M′BMB
q2|AB→B′γ(q)|2 (2.37)
where q is the center of mass momentum, A is the amplitude of the decay (which contains
a factor of√
q), and MB (MB′) is the mass of the decaying (final state) hyperon. The ratio
can then be expressed as
〈∆0|nγ〉2
〈Σ∗0|Λγ〉2=(
Mn
M∆
)(MΛ
MΣ∗0
)−1( qn
qΛ
)3 43
= 1.56. (2.38)
48
The values for the center of mass momentum are qn = 0.259 GeV/c and qΛ = 0.241
GeV/c, [60].
This implies that the U-spin prediction for the partial width of the electromagnetic
decay using the measured width of the ∆0→ nγ is,
1.56−1×Γ(∆0→ nγ) = 1.56−1×660±60 = 423±38 keV,
where the partial width from the ∆→ nγ comes from reference [60].
Similarly a U-spin prediction can be calculated for the ∆+→ pγ partial width to the
Σ∗+→ Σ+γ partial width.
⟨∆
+|pγ⟩
=⟨
12− 1
2|12− 1
20 0⟩
= 1
⟨Σ∗+|Σ+
γ⟩
=⟨
12
+12|12− 1
20 0⟩
= 1,
leading to a ratio of,
Γ(∆+→ pγ)Γ(Σ∗+→ Σ+γ)
=(
Mp
M∆
)(MΣ+
MΣ∗+
)−1( qp
qΣ+
)3
= 2.96.
The values for the center of mass momentum are qp = 0.259 GeV/c and q+Σ
= 0.173
GeV/c, [60].
This implies that the U-spin prediction for the partial width of the electromagnetic
decay using the width of the ∆+→ pγ decay is,
2.96−1×Γ(∆+→ pγ) = 2.96−1×660±60 = 223±20 keV. (2.39)
It is important to note that the U-spin predictions contain no direct meson cloud
calculation but instead implements the empirical information from the ∆ electromagnetic
decays. It is assumed for the sake of these predictions that all meson cloud contributions
are contained in this empirical value. It is not clear yet if the meson cloud contributions
are the same for both the Σ∗0 and Σ∗+ decays. More theoretical calculations are necessary
before this issue can be settled.
49
2.3 Meson Cloud effect
In the work by Sato and Lee [39], an effective hamiltonian of the γN→ ∆ vertex
interaction is developed by applying a unitary transformation to the Lagrangian with
various field components N,∆,π,ρ,ω and γ . It is found that the helicity amplitudes
calculated from the dressed γN→ ∆ vertex are in better agreement to the empirical values
(listed in the Particle Data group) than the bare amplitude. The differences in the bare and
dressed amplitude are due to the nonresonant meson exchange mechanisms refered to as
“meson cloud” effects. The Sato and Lee (SL) model attempts to have a consistent
description of both the πN scattering and the electromagnetic pion production reactions.
The SL model expresses the γN→ πN amplitude as
TπN,γN(E) = tπN,γN(E)+ tRπN,γN(E). (2.40)
The nonresonant component of the amplitude is
tπN,γN(E) = [tπN,γN(E)GπN(E)+1]νπN,γN . (2.41)
where the nonresonant πN scattering amplitude is
tπN,πN(E) = νπN,πN [1+GπN(E)tπN,πN(E)]. (2.42)
Here νπN,πN is the πN potential and the GπN(E) is the πN propagator. The amplitude in
terms of the dressed vertex interactions is then
tRπN,γN(E) =
Γ†∆,πNΓ∆,γN
E−m∆−Σ∆(E), (2.43)
where m∆ is the ∆ mass and Σ∆(E) is the energy shift. The dressed components of the
vertex interaction, described in terms of the bare vertex interaction, are
Γ†∆,πN = [tπN,γN(E)GπN(E)+1]Γ†
∆,πN (2.44)
Γ∆,γN = Γ†∆,γN +Γ∆,πNGπN(E)tπN,γN . (2.45)
50
Using Eq. 2.41 and 2.42, the expression in Eq. 2.45 can be rewritten as
Γ∆,γN = Γ†∆,γN +δΓ∆,γN , (2.46)
where
δΓ∆,γN = Γ∆,πNGπN(E)νπN,γN (2.47)
and
Γ∆,πN = Γ∆,πN [1+GπN(E)tπN,πN ] . (2.48)
Figure 2.3: The diagrams for the dressed γN → ∆ vertex. The meson cloud diagrams arein brackets.
The πN potential contains the direct and crossed nucleon terms, ρ exchange, and the
crossed ∆ term. Each interaction vertex is regularized with a dipole form factor. The
parameters of the model are adjusted along with m∆ to fit the empirical πN scattering
phase shift. The dressed γN→ ∆ term contains the meson loops illustrated in Figure 2.3.
To analyze the N∆ form factors, the model fits all of the available pion
electroproduction data at energies close to the ∆ position. The results, using a simplified
51
parameterization, are used to obtain the dressed M1 form factors G∗(Q2). The magnetic
dipole transition form factor for γ∗∆→ ∆(1232) are normalized to the proton dipole form
factor and compared to experimental data, shown in Figure 2.4. The data is from DESY
and SLAC [42], BATES [43], MAMI [44], and JLAB [45]. The solid curve indicates the
dressed calculation, while the dotted line is without the meson cloud effect.
The results in Ref. [41] reveal that the meson cloud effect can contribute significantly
(∼ 40%) to the overall electromagnetic decay width of the ∆→ Nγ . There are no
calculations yet available for the Σ∗→ Y γ decay amplitude. Assuming the meson cloud
effects are important, more sophisticated calculations are necessary to probe the structure
of the baryon resonances. Few models actually include meson cloud contributions for
electromagnetic decay.
52
Figure 2.4: The magnetic dipole transition form factor G∗(Q2) for γ∗∆→ ∆(1232). Theexperimental points (empty circles) are for the inclusive data from pre-1990 experimentsat DESY and SLAC [42], and exclusive data (filled squares) are from BATES [43], MAMI[44], and JLAB [45]. The Solid curve indicates the dressed calculation, while the dottedline is without the meson cloud effect.
53
3 JEFFERSON LAB, CEBAF AND CLAS DETECTOR
The g11a data were extracted from an experiment conducted in Hall B at Thomas
Jefferson National Accelerator Facility (TJNAF), in Newport News, Virginia. The g11a
data set was collected in 2004 as part of the E04021 experiment Spectroscopy of Excited
Baryons with CLAS: Search for Ground and First Excited States.The initial purpose for
this run was a high-statistics search for the Θ+ pentaquark [54]. A loose trigger was used,
allowing for many other photoproduction studies.
Figure 3.1: The aerial view of CEBAF at Thomas Jefferson National Labs. The“racetrack”-shaped area indicates where the accelerator ring lies underground. Eachexperimental hall is underneath the three grassy sectors near the bottom of the picture.Hall B is the middle hall. Image source:[35].
54
The run conditions for g11a included a tagged photon beam incident on a liquid
Hydrogen target. The CLAS (CEBAF Large Acceptance Spectrometer) detector was used
in the acquisition of multi-particle final states at approximately 60% coverage of the full
4π solid angle. The data used in the following analysis was produced with tagged
bremsstrahlung from a electron beam at 4.023 GeV. The CLAS tagger hodoscope can
measure photons between the energies of 20% and 95% of the electron beam energy. The
proton target was came from liquid Hydrogen maintained by a cryogenic system. The
g11a run resulted in roughly 20 billion triggers stored as 21 TB of raw data.
In the follow chapter details of the experimental apparatus and setup used in data
collection for the g11a experiment are covered. An overview of the Continuous Electron
Beam Accelerator Facility (CEBAF) is seen in Figure 3.1.
3.1 Continuous Electron Beam Accelerator Facility
CEBAF is a 6 GeV end-point energy electron beam accelerator. The facility delivers
electron beam to the experimental halls A,B, and C at Thomas Jefferson Lab. CEBAF
uses radio-frequency (RF) cavities for electron acceleration. The superconductors
employed in the cavities provides three times more power than without the
superconductors in the RF design. Greater efficiency using the superconductors is due to
zero energy lost by electrons in the RF cavity. In addition, the cavity temperature is not
increased by transmission of electrons. This non-resistive superconducting nature of the
cavities allows CEBAF to attain a 100% duty factor being that there in no down-time
required to cool the conduction elements of the accelerator [36]. The continuous delivery
of electrons allows rapid acquisition of high statistics datasets. A schematic diagram of
CEBAF can be seen in Figure 3.2.
Injectors are used to input the electron beam. The injector consists of a polarized
photo-emission electron gun. The three diode lasers of the electron gun, one for each
55
Figure 3.2: A schematic diagram of the Continuous Electron Beam Accelerator Facility.The linear red tubes represent the LINACs made of superconducting RF cavities groupedinto 20 cryomodules. A magnification of the recirculation arcs is shown in the top right.The experimental halls are shown on the bottom left. Image source:[35].
experimental hall, allows each hall to independently control its current and beam
polarization. The lasers are pulsed independently at 499 MHz, incident on a strained
GaAs photocathode. The lasers are pulsed 120 out of phase, matching the 1497 MHz
frequency of the accelerator. Each experimental hall receives electron bunches at intervals
of 2 ns. After extraction from the photo-cathode the electrons are accelerated to 45 MeV
with the 2 1/4 superconducting Radio frequency (RF) cavities. An optical chopper is then
used in the injector system to cleanly separate the bunches prior to sending them to the
CEBAF’s recirculating linear accelerators (LINACs).
56
Figure 3.3: A picture of a pair of superconducting niobium Radio Frequency (RF) cavities.CEBAF uses 338 uperconduction cavities, like the RF cavity shown. Image source:[36].
Figure 3.4: A schematic diagram of the superconducting RF cavity in operation. Theacceleration gradient is provided by establishing a standing wave, leading to a continuouspositive electric force on the electron. The phase of the waves advances the position of theelectron bunch in the cavity creating the gradient. Image source:[35].
Each recirculating LINAC contains 168 superconducting RF Niobium cavities.
Figure 3.3 is a picture of the RF cavity assembly. The superconducting cavity temperature
is cooled to 2 K with liquid Helium. Radio frequency standing waves in the cavities are
used to produce the acceleration gradient for the electron beam, see Figure 3.4. A
57
continuous positive force is made on the beam electrons in the cavity by the use of
standing waves kept in phase with the beam bunches. There are two LINACs located along
the straight portion of the 7/8-mile recirculating accelerator path. Recirculation occurs by
bending magnets in the curved portions of the tracks. Electrons can pass through the pair
of LINACs up to five times before being delivered to an experimental hall. Each pass
through a LINAC adds up to 600 MeV to the beam energy. The maximum energy
presently attainable by five passes is approximately 6 GeV. The beam is extracted by the
experimental halls by using RF separator cavities. Each hall can control the beam energy
delivered by extracting the beam after a given number of passes (no greater than five).
Before delivery, the beams are fanned out according to energy at the end of the south
LINAC. To divert specific electron beam bunches to the desired hall, a 120 phase
separation is used by the RF separator cavities. This style of separation allows
experiments that require different beam to run at the same time. CEBAF is capable of
delivering a five-pass beam to each hall simultaneously, but it cannot provide a single low
energy beam to two halls at the same time.
58
3.2 The Bremsstrahlung Photon Tagger
For this and other photoproduction experiments at CLAS, the CEBAF electron beam
is used to produce bremsstrahlung photons from the beam incident on a target. Photons
are created by passing electrons through a thin radiator, creating bremsstrahlung radiation.
The bremsstrahlung cross section is proportional to Z(Z +1), with Z being the atomic
number of the radiator. Because of the spectral nature of the photon energy produced, a
tagger system is required to measure the recoiling beam electron energy. The energy of
the bremsstrahlung photon is then deduced through conservation of energy. Because the
produced number of bremsstrahlung photons is nγ ∝ 1/Eγ , the production is dominated by
low energy photons.
The high mass and density of the gold radiator is ideal for the reaction
ebeam +Au→ e′beam +Au′+ γbrem. For high momentum incident electrons, the production
photon and the recoil electron will continue in the same relative direction, in the lab
frame, sharing the energy of the incident electron. The beam line then passes through a
dipole magnet which separates out the electrons from the photons. Electrons that have not
lost energy and have not reacted with the radiator are disposed of in a beam dump. The
reacted electrons experience a greater change in direction through the field. Electrons with
energies between 20% and 95% of the original beam energy are passed into the
hodoscope. Two layers of scintillation paddles to determine the electrons energy and
timing. The bremsstrahlung photons, γbrem, associated with the tagged electrons pass
through a collimator, and continue down the beam line to interact with the target. Beam
buckets are delivered every 2 ns with a width of ∼0.02 ns. A schematic diagram of the
photon tagging system is shown in Figure 3.5.
59
Figure 3.5: A schematic of the photon tagging system. The tagger is setup to allow indirectmeasurements of the photon beam energy. The recoil electron is directed into the taggerspectrometer to that is energy can be measured to deduce the photon energy produced.Image source: [56].
3.2.1 The Radiator
The radiator for the tagger consists of several different foils and a “harp” made of
two perpendicular wires. These are mounted on a “ladder” that can move various radiators
into the beam line. For the g11a data-taking runs, the thickest of the radiators used was a
gold foil with a thickness of 10−4 radiation lengths (646 µg/cm2). A thinner foil (10−5
radiation lengths) was used for the normalization run.
The “harp” was used to measure the electron beam profile and position at the
radiator. During the scan, two perpendicular wires pass through the beam, while
scintillators downstream are used measure the intensity of the beam at the wire position.
60
3.2.2 The Magnetic Spectrometer
The tagging system used a magnetic spectrometer to measure the energy of the
recoiling electron after the interaction in the radiator. A 1.75 T maximum field,
normal-conducting dipole magnet is used to direct both recoiling and non-interacting
electrons. The dipole magnet was designed to enable electron beams up to 6.1 GeV to be
used to produce real photons and still bend the non-radiating electrons into the beam
dump. The energy resolution, for operating below 4 GeV, is ∼ 0.2%.
3.2.3 The Hodoscope
The hodoscope is the scintillation device used to measure the energy of the electrons
that radiated, to obtain the photon energy, to determine the timing of the event with
sufficient precision, and to act as part of the trigger. The hodoscope is made with two rows
of scintillating paddles, the E-counters and T-counters for “Energy” and “Timing”. The
top plane of 384 scintillators that are used to determine the momentum of the recoiling
electron, called the E-plane. The E-plane scintillators are 20 cm long, 4 mm think and
between 8 cm - 16 cm wide with a photomultiplier tube (PMT) on one end. These
E-counters are designed to cover areas that represent approximately equal sized energy
bins. The placement of each E-counter improves resolution through the use of an
overlapping formation, increasing the effective number of E-counters to 767, giving to an
energy resolution of ∼ 10−3×Ebeam. The lower scintillator plane determines precise
timing information of the recoiling electrons, called the T-plane. The T-plane sits 20 cm
below the E-plane and consists of 61 scintillators with a PMT on each end. The T-counters
are 2 cm thick, leading to a timing resolution of 110 ps. Both the E-counters and
T-counters are arranged so that the scintillators are normal to the electron trajectory as
they pass through the focal plane. A schematic diagram of the tagger magnet and
hodoscope is provided in Figure 3.6.
61
Figure 3.6: A schematic of the tagger magnet and hodoscope used in the tagging system.The trajectories of the recoil electrons are depicted by the dashed lines. Electrons fromvarious trajectories in the spectrometer correspond to bremsstrahlung photons of a givenenergy. Image source: [56].
3.2.4 The Tagger Readout
The signals from each E-counter PMT are passed to a discriminator. The PMTs from
each end of the T-counters are fed into a constant fraction discriminator (CFD). The
signals that pass the CFD are sent to a Master Or (MOR) and then the time-to-digital
converter (TDC) array. The TDC is stored in raw data bank preserving precise timing
information for each T-plane hit as well as the total number of hits recorded in the tagger.
The total number of recoil electron hits is needed to derive the energy-dependent photon
flux. The electron timing information is used to find the photon interaction time for each
event recorded. The MOR is used to set the trigger and controls the stop signal to the
E-counter TDC array. The E-counter readout in the trigger is written to the data stream
along with the T-counter readout that set the trigger. A schematic of the Hall B tagger
62
logic setup is shown in Figure 3.7. More information on the Photon Tagger can be found
in [56].
Figure 3.7: A schematic of the tagger logic setup. The T-counter hits are also used to setthe event trigger. The common stop to the E-counter TDC array is controlled by the theCLAS Level 1 trigger. Image source: [56].
63
3.3 The CLAS Detector
The CEBAF Large Acceptance Spectrometer (CLAS) Detector is comprised of
various detector subsystems. For the present analysis the subsystems required are the start
counter, the drift chambers, time-of-flight counters, the toroidal magnet, and the
electromagnetic calorimeter. The gas Cerenkov detector is primarily used in
electroproduction experiments and is not discussed here. A photograph and schematic are
shown in Figures 3.8 and 3.9 respectively.
Figure 3.8: A photograph of the CLAS detector from inside Hall B. The Time-of-flightscintillators are pulled away from the drift chambers showing inside. Image source: [35].
64
Figure 3.9: A schematic diagram of the full CLAS detector. The drift chambers are shownin violet, the toroidal magnet is shown in light blue, the Time-of-flight scintillators areshown in red, and the electromagnetic calorimeter is shown in green. Image source: [35].
3.3.1 The g11a Cryotarget
To study reactions of the type γ p→ K+Y , a target of high density atomic protons is
required. Liquid hydrogen was used as the target material in the cell. Many different
target geometries and materials are used during various production runs. The cryotarget
cell geometry used during the g11a run period was a cylindrical Kapton chamber with
dimensions of 40 cm in length and 4 cm in diameter [57]. The density of the liquid
hydrogen is determined using the temperature and pressure inside the cell which were
65
monitored on an hourly basis. The target density averaged over all g11a runs is 0.07177
g/cm3. A diagram of the g11a cryotarget is shown in Figure 3.10.
Figure 3.10: A diagram of the g11a Cryotarget. The dimensions of the target cell usedwere 40 cm long and 4 cm in diameter. The cell was filled with liquid H2. Image source:[35].
3.3.2 The Start Counter
A new start counter was installed for use in the g11a run period. The start counter
was designed to achieve full acceptance coverage using the 40 cm long cryotarget. The
start counter is a segmented scintillation detector surrounding the cryotarget. The start
counter is segmented into six sectors corresponding to each sector of CLAS. Each sector
contains four scintillator strips instrumented with PMTs. The start counter timing
resolution is roughly 400 ps. The number of scintillator paddles was chosen based on the
estimated integrated rate load at the anticipated luminosity. The start counter timing
information is not directly used in the present analysis. However, the start counter is an
integral part of the g11a trigger. Other details on the CLAS start counter can be found in
[58]. A diagram of the start counter used in the g11a run period is shown in figure 3.11.
66
Figure 3.11: A diagram of the new g11a Start Counter. The detector is a six-sectorscintillation device with four mounted photomultiplier tubes for each sector. One sectoris cut away in the diagram to show the target space. Image source: [58].
3.3.3 The Superconducting Toroidal Magnet
CLAS is assembled around six superconducting magnetic coils separated in the
azimuthal angle by 60 around the beam line. The geometry of many of the CLAS
detector elements are determined by the toroidal magnet. The six superconducting coils
required to generate the field create shadow regions in which no particles can be detected,
making a natural segmentation for the other assemblies. The electronics are mounted in
the shadow regions of the detector created by the torus cryostats. The beam line passes
through the center of the coil configuration. A photograph of the toroidal magnet is shown
in Figure 3.12. The toroidal magnet has a maximum operating current of 3861 A, leading
to a maximum magnetic field of 3.5 T. The value of∫
~Bd~l varies from 2 Tm for high
momentum tracks in the forward angles to 0.5 Tm for trajectories where θ > 90. During
67
Figure 3.12: A photograph of the coils of the CLAS toroidal magnet before installation ofthe rest of the detector subsystem. Image source: [59].
the g11a run period the current was set to 1920 A, leading to a maximum field of ∼ 1.8 T
in the anti-clockwise direction about the beam line when looking upstream. The magnetic
coils generate a toroidal magnetic field ~B, of six-fold cylindrical symmetry around the
beam line. The charged particles in CLAS pass through this field and bend in θ , toward
(away) from the beam line if the particle is negatively (positively) charged in the default
magnet configuration. The radius of curvature of a charged particles can be expressed as
r = pprep/qB, where pprep is the component of the particle’s momentum perpendicular to
the magnetic field. The contour and cross section of the CLAS toroid magnetic field is
shown in Figure 3.13. The superconducting magnet has a cryostat temperature regulation
system using liquid helium delivered from the central CEBAF helium refrigerator (CHL).
68
(a) (b)
Figure 3.13: (a) The contours of constant absolute magnetic field of the CLAS toroid in themidplane between two of the coils. (b)The field vectors for the CLAS toroid transverse tothe beam. The field lines represent the field strength. The six coils are shown in the crosssection. Image source: [52].
The coils are regulated to the operate at a temperature of 4.5K. Other details on the
CLAS superconduction toroidal magnet can be found in [59].
3.3.4 The Drift Chambers
The drift chambers are the core of the charged particle tracking system. Effective
charged particle tracking can be achieved by measuring the location of a particle at several
points along its trajectory while minimizing its multiple scattering. The CLAS drift
chambers enable position measurements of charged particles with a precision of a few
hundred microns, while limiting the interference of background radiation. Momentum is
determined by tracking the particles as they travel through the field generated by the
toroidal magnet. Particles are tracked with three drift chamber regions. Region 1 is
mounted to the magnet’s cryostats and lies between the start counter and the innermost
part of the toroidal magnet coils. Region two sits in the middle of CLAS in the area of the
69
Figure 3.14: A diagram look down showing the CLAS drift chamber region relative to theother subsystems. The dashed lines outline the location of the toroidal magnet coils. Imagesource: [59].
strongest magnetic field. Region 3 sit closest to the Time-of-flight paddles outside the
magnet coils, see Figure 3.14.
Drift chambers should have a one-to-one relationship between the time measured and
the distance of closest approach (DOCA) of the track to the wire. This implies that the
electric field should be cylindrically symmetric about the high-voltage sense wire. A
quasi-hexagonal geometry made of six field wires are used to create the needed field
around each of the 35,148 sense wires.
Each drift chamber region has six sectors, each which span roughly 60 of the plane
perpendicular to the beamline. Each region consists of two sublayers or “super-layers.”
The super-layers contain an array of drift cells that have six 140 µm gold-plated field
70
wires around the 20 µm gold-plated tungsten sense wire. The super-layers which are
oriented such that the sense wires are perpendicular to the mid-plane of each sector are
called the “axial” layers. The super-layers which are oriented 6 to the axial wires, to
improve resolution in the φ -direction, are known as the “stereo” layers. There is also a set
of “guard” wires that sit on the edges of each super-layer that hold a low voltage to
simulate a continuous electric field configuration in each cell. This allows the field in each
cell to by roughly independent of the cell’s position in the chamber.
The drift chambers are filled with a 90% Argon and 10% CO2 gas mixture. A
potential difference between the sense wires and the field wires in maintained by
operating the field wires at a negative high voltage. During detection, charged tracks
ionize the gas in the cell around the positively-charged sense wires. As a particle passes
through the cell and ionizes the gas electrons along the particle’s path, these electrons
begin to accelerate towards the sense wire. Electron collisions with the other molecules in
the gas lead to the a fairly constant “drift” velocity. The collection of these electrons
registers as a voltage pulse in the sense wire. The signals from the sense wires pass
through a preamplifier and discriminator before being recorded in the data stream. More
details on the CLAS drift chambers can be found in [53]
3.3.5 The Time-of-Flight Detector
The time-of-flight (TOF) subsystem is an essential component of the CLAS detector.
The TOF detector provides the required timing information for charged tracks that can be
used to develop the particle identification schemes. The typical path length that particle
will travel from the target to the TOF paddles is ∼ 5 m. The TOF shell is a six-fold
segmented array of scintillator strips that covers the outside shadow area of the torus coils
in each sector from 8 to 142. Each sector of the TOF shell is composited of 57
71
scintillator paddles made of Bicron BC-408 scintillation material. Each paddle is 5.08 cm
in thick and has a PMT instrumented at either end.
There are four scintillator panels in each sector, as seen in Figure 3.15 (a). In the first
panel there are 23 paddles that detect particles in the range of 0−45. Each of the 23
paddles is 15 cm wide. The backward angle paddles are 22 cm wide and are
photo-coupled to 3 PMTs using light guides, as seen in Figure 3.15 (b). The length of the
paddles varies from 32 cm to 445 cm as required by the shape of the sectors. The eighteen
least forward paddles are coupled in pairs. This leads to a total of 48 logical counters. The
timing resolution of the TOF detector is between 80 and 160 ps, in which the spread is
defined by the variation in scintillator length.
The TOF subsystem is a critical part of the g11a trigger. Details concerning the
design, construction, and testing of the TOF detector are discussed in [51].
(a) (b)
Figure 3.15: (a) An isolated view of the time-of-flight paddles in one sector. The setdesignated for forward angle detection are on the right. (b) The schematic of the lightguide used to connect the backward angle TOF paddles to the PMTs.
72
3.3.6 The Forward Electromagnetic Calorimeter
The detection of neutral particle like photons and neutrons is made possible during
the g11a run period by used the the Electromagnetic Calorimeter (EC). The EC sits about
5 m from the target on the outside of the TOF system in CLAS. There are six nearly
equilateral triangle EC volumes, one for each CLAS sector. Each EC volume is made of a
sandwich of 39 alternating layers of plastic Bicron BC412 scintillator strips that have
dimension of 1 cm × 10 cm that run between 0.15 to 4.2 m in length, and lead sheets 2.2
mm thick. The calorimeter area of each successive layer increases, minimizing the shower
leakage at the EC edges.
Each scintillator layer consists of 36 scintillator strips that run parallel to one side of
the triangle. Each successive layer is oriented 120 relative to the previous layer,
following the orientation of the triangle. This structure results in three stereo views
labeled as the U-plane, V-plane and the W-plane, see Figure 3.17. Each stereo view is
subdivided into an inner (5 layers) and outer (8 layers). The inner set of layers sit closest
to the target. There are 216 PMTs required for each stereo view, leading to a total of 1296
PMTs and 8424 scintillator strips in the whole CLAS EC.
The signal from the scintillator is transmitted to the PMTs through green wavelength
shifting fibers, see Figure 3.16 (a). All scintillators from one vertical line, see Figure 3.17
(a), are connected to a single PMT so that a shower created by a particle passing through
has three such lines (U,V,W) associated with its cluster reconstruction position. The
reconstruction of a typical electromagnetic shower is determined by identifying groups of
strips in the three stereo views. The reconstruction algorithm collects groups that have a
PMT response above an energy deposition threshold for contiguous strips. The algorithm
sorts the groups according to the sum of the scintillator strip energy. An electromagnetic
shower reconstruction event is shown if Figure 3.17 (b).
73
Figure 3.16: A vertical slice of the EC light readout system. PMT - Photomultiplier TubeLG - Light Guide, FOBIN - Fiber Optic Bundle Inner, FOBOU - Fiber Optic Bundle Outer,SC - Scintillators, Pb - 2.2 mm Lead sheets, IP - Inner Plate (closest to target or face of EC)Image source: [55].
74
(a)
(b)
Figure 3.17: (a) View of one of the six CLAS electromagnetic calorimeter triangularmodules showing the three projection planes. (b) The diagram of event reconstructionin the EC. Energy deposition profile is shown along each stereo view. Image source: [55].
75
3.4 The Beamline Devices
There are additional instruments used in the beamline for diagnostic studies. The
Beam Position Monitors (BPM) system uses RF-cavity beam-position monitors located
36.0 and 24.6 m from the target. The beam profile was measured with the Hard Scanners
which use two orthogonal tungsten and iron wires that pass through the beam. The
electrons scatter off of the harp wires and are detected by the PMT arrays upstream. The
harp scanners are located at 36.7, 22.1 and 15.5 m upstream from the target. The Total
Absorption Shower Counter (TASC) sits at the end of the beamline and is used for
measuring real photon flux. The TASC is a lead glass scintillator array, with close to 100%
photon detection efficiency. This device is only used during the g11a “normalization”
runs, when beam current was reduced. The Pair Spectrometer (PS) is a diagnostic device
used to measure the rate of production of e+e− pairs. It consists of a dipole magnet and a
thin aluminum radiator located within the magnet and an array of eight scintillator
paddles. When photons hit the radiator e+e− pairs are produced. The magnetic field
deflects the pair in opposite directions as they are directed into the scintillators. The
photon can then be reconstructed from the left-right coincidence from the scintillators.
The Pair Counter (PC) is used to study the photon beam near the beam dump. The PC is
made of a four-scintillator array used to detect e+e− pairs that are created when the beam
photons strike a thin aluminum foil. Figure 3.18 shows a schematic of the general layout
of these devices. More information about these devices can be found in Ref. [56].
3.5 The g11a Trigger and Data Acquisition
Each component of CLAS is setup to actively run and monitor the signals created in
the subsystem. Naturally, not all signals are worth recording for analysis. The trigger is
designed to determine which sets of signals pertain to the physics of interest and to turn on
76
Figure 3.18: A schematic layout of the beamline and flux monitoring devices. The beamlineenters from the left. Image source: [56].
and off data recording. Once triggered, the data acquisition system (DAQ) writes them to
magnetic tape to be analyzed.
The trigger for the g11a run period was setup to optimize the data collection for
studies on the existence of the pentaquark state. During the g11a data collection, events
are recorded when both the tagger Master OR (MOR) and the CLAS Level 1 trigger fall
within a coincidence window of 15 ns. The full tagger focal place was kept functional and
recorded data. However, only the first 40 of the total 61 T-counters were included in the
MOR. This effectively keeps event that are below the desired center-of-mass range from
being triggered by taking low energy photons out of the trigger. Still some low energy
photon events were recorded when a high energy photon accidentally fell in the same
timing window. The CLAS Level 1 trigger required a coincidence between any of the four
start counter paddles and the TOF paddles, from two separate sectors, in a window of 150
ns. This requirement leads to data collection of events that all have at least two charged
tracks. The tagger, the start counter, and the TOF paddles all have multiple detection
77
elements so the logic requires a pre-trigger OR of the discriminated signal in each system
to generate one signal from each control system that could be used in the trigger module
coincidence. Prior to the pre-trigger, the signals in each detector system are split to go to
the analog-to-digital (ADC) and the time-to-digital-converter (TDC) boards. During a
trigger the ADC and TDC from all detector systems are read into the data stream and the
data banks are assembled into an event and recored.
During the reconstruction of g11a, the cooking scripts used a prlink of
prlink g11 1920.bos and the hybrid magnetic field map bgrid T67to33.fpk resulting in
∼67% of the “new” or cold magnetic field map and 33% of the “old” or warm magnetic
field map. Approximately 20 billion event triggers were recorded by the data acquisition
system.
3.6 Summary
A general description has been given of the Continuous Electron Beam Accelerator
Facility and the CEBAF Large Acceptance Spectrometer at the Thomas Jefferson National
Accelerator Facility. The subsystems specific to this analysis are covered along with some
minor discussion of how they are used in each case. Though initially intended for other
purposes, the g11a dataset collected by the CLAS detector in Hall B can be used to study
other types of hadron photoproduction. In the next several chapters, the details of the
extraction of the Σ∗0, and Σ∗+ radiative decay branching ratios are given.
78
Part I
Electromagnetic decay of the Σ∗0
79
The details of the present study to extract the electromagnetic (EM) decay to the Σ∗0
are now presented. As a check on previous results, an attempt is made to reproduce the
original CLAS analysis of Taylor [10]. A rigorous approach is undertaken to understand
the method of kinematic fitting and how best to apply this method to the general
investigation of hadronic electromagnetic decays. From this, an analysis method for the
more challenging topology γ p→ K0Σ∗+ is proposed.
A detailed study of the Monte Carlo used for the Σ∗0 electromagnetic decay is also
presented. This provides guidance on the nature of the corrections and the systematic
effects of the EM decay ratio in the CLAS data.
Some critical details on the overall systematic uncertainties for the physics and
method of extraction of the EM decay ratio are presented in the next chapters. These
studies lead to a relatively good procedure to optimize the statistical and systematic
uncertainties in the final results.
80
4 EVENT SELECTION
Using the g11a data set, it is necessary to select events of the channel γ p→ K+Σ∗0.
The Σ∗0 resonance decays 87.0±1.5% of the time into Λπ0 and about 1% into the radiative
decay Λγ [60]. Roughly 63.9±0.5% of the time the Λ decays to pπ−, leading to the final
states γ p→ K+pπ−π0 and γ p→ K+pπ−γ , respectively [60]. The charged particles can
easily be detected with use of the CLAS drift chambers and Time-of-flight scintillators,
whereas the π0 and γ must be deduced using conservation of energy and momentum. The
analysis was done using the previously skimmed g11a data set for two positively charged
tracks and one negatively charged track with no other skim conditions. For checking and
testing corrections, the particle identification and event selection of Section 4.6 is used.
4.1 Run Inclusion
The CLAS runs for g11a include 43490 to 44133, of which only 43490 to 44107
were taken at beam energy 4.019 GeV. The smaller portion at beam energy 5.021 GeV
would need to be analyzed separately, and was excluded. The commissioning runs 43490
to 43525 were excluded as well. Runs 43675, 43676, 43777, 43778, and 44013 were
taken to test different trigger configurations and were also not included. Runs 43981 and
43982 had drift chamber issues, while runs 43989-43991 had data acquisition problems.
These runs were not included to minimize systematic differences in the sequence of runs
analyzed.
4.2 Energy Loss Corrections
Energy loss for charged particles as they pass through various materials in the CLAS
detector occurs, and a systematic adjustment to the particle’s energy is necessary. Energy
losses occur in the target material, target walls, the scattering chamber, the start counter,
and the various air gaps between the detector elements. These corrections are dealt with
81
by using the eloss software package, which can be set for the target parameters and start
counter used in g11 [67].
Figure 4.1: ∆E with respect to momentum for left: proton, and right: K+
In the three-track topology for the K+Λ final state, the detached vertex of the Λ is
reconstructed using tracking information from the p and π−. The Λ decay vertex is the
starting point for the charged tracks to which an energy loss correction would be applied.
The energy loss package was tested, to ensure it was making reasonable corrections,
by studying the energy of the π−, K+, and proton before the correction and then after. The
difference ∆E = Ea f ter−Ebe f ore for the proton and K+ with respect to particle momentum
is shown in Figure 4.1. The program is made to work for all charged particles with charge
equal to 1 (except for electrons). The valid momentum range is 0.05 < p/m < 50. It was
observed that for high momentum π+ the energy loss is quite small, and for low
momentum the correction reaches up to ∼50 MeV in accordance with dE/dx ∼ 1/β 2. For
heavier particles, such as the proton, the effects are more easily demonstrated as seen in
Figure 4.1. These plots were made using the particle id given in the PART bank (described
in Section 4.6, also see Appendix B for BOS structure) leading to some irregular bands
and edges for a small number of counts. The vast majority of the counts lead to the
expected trend. This indicates that the energy loss correction is working correctly.
82
4.3 Tagger Corrections
The tagged-photon energy comes from the E-counters of the CLAS photon tagger.
There are known inaccuracies in the tagger energy found in the focal plane. This effect
was originally studied using inclusive (γ)p→ pπ+π− events with kinematic fitting. The
three charged particles are detected and kinematically fit to the missing photon. The
events that pass a 10% confidence level cut were used to find the difference between the
measured and the kinematically fit photon energy.
∆Eγ = Ek f itγ −Emeas
γ
The corrections are binned for each tagger E-counter, and from each bin a Gaussian mean
is extracted. For each run the beam offset Br was calculated and used in the correction.
For an event from run number r with photon from E-counter e, the correction applied to
the photon energy becomes,
∆Eγ,e,r = ∆Ek f itγ,e +Br,
with ∆Eγ,e being the Gaussian mean from E-counter e. This correction has been
previously derived for g11a [64], with additional studies on the need for the correction due
to the sagging of the focal plane between its four support yokes provided in [56]. The sag
shifts the narrow E-counters from the set locations, which can lead to detecting electrons
with slightly distorted energies. To confirm the results of these previous studies, the same
topology (γ)p→ pπ+π− was used to kinematically fit the missing photon, giving the
same trend in ∆E as in the original work [64]. The results from this check are shown in
Figure 4.2, which shows the relative tagger correction needed for the set of E-counters
found by fitting the ∆Eγ = Ek f itγ −Emeas
γ distribution with a Gaussian for each E-counter
bin. The Gaussian mean over the full range is used to apply a systematic correction for
each E-counter. There are several points seen in Figure 4.2 (Left) that fall off the general
83
trend. These are due to swapped cables [64]. A correction can still be applied to these
E-counter so that no E-counters are excluded. There are also no T-counters excluded.
Figure 4.2: Left: ∆Eγ/Ebeam vs. E-counter for the reaction (γ)p→ pπ+π− used to find thetagger correction. Right: The extracted Gaussian mean from the ∆Eγ/Ebeam vs. E-counterfits.
4.4 Momentum Corrections
The actual momentum of charged particles can differ from the corresponding CLAS
tracking information. These differences can be due to faulty drift chambers that may be
misaligned or distorted, as well as inaccuracies in the magnetic field map [56].
The g11a momentum correction for all runs has been developed using the test
channel γ p→ pπ+π−. The pπ+π− are detected and the energy loss and tagger
corrections are applied. Three kinematic fits done, each having a different final-state
particle as the “missing” (or undetected) particle from energy and momentum
conservation. The hypotheses for the corresponding fits are γ p→ pπ+(π−),
γ p→ pπ−(π+), γ p→ π+π−(p). The measured magnitude of the momentum and the
directional components were then compared with the “missing” momentum vector from
the kinematic fit such that ∆p = pk f it− pmeas, ∆λ = λ k f it−λ meas, and
∆φ = φ k f it−φ meas. These distributions were binned in the relevant kinematic variables
84
and fit with Gaussians. Here λ and φ are the directional tracking parameters. An
expression for each correction needed is found as a function of momentum and orientation
(θ ,φ) for each particle species. Corrections were found for each CLAS sector, with each
sector broken up into twelve 5 degree bins in the azimuthal angle φ , and then each φ bin
into fifteen polar angle θ bins. The momentum correction is tabulated after all kinematic
regions have been considered [64].
Figure 4.3: Left: ∆p+ found for the K+ in the topology γ p→ K+Λ→ K+pπ− for Sector1, θ ∈ (20,25) and φ ∈ (−15,−10). Right: The Gaussian fit for ∆p+.
To check the corrections, the topology γ p→ K+Λ→ K+pπ− is used, with the tagger
and energy loss corrections previously implemented. Because the momentum correction is
small, the bin for Sector 1, θ in (20,25), and φ in (−15,−10) is shown in Figure 4.3,
which is known to have a visible correction. This result compares well with the work
previously done [64]. It is found that the magnitude and trend of the correction are the
same for all positively charged particles, and differ from all negatively charged particles
but mostly just by a sign change, such that ∆p+ ≈−∆p−.
4.5 Effectiveness of Corrections
Some studies were performed to check the effectiveness of the corrections using the
channel γ p→ K+Λ from the final state decay products K+pπ−. The variables are the
invariant mass of the pπ− and the missing mass off the K+. The PDG value of the Λ mass
85
is ∼ 1.1156 GeV. The particle identification outlined in the following section is employed
to isolate the decay products with no other kinematic cuts applied. Only the detected
particles that fall within |∆t|< 2 ns are considered, where ∆t is the difference between the
time the particle struck the start counter and the time at which the photon was at the
interaction vertex for that particle. The energy loss correction, the tagger correction, and
the momentum correction are applied and the spectrum is studied before and after these
corrections. A comparison for each distribution is seen in Figure 4.4. A Gaussian fit to the
invariant mass of pπ− before the corrections gives a mean of 1.115 GeV with a σ of 1.4
MeV, which is then improved to a mean of 1.116 GeV and a σ of 1.35 MeV, as seen in
Figure 4.5. The mass off the K+ in the region 1.1 to 1.21 GeV is used to find the Λ
candidates and also fit with a Gaussian. The mean of the fit improved from 1.110 GeV to
1.116 GeV as seen in Figure 4.6.
Figure 4.4: Left: Invariant mass of (pπ−), Right: Missing mass off the K+. Beforecorrections is shown in black and after is shown in red.
4.6 Particle Identification
In the present analysis, the PART identification scheme was used as an initial starting
point. The PART id uses the start counter to find an interaction vertex time for each
86
Figure 4.5: Left: Invariant mass of (pπ−) with Gaussian fit before corrections, Right:Invariant mass of (pπ−) with Gaussian fit after corrections.
Figure 4.6: Left: Missing mass off the K+ with Gaussian fit before corrections, Right:Missing mass off the K+ with Gaussian fit after corrections.
charged particle and matches it up with photons in the tagger, where there are up to 10
photons for a given event. The photon with the closest time to any track is selected as the
photon that caused the event. Specifically, the time of interaction is acquired using the
electron beam bucket (RF time) that produced the event. To correlate the interaction time
with the photon production time, a coincidence of the tagger T-counter with the Start
Counter is used. The RF time for the photon is then used to get the vertex time (photon
interaction time) for the event. Using the time of flight from the event vertex to the
scintillator counter, the velocity β is calculated for each particle. From β and the
particle’s measured momentum, a mass is calculated. Each track does not need to have a
87
hit registered in the start counter for its mass to be calculated, only one track in the event
needs a Start Counter hit. The PART id is made during the “cooking” of the data.
The restrictions for PART from the mass time of flight for kaons is
0.35≤MK+ ≤ 0.65 GeV, for pions is 0.0≤Mπ− ≤ 0.3 GeV, and for protons is
0.8≤Mp ≤ 1.2 GeV. The mass calculated from time of flight is
mcal =
√p2(1−β 2)
β 2c2 , (4.1)
where
β = L/ctmeas. (4.2)
From this initial identification it is possible to incorporate additional timing information to
improve event selection with quality constraints. The measured time-of-flight and
calculated time-of-flight can be used for an additional constraint. The measured
time-of-flight is tmeas = tsc− tγ , where tsc is the time at which the particle strikes the
time-of-flight scintillator wall and tγ is the time at which the photon was at the interaction
vertex. ∆t is then,
∆t = tmeas− tcal, (4.3)
where tcal is the time-of-flight calculated for an assumed mass such that
tcal =Lc
√1+(
mp
)2
, (4.4)
where L is the path length from the target to the scintillator, c is the speed of light, m is the
assumed mass for the particle of interest and p is the momentum magnitude. Cutting on
∆t or mcal should be effectively equivalent.
Using ∆t for each particle it is possible to reject events that are not associated with
the correct RF beam bucket. This is done by accepting only |∆t| ≤2 ns in the initial
analysis. This cut is loose enough to minimize signal loss, while providing a large enough
range in ∆t to study systematic variations down to |∆t| ≤1 ns. Figure 4.7 shows ∆t for the
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protons versus the ∆t for the K+. The main cluster of events at (0,0) comes from the
events that best satisfy the mass hypothesis for that particle. In all of the plots shown, the
bad TOF paddles have already been removed, see Section 4.9. The clusters of events at
(±2 ns,±2 ns),(±4 ns,±4 ns), etc. indicate the photon was from the wrong beam bucket.
The dashed lines represent the time cut used in the initial look at the data. For the final
results reported, a |∆t| ≤1 ns cut is used.
Figure 4.7: ∆t giving the PID timing cut. Clusters of events at (±2 ns,±2 ns),(±4 ns,±4ns), etc. are from photons from a different beam bucket. The dashed lines represent thetiming cut used.
The kaon mass distribution directly acquired from the PART bank is seen in Figure
4.8. The pion background from other beam bunches, seen as diagonal bands, is reduced
after the timing cut. It is also possible to see this reduction in accidentals in the β vs. p in
89
Figure 4.6 (left), which shows the β distribution before any cuts. Figure 4.6 (right) shows
the same distribution after the timing cut implemented in Figure 4.7.
Figure 4.8: Calculated mass based on the PART bank information with no cuts used.Visible accidentals result from other RF buckets that are present.
There are also some counts seen at low momentum in the kaon band that are reduced
by the restriction on the lower bound of the kaon momentum, along with a ∆β cut
described next. There is also a pion and proton low-momentum restriction implemented.
A ∆β cut is used to clean up the identification scheme. ∆β is the difference between
time-of-flight β = L/(ctmeas), and the calculated β = p/√
p2 +m2, where L is the particle
path length and tmeas = tsc− tγ is the difference in the time at which the particle strikes the
time-of-flight scintillator counter and the time at which the photon was at the interaction
vertex. p is the particle momentum and m is the known PDG particle mass. The ∆β
90
Figure 4.9: Left: K+ β distribution before any cuts, Right:K+ β distribution after timingcut.
distribution for the kaon is shown in Figure 4.10 before any cuts or photon time restriction.
Figure 4.11 shows the ∆β for the K+ after a ±1 ns timing cut, showing a much cleaner
distribution. The good events were taken within a cut of −0.02≤ ∆β ≤ 0.02 for the kaon.
For the sake of demonstrating how well a ±1 ns cut can clean up the particle
selection, the calculated mass without any ∆β cut is shown for the kaon, pion, and proton
in Figures 4.12, 4.13, and 4.14. For the kaon, Figure 4.12 can be compared to Figure 4.8
that has no timing cut yet implemented.
91
Figure 4.10: ∆β for the K+ before any cuts.
Figure 4.11: ∆β for the K+ after a ±1 ns timing cut.
92
Figure 4.12: K+ calculated mass versus momentum after a ±1 ns timing cut.
Figure 4.13: π− calculated mass versus momentum after a ±1 ns timing cut.
93
Figure 4.14: Proton calculated mass versus momentum after a ±1 ns timing cut.
94
4.7 Vertex Information
The reaction of interest γ p→ K+Σ∗0→ K+Λ(X) has two detached vertices: one
from the Σ∗0 and one from the Λ. Using tracking information from the TBER bank (see
Appendix B) it is possible to find a distance of closest approach (DOCA) from the p track
and π− track to obtain a vertex and cut at a reasonable distance. The cut can help to
achieve a good Λ. An initial cut of 5 cm is used as a starting point for the DOCA cut as is
seen in Figure 4.17. Figure 4.15 shows the p,π− invariant mass before and after the
DOCA cut, showing a reduction in background and improvement in the peak width. This
loose DOCA cut is used only to reduce the background before the kinematic fitting
procedure. A study of the effects of this cut on the final result is performed later in Section
9.
Figure 4.15: Left: Invariant mass of p,π− before DOCA cut. Right: Invariant mass ofp,π− after DOCA cut.
It is expected that of the final state decay products, only the kaons are produced at the
primary vertex. To reduce the likelihood of considering accidentals, a cut is applied to the
kaon vertex so that only kaons produced in the target volume are considered. The kaon
vertex z-coordinate distribution is shown in Figure 4.16. The kaon vertex was also
obtained using the kaon track TBER vertex information and the kaon track with the
95
Figure 4.16: Kaon vertex distribution. The dashed lines are the implemented cuts.
closest approach to the beam line. The dashed lines in Figure 4.16 indicate the cuts at -30
cm and 10 cm on the z-coordinate of the vertex position.
Figure 4.17: Distance of closest approach; the dashed line is the implemented cut.
96
4.8 Beam Photon Selection
The status of the tagger hits was obtained directly from the TARG bank (see
Appendix B) by checking for good events with tagger bank status 7 or 15. For bank status
7 and 15, tagger hits with no more than one E-counter hit is associated with a T counter
hit. These tagger events were accepted unless there were two type 15 at once, which were
eliminated. The resulting good photons were kept until the end of the analysis. If there
was more than one photon tagged with the same event number that survived all cuts, the
event would also be eliminated. The number of events with more than one photon tagged
after all cuts is negligible at less than 0.001%.
4.9 Detector Performance Cuts
Cuts were applied to take into account both the regions where there are known
obstructions to the acceptance and regions of CLAS that are not well simulated. This
includes tracks at extremely forward or backward angles, areas near the torus coils, and
regions where the drift chambers and scintillator counter efficiencies were not well
understood. Some minimal fiducial cuts are applied to account for some angular regions
that are shadowed by the toroidal coils and the Region 2 end-plates. As θ decreases, the
shadowed region of detection increases as seen from the center of the target. Tracks that
point near these shadow regions are less likely to reflect an accurate reconstruction. These
regions can be eliminated by identifying the tracks affected and applying a general fiducial
volume cut. Figure 4.18 shows the angular distribution for the kaon before the fiducial
cuts and Figure 4.19 shows the same distribution after the fiducial cuts.
The azimuthal angle φ is within the range of ±30 relative to the sector mid-plane.
The restriction cutting out the edge of the acceptance was developed using the engineering
drawings for CLAS showing the width of the shadow region due to the coils as a function
of θ found after initial installation. There is an azimuthal cut of |φ |< 26 implemented
97
with the θ restriction limiting the fiducial volume. The functional form of the cut used to
define the fiducial volume is
θ > 4.0+510.58
(30−φ)1.5518 . (4.5)
A minimum momentum of 0.125 GeV, after the energy loss correction, was
implemented for both positive and negative particles. For cleaner momentum
reconstruction, tracks with a minimum proton momentum of 0.4 GeV were required as
well. These restrictions were studied previously for the g11 run period [63].
Some dead or unreliable Time-of-flight (TOF) scintillators were removed. These
TOF’s were originally identified by the g11 run group by examining occupancy plots of
both the data and Monte Carlo. All TOF’s with noticeable discrepancies for pions or
protons were flagged as unreliable [63]. Table 4.1 shows the list of bad TOF’s in each
sector. Particles hitting these TOF’s are rejected in the analysis.
Figure 4.18: Angular distribution of the kaons before fiducial cuts.
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Figure 4.19: Angular distribution of the kaons after fiducial cuts.
Table 4.1: Bad Time-of-flight scintillators.
Sector BAD TOF’s
1 33
2 8,17,27
3 18,19
4 8,13,14
5 23
6 30
Several runs in the g11a data set have been identified in which the forward part of the
TOF array in sectors 2 and 3 have been determined to be either not operating or running at
99
very low voltage. Rather than setting additional cuts to the analysis, these runs were
simply eliminated. A list of these runs is given in Table 4.2.
Table 4.2: The list of run numbers that have problems with the forward part of the TOF.
Sector 2 Sector 3
43989 43586
43990 43587
43991 43588
44000 43589
44001 43590
44002 43591
44007 43592
44008 43593
44010 43594
44011 43595
44012 43596
100
5 ANALYSIS PROCEDURE
The approach used in this analysis is to remove as much identifiable background as
possible while preserving the π0 and radiative signals. Because of the closeness of the
radiative signal to the π0 peak in the mass spectrum from Σ∗0→ Λπ0 decay, the radiative
signal extraction requires a certain degree of finesse with a kinematic fitting procedure.
To clean up the kaon for the analysis, there is a cut made on identified kaon
candidates that are truly π+. By reestablishing the energy of the PART identified kaon
with the mass of the pion, one can test for possible contamination. The missing mass
squared is studied for the reaction γ p→ pπ+π−(X). A spike at zero mass squared
indicates the reaction γ p→ pπ+π−. The missing mass squared (see Section 1.7) is shown
for the reaction γ p→ pπ+π−(X) in Figure 5.1. This particle identification contamination
can be removed by cutting slightly above zero but not over the region where the π0 peak
should be. A cut at 0.01 GeV2 is chosen so as to not cut into the good K+ events.
Reactions such as ρ → π+π− are also eliminated by this cut.
It is also possible to do a similar study using γ p→ π+π−(X). A peak below zero
would indicate extra pion contamination that could be cut out, as well as anything around
the mass of the proton. The γ p→ π+π−(X) spectrum after the cut on γ p→ pπ+π− is
seen in Figure 5.2. With no peak at the proton and nothing to speak of below zero, a clean
kaon is achieved from the cut in Figure 5.1.
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Figure 5.1: Pion contamination: Mass squared (M2X ) of any missing particle for the
γ p→ pπ+π−(X) reaction where the π+ was a potentially mis-identified kaon. Eventswith M2
x < 0.01 GeV2 were removed.
Figure 5.2: Pion contamination: Mass squared (M2X ) of any missing particle for the
γ p→ π+π−(X) reaction, where the π+ was a potentially mis-identified kaon.
102
For the hyperon excited-state region, the Λ(1116) can be seen from the invariant
mass of the p-π− pair in Figure 5.3. The instrumental resolution (σ ) of the peak is about
1.3 MeV, see Figure 5.3. The p-π− pairs with invariant masses between ±0.005 GeV of
the PDG Λ mass are kept for analysis. This is intended to be a loose cut at around ±3σ to
keep as many candidates as possible that will be further cleaned up with the kinematic
fitting procedure discussed later. For this same reason it is not necessary to be concerned
about the background the peak sits on.
Figure 5.3: Λ peak from the invariant mass of the π− and proton.
The various spectra are studied before and after each cut. Before any kinematic cuts
are included, the plots in Figure 5.4 show the invariant mass of the p-π−, the missing mass
spectrum for all detected particles, the mass off of the p-π−, and the missing mass off the
103
K+. The next set of plots shown in Figure 5.5 include the cut on the invariant mass of the
p-π− (shown in Figure 5.3) at ±0.005 GeV around the Λ. There is a plot added that shows
the range of missing energy of the detected particles. There is clearly additional structure
around 0 and 0.08 GeV that is removed by the restriction on the excited state hyperon
mass discussed next. There is also a plot of the transverse component of the missing
momentum. Pxy close to zero indicates that the missing particle is traveling nearly straight
down the beam line.
After the Λ cut, the missing mass off the K+ is investigated. The Σ∗(1381) and
Λ(1520) are clearly seen in Figure 5.6. The Λ(1405), which decays almost 100% to Σπ ,
would show up as a shoulder on the right of the Σ∗ peak, however no evidence is seen in
the spectrum. The plot in Figure 5.6 shows a fit of a relativistic Breit-Wigner to the Σ∗0
with a quadratic background. The mass and width from the fit can be seen from the fit
parameters. The mass from the fit without any constraints on the parameters is 1.386 GeV.
The width from the fit is 39.8 MeV. These values compares well with the Particle Data
Group value centroid and width of 1.3837 and 36 ±5 MeV respectively. The cut on the
missing mass of the K+ is also shown in the plot.
To study the excited hyperon region of interest before the Λ cut, an additional set of
plots was made without the ±0.005 GeV cut around the Λ, but instead with a cut from
1.34 GeV to 1.43 GeV on the missing mass of the K+. This cut selects the potential Σ∗
candidates. Figure 5.7 shows the same plots but only with the hyperon mass restriction.
Figure 5.8 show these distributions after both kinematic cuts are implemented. In the
missing mass off the p-π−, the visible peak at the mass of the K∗(892) implies that there
will be a presence of a background that can be studied using Monte Carlo simulations.
The immediate issue is the presence of a peak at the mass of the K+; this is due to leakage
of ground-state hyperon production into the excited-state region. To eliminate this
background the requirement of MX > 0.55 GeV is set.
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Figure 5.4: No kinematic cuts applied. Top left: Invariant mass of the p-π−. Top right:Missing mass squared of all detected particles. Bottom left: Missing mass off the p-π−.Bottom right: Missing mass off the K+. All units are (GeV).
From Figure 5.8 (upper right plot) one can see the missing mass squared of the
reaction γ p→ K+Λ(X) with all the afore mentioned cuts applied. A peak is visible for the
M2π0 as well as at zero missing mass squared. The counts above the π0 peak are mostly
105
Figure 5.5: Plots after Λ cut only. Top left: Missing energy of all particles detected. Topright: Missing mass squared of all detected particles. Bottom left: Missing mass off the Λ.Bottom right: Transverse missing momentum. All units are (GeV).
due to γ p→ K+Σ0(X), which can also be taken into consideration with the Monte Carlo.
As seen in the progressive cuts, the peak at zero missing mass squared is greatly
diminished; this is primarily due to the elimination of the γ p→ K+Λ channel. In principle
one can now use the distribution of the missing energy and the perpendicular component
of the missing momentum to check against what is expected. In the missing energy all of
106
Figure 5.6: Missing mass off the K+ after the cut on the Λ. The fit to the Σ∗ uses arelativistic Breit-Wigner and quadratic background. The dashed line indicates the cutaround the Σ∗0. The Λ(1520) is also clearly visible.
the other structures have more or less disappeared, leaving a peak at the missing energy
between 0.2 and 0.3 GeV. The counts at lower energy are from the K∗Λ and Λ(1405),
which will also be managed though the simulation’s acceptance of these channels, see
Figure 7.18. The missing energy is not used to separate the radiative signal from the π0,
but it can be used qualitatively to indicate that other background channels have been
reduced since the expected missing energy for the π0 channel is ∼0.2 GeV as seen from
the simulations in Figure 5.9. Clearly, the remaining peak in Figure 5.8 (upper left)
primarily comes from the missing energy from the π0.
The missing transverse momentum is defined as P2xy = P2
x +P2y , where Px and Py are
the x and y components of the missing momentum. One would expect a relatively rounded
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Figure 5.7: Plots after Σ∗ cut only. Top left: Missing energy of all particles detected. Topright: Missing mass squared of all detected particles. Bottom left: Missing mass off the Λ.Bottom right: Transverse missing momentum. All units are (GeV).
and featureless distribution for Pxy with no sharp peaks for either the radiative or π0
channels, such as the distributions seen in Figure 5.11 from the simulations of these
channels. The small peak close to zero in the Pxy distribution in Figure 5.8 (bottom right)
is reduced once only one tagged photon is required, but as seen in Figure 5.10, there is
still a spike in the distribution close to zero that can be seen in the P2xy distribution that is
108
Figure 5.8: Plots after Σ∗ and Λ restrictions. Top left: Missing energy of all particlesdetected. Top right: Missing mass squared of all detected particles. Bottom Left: Missingmass off the Λ (counts below the dashed line are cut). Bottom right: Transverse missingmomentum. All units are (GeV).
important to handle. This peak is likely caused by double bremsstrahlung in the radiator
such that γ1 + γ2 p→ K+Λ+ γ1. The ground-state Λ production events for which a photon
readout from the tagger can contain an extra photon in the event. These extra photons will
be directed straight down the beam line and show up as a spike at P2xy = 0. This may also
109
Figure 5.9: Missing energy produced from simulations for the reaction γ p→ K+Λπ0.
occur if the event is an accidental or the wrong electron is selected due to inefficiencies in
the tagger plane. A final cut is then used to eliminate this contamination. Based on the
plot seen in Figure 5.10, candidates with P2xy > 0.0009 GeV2 were preserved for further
analysis. It is important to remove 100% of this forward peak because any leakage could
show up as a radiative signal. The cut is clearly seen to be far enough from the peak that
essentially no tail can pass through. Simulations were used to make sure that the
γ p→ K+Λγ channel was not significantly reduced. The simulations shown in Figure 5.11
were used to estimate the amount of radiative signal lost by assuming equal counts in both
the π0 and radiative channel and finding the percent of signal lost with a P2xy > 0.0009
GeV2 cut. About 1% of the simulated signal events lost by this cut. Figure 5.11 shows the
transverse missing momentum distribution for the hypothetical radiative decay and the π0.
Figure 5.12 shows a magnification of the area of interest and the cut implemented. Neither
channel is greatly affected by this cut, whereas the γ1 + γ2 p→ K+Λ+ γ1 contribution is
110
eliminated. Systematic variation of this cut is studied in Section 9. Full details of the
Monte Carlo are provided in Section 7.
Figure 5.10: Left: Transverse missing momentum Pxy after only one tagged photon isselected. Right: P2
xy after only one tagged photon is selected with cut implemented, seen asthe dotted line.
The missing mass spectrum can be seen after all required cuts, including the Pxy cut,
in Figure 5.13 (top right). The radiative signal is small and not directly visible. Fitting
with a double Gaussian is not a viable option because the tail of the π0 bleeds into the
space for the radiative signal. The extraction of such a signal prompts a challenge
specifically because the phase space of the γ p→ K+Λγ and the γ p→ K+Λπ0 reactions
are so similar and the radiative decay branching ratio is so small relative to the π0 branch.
A decay of π0→ γγ can easily look like a radiative decay directly from the Σ∗,
overwhelming the channel of interest. Next, a detailed understanding of the uncertainties
in tracking of all the detected particles will be used in order to establish a well-defined
confidence level that the missing particle is either a π0 or a γ .
111
Figure 5.11: Simulations for the perpendicular momentum Pxy of missing mass candidates.The blue distribution is for the simulated γ p→ K+Λ(π0) reaction and the yellow is for theγ p→ K+Λ(γ) reaction.
112
Figure 5.12: Magnification of the simulation of the perpendicular momentum Pxy ofmissing mass candidates and the implemented cut showing little effect on signals.
113
Figure 5.13: Plots after all mentioned cuts. Top left: Missing energy of all particlesdetected. Top right: Missing mass squared of all detected particles. Bottom left: Missingmass off the Λ. Bottom right: Transverse missing momentum. All units are (GeV).
114
6 KINEMATIC FITTING
A kinematic fitting package was written to improve the resolving tools for working
with CLAS measured variables. Considerable effort had been taken to improve TBER
information for the drift chamber tracking errors for G11 [66]. Additional energy loss as
well as multiple scattering effects were also considered in this package.
Track reconstruction in CLAS is done in a sector-dependent coordinate system. This
same coordinate system is used in developing the kinematic fitting and performing the fits.
During tracking, a covariance matrix is produced, containing the resolution uncertainties
and correlating coefficients of the tracking parameters for each track, which are located in
the BOS TBER bank (see Appendix B). After achieving an accurate covariance matrix,
the process of kinematic fitting can be used to greatly improve an analysis procedure.
The kinematic fitting technique can take advantage of a number of types of
constraints such as energy and momentum conservation, common vertices, or physical
limits to improve the measured quantities that are used in the analysis. The method of
Lagrange multipliers is a common way to handle the constraints with a least squares
criteria. Here some highlights are discussed from the methods outlined in [68], and [69].
Assume there are n independently measured data values y, which in turn are
functions of m unknown variables qi, with m≤ n. The condition that y = fk(qi) is
introduced where fk is a function dependent on the data points that are being tested for
each k independent variable at each point.
Because each yk is a measurement with corresponding standard deviation σk, the
equation yk = fk(qi) cannot be satisfied exactly for m < n. It is possible to require that the
relationship be closely numerically satisfied by defining the χ2 relation such that
χ2 = ∑
k
(yk− fkQ−∑k Qkiηi)2
σ2k
≡∑k
(∆yk−∑i Qkiηi)2
σ2k
. (6.1)
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The resulting values ηi = qi−q−Q found through the minimization of the linearized χ2
function are unbiased and have minimum variance σ2qi
. This implies the best possible
parametrization.
The linearized minimization conditions ∂ χ2/∂qi = 0 can then be expressed as
∑k
QkiQk j
σ2k
η j = ∑k
Qki
σ2k
∆yk,
where 1≤ i, j ≤ m, defining the covariance matrix
VQ i j =
(∑k
QkiQk j/σ2k
)−1
,
with solution
ηi = ∑j,k
VQ i jQk j∆yk
σ2k
.
The linearized χ2 equation can now be written in matrix form:
χ2 = (∆y−Qη)T V−1
y (∆y−Qη),
where Q is the matrix of coefficients and V−1y is the inverse of the covariance matrix.
The partial derivative with respect to the η parameters leads to the m equations
QT V−1y (∆y−Qη) = 0.
resulting in
QT V−1y Qη = QT V−1
y ∆y.
Using the relation VQ = (QT V−1y Q)−1, one obtains the m equations
η = VAQT V−1y ∆y.
The parameters q are then determined from q = qQ +η . The matrix V−1y necessarily
has dimensions of n×n, while Q is n×m, and VQ has the size m×m.
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To impose constraints a more generalized expression for χ2 is required. First the set
of r constraints are described in the from M(q) = 0. A set of linearized equations are
obtained by expanding around qQ, M(qQ)+(q−qQ)∂M(qQ)/∂q≡ Dη +d = 0. Here D
is a r×m matrix and d is a vector of length r. This matrix form of the constraints are then
adding to the matrix form of the χ2 equation resulting in,
χ2 = (∆y−Qη)T V−1
y (∆y−Qη)+2L T (Dη +d).
The vector L is of length r and is the corresponding set of Lagrange multipliers. Again a
solution for η is found by taking the partial derivatives with respect to η and L and
setting them equal to 0. The ∂/∂L = 0 equation provides all the constraint conditions.
Solving the equations lead to an expression of η , where η is equal to the initial
unconstrained values η0 added to a term proportional to the Lagrange multiplier. The
unmeasured variables are then improved by stretching the values of η from the initial
unconstrained values η0, within the kinematically defined resolutions, to meet the
imposed constraints.
For the purpose of implementation one can now simplify the notation to build an
algorithmic outline. In the case of a linear nature in the constraint equations, it is possible
to use an iterative approach.
The unknowns again are divided into a set of measured (~η) and unmeasured (~u)
variables. The Lagrange multipliers, Li, are introduced to be used in each constraint
equation. These terms can be used to rewrite the equation for χ2,
χ2(~η ,~u, ~L ) = (~η0−~η)TV−1(~η0−~η)+2 ~L T ~F . (6.2)
The χ2 differentiation with respect to all the other variables linearizes the constraint
equations. The new constraint can then be used in multiple iterations. The vector ~η0 is the
content of the initial tracking information for the measured variables. V is the covariance
matrix containing all the kinematic resolutions and correlation of the measured variables.
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Naturally the result for each new iteration ν +1 depends on the numerical values found in
the previous iteration (ν):
~uν+1 =~uν − (ATu S−1Au)−1AT
u S−1~r, (6.3)
~L ν+1 = S−1[~r +Au(~uν+1−~uν)], (6.4)
~ην+1 =~ηo−VATη
~L ν+1, (6.5)
with definitions of,
(Aη)i j ≡∂Fi
∂η j, (6.6)
(Au)i j ≡∂Fi
∂u j, (6.7)
~r ≡ ~Fi +(Aη)ν(~η0−~ην), (6.8)
S≡ (Aη)νV (ATη)ν , (6.9)
for the ν th iteration. The change in χ2 from each iteration ∆χ2 = |χ2ν −χ2
ν+1| is used to
find the termination in the iteration procedure such that ∆χ2 > 0.001 for each recursion.
There is also an exit in the loop for any confidence level below 1×10−6 to which these
events are not expected to improve by a measurable value and can be passed. The
iterations were counted and it was found that this convergence occurs within less than 15
iterations.
For clarity it is important to discuss which coordinate system was used for the
constraint equations and applied covariance corrections. There are three main coordinate
systems used for CLAS analysis. One can use the lab system, the sector system and the
tracking system. For analysis the lab system is most useful. In this system the beam line
sits along the zlab-axis; the xlab-axis goes right through sector 1 and the ylab-axis points
vertically up in between sector 2 and 3.
The track reconstruction in CLAS is done in a sector-dependent scheme. The tracking
coordinates in each sector are defined so that the xtrack-axis lies along the beam line
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whereas the ztrack-axis is aligned with the average Poynting vector of the magnetic field in
the given sector. Then ytrack simply passes through the center of that sector orthogonal to
the other axes. The tracking coordinates transform into the lab coordinates likextrack
ytrack
ztrack
=
zlab
cos(α)xlab + sin(α)ylab
−sin(α)xlab + cos(α)ylab
(6.10)
and α is sector dependent such that α = π
3 (Nsector−1).
The resulting tracking parameters of interest are given by q/p, λ , φ , d0 and z0. The
parameters d0 and z0 are variables giving the vertex information in the projection of the
“vertex” plane. Momentum in the laboratory frame for each track can be described as
px = p(cosλ sinφ cosα− sinλ sinα), (6.11)
py = p(cosλ sinφ sinα + sinλ cosα), (6.12)
pz = pcosλ cosφ , (6.13)
where α is the same as in Eq 6.10.
The covariance matrix V from tracking found in the TBER bank is limited and does
not contain the needed effects of multiple scattering and the energy loss in the target cell,
the carbon epoxy pipe, or the start counter. Corrections to the diagonal elements of the
covariance matrix V are systematically applied, as well as the correction needed for the
tracking uncertainties found previously for g11 [66].
Using the tracking variables and the geometry of CLAS one can take reasonable
steps in the determination of uncertainties due to multiple scattering. Following the
observationally derived approach employed by Lynch and Dahl [70], the RMS scattering
angle in a material can be described in terms of material density ρ , mean atomic number
Z, mean atomic weight A, speed of traversing particle βc, momentum p, and thickness x.
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The expression for the RMS scattering angle is then,
⟨θ
2⟩= 2χ2
c1+F2
[1+ v
vln(1+ v)−1
], (6.14)
χ2c = 0.157
Z(Z +1)A
ρxp2β 2 , (6.15)
χ2a = 2.007×10−5 Z2/3
p2
(1+3.34
Z2α2
β 2
), (6.16)
Ω =χ2
cχ2
a, (6.17)
v =Ω
2(1−F). (6.18)
The α here is the fine structure constant. F is the fraction of the full multiple-scattering
distribution that is estimated for the process of interest. The value of F = 0.99 was chosen
based on the notion that F varies anywhere in the range of 90% to 99.5% for the material
of interest. It is estimated that the formula yields results within 2% of experimental values
[70]. This expression can then be used to approximate the variance of the dip angle λ due
to multiple scattering in the target, scattering chamber, and Start Counter such that,
∆V (λ ,λ ) =⟨θ
2⟩target +
⟨θ
2⟩scattering chamber +
⟨θ
2⟩start counter . (6.19)
This term can then be used to find the correction for the φ diagonal element,
∆V (φ ,φ) =∆Vλλ
61+ sin2
λ
cos2 λ. (6.20)
Taking into consideration the effects of the energy loss when charged particles
traverse these materials, a standard thick absorber particle approximation was used. This
method is also found in Leo [70]. In this method the energy spread is Gaussian with a
variance written as,
σ2E = 4πNAr2
em2ec4
ρxZA
γ2(
1− β 2
2
), (6.21)
where NA is Avogadro’s number, re represents the classical electron radius, and me is the
electron mass. The adjustment to the momentum term in the covariance matrix can then
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be described as,
∆V (1/p,1/p) =σ2
Ep4β 2 . (6.22)
The vertex information is extracted after vertex corrections are made for the K+. The
vertex information for the proton and π− are from the secondary vertex and are not
applied until the Λ decay. In general if a channel is to be kinematically fit that contains a
neutral particle that decays, the energy and multiple scattering effects will be based on the
position of the secondary vertex.
In completing construction of the kinematic fitting program and the development of
an accurate covariance matrix, it is necessary to test and check usability. Various
techniques were studied to investigate procedure efficiency and systematic uncertainties.
This work is discussed in reference [71].
After developing and testing the software one is able to produce a flat Confidence
Level distributions after fitting Monte Carlo and data from g11 for reactions such as
γ p→ K+π−p(π0) channel. The pull distributions obtained from fitting are approximated
by a Gaussian with σ ranging from 0.966 to 1.064. Kinematic quantities, such as missing
mass spectra calculated from the fits, are significantly improved, while background is
significantly reduced. These studies are discussed in reference [71].
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7 SIMULATIONS
For the Monte Carlo setup, the relative contribution of different reaction channels
were taken into account. The experimental photon energy distribution for the subset of
g11a at 4.0186 GeV was used to determine the energies of the incident photons in the
simulation. The weighting of each channel and its contribution to the variation in the final
result is discussed in Section 9.
7.1 Monte Carlo Generator
The event generator FSGEN (Full Spectrum Generator) was used with a variable
t-dependence such that a channel with a kaon is generated uniformly in the center-of-mass
frame in φ with a t-dependent distribution in θcm according to P(t) ∝ e2.0t where t < 0.
The decay products of excited state hyperons are produced isotropically in the rest frame
of the hyperon and boosted back to the lab frame.
The incident electron energy was set to 4.0186 GeV and the real photon energy range
setting used was 1.0-4.2 GeV. Gaussian distributions in x and y with σ = 0.5 cm were used
to approximate the beam width in the target. A target length of 40 cm, from 30 cm to -10
cm, was used to generate events uniformly along the length of the target on the beam axis.
FSGEN produces the BOS MCTK track bank and the MCVX vertex bank (see
Appendix B) that contain the needed information to be thrown into CLAS. The generated
events were then fed into GSIM, the CLAS Geant-based Monte Carlo program. GSIM
simulates each CLAS detector portion and creates the reconstructed information of the
simulated hits. The hit information is then written out in the same BOS bank format as for
the real data.
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The generated events were then fed into GSIM with the settings for the ffread card
for processing as:
AUTO 1
LIST
KINE 1
BEAM 4.016
MAGTYPE 2
MAGSCALE 0.4974 0.0
FIELD 2
GEOM ’ALL’
NOGEOM ’EC1’ ’CC’ ’MINI’ ’PTG’
NOSEC ’OTHE’
TARGET ’g11a’
TGPOS 0.0 0.0 4.06
STZOFF -14.06
STTYPE 1
RUNG 43582
CUTS 5.e-3 5.e-3 5.e-3 5.e-3 5.e-3
DCCUTS 1.e-4 1.e-4 1.e-4 1.e-4 1.e-4
ECCUTS 5.e-4 5.e-4 5.e-4 5.e-4 5.e-4
SCCUTS 1.e-4 1.e-4 1.e-4 1.e-4 1.e-4
STCUTS 5.e-5 5.e-5 5.e-5 5.e-5 5.e-5
FASTCODE
TRIG 500000
STOP
The RUNG flag specifies a known good run with correct parameter configurations.
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7.2 Monte Carlo Smearing
The GSIM output files were then fed into GPP. GPP (GSIM post-processor) is
primarily used to smear timing and remove some bad wires using the data base wire map.
The GPP command and flag used is:
gpp -P0x1f -R43582 -Y -f0.50 -a1.0 -b1.0 -c1.0 -o<output> <input>
where the GPP parameters used were:
• f = 0.50 (time smearing)
• a = 1.0 (DOCA smearing Region 1)
• b = 1.0 (DOCA smearing Region 2)
• c = 1.0 (DOCA smearing Region 3)
Figure 7.1: Left: momentum resolution from data used to match Monte Carlo. Right: thematching of Monte Carlo to data by smearing out the measured resolution, red is MonteCarlo and blue is data.
The scintillator times are smeared according to the length of the scintillator. The time
smearing from flag -f uses a 0.50 scale factor for SC tdcl/tdcr smearing and comes directly
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from the work in reference [72]. The 1.0 scale factor for DOCA smearing for Regions 1, 2,
and 3 are default values that introduce only minimal smearing. The wire map of run
number 43582 is uses are the wire map for the Monte Carlo. This run has a set of bad
wires consistently down in the g11a run period, this run is indicated with the -R flag. The
-Y flag drops the DC hits according to the efficiency in the GPP map, and the DC wire
map in the data base. Dead wires are removed with the -P Bitwise Process flag.
The GPP settings are selected by studying and comparing resolutions of data and
Monte Carlo and implementing a degree of smearing that leaves space for additional
kinematic dependent smearing. To more accurately reflect the resolution of the data for
each measured variable empirical smearing was also applied to each reconstructed vector
in the simulations. During the construction of the Covariance Matrix it was necessary to
obtain simulations with near identical resolutions to data so that the same Covariance
Matrix could be used for both. This was done by first applying the minimal smearing
mentioned, from GPP, to the Monte Carlo then sampling a Gaussian distribution with
mean value λ , and φ obtained from the recovered tracking angles. The width of the
Gaussian was provided by the resolutions in λ , and φ in the kinematic range of the data.
This effectively smears out the directional tracking variables by an additional factor of
∼1.91 of the resolution obtained from the tracking code. The value 1.91 is averaged over
all kinematic ranges. Momentum was smeared in a similar fashion to match data. Figure
7.1 shows the δ p as a function of momentum, where δ p = pk f it− pdet . The pk f it is the
momentum from the missing proton using the test channel γ p→ K+Λ. The pdet is the
proton detected momentum. This distribution is then sliced up and fit with a Gaussian in
several momentum bins and plotted and shown in Figure 7.1 (right). For each momentum
bin a different width of the smear term is applied. Without this type of careful smearing to
all tracking parameters, the confidence level obtained from any kinematic fit of the Monte
Carlo could not have a flat distribution for the appropriate hypothesis.
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To check that the smearing is working appropriately, the missing mass peak is
checked against data. A peak that is too narrow could lead to a misrepresented acceptance
in both the Λπ0 and Λγ channels. Figure 7.2 shows the change in the missing mass peak
width from the Monte Carlo reaction γ p→ K+Σ∗0→ K+Λ(X).
Figure 7.2: Comparison between data and Monte Carlo width of the missing mass squareddistribution. The left plot show the data with a Gaussian fit, while the middle is the MonteCarlo before smearing and the right is the Monte Carlo after smearing.
7.3 Monte Carlo Processing
After GPP, the output files were passed through RECSIS for processing and
reconstruction to prepare for analysis.
The RECSIS tcl configuration file was set such that:
source /u/group/clas/builds/release-4-8/packages/tcl/recsis proc.tcl;
turnoff ALL;
global section off;
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turnon seb trk tof egn user pid;
setc outbanknames(1) ”all”;
outputfile clas.out PROC 2047;
setc prlink file name ”prlink g11 1920.bos”;
setc bfield file name ”bgrid T67to33.fpk”;
set torus current 1920;
set mini torus current 0;
set poltarget current 0;
set TargetPos(3) -10.;
set trk maxiter 8;
set trk minhits(1) 2;
set trk lrambfit chi2 50.;
set trk tbtfit chi2 70.;
set trk prfit chi2 70.;
set trk statistics 3 ;
set dc xvst choice 0;
set def adc -1;
set def tdc -1;
set def atten -1;
set def geom -1;
set st tagger match 15.;
set lst do -1;
set lpid make trks 0;
set trigger particle 2212;
where the frozen version of recsis proc.tcl for g11 processing was sourced.
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7.4 Trigger Simulations
The GSIM package helps to systematically account for inefficiencies in the various
CLAS detection hardware for a given set of tracks triggers an event. Extensive efforts have
been done to develop a procedure to correctly simulate the g11a trigger. The investigation
by Krahn [72] utilized the trigger word written into the data stream for each recored event.
The CLAS Level 1 trigger requires a coincidence between the start counter time and TOF
scintillator time within a time window for two charged tracks in two different sectors. The
trigger word contains information on which sectors met the CLAS Level 1 trigger
requirements. The study uses the test channel γ p→ pπ+π− to check how many times one
of the three final state particles did not meet the trigger condition. This work lead to a
“trigger map” for each particle type (proton, π+, π−) as a function of sector, TOF paddle
and azimuthal angle φ . The same map is used for both the K+ and the π+ [72].
To simulate the trigger in Monte Carlo, a random number between 0 and 1 is
generated and if this number is less than the efficiency in the map, then the trigger word is
set to positive for that track. The event is accepted if two or more tracks are triggered by
this method.
Because of the nature of the present analysis, no effect is seen from the trigger
efficiency correction. The change in effective triggers in the Monte Carlo decreases
equally in the dominating acceptance terms in the numerator and denomination of the final
results, due to similar topology. The procedure is reproduced only as a check to verify that
the final ratio is not changed.
7.5 Matching Data and Tuning
The FSGEN generator is based off the CERN generator GENBOD, with refined
modification to do physics at the CLAS energy scale and the added features for generating
t-channel production events. These type of generators make multi-particle weighted
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events according to Lorentz-invariant Fermi phase space. The total center-of-mass energy,
as well as the number and masses of the outgoing particles are specified by the user, but
may be changed from event to event. The center-of-mass vector momenta (and energies)
of the outgoing particles are generated with the weight that must be associated with each
event. The average proper lifetime can be specified by the user for each ground state
decay product. To demonstrate, the lifetime of the λ is reconstructed from the Monte
Carlo channel γ p→ K+Σ∗→ K+Λπ0 using the distance between the primary interaction
vertex and the reconstructed vertex from the proton and π−. The mean life from the PDG
[60] is entered into the generator as cτ = 7.89 cm, while the decay time from the fit is
cτ = 7.82 cm, as seen in Figure 7.3, which is within reason.
Figure 7.3: Decay time for the Λ from the Monte Carlo generated for γ p→ K+Σ∗ →K+Λπ0.
The generator uses a relativistic Breit-Wigner shape for all resonance mass
distributions. The width of the resonance needs to by specified for each mass. The K∗,
Λ(1405), and Σ∗ Monte Carlo mass distributions are shown if Figure 7.4. These
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distributions are from the γ p→ K∗+Λ, the γ p→ K+Λ(1405), and the γ p→ K+Σ∗0,
respectively.
The analysis relies on small corrections from the acceptance of the detector and
leakage of various channels into the signal being extracted. Because of the minimal size of
the leakages of the background, as shown by the acceptance tables in section 7, the effect
of individual cross sections on the final result is very small. The channels studied here
with Monte Carlo are listed in Table 8.4.
Figure 7.4: Monte Carlo distributions for the left: K∗, middle: Λ(1405), and right: Σ∗0. ABreit-Wigner fit is used to demonstrate accurate width and mass in each case.
To tune the Monte Carlo for the Σ∗→ Λπ0 channel, first a 1/Eγ photon energy
distribution was used. Figure 7.5 show the bremsstrahlung distribution for both data and
Monte Carlo. The photon energy distribution for the data in the figure was obtained using
unskimed data without any cuts and iterating over all photons in the TAGR bank. The
Monte Carlo photon energy distribution in the figure come directly from the generator,
also without any cuts. The acceptance for the Σ∗→ Λπ0 channel was determined by the
ratio of accepted events to thrown events in each Eγ bin. The data and Σ∗→ Λπ0 Monte
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Table 7.1: The set of Monte Carlo channel generated for acceptance studies.
Reaction Generated number of counts produced
Λ(1405)→ Σ0π0 4×106
Λ(1405)→ Σ+π− 4×106
Λ(1405)→ Λγ 4×106
Λ(1405)→ Σ0γ 4×106
Σ(1385)→ Λπ 1.8×107
Σ(1385)→ Σ+π− 4×106
Σ(1385)→ Λγ 1.8×107
Σ(1385)→ Σ0γ 4×106
ΛK∗+→ K+π0 4×106
ΛK∗+→ K+γ 4×106
Figure 7.5: Left: Photon energy distribution for data using unskimed data without anycuts. Right: Photon energy distribution taken from the Monte Carlo generator showing thebremsstrahlung distribution.
Carlo were cut on the Y ∗ mass range of 1.34-1.43 GeV, the Λ invariant mass at ±0.005
GeV around the PDG value of the Λ mass, and the missing mass squared of the reaction at
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0.017-0.022 GeV2, to isolate the Σ∗→ Λπ0 channel in the data. The yield was determined
by the ratio of the raw Λπ0 events to the number of incident photons in each Eγ bin, so as
to normalize with the Bremsstrahlung spectrum. Corrections were made for each bin with
the newly obtained acceptance. The density of the target was assumed to be constant, and
no background subtraction was preformed. The resulting cross section was then fit with a
sixth-order polynomial. The polynomial curve was then implemented in the generator
using a Von Neumann rejection method to improve the photon energy distribution to
better match the form of the Λπ0 cross section from the data. This new Λπ0 Monte Carlo
is then used to find a more accurate acceptance and the process is done again. Using the
momentum distributions of each decay product, shown in Figure 7.8, no further
improvement in Monte Carlo to data matching is seen after one iteration. Figure 7.5
Figure 7.6: Left: Cross section and fit function used in Monte Carlo generation for thereaction Σ∗ → Λπ0. Right: Comparison between data and generated cross section afterusing correction to the photon energy distribution.
shows the cross section with respect to photon energy from the data channel Λπ0 used to
obtain the line function used in event weighting. The last point at high photon energy was
not used in the fit. A comparison between Monte Carlo cross section and data cross
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section with respect to photon energy is shown in Figure 7.5 (right) once the modification
to the Monte Carlo cross section was made.
From the adjusted Monte Carlo, acceptance corrections are found for bins in the kaon
cosine center-of-mass angle and the approximated differential cross sections are used to
adjust the Λπ0 generator to fill each corresponding bin according to the form of the
distributions from the data. Each angle bin is broken in to Eγ bins and represented
accordingly in the new event weighting scheme of the generator. The differential cross
section used in this modification are shown in Figure 7.7 for energies 1.5-3.1 GeV.
Figure 7.7: Approximated differential cross section for γ p→ K+Σ∗0→ K+Λπ0 in cosθKin the center-of-mass frame.
133
After these modifications were made the resulting Monte Carlo can be checked with
the data using the momentum distributions for the kaon, pion, and proton tracks as well as
the kaon lab frame angle distribution, see Figure 7.8. The change in the branching ratio
from these corrections is very small and discussed in Section 9.
The form of the differential cross sections found for the reaction
γ p→ K+Σ∗0→ K+Λπ0 was also used to generate the other Σ∗0 channels. This includes
the radiative channel as well as the Σ+π− channel. It is shown later in Section 9 that
because of the low acceptance of the Σ+π− channel, these corrections are very small, and
make no difference in the resulting ratio.
In method-1 (section 8.1), a flat distribution with no t-dependence was used to
produce the K∗Λ and Λ(1405) decays, which was also used in previous studies [10]. In
method-2 (section 8.2), a t-slope of 2.0 GeV2 is used for all the Λ(1405) channels. The
corrections based on the acceptances of these contributions are small and tuning the
Monte Carlo generator make changes that are negligible in the final branching ratio. This
is discussed further in Section 9.
The simulations for γ p→ K+Σ∗0→ K+Λγ and γ p→ K+Σ∗0→ K+Λπ0 were
produced and studied. The same cuts as used for data are applied, and the missing mass
spectrum is shown along with the missing mass off the p-π−, seen in Figure 7.9 for the
radiative channel and Figure 7.10 for the π0 channel. The same distributions are studied
for the various Λ(1405) decays, as well as for the γ p→ K+Σ∗0→ K+Σ+π− channel (see
Figures 7.11-7.14). For the Λ(1405)→ Σ+π− decay, very few events should pass all the
cuts, so the distribution is shown before kinematic constraints are used. The various
Monte Carlo distributions for the γ p→ K+Σ∗0→ K+Λ(X) give a basis for comparison
when studying the missing mass for the data. One can also gain perspective on the form of
the anticipated background using the Monte Carlo simulations. For example, the
K∗→ K+Λ channel is expected to add a peak to the missing mass off the Λ around 0.9
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GeV, as seen in Figure 7.17. The radiative decay of the K∗ is also considered, though that
signal is very small.
Figure 7.8: Final comparison between Monte Carlo (lines) and data (points with errors) forthe reaction γ p→ K+Σ∗0→ K+Λπ0. Top left: cosθ of kaon in the lab frame. Top right:kaon momentum distribution. Bottom left: pion momentum distribution. Bottom right:proton momentum distribution.
To help demonstrate the possible effects of the various backgrounds to the missing
mass distribution and the missing mass off the Λ, the Monte Carlo distributions are shown
in Figures 7.11-7.17 for the γ p→ K+Λ(1405), the γ p→ K+Σ∗0→ K+Σ+π−, and the
γ p→ K∗+Λ channels. Each channel is used to obtain a correction term for the
corresponding background channel that can be used to correct for the contamination of
that channel in the final ratio. This is discussed in Section 8.1.
135
Figure 7.9: Monte Carlo for the γ p→ K+Σ∗0→ K+Λγ channel. Left: the K+pπ− missingmass squared. Right: the missing mass off the Λ.
Figure 7.10: Monte Carlo for the γ p→ K+Σ∗0 → K+Λπ0 channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
Figure 7.11: Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ0π0 channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
136
Figure 7.12: Monte Carlo for the γ p→ K+Λ(1405)→ K+Λγ channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
Figure 7.13: Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ0γ channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
Figure 7.14: Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ+π− channel (Before Cuts).Left: the K+pπ− missing mass squared. Right: the missing mass off the Λ.
137
Figure 7.15: Monte Carlo for the γ p→ K+Σ∗ → K+Σ+π− channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
Figure 7.16: Monte Carlo for the γ p→K∗+Λ→ΛK+γ channel. Left: the K+pπ− missingmass squared. Right: the missing mass off the Λ.
Figure 7.17: Monte Carlo for the γ p → K∗+Λ → ΛK+π0 channel. Left: the K+pπ−
missing mass squared. Right: the missing mass off the Λ.
138
Figure 7.18: Left: Monte Carlo missing energy for the γ p→ K∗+Λ→ ΛK+π0 channel.Right: Monte Carlo missing energy for the γ p→ K+Λ(1405)→ K+Σ0π0 channel.
139
8 EXTRACTION METHODS AND CONSTRAINTS
In the following sections, three different methods are laid out based on different
constraints required in the kinematic fitting procedure. The steps required to obtain the
radiative branching ratio for the excited-state hyperon Σ(1385) region relative to the
well-known Λπ0 decay channel of the Σ0(1385) are described. The procedure for
obtaining the Λ(1405) counts and K∗ counts that are used to correct the branching ratio is
outlined in the discussion of method-1 in Section 8.1. The procedure for obtaining these
background contributions is exactly the same as for the other two methods using the
acceptance terms for the corresponding method.
The tactic behind the use of constraints in an investigation of the branching ratio is to
maximize the usage of the known physics contained in the measured values without
significantly depleting statistics. In the original CLAS analysis [10] of this ratio, the p-π−
was constrained to be a Λ in a separate (2-C) fit prior to the final kinematic fitting
procedure used to separate the radiative signal from the π0. Using cuts based on
covariance early in the analysis can introduce biases if there is further reliance on the
covariance matrix at later stages in the analysis, as is discussed in Ref. [71]. Method-1 in
Section 8.1 attempts to reproduce the previous work by Simon Taylor [10] without a prior
kinematic fit of the p-π− to the invariant mass of the Λ. The Monte Carlo generation in
method-1 is also handled in attempt to be consist with this previous study.
Even with an excellent quality covariance matrix, the size of the radiative signal to
the π0 and the similar topology makes it difficult to cleanly separate them with a single
kinematic fit. This lead to the use of a two-step kinematic fitting procedure that first fits
with a missing π0 hypothesis, preserving only the low confidence level candidates, then
fits these remaining events with a radiative hypothesis and preserves the high confidence
level candidates. This two-step kinematic fitting procedure requires all other background
channels to be previously minimized so that there is a high probability that the low
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confidence levels event from the missing π0 fit are from the radiative decay of the
Σ0(1385). The resulting counts after the final fit are then corrected by the acceptance and
background subtraction to result in the true radiative counts. The same is done for the π0
channel. Once the counts for each channel and all the acceptances for each leakage
channel are known, a branching ratio can be calculated. This two-step kinematic fitting
technique is used in all three methods.
After the first kinematic fit, it is necessary to test the quality of the radiative signal
going into the final kinematic fit. This is done by studying the χ2 probability density
function of the events in the second kinematic fit. If the fit χ2 distribution emulates the
theoretical distribution for that particular set of constraints then the majority of events are
matching the radiative hypothesis. This “quality test” can be used to place the numerical
value of the first confidence level cut to get the best χ2 distribution. When the final
kinematic fit is performed, the resulting events will have a maximized probability of
satisfying the radiative hypothesis. There should then be little systematic variation in the
ratio based on the final confidence level cut as long as the cut is beyond the noticeable rise
around zero in the confidence level distribution from the poor quality candidates.
In the first method (Section 8.1), the χ2 probability density function shape is used to
regulate the confidence level cuts in the (1-C) fits. Due to the nature of the singularity in
the χ2 distribution of the (1-C) fit, only qualitative information can be deduced, leading to
a more arbitrary placement of the first confidence level cut. One can reduce this
uncertainty by using a (2-C) fit in the separation procedure, which has a χ2 distribution
that is easier to fit, putting a more quantitative regulation on the confidence level cuts. To
do this, a constraint is placed on the invariant mass of the Λ in the final kinematic fitting
procedure taking it from a (1-C) to a (2-C) fit. This is shown in method-2 and discussed in
Section 8.2.
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Kinematically fitting the p-π− and the missing particle all together to the mass of the
Σ∗0 is also a (2-C) fit. This is the approach of method-3. However, a greater number of
good candidates would be excluded from the same confidence level cuts due to the
Breit-Wigner shape of the resonance (discussed in reference [71]) leading to an increase in
systematic uncertainty at the same confidence level cuts or a increase in statistical
uncertainty at more effectively placed confidence level cuts. Method-3 is completed only
as a consistency check and discussed in Section 8.3.
Method-2 is then the most suitable approach, with reasonable constraints that
preserve the highest number of good quality candidates. This method is used in the final
reporting of the branching ratio as well as the systematic studies discussed in Section 9.
Each method assumes the cuts previously outlined to minimize background before
kinematic fitting.
8.1 Method-1
The first method shown is meant to emulate the original analysis for this channel as
done by Taylor [10]. Some differences in the constraints and how they are applied are
introduced to reduce any systematic bias in the fitting procedure. Details in kinematic and
vertex fitting that can introduce systematic bias are covered in Ref. [71]. A suitable
kinematic fitting procedure is used to fit each track of all detected particles to a particular
missing particle hypothesis. This is done by using the undetected particle mass in the
constraint equation. The detected particle tracks are kinematically fit as a final stage of
analysis and filtered with a confidence level cut. There are three unknowns (~px) and four
constraint equations from conservation of momentum: this is a simple (1-C) kinematic fit.
In the attempt to separate the contributions of the Σ∗0 radiative decay and the decay to
Λπ0, the events were fit using different hypotheses for the topology:
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γ p→ K+pπ−(π0) (1-C)
γ p→ K+pπ−(γ) (1-C).
The constraint equations are
F =
Ebeam +Mp−EK−Ep−Eπ −Ex
~pbeam−~pK−~pp−~pπ −~px
=~0. (8.1)
~Px and Ex represent the missing momentum and energy of the undetected π0 or γ .
Because of such similar topologies and the size of the radiative signal, the kinematic
fitting procedure cannot be expected to cleanly separate the “Λγ” events from the “Λπ0”
events in just one fit. This is handled with a two-step kinematic fitting procedure, making
first a kinematic fit to a missing π0 hypothesis and then checking the quality of the fit of
the low confidence level candidates in a second kinematic fit to the actual radiative
hypothesis.
One way to check the quality of the fit to a particular hypothesis is to use a fit to the
χ2 function from the fitting procedure. Because a (1-C) fit is used, the χ2 distribution for
one degree of freedom is used [71] and a fit function with a flat background (represented
by P2) can be applied,
f (χ2) =
P0√2Γ(1/2)
e−P1χ2/2√(χ2)
+P2. (8.2)
By setting an upper bound on parameter P0 it is possible to fit around the singularity
and get a quantitative measure for the quality of the fit from parameter P1. The bound on
P0 is chosen so that χ2 > 1.0×10−6, which is reasonably close to zero. Because of this
restriction around the singularity, it is difficult to get a very good fit, but in principle the
closer P1 is to unity, the closer the data matches the ideal χ2 distribution for one degree of
freedom. Naturally the distribution is not expected to be ideal due to the present
background, but it is possible to use the χ2 distribution fit in conjunction with the
confidence level from the kinematic fit as an additional tool for testing the resolving
capacity of the fitting procedure.
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Figure 8.1: Left: χ2 distribution for the π0 hypothesis with a fit using Eq. 8.2. Right:confidence level distribution for the π0 hypothesis.
Figure 8.2: Left: χ2 distribution for the γ hypothesis with a fit using Eq. 8.2. Right:confidence level distribution for the γ hypothesis.
Considering the π0-hypothesis for the data in the fit, the confidence level in Figure
8.1 is flat (right) and the χ2 distribution (left) looks as expected for one degree of freedom.
The P1 parameter is close to 1 and the expected shape is quite reasonable.
For the γ-hypothesis the confidence level also seems relatively flat in Figure 8.2
(right), but the χ2 distribution (left) has a different shape than expected. The fit to a
distribution for one degree of freedom shows a P1 parameter that is particularly small at
0.15. This is an indication that the vast majority of data being kinematically fit at this
stage are not satisfying the base assumption that the missing particle is massless. This
144
implies that even with a high confidence level cut there will still be an overwhelming
amount of π0 contamination passing through. It is possible to take an additional step in
the kinematic fitting procedure for cleaner separation.
The two-step kinematic fitting procedure is used to first fit to a π0-hypothesis and
take all low confidence level candidates, and then fit to a γ-hypothesis and take a high
confidence level cut. Because of the previous cuts outlined earlier in the analysis, there
should now be primarily π0 background and the true radiative signal. By first fitting to π0
and taking the low confidence level candidates, one reduces the measured probability that
the surviving candidates will have a missing mass of the π0 before they are finally fit to a
γ-hypothesis.
The initial selection of the confidence level cut was chosen with the intention of
separating the γ p→ K+Σ∗0→ K+Λπ0 and γ p→ K+Σ∗0→ K+Λγ channels, while
maximizing the statistics. A confidence level P(χ2) of 10% is estimated using Figure 8.1
(left) to be able to minimize contamination and is used as a starting point. It is important
to start at a point before the distribution begins to rise on the low end of the confidence
level distribution. These events represent events that do not meet the hypothesis in the fit
at all. Various confidence level cuts can be used to study the systematic variation of the
ratio, see Section 15.2.
The two-step kinematic fit is implemented by a requirement of Pπ0(χ2) ≤ 10% and
Pγ(χ2) > 10% to isolate the radiative channel, where Pπ0 and Pγ are the confidence levels
of the π0 missing mass hypothesis and the γ missing mass hypothesis, respectively. A
requirement of Pγ(χ2) ≤ 10% and Pπ0(χ2) > 10% is then used to isolate the π0 events.
The candidates that pass all other cuts, but do not pass both secondary cuts for each
hypothesis Pγ(χ2) > 10% and Pπ0(χ2) > 10%, are used in the calculation of how much
background is present from such channels as the Λ(1405). An acceptance from Monte
145
Carlo is needed for all background channels to be able to determine the leakage and
corrections needed.
Figure 8.3: Left: χ2 distribution for γ hypothesis with a fit using Eq.8.2. Right confidencelevel distribution for γ hypothesis.
After the two-step kinematic fitting procedure, one can again study the γ-hypothesis
confidence level, which seems relatively flat in Figure 8.3, as well as the χ2 fit, which now
looks more like a standard distribution for one degree of freedom. The fit parameter P1 has
increased much closer to 1. This is an indication that an improvement has been made on
the quality of candidates going into the fit. This gives some assurance that the candidates
that come out of the secondary fit can accurately be filtered with the confidence level cut.
Figure 8.4 shows the missing mass squared distribution after all cuts and the first
kinematic fit. The radiative signal defined by the first confidence level cut is seen around
zero, separated from the π0 peak. The π0 peak as defined by the first confidence level cut
is seen as the predominant peak separated from the radiative signal to the left of the peak
and other rejected background to the right of the peak.
8.1.1 Acceptance
The π0 leakage into the γ channel is the dominant correction to the branching ratio.
To properly calculate the ratio the leakage into the π0 region from the γ channel is also
146
Figure 8.4: Missing mass squared distribution for the events that are going into the secondstep of the kinematic fitting procedure. The kinematic fit to π0 satisfying P(χ2) < 10%shows the radiative candidates (yellow), as well at the rejected background from theΛ(1405) (green). The white region shows the π0 candidates from a P(χ2)≥ 10% cut.
used. Taking just these two channels into consideration the number of true counts can be
represented as N(Λγ) for the Σ∗→ Λγ channel, and N(Λπ) for the Σ∗→ Λπ0 channel.
The acceptance under the Σ∗→ Λγ hypothesis can be written as Aγ(X), with the subscript
showing the hypothesis type and the actual channel of Monte Carlo input to obtain the
acceptance value is indicated in the parentheses. For the calculated acceptance of the
Σ∗→ Λγ channel, under the Σ∗→ Λγ hypothesis, the acceptance is Aγ(Λγ), and for the
Σ∗→ Λπ0 hypothesis it is Aπ(Λγ). It is now possible to express the measured values for
each channel nγ and nπ as
nγ = Aγ(Λγ)N(Λγ)+Aγ(Λπ)N(Λπ) (8.3)
147
nπ = Aπ(Λπ)N(Λπ)+Aπ(Λγ)N(Λγ). (8.4)
The desired branching ratio of the radiative channel to the π0 channel using the true
counts is then R = N(Λγ)/N(Λπ). This can be obtained by dividing Equation 15.2 by 8.4
and expressing in terms of R such that;
nγ
nπ
=RAγ(Λγ)+Aγ(Λπ)Aπ(Λπ)+RAπ(Λγ)
, (8.5)
then solving for R to get the branching ratio expressed in terms of measured values and
acceptances,
R =nγAπ(Λπ)−nπAγ(Λπ)nπAγ(Λγ)−nγAπ(Λγ)
. (8.6)
Equation 8.6 is under the assumption that contributions from the Σ(1385) will only
show up as Λγ or Λπ0, neglecting the Σ(1385)→ Σπ channel. The estimate of the total
number of Σ(1385)’s produced using the Λπ0 channel is:
N(Σ∗) =N(Σ∗→ Λπ0)
R(Σ∗→ Λπ0)A(Σ∗→ Λπ0), (8.7)
where R(Σ∗→ Λπ0) is the branching ratio of the Σ(1385) decay to Λπ0 and A(Σ∗→ Λπ0)
is the acceptance for that channel. It is then possible to deduce an estimate of the number
of Σ(1385)→ Σ+π− counts that would contribute to the π0 peak:
N(Σ∗→ Σ+
π−) = R(Σ∗→ Σ
+π−)A(Σ∗→ Σ
+π−)N(Σ∗) (8.8)
=R(Σ∗→ Σ+π−)A(Σ∗→ Σ+π−)
R(Σ∗→ Λπ0)A(Σ∗→ Λπ0)N(Σ∗→ Λπ
0), (8.9)
where R(Σ∗→ Σ+π−) is the branching ratio of the Σ(1385) to decay into Σ+π− and
A(Σ∗→ Σ+π−) is the corresponding acceptance after all cuts. It is possible to simplify the
expression by using,
RΣπΛπ =
R(Σ∗→ Σ±π∓)R(Σ∗→ Λπ0)
= 0.135±0.011,
using the PDG average value [60]. The two charged combinations of the Σπ have equal
probability of decay. The Clebsch-Gordon coefficient for the Σ∗→ Σ0π0 decay is zero,
148
assuming isospin symmetry. The π0 peak counts can then be more accurately depicted,
while the γ peak counts remain unchanged. The observed counts expressed in terms of
true counts and corresponding acceptances for each hypothesis can then be expressed as
nγ = Aγ(Λγ)N(Λγ)+(Aγ(Λπ)+RΣπ
Λπ
2Aγ(Σπ))N(Λπ) (8.10)
and
nπ = (Aπ(Λπ)+RΣπ
Λπ
2Aπ(Σπ))N(Λπ)+(Aπ(Λγ))N(Λγ). (8.11)
Solving for R will result in a branching ratio that includes all needed information from the
Σ(1385). Though the corrections to R should be small for other contamination it is
necessary to include them in the calculation. The variation in the ratio based on the
background contamination is studied in Section 9 and shown to be quite small. The
distributions seen in Figures 7.11-7.17 indicate that there is some probability that
contamination for these other channels can leak through and acceptance studies need to be
done for all channels under both the Λγ and Λπ0 hypotheses. Results from the acceptance
for each hypothesis can be seen in Table 8.3. The branching ratio must include corrections
for the K∗+→ K+X and the Λ(1405)→ Σ+π− contamination, as well as the contribution
to the numerator from the Λ(1405)→ Λγ decay. The leakage of the Σγ channel is
assumed to be small relative to the Λγ signal. However, this channel is still considered in
the acceptance studies, see Table 8.3.
The formula for the branching ratio to take these backgrounds into consideration is
R =∆nγ
(AΣ
π(Λπ)+ RΣπΛπ
2 AΣπ(Σπ)
)−∆nπ
(AΣ
γ (Λπ)+ RΣπΛπ
2 AΣγ (Σπ)
)∆nπAΣ
γ (Λγ)−∆nγAΣπ(Λγ)
,
(8.12)
∆nπ = nπ −Nπ(Λ∗→ Σ+
π−)−Nπ(Λ∗→ Σ
0π
0)−Nπ(Λ∗→ Σ0γ)
−Nπ(Λ∗→ Λγ)−Nπ(K∗→ Kπ0),
(8.13)
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∆nγ = nγ −Nγ(Λ∗→ Σ+
π−)−Nγ(Λ∗→ Σ
0π
0)−Nγ(Λ∗→ Σ0γ)
−Nγ(Λ∗→ Λγ)−Nγ(K∗→ Kγ). (8.14)
The nγ (nπ ) terms come directly from the yield of the kinematic fits and represent the
measured number of photon (pion) candidates. In the notation used, lower case n
represents the measured counts, while upper case N represents the acceptance corrected or
derived quantities. The Nγ,π terms are corrections needed for the leakage from the
Λ(1405) and K∗ channel. These corrections are necessary to take into account due to the
fact that the structure of the background underneath the Σ(1385) is not flat, which could
lead to over estimating the Σ(1385) contribution. The structure of the various
contributions can be seen in the simulation plots in Figures 7.11-7.17. The notation
utilized is such that the pion (photon) channel identifications are denoted AΣπ(Σ+π−)
(AΣγ (Σ+π−)) so that AΣ
γ (Λπ) denotes the relative leakage of the Λπ channel into the Λγ
extraction and AΣπ(Λγ) denotes the relative leakage of the Λγ channel into the Λπ
extraction, where AΣ denotes acceptance strictly for Σ(1385) compared to the AΛ which is
the acceptance for the Λ(1405).
Table 8.3 lists all channels taken into consideration and the value of the acceptance
for the Pπ0(χ2) ≤ 10% with Pγ(χ2) > 10% and the Pγ(χ2) ≤ 10% with Pπ0(χ2) > 10%
cuts used. The table list three columns sorted by hypothesis Aγ , Aπ , and the counts that
made all other cuts but did not satisfy either the γ or π0 hypothesis denoted as Aγπ .
The number of N(Λ∗) is dependent on the number of observed counts n(Σ0π0) above
the π0 peak seen in green in Figure 8.4, which can be expressed as
N(Λ∗) =nΛ
R(Λ∗→ Σ0π0)AΛ(Σ0π0). (8.15)
The notation nΛ here is short hand for n(Σ0π0) while R(Λ∗→ Σ0π0) is the probability that
the Λ(1405) will decay to the Σ0π0 and AΛ(Σ0π0) is the probability that this decay
channel will be observed after all the applied cuts. Isospin symmetry is assumed so that
150
R(Σ0π0) = R(Σ+π−) = R(Σ−π+)≈ 1/3 for the Λ(1405) decay channels. An estimate of
the number of counts in the π0 peak coming from the reaction Λ∗→ Σ+π−, using Eq.
8.15, can be calculated;
Nπ(Λ∗) = R(Λ∗→ Σ+
π−)AΛ
π (Σ+π−)N(Λ∗) =
AΛπ (Σ+π−)nΛ
AΛγπ(Σ0π0)
. (8.16)
A small adjustment can be made to ensure that the Λ∗→ Σ+π− contributions to the
green mound in Figure 8.4 are also included by adding in the relative acceptance
AΛ(Σ+π−) to the denominator. These acceptance terms are achieved by independently
using Monte Carlo for the γ p→ K+Λ(1405)→ K+Σ0π0 and
γ p→ K+Λ(1405)→ K+Σ+π− reactions and using the counts that survive all cuts but did
not satisfy either the γ or π0 hypothesis also denoted by Aγπ . The leakage for the
γ p→ K+Λ(1405)→ K+Σ+π− channel is very small but is included for completeness.
The final result is,
Nπ(Λ∗) =AΛ
π (Σ+π−)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−). (8.17)
From this example it becomes transparent how to express all other associated Λ(1405)
corrections using only the observed nΛ counts and the corresponding acceptance for that
channel. The corrections for the γ channel are obtained from,
Nγ(Λ∗→ Λγ) =AΛ
γ (Λγ)R(Λ∗→ Λγ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−), (8.18)
Nγ(Λ∗→ Σ0γ) =
AΛγ (Σ0γ)R(Λ∗→ Σ0γ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−), (8.19)
Nγ(Λ∗→ Σ0π
0) =AΛ
γ (Σ0π0)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−),
Nγ(Λ∗→ Σ+
π−) =
AΛγ (Σ+π−)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−). (8.20)
151
For the π0 channel they take the form,
Nπ(Λ∗→ Λγ) =AΛ
π (Λγ)R(Λ∗→ Λγ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−), (8.21)
Nπ(Λ∗→ Σ0γ) =
AΛπ (Σ0γ)R(Λ∗→ Σ0γ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−),
Nπ(Λ∗→ Σ0π
0) =AΛ
π (Σ0π0)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−), (8.22)
Nπ(Λ∗→ Σ+
π−) =
AΛπ (Σ+π−)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−), (8.23)
where R is used for the corresponding branching ratio in each case. The value of 27 keV is
used for the width of the Λ(1405)→ Λγ decay, and the value of 10 keV is used for the
width of the Λ(1405)→ Σ0γ decay which come from reference [74]. The choice in these
values is discussed in Section 9. The terms for Aγ (Aπ ) are indicative of the acceptance
terms under a radiative (π0) hypothesis in the kinematic fit.
In order to find nΛ, one can look at the events for which neither the γ nor the π0
hypothesis is satisfied, seen in green in Figure 8.4. The value of nΛ is difficult to
determine due to the non-Breit-Wigner shape of the Λ(1405) decay. Because the
contribution of the Λ∗ to the overall ratio is small, an exact fit is not required. To establish
an upper and lower limit of the contamination affects on the ratio, two methods of
extracting the nΛ counts are employed and the variation in the ratio based on these two
methods is discussed later in Section 9. To obtain the maximum possible counts for nΛ the
raw counts rejected from the kinematic fit can be used. Because it is assumed that the
counts that survive all other cuts but do not satisfy the radiative hypothesis or the π0
hypothesis are primarily from the Λ(1405)→ Σ0π0 channel it is possible to use these
counts directly as an upper limit of the value of nΛ. This should be the maximized value of
nΛ because there should still be a notable amount π0 events that did not pass the high
confidence level cut of the π0 hypothesis.
152
A second method was also employed. Monte Carlo was used to fill the background
according to internal decay kinematics and normalized to the data so that the Monte Carlo
mass spectrum level matches the that of the data providing an estimate for nΛ count. This
was done by using the Λπ0 channel Monte Carlo and normalizing it to the data, then
adding the full spectrum of the Λ(1405) Monte Carlo so that it would decay to all known
channels with appropriate branching ratios, under the same normalization as the Λπ0
channel Monte Carlo. The Λ(1405) reaction Monte Carlo was added until it matched the
data distribution relative to the Λπ0 channel.
Figure 8.5: Left: data (error bars) with Monte Carlo (line) from the Λπ0 channel and thefull spectrum of Λ(1405) filled to match the data. Right: data (error bars) with Monte Carlo(line) from the Λ(1405) only.
Figure 8.5 (left) shows the matching of the data and the Monte Carlo. The Monte
Carlo used is γ p→ K+Σ0→ K+Λπ0 mixed with the full spectrum of the
γ p→ K+Λ(1405) according to PDG [60]. By adding more and more Λ(1405) until the
region M2x (K+pπ−) > 0.027 GeV begins to match the data, an estimate of the amount of
Λ(1405) present relative to the Λπ0 channel is found. Figure 8.5 (right) shows the added
counts from the Λ(1405) without the Λπ0 channel, matched the data points in that region.
These counts are used to correct for all other contamination except for the K∗.
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Table 8.1: Counts for nΛ found though the different methods; the raw counts rejected fromthe π0 hypothesis, and the estimated number of nΛ from Monte Carlo. The uncertaintiesare fit uncertainties combined with statistical uncertainties.
Method Counts
Raw Counts 5429±82.45
MC Data Match 5112±69.95
Table 8.1 show the results of the nΛ counts for the two methods. The counts from the
Monte Carlo data matching are used in the final result, however the variation in the final
branching ratio is small based on this background as seen is Section 9.
Few counts from the K∗+ survive all the cuts, but to include corrections that consider
the few that do, an estimate of the K∗+ counts present must be established. With this
estimate it is possible to find the acceptance corrected number of K∗+ events,
N(K∗+) =n(K∗+→ K+π0)
R(K∗+→ K+π0)A(K∗+→ K+π0), (8.24)
where n(K∗+→ K+π0) is the estimated number of K∗+→ K+π0 events present.
R(K∗+→ K+π0) = 2/3 is the probability for the decay and A(K∗+→ K+π0) is the
acceptance of the K∗+→ K+π0 channel. A(K∗+→ K+π0) is determined by developing
an extraction method to obtain the K∗+→ K+π0 counts and observing how many thrown
events survive all cuts for the extraction method used. Three different extraction methods
of obtaining the K∗+→ K+π0 counts are discussed next. Once N(K∗+) is established, the
number of K∗+ that would be present under the π0 hypothesis could be expressed as,
Nπ(K∗+→ K+π
0) =Aπ(K∗+→ K+π0)n(K∗+→ K+π0)
A(K∗+→ K+π0), (8.25)
where Aπ(K∗+→ K+π0) is the acceptance for the K∗+→ K+π0 channel under the Λ(π0)
hypothesis. It is also possible to consider the radiative decay of the K∗ using N(K∗+) such
154
that,
Nγ(K∗+→ K+γ) = R(K∗+→ K+
γ)Aγ(K∗+→ K+π
0)N(K∗+), (8.26)
Nγ(K∗+→ K+γ) =
32
R(K∗+→ K+γ)
Aγ(K∗+→ K+γ)A(K∗+→ K+π0)
n(K∗+→ K+π
0). (8.27)
Here the branching ratio R(K∗+→ K+γ) is small at 9.9×10−4 [60]. The change in the
ratio from the N(K∗+) events is quite small and was neglected in the previous analysis
[10], so the procedure to obtain the N(K∗+) counts is laid out here but adjustment to the
ratio is used only in method-2.
To acquire an estimate of the K∗ events present, again more than one method is
employed. The methods include: a straight Gaussian fit with quadratic background, an
estimate from MC matching to data, and an extrapolation using a series of Gaussian fits at
various excited state hyperon mass ranges.
Relatively few K∗ candidates survive the 1.34-1.43 GeV hyperon mass cut used in
the analysis as seen in Figure 8.6 (left). This make it difficult to get an accurate Gaussian
fit on the bump is the missing mass off the Λ spectrum. A simple Gaussian fit with
quadratic background is used to obtain a crude estimate of counts. The Gaussian fit,
shown in green is seen Figure 8.6 (left), with a quadratic background, shown in red, and
the final Gaussian, shown in blue.
It is much easier to get a minimized χ2 fit over the range that allows the higher part
of the excited state mass spectrum to pass through. Fits are done at ranges 1.34-1.5 GeV,
1.34-1.55 GeV, 1.34-1.6 GeV, 1.34-1.65 GeV, 1.34-1.7 GeV, 1.34-1.75 GeV, and 1.34-1.8
GeV. A Gaussian fit is acquired at each hyperon mass range. These resulting counts from
each Gaussian fit are then plotted for each mass window to produce the trend seen in
Figure 8.6 (right). The mass window is defined as the difference from the starting hyperon
mass range to the cut off. The range of interest from 1.34-1.43 GeV is 0.09 GeV. A
polynomial fit to the mass window bins is applied and then a value at the low range of
interest is extrapolated. Figure 8.6 (right) is used to extract a value of 1983 for the K∗
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contribution for the mass window of 0.09 GeV. The data matching method for obtaining
Figure 8.6: Left: fit with a Gaussian and quadratic background for the hyperon mass range1.34-1.43 GeV. Right: fit with polynomial to the point derived from Gaussian fits of variousmass windows.
Figure 8.7: Missing mass off of the Λ for the match of data and Monte Carlo to obtain thecounts from K∗. The lines is data and the points are from Monte Carlo after adding in theK∗ and Λ(1405).
the K∗ counts is the same as for the data matching for nΛ. After using the Monte Carlo to
first match the π0 peak, and then the Λ(1405) spectrum, the missing mass off of the Λ is
used to fill the full spectrum of the K∗ to get the Monte Carlo mass spectrum to
appropriately match that of data, see Figure 8.7.
156
The estimated counts from various methods are reported in Table 8.2. The
extrapolated value, being the median of the three methods, is used in the final result, again
the variation in the final reported ratio due to this contribution is discussed in Section 9.
Table 8.2: Counts for nK∗ found though different methods. Uncertainties are fituncertainties combined with statistical uncertainties.
Method Counts
Gaussian Fit 2121±60.57
Extrapolation 1983±55.45
Data Match 1888±43.95
Once the estimated K∗ and Λ∗ counts are determined, all of the contamination can be
subtracted out of the calculated branching ratio in Eq. 15.5 using the form in Eq. 8.13 for
contamination in the Λπ0 channel and Eq. 8.14 for the contamination to the radiative
signal. The acceptance values required for the calculation of each contamination term in
Equations 15.12-8.23 are listed in Table 8.3. The table includes the acceptance values for
each Monte Carlo channel taken into consideration for the radiative hypothesis Aγ , the
missing π0 hypothesis Aπ0 , and the counts that survive all other cuts but do not satisfy
either of the radiative or missing π0 hypotheses Aγπ . The confidence level cuts used for
the two-step kinematic fitting procedure to produce the table and the final result where
Pγ = Pπ0 = 10%. Table 8.3 shows more Aγπ acceptance terms than are required. Only the
AΛγπ(Σ0π0) and the AΛ
γπ(Σ+π−) terms in the Aγπ column are needed for the branching ratio
calculation. The rest of the column is produced for completeness. The statistical
uncertainty reported for each term is based on the uncertainty from the counts
δnΛ ∼√
nΛ, the acceptance terms and uncertainties and ratio and associate uncertainty all
propagated accordingly.
157
Table 8.3: Acceptances (in units of 10−3) for the channels used in the calculation of thebranching ratios. Here there is a 10% confidence level used as upper and lower P(χ2) cuts;the Pxy cut was 0.03 GeV. The uncertainties are statistical only. The three columns containthe acceptance for each hypothesis Aγ , Aπ , and the counts that made all other cuts but didnot satisfy either γ or π0 hypothesis denoted as Aγπ .
Reaction Aπ Aγ Aγπ
Λ(1405)→ Σ0π0 0.0495±0.0010 0.0009±0.0004 1.491±0.029
Λ(1405)→ Σ+π− 0.021±0.0008 0.001±0.0004 0.0078±0.0009
Λ(1405)→ Λγ 0.0051 ±0.002 1.61±0.025 0.0632±0.0012
Λ(1405)→ Σ0γ 0.291±0.009 0.190±0.007 0.731±0.022
Σ(1385)→ Λπ 1.98±0.0396 0.0561±0.002 0.0442±0.002
Σ(1385)→ Σ+π− 0.191±0.0004 0.0058±0.001 0.0018±0.0008
Σ(1385)→ Λγ 0.0179±0.002 2.131±0.042 0.0728±0.006
Σ(1385)→ Σ0γ 0.407±0.012 0.162±0.095 0.282±0.017
ΛK∗+→ K+π0 0.201±0.015 0.001±0.0004 3.261±0.051
ΛK∗+→ K+γ 0.0028±0.0001 0.162±0.003 2.230±0.046
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8.1.2 Results
To calculate the branching ratio Eq. 15.5 is employed. The terms that take the
Σ∗→ Σ+π− into account are relatively small. Using the acceptance values in Table 8.3,
these terms become;RΣπ
Λπ
2AΣ
π(Σπ) = 1.29×10−5, (8.28)
andRΣπ
Λπ
2AΣ
γ (Σπ) = 3.92×10−7. (8.29)
The numerator of Eq. 15.5 can then be expressed as,
∆nγ(1.98×10−3 +1.29×10−5)−∆nπ(0.0561×10−3 +3.92×10−7), (8.30)
and the denominator,
∆nπ(2.13×10−3)−∆nγ(0.0179×10−3). (8.31)
Each ∆n contains the raw counts extracted from the kinematic fit and the background
counts that should be removed.
The results for the various background contributions are listed in Table 8.4. Though
the largest background in the γ channel is from the contribution of the π0 tail, it is not
listed here because the correction is included in the acceptance terms seen if Eq. 8.30.
This is also true for the γ background leakage into the π0 channel, seen in Eq. 8.31. The
final values for the subtracted contamination terms come from Equations 15.12-8.23. The
raw counts for the radiative and π0 extraction using Pγ = Pπ0 = 10% are also listed.
After subtracting out the values from the table the true signal counts become
∆nγ = 579.42 and ∆nπ = 13281.38.
The final ratio, using Eq. 15.5, RΛγ
Λπfor the Σ(1385) for this method is then,
RΛγ
Λπ=
Γ[Σ0(1385)→ Λγ]Γ[Σ0(1385)→ Λπ0]
= 1.43±0.11% (8.32)
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Table 8.4: Breakdown of statistics for each term in Eq. 15.5 for the Λ(γ) and Λ(π0)hypothesis. Each listed channel is subtracted from the raw counts directly from thekinematic fit to obtain the final ratio. The uncertainties listed are statistical only. TheK∗+ counts are not used in method-1 in order to follow Taylor [10].
Hypothesis Λ(γ) Λ(π0)
Raw counts 589±24.27 13522±116.28
Λ(1405) : N(Σ0π0) 3.07± 1.37 168.8± 5.28
Λ(1405) : N(Σ+π−) 3.41± 1.36 71.62± 4.26
Λ(1405) : N(Λγ) 2.97± 0.60 0.01± 0.07
Λ(1405) : N(Σ0γ) 0.13± 0.053 0.19± 0.054
K(892) : N(ΛK∗+) 0.100± 0.011 14.8± 3.7
Adjusted counts 579.42±24.35 13281.38±116.48
The uncertainty presented is statistical only. The statistical uncertainty in the counts nπ
and nγ used is the square root of the counts. The overall statistical uncertainty is
propagated accordingly through Eq. 15.5 using the corresponding uncertainty for each
acceptance term that is listed in Table 8.3. The branching ratio 1.43±0.11% is
comparable within the statistical uncertainty to previous results 1.53±0.39% from Taylor
[10]. The essential differences in Taylor’s method using the G1C data set included using a
kinematic fit to the Λ hypothesis prior to the final missing π0 or radiative hypothesis, the
method of extracting background counts, and using a Gaussian fit after the final kinematic
fit to obtain the counts and uncertainties. In simulations, Taylor used an initial angular
dependence of 5−3cos2θ for the γ p→ K+Σ∗0→ K+Λπ0 channel. In the present analysis
the angular distribution from the data are used. Like Taylor, the present analysis uses a
zero t-slope for the Λ(1405) and the γ p→ K+Σ∗0→ K+Λγ channel; a more physical
representation is used in section 8.2.
160
The method of using a two-step kinematic fitting procedure to extract the ratio has a
great deal of systematic dependency on the choice of the confidence level cut. This is first
studied by varying the overall range of the confidence level while keeping the relative
probability from the higher and lower bound consistent with each other such that Pπ0(χ2)
= Pγ(χ2) for isolating both the π0 and radiative signal. This is done in a range from 0.1%
to 40%, see Figure 8.8. At high confidence levels too much π0 background will pass the
first kinematic fit and the χ2 distribution will again be distorted. In addition to this issue,
for very low confidence level cuts in the initial fit, the π0 background is minimized, but
the same cut in the second kinematic fit would allow almost all candidates to pass, again
letting too much background in.
To correctly study the systematic variation of the confidence level cut, the
distribution of the χ2 should be first tested for each variation to make sure the cuts to
Pπ0(χ2) and Pγ(χ2) correspond to a good signal. To find the best range of variation, the
cuts are staggered so that Pπ0(χ2) and Pγ(χ2) are no longer equivalent. Table 8.5 shows
the selected range after an attempt to achieve the best quality of the radiative signal based
on a qualitative analysis of the χ2 distribution fit for one degree of freedom and the
resulting P1 fit parameter. Cuts are selected to keep P1 within 20% of P1 = 1. Also in the
table is a column for Pπ0(%)/Pγ(%) enabling an independent point for each value of R to
be plotted against for each cut selection. This is done over a range of confidence level
cuts, see Figure 8.9. In conclusion, the cuts 10% = Pπ0(χ2) = Pγ(χ2) used to obtain the
branching ratio is a reasonable choice.
Further systematic studies are not performed for this method. The χ2 probability
density function shape used to regulate the confidence level cuts for the (1-C) fits is too
difficult to fit, leading to no more than qualitative conclusions drawn from the fit
parameter P1. Because the uncertainty of the systematics in this approach can not be well
defined, only a statistical uncertainty is reported for the first method. Method-2 attempts
161
to provide a clearly defined systematic uncertainty by using a (2-C) fit in the separation
procedure, which has a χ2 distribution that is easier to fit, putting a more quantitative
regulation on the confidence level cuts.
Table 8.5: Dependence of the corrected branching ratio on the confidence level cuts.
Pπ0(%) Pγ(%) R(%) Pπ0(%)/Pγ(%) P1
0.1 5 1.57± 0.13 0.02 1.128
1 5 1.52± 0.12 0.2 1.166
1 10 1.45± 0.11 0.1 1.166
10 10 1.43± 0.11 1.0 1.187
0.01 1 1.44± 0.14 0.01 1.101
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Figure 8.8: A study over a variation in confidence level; each cut corresponds to Pπ0(χ2) =Pγ(χ2).
163
Figure 8.9: Only the selected confidence level cuts with the best quality signal based on theP1 parameter in the χ2 fit (values seen in Table 8.5). Pπ0(χ2)/Pγ(χ2) is used on the x-axisto obtain a distinguishable point for each ratio value.
164
8.2 Method-2
It is possible to reduce the uncertainty in the ratio that comes from the initial Λ cut of
±0.005 GeV relative to the PDG value by using an additional constraint in the kinematic
fit. The additional constraint is introduced into the kinematic fitting procedure already
outlined in Section 8.1 but with the further requirement that the proton and π− track have
an invariant mass of the Λ in the hypothesis. To ensure that no systematic bias is
introduced, it is best to fit the Λ and missing mass hypothesis together, see reference [71].
This fitting procedure involves fitting each track of all detected particles to a particular
missing particle hypothesis, while requiring the proton and π− be constrained to have the
invariant mass of the Λ. This is done by using the undetected particle mass in the
constraint equation, while the invariant mass of the Λ constraint is also met. The detected
particle tracks are kinematically fit as a final stage of analysis and again filtered with the
confidence level cut. In this fit, there are three unknowns (~px) and five constraint
equations, four from conservation of momentum and then the additional invariant mass
condition. This makes a (2-C) kinematic fit. In the attempt to separate the various
contributions of the Σ∗0 radiative decay and the decay to Λπ0, the events were again fit
using the hypotheses for the topology:
γ p→ K+pπ−(π0) (2-C)
γ p→ K+pπ−(γ) (2-C).
The constraint equations are
F =
(Eπ +Ep)2− (~pπ +~pp)2−M2
Λ
Ebeam +Mp−EK−Ep−Eπ −Ex
~pbeam−~pK−~pp−~pπ −~px
=~0. (8.33)
Again, ~Px and Ex represent the missing momentum and energy of the undetected π0 or γ .
The same two-step kinematic fitting procedure is used to resolve the radiative signal
from the π0. An additional advantage of this approach is that it is possible to get a much
165
better fit to the χ2 distribution for a 2-C fit because there is no singularity in the form of
f (χ2), which was the case for a 1-C fit. The distribution for a 2-C fit follows the from Ref.
[71],
f (χ2) =
P0
2e−P1χ2/2 +P2. (8.34)
This fit function still has a flat background term, P2, but P1 now becomes the measure of
how close the distribution in the histogram is to the ideal (P1 = 1.0) theoretical χ2
distribution for two degrees of freedom.
The more accurate nature of the fit to the theoretical probability density function for
two degrees of freedom enables a more systematic approach to defining the placement of
the confidence level cuts. Because there are two kinematic fits for both the π0 channel and
radiative signal, some new notation is introduced. The first confidence level cut used to
filter out the larger π0 signal from the radiative signal by using a kinematic fit to Λ(π0)
and taking only the low confidence level candidates is denoted as Paπ (χ2). The final
kinematic fit used to isolate the radiative signal, using a Λ(γ) hypothesis has a confidence
level cut denoted as Pbγ (χ2), taking only the high confidence level candidates. These two
confidence level cuts need not be at the same value, as was assumed in method-1
(following Taylor).
The key question here is how to select the cut points for Paπ (χ2) and Pb
γ (χ2). The
Λπ0 channel will be reduced for a lower value of Paπ (χ2), which is desirable for extracting
the radiative decay signal. On the other hand, this cut cannot be made arbitrarily small,
since it reduces the statistics (i.e., increases the statistical uncertainty). Similarly, the Λγ
signal will be purified by a higher cut on Pbγ (χ2), but again the higher the cut, the lower
the statistics. Of course, Monte Carlo can be used to examine the acceptance of these cuts
for various branching ratios (Λγ/Λπ0). In the end, the branching ratio extracted from the
data should not depend on the cut points chosen (assuming the Monte Carlo gives accurate
cut acceptances). As was shown for method-1, this was only the case after careful
166
consideration of the P1 parameter. In method-1 the P1 parameter can only be studied
qualitatively. Furthermore, for method-2, the Monte Carlo can be used to optimize the
trade-off between statistical uncertainty and systematic uncertainty (due to the choice of
confidence level cuts based on a more quantitative analysis of P1).
Details of the optimization method of the confidence level cuts, using the Monte
Carlo, are described in [71]. Here, an overview of the optimization method is described.
In the Monte Carlo, only Λγ events can be generated, and the value for “pure” P1
signal is obtained with a fit to equation (8.34). This pure P1 can be compared with the P1
value from the fit of signal and Λπ0 contaminated events. The degree of contamination can
then be regulated with a low Paπ cut, giving the capacity to find an ideal Pa
π based on the
change in P1. This step is essential because the resolving capacity of the final kinematic fit
to Λγ is limited depending on the amount of signal to background present in the fit.
Examples of fits to the χ2 distributions for different amounts of Λγ and Λπ0 Monte
Carlo events present in the final kinematic fit to Λγ are shown in Figure 8.10. The low
statistics of this Monte Carlo study is intentional to emulate the statistics of the radiative
signal in the g11a data. A cut of Paπ (χ2) < 1% has been implemented before the final
kinematic fit in each case.
At the top (far left), the unphysical case of Λγ dominance is shown to demonstrate
that the fit gives a P1 value higher than those fits including events from the Λπ0 channel.
With an increasing number of Λπ0 events in the kinematic fit, the P1 parameter decreases
from the initial P1 values. In other words, the change of the P1 parameter from its initial
pure value is an indication of how much π0 background is present. The fit to the χ2
distributions for the different amount of Λπ0 background gives a basis of comparison
when analyzing the g11 data while using a Paπ (χ2) < 1% cut.
167
A quantitative measure of the deviation of the P1 parameter can be defined (notation
σγ in [71]) and the dependence of this P1 deviation has been mapped out as a function of
these cut points [71].
Figure 8.10: The fit to the χ2 distribution from Monte Carlo for various mixtures ofradiative signal and Λπ0 events for small statistics, after cutting events with Pa
π (χ2) < 1%.The amount of π0 background still present in each case can be determined by looking atthe number of events added from the pure singal case in the top left plot.
At the heart of the optimization method, the Monte Carlo is used to determine both
the statistical uncertainty and the “recovery” uncertainty. The recovery uncertainty (given
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by δR in [71]) is given by the deviation of ratio R (see equation 8.6) from the full analysis
procedure as compared to the known ratio used in the Monte Carlo. As the confidence
level cuts are made tighter, the amount of Λπ0 events are reduced, which reduces the ratio
deviation (δR) and the deviation of the P1 parameter from its initial pure radiative value.
On the other hand, tighter cuts increases the statistical uncertainty in RΛγ
Λπ, so the
confidence level cuts are placed independently to improve statistics without introducing
uncertainty from δR.
The optimum cuts occur when the fractional deviation in the ratio δR equals the
fractional error due to statistics. The optimum cuts occur at different values for a given
mixture of Monte Carlo Λπ0 and Λγ events. Using the g11 data, the Monte Carlo has been
tuned to have approximately the same ratio and the same statistics as the real data, and the
optimum cuts are thereby determined quantitatively. Again, the details are given in Ref.
[71]. The optimum cuts, along with the value of δR, are given in Table 8.6. Note that the
value of δR is not the systematic uncertainty in RΛγ
Λπ. Rather, the systematic uncertainty
comes from the data, based on the variation in the extracted ratio of Eq. (15.5) after all
Monte Carlo corrections for each set of optimized cuts. The systematic uncertainty is
presented in Section 9.
Table 8.6: Optimization points for each Paπ0 and Pb
γ from Ref. [71].
Paπ0(%) Pb
γ (%) δR
7.5 25 0.097
6.25 20 0.09
5.0 15 0.087
1.0 10 0.085
0.075 5 0.088
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Equally important to extracting the true radiative signal is extracting the true Λπ0
signal, which is the leading term in the denominator of Eq. (15.5). To do this, we reverse
the above procedure, and first take the low confidence level events for the Λ(γ)
hypothesis, denoted by the Paγ (χ2) cut, and then take the high confidence level events for
the Λ(π0) hypothesis, denoted by the Pbπ (χ2) cut. (Note the reversal in the use of a and b
as compared with the radiative signal given previously.)
As a starting point, the cuts Pbπ (χ2) > 10% and Pa
γ (χ2) < 1% are used to extract the
Λπ counts. Variations in these cuts are studied in Section 9. Figure 8.11 shows the χ2 and
confidence level distributions of the π0 candidates after the Paγ (χ2) < 1% cut and before
any cut on Pbπ (χ2). The fit to the χ2 distribution shows a P1 parameter of 1.117, close to
the ideal of unity, indicating a very small portion of background from the radiative decay
is present in the kinematic fit. In fact, the cut on Paγ (χ2) removes very few events; the
plots in Figure 8.11 would look very similar even without this cut, since there is so little
Λγ background.
For comparison, the corresponding χ2 and confidence level distributions for the
radiative hypothesis are shown in Figure 8.12 before any cuts are applied. Here, the χ2
distribution (left plot in Fig. 8.12) is far from the ideal, indicating that the vast majority of
events are not consistent with the radiative hypothesis, as expected since the Λπ0 events
are dominant.
After regulating the background of the π0 with the Paπ (χ2) < 1% cut, the χ2 and
confidence level distributions improve as seen in Figure 8.13. The fit to the χ2 distribution
shows a P1 parameter of 0.868 indicating that a much greater percentage of events in the
fit are now consistent with the radiative hypothesis. The value P1 can be compared to the
values seen in Figure 8.10 giving an estimate of the amount of Λπ0 background still
present in the final Λγ fit. The placement of the Pbγ is determined within 10% by using δR
from Table 8.6.
170
Figure 8.11: Left: χ2 distribution and fit with two degrees of freedom for the π0 candidatesfrom the g11a data after the Pa
γ (χ2) < 1% cut and before the Pbπ (χ2) > 10% cut; Right: the
corresponding confidence level distribution for the π0 candidates.
Figure 8.12: Left: χ2 distribution and fit with two degrees of freedom for the γ candidatesfrom the g11a data before the Pa
π (χ2) < 1% cut; Right: the corresponding confidence leveldistribution for the γ candidates.
171
Figure 8.13: Left: χ2 distribution and fit with two degrees of freedom for the γ candidatesfrom the g11a data after the Pa
π (χ2) < 1% cut and before the Pbγ (χ2) > 10% cut; Right: the
corresponding confidence level distribution for the γ candidates.
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8.2.1 Results
The branching ratio is calculated using Eq. 15.5 as outlined in method-1. ∆β is
selected using a 1 ns timing cut as described while keeping ∆β < 0.02 for the kaon but
using a looser cut of the ∆β < 0.05 for the proton and pion. This choice is discussed in
Section 9. All other cuts remain the same as previously outlined. The procedure to obtain
the acceptance terms is the same but must be done for the new constraints and cuts
implemented. The acceptance used in listed in Table 8.7.
Table 8.7: Acceptances (in units of 10−3) for the channels used in the calculation ofthe branching ratios. Here Pb
γ (χ2) > 10% and Paπ (χ2) < 1% while Pb
π (χ2) > 10% andPa
γ (χ2) < 1% and the Pxy cut was 0.03 GeV. The uncertainties are statistical only. The threecolumns contain the acceptance for each hypothesis Aγ , Aπ , and the counts that made allother cuts but did not satisfy either the γ or π0 hypothesis denoted as Aγπ .
Reaction Aπ Aγ Aγπ
Λ(1405)→ Σ0π0 0.0495±0.0031 0.001±0.0001 1.189±0.019
Λ(1405)→ Σ+π− 0.029±0.002 0.0013±0.0001 0.0078±0.001
Λ(1405)→ Λγ 0.0011±0.0001 1.65±0.031 0.0223±0.002
Λ(1405)→ Σ0γ 0.170±0.012 0.191±0.009 0.437±0.013
Σ(1385)→ Λπ 1.421±0.0278 0.0321±0.002 0.0312±0.002
Σ(1385)→ Σ+π− 0.161±0.01 0.00254±0.001 0.00138±0.0006
Σ(1385)→ Λγ 0.0184±0.002 2.335±0.039 0.0704±0.005
Σ(1385)→ Σ0γ 0.191±0.011 0.058±0.0001 0.225±0.015
ΛK∗+→ K+π0 0.213±0.010 0.010±0.006 2.931±0.051
ΛK∗+→ K+γ 0.0022±0.0001 0.158±0.003 2.351±0.046
The methods used for obtaining the background counts for the γ p→ K+Λ(1405) and
the γ p→ K∗+Λ outlined in method-1 are the same, but with the exception of using a more
173
realistic t-slope of 2 GeV2 for the Λ(1405) channels [75]. The differential cross section
from the data from the γ p→ K+Σ∗0→ K+Λπ0 are used to modify the generator for all
Σ∗0 channels.
Because the amount γ p→ K+Λ(1405) and γ p→ K∗+Λ background counts depend
on the hypothesis and confidence level cuts used, it is necessary to obtain new values.
Table 8.8 shows the result for the three methods used to obtain the Λ(1405) counts and
Table 8.9 shows the result for the three methods used to obtain the K∗+Λ counts.
Table 8.8: Counts for nΛ found through the different methods; the raw counts rejected fromthe π0 hypothesis, and the estimated number of nΛ from Monte Carlo. The uncertaintiesare fit uncertainties combined with statistical uncertainties.
Method Counts
Raw Counts 4172±64.59
MC Data Match 4085±63.91
Table 8.9: Counts for nK∗ found through the different methods. Uncertainties are fituncertainties combined with statistical uncertainties.
Method Counts
Gaussian Fit 1352±36.77
Extrapolation 1289±35.90
Data Match 1207±34.74
The counts nγ and nπ are again taken directly from the kinematic fit, see Figure 8.14.
The newly calculated acceptance terms from Table 8.7 are used with the nΛ and
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Figure 8.14: Left: the nπ counts extracted using the confidence level cuts Paγ < 0.01 and
Pbπ > 0.1. Middle: the nγ counts extracted using the confidence level cuts Pa
π < 0.01 andPb
γ > 0.1. Right: the counts nπ and nγ shown in the spectrum before any kinematic fit.
n(K∗+→ K+π0) estimates from Table 8.8 and Table 8.9 to obtain a new ratio calculated
again using Eq. 15.5,
RΛγ
Λπ=
Γ[Σ0(1385)→ Λγ]Γ[Σ0(1385)→ Λπ0]
= 1.42±0.12(stat)% (8.35)
The details of the calculation of R and statistical uncertainty are presented in
Appendix A. The systematic uncertainty for this method is reported in Section 9.
175
8.3 Method-3
It is also possible to put a constraint on the missing particle in combination with the
proton and π− tracks. It is possible to kinematically fit each track of all detected particles
to a particular missing particle hypothesis, while requiring the proton, π−, and missing
particle be constrained to have an invariant mass of the Σ∗0. The fit is achieved by using
the undetected particle mass in the constraint equation, while the decay products from the
Λ in combination with the missing particle are confined to be the mass of the Σ∗0 for each
hypothesis. The detected particle tracks are kinematically fit as a final stage of analysis
and again filtered with the confidence level cuts. In this fit there are three unknowns (~px),
five constraint equations from conservation of momentum, and the additional invariant
mass condition similar to method-2. This type of fit is also a (2-C) kinematic fit. The
constraint equations are
F =
(Eπ +Ep +Ex)2− (~pπ +~pp +~px)2−M2
Σ∗0
Ebeam +Mp−EK−Ep−Eπ −Ex
~pbeam−~pK−~pp−~pπ −~px
=~0. (8.36)
Again ~Px and Ex represent the momentum vector and energy for the missing particle in the
hypothesis.
8.3.1 Results
The same two-step kinematic fitting procedure is used to resolve the radiative signal
from the π0. The χ2 distribution should again follow the form of the distribution for two
degrees of freedom. This method not only restricts the proton and π− in the constraint, but
also effectively uses the additional restriction of the missing energy to achieve a
separation of the two topologies. The disadvantage of this method is that the three part
invariant mass of Σ∗0 should be a 36 MeV wide Breit Wigner shaped peak which means
that an invariant mass from poorly measured events can look the same as a well measured
176
event in the tail of the peak. In other words the invariant mass constraint is sensitive only
to detector resolutions and not peak width. All events from the outer edges of a wide peak
receive a χ2 low weighting in the probability distribution of the measured events in the
kinematic fit. The additional restriction ultimately makes separation of signal and
background more challenging by removing the freedom in the three invariant mass to be
the form of a resonance. This can be seen in the confidence level distribution of the π0
channel in Figure 8.15.
Figure 8.15: The confidence level distribution with the additional constraint on the invariantmass of the Σ∗0 using a missing π0 hypothesis.
Notice that the confidence level distribution in Figure 8.15 is no longer flat around
10%. This means that a cut at this point would have an indeterminate systematic error
introduced. In fact the distribution does not flatten out until much higher in the
distribution.
Instead of proceeding with a cut on the distribution in Figure 8.15 the previous
method is used in the initial step of the two step kinematic fitting procedure. By using the
kinematic fit to a missing π0 with only the Λ constraint it is possible to use the previous
initial cut of 1% to filter out the π0 background. This allows the radiative signal to pass
177
through to be tested with the Σ∗0 invariant mass constraint. The χ2 distribution and
confidence level distribution for the second kinematic fit are shown in Figure 8.16.
Figure 8.16: The χ2 and confidence level distribution with the additional constraint on theinvariant mass of the Σ∗0 with the missing γ hypothesis.
Figure 8.16 indicates the correct general shape for both distributions, however the
same issue arises: the confidence level distribution is not flat in the lower region. In
addition, the form of the χ2 distribution would be very difficult to investigate due to low
statistics. There is no direct way to optimize with the choice of confidence level cuts in
this method. To obtain a result the confidence level cuts are taken from Method 2, using
the Σ∗0 invariant mass constraint only with the missing γ hypothesis.
The ratio achieved is
RΛγ
Λπ=
Γ[Σ0(1385)→ Λγ]Γ[Σ0(1385)→ Λπ0]
= 1.37±0.20(stat)%. (8.37)
This result is with Pγ = Pπ0 < 1% on the initial confidence level cut and Pγ = Pπ0 >
10% on the the second cut.
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9 SYSTEMATIC STUDIES
Method-2 is selected for a set of systematic studies. The value of each cut is varied to
study the effect on the final acceptance corrected ratio. For each variation the new
acceptance terms in Equation 15.5 are recalculated with the corresponding Monte Carlo.
Each major systematic uncertainty contribution is collected together in a table at the end
of this section. Each term is numbered as it is discussed to tie the discussion to the
corresponding term listed Table 9.3.
Several ∆β cut variations are checked starting with ∆β < 0.02 for all charged
particles leading to a branching ratio of 1.49±0.15%. There is also a check at ∆β < 0.1
giving a ratio of 1.38±0.11%. The ∆β selected is using a 1 ns timing cut while keeping
the ∆β < 0.02 for the kaon but a looser cut of ∆β < 0.05 for the proton and π−. This
variation is presented at the top of Table 9.3 as number (1).
To clean up the selected kaon events there is a cut made on identified kaon candidates
that are truly π+. The missing mass squared is studied for the reaction γ p→ π+π−(X).
This contamination is easily removed with a cut slightly above zero but below the mass
region of the π0. A cut of 0.01 GeV2 was used with the intention of not cutting into the
good K+ events. This cut is seen in Figure 5.1. The ratio was calculated with and without
this cut, and there is no measurable difference in the resulting branching ratio or statistical
uncertainty. The ratio with and without this cut is presented in Table 9.3 as number (2).
The distance of closest approach cut for the proton and π− is varied from a cut of
DOCA < 1 cm to DOCA < 20 cm and it is found that there is some variation, however the
statistical error increases significantly for smaller cut values. The cut at DOCA < 5 cm
marks the beginning of the most stable region in Figure 9.1. The various DOCA cuts
shown in Figure 9.1 are in centimeters while the acceptance corrected branching ratio is
shown in (%). The cut at DOCA < 5 discussed in the analysis is preserved and used in the
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final quoted ratio. The variation based on the DOCA cut is presented in Table 9.3 as
number (3).
Figure 9.1: The variation in the DOCA cuts in centimeters with the acceptance correctedbranching ratio shown in (%).
The perpendicular momentum cut is required in the analysis due to its exact
topological similarity to the radiative signal. To find the most efficient Pxy cut, the ratio is
calculated at various cut values until the ratio stabilizes. The lowest stable cut is selected
to preserve statistics. The final cut is the cut reported in the analysis previously at
Pxy < 0.03 GeV. Figure 9.2 shows the acceptance corrected ratio for various Pxy cuts with
a flat line fit to the stable points used to obtain the uncertainty in the Pxy cut. This variation
is presented as number (4).
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Figure 9.2: The variation in the perpendicular momentum cuts with the acceptancecorrected branching ratio shown in (%).
There is no direct way to estimate the systematic uncertainty in the Monte Carlo’s
capacity to match the data. However, it is possible to check the affects of various tuning
parameters to see the variation in the branching ratio. The systematic effects of the Monte
Carlo are studied by making adjustments to the generator, producing a new simulation and
then re-evaluating the acceptance terms. It is possible to check variation in the ratio (based
on the new acceptance) by modifying only the generator to match the production cross
section seen in the photon energy distribution from the data and then checking various
exponential t-dependences. Prior to using the approximated differential cross section, the
best data matching t-slope value in the Monte Carlo is found to be 0.65 GeV. To study the
generator’s effect on the ratio a t-slope in the range of 20% lower and 20% higher than the
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ideal matching value of 0.65 GeV is selected to test the effects on the resulting acceptance
corrected branching ratio. The change in t-slope is done simultaneously for the radiative
and π0 channels. The acceptance for both channels are recalculated and the corresponding
ratio at various t-slopes is plotted in Figure 9.3. There is very little variation seen. The
branching ratio using a t-slope of 0.65 is 1.422%, and using the differential cross sections
to match the data, the branching ratio is 1.421%. In the final ratio reported, the
modification in the generator used to match the differential cross sections is implemented
after the production cross section modification to the photon energy distribution. The
t-slope is implemented only in these variation studies. This is listed as number (5) in the
table.
Figure 9.3: Left: Variation in the acceptance corrected branching ratio for various t-slopesetting in the generator for the π0 channel; right: for the radiative channel.
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To demonstrate the effect of each acceptance term on the final branching ratio, all
Monte Carlo channels that are included in the branching ratio are tested by setting the
yield for the contamination found for that channel to zero in Eq. 15.5. This is done for
every included contamination one at a time. The result for each channel is shown in Table
9.1. The branching ratio with all channels included is 1.420%.
Table 9.1: Branching ratio for excluded channels in (%).
Counts Excluded Branching Ratio
Λ(1405) : N(Σ0π0) 1.448±0.116
Λ(1405) : N(Σ+π−) 1.424±0.116
Λ(1405) : N(Λγ) 1.446±0.116
Λ(1405) : N(Σ0γ) 1.422±0.116
Σ(1385) : N(Σ+π−) 1.425±0.116
Σ(1385) : N(Σ0γ) 1.421±0.116
Each term is Table 9.1 is not separately included in the systematic uncertainty
calculation. Instead terms (6) and (7) in Table 9.3 show the branching ratio with and
without the total background subtraction taken into account for the Λ(1405) and the K∗
respectively. These variation are found by simply setting nΛ = 0 or n(K∗+→ K+π0) = 0
and comparing the resulting ratio with the nΛ and n(K∗+→ K+π0) previously extracted.
Prior analysis using the g11a data set shows that there are substantial inefficiencies in
the production trigger used to collect the data [72]. The inefficiencies are deduced to be
caused by inefficiencies in the TOF array and timing in the start counter in the trigger.
This leads to sector, θ , φ and particle dependent efficiencies in the trigger that need to be
accounted for when extracting finite cross sections from the data. A study was preformed
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to test the impact on the ratio due to the trigger efficiency corrections to the acceptance
from the Monte Carlo. The change seen in the ratio is of the order of ∼ 0.05% of the ratio.
This is a negligible uncertainty relative to the other uncertainty discussed in the Monte
Carlo.
To look at the systematic dependency on the choice of the confidence level cuts, the
range defined by the σγ [71] uncertainty for Pbπ (χ2) should be checked. As previously
described the Monte Carlo studies leads to the set of optimal Paπ0 cuts for a given Pb
γ , as
seen in Table 8.6. To maximize statistics and minimize δR the optimal cuts chosen for the
analysis were Pbγ > 1% and Pb
γ > 10%. The variation in the branching ratio is studied by
selecting the confidence level cuts that lie slightly outside the optimization region in
Figure 27 in the CLAS note [71]. In this way the largest range for Pbγ and Pb
γ can be tested
while still respecting the information derived in the optimization map. This leads to a
systematic uncertainty that can be assigned to the resolving procedure used in method-2.
Table 9.2 lists the cut defined by the Monte Carlo derived optimization map in CLAS note
[71]. Three points in Paπ are used in the optimization region for the Pb
γ > 10% cut. There
are then two cuts used for both Pbγ > 5% and Pb
γ > 15% that lie just outside the
optimization region.
Using the full range of ratios in Table 9.2, the largest and smallest values are used to
find the deviation of the ratio defined by the new set of confidence level cuts and the
quoted ratio R = 1.42% in method-2. This deviation is used as the uncertainty in the
resolving procedure and added in with the other systematic uncertainties. The variation
from the choice of Confidence level cut is listed in Table 9.3 as (8).
The exponential t-slope is also varied for the Λ(1405) and found to have negligible
effects on the branching ratio. The variation in t-slope tested is 0-2 GeV2, resulting in a
change of less than 0.1% of the branching ratio.
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Table 9.2: Dependence of the corrected branching ratio on the confidence level cuts for theselected systematic range.
Pbγ (%) Pa
π (%) R(%)
15 7.5 1.388± 0.12
15 5 1.390± 0.12
10 5 1.422± 0.12
10 1 1.420± 0.12
10 0.5 1.421± 0.12
5 0.1 1.448± 0.12
5 0.05 1.436± 0.12
The branching ratio of the electromagnetic decays of the Λ(1405) are not known.
There has been experimental work by Burkhardt and Lowe [74] which are the values used
in the present analysis of 27 keV for the Λ(1405)→ Λγ channel and 10 keV for the
Λ(1405)→ Σ0γ . These values are not included in the PDG official listing. The
Λ(1405)→ Σ0γ has such small acceptance that little effect is seen in the ratio. However,
the Λ(1405)→ Λγ acceptance can lead to a noticeable affect on the ratio.
There is some work by Workman and Fearing [76], which is a very similar analysis
to Burkhardt and Lowe, but before the experiment on kaonic atoms had a result. They also
find an interference between Born and resonance diagrams, giving a value of 23 keV for
the EM decay width to Λ(1405)→ Λγ .
The Chiral bag model work of Umino and Myhrer [77], presented a “cloudy bag”
model, where the bag parameters have been tuned using other data (mass spectrum of the
negative parity hyperons). They also find an interference (cancellation), which makes a
smaller EM decay width, but from a different set of diagrams. They get 75 keV for the
EM decay to Λ(1405)→ Λγ .
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Other work by Darewych, Horbatsch and Koniuk, [78], use a straight-forward mixing
model based on the SU(6) wave-functions. No interference between diagrams is found,
leading to a prediction of 143 keV for the EM decay to Λ(1405)→ Λγ . Using this width
in the calculation of the present analysis would lead to a ratio of 1.33±0.12.
The width from the Darewych calculation is large in comparison to all other models,
and especially large compared with the Burkhardt and Lowe result, so it is excluded from
the systematic studies. This leads to the largest width of 75 keV which leads to a ratio in
the present analysis of 1.39±0.12. This result is included in line (9) in Table 9.3.
Table 9.3 shows a summary of the systematic studies and the higher and lower range
of the ratio based on the variations mentioned for each type of uncertainty.
Table 9.3: Ranges of systematic variation in resulting ratio in (%) showing L(Low)-contribution and H(High)-contribution and rang in each case.
Type of Error Low Range L-Contribution High range H-Contribution
(1)∆β 1.380±0.12 -0.040 1.490± 0.15 +0.070
(2)Miss ID 1.420±0.12 -0.000 1.422±0.12 +0.002
(3)DOCA 1.350±0.12 -0.007 1.480± 0.12 +0.060
(4)Pxy 1.415±0.12 -0.005 1.433± 0.12 +0.013
(5)t-slop 1.380±0.12 -0.040 1.440± 0.12 +0.020
(6)Λ∗-BG 1.420±0.12 -0.000 1.470± 0.12 +0.050
(7)K∗-BG 1.420±0.12 -0.000 1.431± 0.12 +0.011
(8)P(χ2) cuts 1.388±0.12 -0.032 1.448± 0.12 +0.028
(9)Λγ cuts 1.390±0.12 -0.030 1.420± 0.12 +0.000
Total Uncertainty -0.072 +0.112
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To calculate the final systematic uncertainty for method-2, the difference in the ratio
R = 1.42 and the high range of the ratio for each case in Table 9.3 is added in quadrature
to obtain a value for the uncertainty of 0.11 greater than the ratio. The lower systematic
uncertainty is based on the difference between the ratio R = 1.42 and the low range of the
ratio for each case, resulting in a value of 0.07 less than the ratio.
9.1 Conclusion
Because of the increase in the statistical uncertainty in method-3, the branching ratio
reported in method-2 of 1.42±0.12(stat)+0.11−0.07(sys) is favored. Previously published work
on this channel yielded a ratio of 1.53±0.39(stat)+0.15−0.17 [10]. The value in method-2 is
consistent within uncertainties.
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Part II
Electromagnetic decay of the Σ∗+
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10 EVENT SELECTION
Using the g11a data set, events are selected for the channel γ p→ K0Σ∗+. The present
PDG branching ratios lists the decay Σ∗→ Σπ to be 11.7±1.5% and assuming isospin
symmetry this leads to a branching ratio of 5.85±0.75% for the Σ∗+→ Σ+π0 decay [60].
This channel will be used to normalized the radiative signal that comes from the channel
Σ∗+→ Σ+γ . The radiative signal is assumed to be quite small, in the range of 0.1−0.5%
of the full Σ∗ width. However, there have yet to be any measurements. For both channels
the topology of the decay is γ p→ K0Σ+(X) where X is not directly measured such that
the π0 and γ are differentiated using conservation of energy and momentum. This
topology leads to the final set of decay products γ p→ K0Σ+(X)→ π+π−π+n(X). The
charged particles can easily be detected with the use of the CLAS drift chambers and
Time-of-flight. The neutron must be detected with the CLAS electromagnetic calorimeter.
The analysis was done using the previously skimmed g11a data set for two positively
charged tracks and one negatively charged track with no other skim conditions.
The previously listed CLAS runs form Part-1 of this thesis were used for the present
analysis. The same energy loss, tagger corrections, and momentum corrections were also
used. The same beam photon selection and detector performance cuts for all charged
particle were used as previously described. The same timing cut of 1 ns is used. All good
photons are kept until the final cuts are implemented and then a check is done to ensure
only one photon survives.
Only pions above 0.125 GeV/c momentum are studied. The beam photon is required
to be between 1.5 and 3.8 GeV.
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10.1 Particle Identification
Figure 10.1: The ∆β distribution for the π− and the cut applied to clean up identification.
Again the PART bank is used to make the initial selection of the charged particles.
Using the time of flight from the event vertex to the scintillator counter, β is calculated for
each pion. The ∆β can then be calculated using the difference between the time-of-flight
β and calculated β . Figure 10.1 shows an example of the ∆β distribution for the π− and
the cut used. The same ∆β < 0.025 is cut used for all pions.
In the reaction of interest, γ p→ K0Σ∗+(X), it is necessary to determine which π+
comes from the K0. It is possible to check both π+ with the detected π− to see the kaon
candidates in each case by using the invariant mass. A preliminary invariant mass plot for
each combination is shown is Figure 10.2.
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Figure 10.2: Invariant mass of the π+-π− combination for the two different π+ detected,prior to any π+ organization to optimize the K0 cut.
Figure 10.3: Invariant mass of the π+-π− combination for the two different π+ after π+
organization to optimize the K0 peak.
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Figure 10.4: The confidence level distribution for the (2-C) kinematic fit with constraint onπ+ and π− to be the mass of the K0 with a missing mass off the K0 of the Σ∗+.
After the K0 selection, a kinematic fit can be used on just the final K0 pions with the
constraint as the invariant mass of the K0. This ensures a quality K0 for a certain
confidence level cut. Prior to the kinematic fit for each event the invariant mass is checked
for each π+ pair. Whichever π+ leads to the invariant mass that is closest to the PDG
mass for the K0 is used in the kinematic fit. Figure 10.3 shows the invariant mass plot for
each combination after the π+ selection.
A (2-C) fit is used with only the selected π+ and π− tracks and an additional
constraint of the missing mass of the Σ∗+. This allows systematic control on the quality of
the K0 and Σ∗+ identification through a single confidence level cut. It is shown in a CLAS
Note [71] that using resonances in missing mass constraints in a kinematic fit can lead to a
significant loss of counts around the tail. Wide resonances can suffer greatly from this
type of fit. However, these types of cuts are only problematic when studying structure
dependent physics such as cross sections. Because the study presented here is on a
branching ratio, the preservation of the Σ∗+ shape is not required. In other words the Σ∗+
counts in the tail of the peak will be lost in the both the numerator and denominator. The
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kinematic fit then helps to ensure only good quality Σ∗+ events survive. An initial
confidence level cut of P(χ2) > 0.1% is chosen, which is quite small, to let in as many
reasonable counts as possible. The systematic variation of this cut is then studied in
Section 15.3. Figure 10.4 shows the confidence level distribution from the kinematic fit
with constraint on π+ and π− to be the mass of the K0 with a missing mass constraint off
the K0 to be the Σ∗+ mass. The distribution is not expected to be flat since there is a lot of
background present. Figure 10.5 shows the invariant mass of the π+, π− combination
before and after the P(χ2) > 0.1%confidence level cut. Figure 10.6 shows mass of the π+,
π− combination before and after the P(χ2) > 0.1% confidence level cut.
(a) (b)
Figure 10.5: (a) The invariant mass of the π+ π− after the best π+ is selected. (b) Theinvariant mass of the π+ and π− after the P(χ2) > 0.1% confidence level cut.
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(a) (b)
Figure 10.6: (a) The missing mass off the K0 before the confidence level cut. (b) Themissing mass off the K0 after the P(χ2) > 0.1% confidence level cut.
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11 NEUTRON IDENTIFICATION
Neutral particles are detected in CLAS as clusters in the EC not associated with any
reconstructed charged track from the drift chambers. Because the g11a run period was not
intended for neutral detection, precision neutron momentum reconstruction requires some
additional steps. The EC timing calibration was done using fast pions such that β ∼ 1
using the TOF as a reference time. This is far from ideal for neutrons where β < 1. The
directional components of the neutral track are found by using the neutron vertex and the
cluster position on the EC for that hit. The momentum of the neutral particle is calculated
from the EC time-of-flight and the path length from the neutron vertex to the cluster
position. The neutrons are differentiated from photons using a β < 0.9 cut eliminating fast
neutrals.
The neutron detection is essential for the reaction of interest. The neutron momentum
can be obtained and then used in combination with the π+ to study the kinematics of the
Σ+. Having clean constraints on the K0 and Σ+ is important when considering the event
topology γ p→ K0Σ∗+→ K0Σ+(X).
A thorough study of the accuracy of the EC for neutron momentum reconstruction in
all kinematic ranges has not been achieved previously; it is an essential part of the present
analysis. Correlations between each measured variable in the EC have also not been
studied well. The covariance of the neutron can give a lot of information about the quality
of the kinematic variables in various areas of the EC. These values can then be used to
weight the neutron measurements appropriately in kinematic constraints that depend on
maximum likelihood methods, see Section 13.
There are resolution changes that are related to the acceptance of the EC. Hits from
the center of each triangular sector have better measurements over those on the edges.
There are also effects from the torus coils and forward angles to consider. The EC consists
195
of two main blocks. The inner and outer blocks can have different resolution parameters
for each measured variable.
Finally, there are some kinematic ranges that have better resolution than others due to
geometry and the phase space of the reaction. All possible combinations of momentum
and position are studied to develop a complete understanding of the neutron variance and
covariance in the electromagnetic calorimeter.
11.1 Neutron Detection Test
The test reaction γ p→ π+π−π+n is isolated in the g11a data set by selecting a π−
and two π+ and kinematically fitting to a missing neutron hypothesis and then taking a
10% confidence level cut. This channel is selected because the final decay products are
identical to the reaction of interest γ p→ K0Σ∗+→ K0Σ+(X). In addition the momentum
range of the detected particles is the same. The simplification made by working with the
test channel is that in the γ p→ π+π−π+n reaction, there is only one interaction vertex.
This implies that the neutron comes from the primary interaction vertex, which can be
easily determined using the charged pions.
It is possible to study the neutron momentum reconstruct and resolutions by requiring
each event to have one detected neutron and then compare this with the missing
momentum. Assuming a high quality missing neutron four vector, this procedure can be
used to find the dynamic resolution over the EC face.
The neutron vertex is found from the MVRT bank (see Appendix B), which uses a
multi-track-vertex fitting routine giving a very accurate vertex for multiple final state
particles, all coming from the same vertex [32]. Because the neutron comes from the
primary interaction vertex in this study, the vertex of the neutron is accurately know.
However, for any tracks in which the neutron comes from secondary vertices, this neutron
vertex is not a easily obtained. Because the neutron vertex information can effect the
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direction vector of the neutron and its momentum, these differences can be important
when studying resolutions.
11.2 Neutron Path
The distance that the neutron travels in CLAS is used with the time-of-flight to
determine the momentum of the neutron. The distance dependent on which EC block
layer and the position of the cluster reconstruction. This is determined using the ECHB
bank (see Appendix B) by looking at the integer value of matchid1 and matchid2. If
matchid2 is between 1 and 5 while matchid1 is zero, then the hit is only from the outer
block of the EC. The other hits, with matchid1 and matchid2 being between 0 and 5
(excluding 0 for matchid1), means that the hit was either in the inner block or both. If
there is a hit in both, the cluster reconstruction position in the inner block is used. If there
is a hit only in the outer EC block the first layer (layer closest to the target) of the outer
block gives the plane of the EC cluster coordinates. If there is a hit in the inner block or
both, the first layer of the inner block is used as the plane of the EC cluster coordinates.
The distance in which the neutron travels inside each block before a cluster is formed
cannot be determined, so there is naturally some uncertainty in path length associated with
each hit of the order of the width the EC block. Because of differences for the inner and
outer EC, the resolutions of each block are studied separately.
11.3 Neutron Time
The TOF for the neutron comes from the start time of that event to the EC cluster
time. Because the EC timing calibration was done using high β particles, there is
additional uncertainty introduced to the reconstructed neutron momentum associated with
the poor time-of-flight. The time-of-flight must be corrected to achieve an accurate
magnitude of the neutron momentum.
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11.4 Neutron Momentum Correction
It is possible to make the corrections needed for the neutron path and time-of-flight
by empirically adjusting the magnitude of the neutron momentum using a correction
function found by studying the trend in the momentum resolution over a kinematic range.
This is done by plotting ∆p = pk f it− pmeas against p for its measured range. The value of
pk f it is the magnitude of the missing neutron momentum from a (1-C) kinematic fit of the
pions detected in the drift chambers. The pmeas is the magnitude of the momentum
reconstructed using the EC cluster position and EC time-of-flight. The trend should be
evenly distributed around ∆p = 0. If it is not then the distribution will display a trend that
can be used to correct the neutron momentum. Once the neutron momentum resolution is
evenly distributed around zero the plot of the neutron detected momentum and the missing
momentum should be approximately one to one. This implies that for the majority of
events, the detected neutron momentum is comparable to the missing neutron momentum
from the kinematic fit. Figure 11.1 shows the magnitude of the momentum distribution of
∆p and the plot of pk f it vs pmeas for both EC block layers, where the momentum
correction has not yet been applied for the outer EC block. Figure 11.2 shows the same
distributions where the outer EC block momentum correction is now applied.
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(a) cosθ vs φ (missing n) (b) cosθ vs φ (detected n)
Figure 11.1: (a) ∆p before the correction, showing EC counts for inner and outer blocklayers. (b) The pk f it vs pmeas for EC counts for inner and outer block layers beforecorrections.
(a) cosθ vs φ (missing n) (b) cosθ vs φ (detected n)
Figure 11.2: (a) ∆p after correction showing EC counts for inner and outer block layerstogether. (b) The pk f it vs pmeas for EC counts for inner and outer block layers aftercorrections.
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To ensure a high quality missing neutron vector to study the corrections and
resolutions of the neutron, a 10% confidence level cut on the kinematic fit is used to obtain
pk f it . Only the detected neutrons found in a direction less than 3 from the missing
neutron are used. A cut in polar lab angle θ is also used between 10-40. Figure 11.3
shows the cosθ vs φ form the missing neutron vector from the kinematic fit. Figure 11.3
shows the same for the reconstructed neutron vector. This same procedure is repeated with
the Monte Carlo of the test channel.
Figure 11.3: Demonstration of cosθ vs φ for the neutron candidates.
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Figure 11.4: Demonstration of cosθ vs φ for the detected neutrons.
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11.5 Neutron Resolutions
The structure of CLAS prevents neutrons from being detected with the same
precision in all directions. The six superconduction coils in front of the EC limit the
capability of the calorimeter detection. The resolution for the neutron around these
blocked regions can be irregular. The EC neutron resolutions give an indication as to the
quality of the reconstructed four-vector for both Monte Carlo and data. The variance and
covariance will be used in the kinematic fitting procedure discussed in Section 13.
Again the missing neutron four-vector is used to study ∆p with respect to p, φ , and θ
as defined in the target frame. Figure 11.5 shows ∆p over a range of the momentum
magnitude, θ , φ and the path used in the momentum reconstruction for the g11a data.
A similar study of ∆p is done for Monte Carlo as well. Figure 11.6 shows ∆p over a
range of the momentum magnitude, θ , φ and the path used in the momentum
reconstruction for simulations.
Once ∆p is obtained for data and Monte Carlo, the distributions with respect to p, θ
and φ are sliced in binned in each dynamic variable. Each bin is projected on to the ∆p
axis and fit with a Gaussian to find the σp resolution for that bin. The Gaussian mean for
each fit is also checked to be centered around zero implying good neutron reconstruction
and correction. Figures 11.7-11.9 shows the results of the resolution extraction for the
g11a data. Figures 11.10-11.12 shows the results of the resolution extraction for the
Monte Carlo. In each case the upper left plot shows the number of events per bin used.
The upper right plot shows the mean from the Gaussian fits. The lower left plot shows the
values of σ from the Gaussian fits over the required a dynamical range. The lower right
plot shows the χ2 from each Gaussian fit for every bin used.
Changes in the momentum resolution also occur in the directional components and
can be studied in a similar manner. A ∆θ and ∆φ can also be obtained using the deviation
of the detected neutron direction from the missing neutron direction. Once ∆θ (∆φ ) is
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obtained for data and Monte Carlo, the distributions with respect to p, θ and φ are sliced
and binned in each dynamic variable. Each bin is projected on to the ∆θ (the ∆φ axis and
fit with a Gaussian to find the σθ (σφ ) resolution for that bin). Figures 11.13 (11.14) show
the results of the resolution extraction for the g11a data. Figures 11.15 (11.16) show the
results of the resolution extraction for the Monte Carlo.
Figure 11.5: The g11a data set is used to obtain ∆p over a range of p (upper left), ∆p overa range of φ (upper right), ∆p over a range of θ (lower left), ∆p over a range of nuetronpath length in (cm) (lower right).
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Figure 11.6: Monte Carlo is used to obtain ∆p over a range of p (upper left), ∆p over arange of φ (upper right), ∆p over a range of θ (lower left), ∆p over a range of neutron pathlength in (cm) (lower right).
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Figure 11.7: The g11a study of σ(p). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lower left plotshows the values of σ from the Gaussian fits over the required range of p. The lower rightplot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.8: The g11a study of σ(θ). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lower left plotshows the values of σ from the Gaussian fits over the required range of θ . The lower rightplot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.9: The g11a study of σ(φ). The upper left plot shows the number of events perbin used. The upper right plot show the mean from the Gaussian fits. The lower left plotshows the values of σ from the Gaussian fits over the required range of φ . The lower rightplot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.10: The Monte Carlo study of σ(p). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required range of p. Thelower right plot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.11: The Monte Carlo study of σ(θ). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required range of θ . Thelower right plot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.12: The Monte Carlo study of σ(φ). The upper left plot shows the number ofevents per bin used. The upper right plot show the mean from the Gaussian fits. The lowerleft plot shows the values of σ from the Gaussian fits over the required range of φ . Thelower right plot shows the χ2 from each Gaussian fit for every bin used.
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Figure 11.13: The g11a data study of σ(θ). The upper left plot shows ∆θ over a range ofmomentum. The upper left shows the resolutions from the Gaussian fits of ∆θ as a functionof momentum. The middle right shows the ∆θ over a range of φ . The middle left showsthe resolutions from the Gaussian fits of ∆θ as a function of φ . The bottom left plot shows∆θ over a range of θ . The bottom left shows the resolutions from the Gaussian fits of ∆θ
as a function of θ . A cut on the cosine of the angle between the missing neutron vector andthe detected neutron vector of 3 is used.
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Figure 11.14: The g11a data study of σ(φ). The upper left plot shows ∆φ over a range ofmomentum. The upper left shows the resolutions from the Gaussian fits of ∆φ as a functionof momentum. The middle right shows the ∆φ over a range of φ . The middle left showsthe resolutions from the Gaussian fits of ∆φ as a function of φ . The bottom left plot shows∆φ over a range of θ . The bottom left shows the resolutions from the Gaussian fits of ∆φ
as a function of θ . A cut on the cosine of the angle between the missing neutron vector andthe detected neutron vector of 3 is used.
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Figure 11.15: The Monte Carlo study of σ(θ). The upper left plot shows ∆θ over a rangeof momentum. The upper left shows the resolutions from the Gaussian fits of ∆θ as afunction of momentum. The middle right shows the ∆θ over a range of φ . The middle leftshows the resolutions from the Gaussian fits of ∆θ as a function of φ . The bottom left plotshows ∆θ over a range of θ . The bottom left shows the resolutions from the Gaussian fitsof ∆θ as a function of θ . A cut on the cosine of the angle between the missing neutronvector and the detected neutron vector of 3 is used.
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Figure 11.16: The Monte Carlo study of σ(φ). The upper left plot shows ∆φ over a rangeof momentum. The upper left shows the resolutions from the Gaussian fits of ∆φ as afunction of momentum. The middle right shows the ∆φ over a range of φ . The middle leftshows the resolutions from the Gaussian fits of ∆φ as a function of φ . The bottom left plotshows ∆φ over a range of θ . The bottom left shows the resolutions from the Gaussian fitsof ∆φ as a function of θ . A cut on the cosine of the angle between the missing neutronvector and the detected neutron vector of 3 is used.
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Each Gaussian mean from these studies is centered around zero, which indicates the
corrections used on the neutron momentum are working reasonably well. A non-zero
mean would indicate an unreliable resolution measurement. The resolutions σ seen for
data match the trend and magnitude of those seen in the Monte Carlo. This is a
confirmation that the corrections for the Monte Carlo are working well and that the trends
seen in the σ distribution are indeed based on geometric factors. The resolutions also
match the data, which is a confirmation that the neutron in the Monte Carlo is well
represented and no additional smearing to the neutron is required.
11.6 Summary
The g11a run period was not intended for neutral detection leading to a momentum
correction giving a more precise neutron momentum. A study of the accuracy of the EC
for neutron momentum reconstruction with kinematic dependence has been done. Some
kinematic ranges have better resolution than others due to geometry and the phase space
of the reaction. All possible combinations of momentum and position have been studied to
develop a complete understanding of the neutron variance and covariance in the
electromagnetic calorimeter. The test reaction γ p→ π+π−π+n is isolated in the g11a
data set by selecting a π− and two π+ and then kinematically fitting to a missing neutron
hypothesis with a 10% confidence level cut. The resulting residuals are studied by slicing
each distribution and fitting the bins with a Gaussian. Each Gaussian mean from these
studies is centered around zero, as expected. The resolutions σ seen for the data match
those seen in the Monte Carlo, so no additional smearing to the momentum is required.
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12 ANALYSIS PROCEDURE
In the following analysis, an attempt is made to remove as much identifiable
background as possible while preserving the counts from the channels
γ p→ K0Σ∗+→ K0Σ+(π0) and γ p→ K0Σ∗+→ K0Σ+(γ). As in Part 1, the radiative
signal is buried by the π0→ γγ decay. The radiative signal extraction again requires a
kinematic fitting procedure. This requires detection of all other particles to differentiate
between the missing π0 and missing γ . This can be done with a kinematic fitting
procedure using the neutron covariance in the EC along with the standard DC covariance
for the detected pions.
The immediate goal is then to achieve clean hadron identification before using the
kinematic fitting procedure for the competing π0 and radiative signal. The best way to
achieve a clean K0 and Σ∗+ identification is with the initial kinematic fit using two pion
tracks to a K0 hypothesis with total missing mass of the Σ∗+. Before proceeding with this
step in the analysis, a preliminary look at the mass spectrum is required to check that the
neutron provides useful kinematics constraints. This check is three fold. First, it allows
testing of the corrected neutron momentum. Second, the final distribution of the missing
mass squared of all particles detected can be compared with the results of the initial
kinematic fit (to the K0 mass). Third, the distribution can be compared with the Monte
Carlo distribution, in order to identify background leakage.
For the sake of notation, let π+1 indicate the π+ used in the K0 invariant mass
selection. Hence π+2 is the other π+. For an initial investigation of the hyperon
excited-state region, ∼ 25% of the g11a is used. Figure 12.1 shows the invariant mass of
the π+1 -π−, (upper left), missing mass off the π
+1 -π− (upper right), the n-π+
2 invariant
mass (lower left), and the missing mass squared of all the detected particles (lower right).
The distributions in Figure 12.1 are before any kinematic constraints and after the π+
organization is done. The K0 is cut on at ±0.005 GeV of the PDG K0 mass to eliminate
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the π+-π− background and the spectra are plotted again. Figure 12.2 shows the invariant
mass of the n-π+2 , (upper left), missing mass off the K0 (upper right), the missing mass off
the n-π+2 combination (lower left), and the missing mass squared of all the detected
particles (lower right), after the K0 cut. Clearly the Σ+ peak is visible and in the correct
place shown in the (upper left) plot. The clear visible peak in the missing mass squared
where the π0 should be is also an indication that the neutron measurement is effective. A
cut on the invariant mass of the n-π+2 combination cleans up the excited-state hyperons,
making the Σ∗+ quite visible. Figure 12.3 shows the missing mass off the K0 (upper left),
and the missing mass off the Σ+ (upper right) after a ±0.05 cut around the Σ∗+ peak. The
lower plots are the missing mass of all detected particles and a magnified Gaussian fit to
the π0 peak.
Finally, Figure 12.4 shows a comparison between the final missing mass squared
distribution from the set of cuts described with a ±0.05 cut around the Σ∗+ (upper left)
compared with a P(χ2) > 0.1% confidence level cut used to obtain the K0 and Σ∗+
candidates (upper right). The same ±0.02 cut around the Σ+ peak is used in both. Also
shown is the missing energy distribution in the Σ∗+ frame, and the P2xy distribution. The
Pxy is shown only as a check against double bremsstrahlung. Clearly there is no peak near
zero, so no further check is required. This final comparison between the two missing mass
squared distributions give a consistency check. In principle the kinematic fit has a well
define confidence level associated with the candidates that survive the cuts. Hence, the
kinematic fit will be used for the rest of the analysis.
The distributions in Figure 12.1-12.4 can be compared with simulations. After
producing the Monte Carlo for each channel, the structure of the distribution in each case
were compared with the data distributions to determine which channels are relevant.
Figure 12.5 shows the missing mass off the Σ+. In the case of the
γ p→ K0Σ∗+→ K0Σ+(π0) and γ p→ K0Σ∗+→ K0Σ+(γ) channels, there should be no
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definite peak or structure present. However, a reaction like γ p→ K∗0Σ+→ K0Σ+(π0)
would show a peak at the K∗ mass.
The distribution in Figure 12.1-12.4 and the result from the constraint on the π+2 -n
indicates that the neutron detection and the corrections to its momentum are working.
From Figure 12.4, it is not possible to resolve the radiative single from a standard fitting
technique. The procedure of kinematically fitting of all the detected particles will be used
to again make this separation. The necessary neutron resolutions have already been
outlined in Section 11.5. Now it is possible to use these resolutions to build a full
covariance matrix for this analysis.
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Figure 12.1: The invariant mass of the π+1 -π−, (upper left), missing mass off the π
+1 -π−
(upper right), the n-π+2 invariant mass (lower left), and the missing mass squared of all the
detected particles (lower right). All distributions are before any kinematic constraints.
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Figure 12.2: The invariant mass of the n-π+2 , (upper left), missing mass off the K0 (upper
right), the missing mass off the n-π+2 combination (lower left), and the missing mass
squared of all the detected particles (lower right), after the ±0.005 K0 peak.
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Figure 12.3: The missing mass off the K0 (upper left), and the missing mass off theΣ+ (upper right), the missing mass squared of all detected particles (lower left), and amagnification of the missing mass squared with Gaussian fit to the π0 region (lower right) .These distribution are made after take a ±0.02 cut around the Σ+ peak as well as the priorcut of ±0.005 on the K0 peak.
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Figure 12.4: The comparison between the final missing mass squared distribution fromthe set of cuts discribed after a ±0.05 cut around the Σ∗+ mass (upper left) and from theP(χ2) > 0.1% confidence level cut used obtain the previous K0 and Σ∗+ candidates. Thesame ±0.02 cut around the Σ+ peak is used in both.
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Figure 12.5: The missing mass off the Σ+ after all cuts described (no kinematic fit). Thedistribution is similar for the the kinematic fit (not shown).
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13 NEUTRON KINEMATIC FITTING
Neutral particles are identified in CLAS as clusters in the Electromagnetic
Calorimeter (EC) that are not associated with any charged track reconstructed from the
drift chambers. It is then possible to differentiate between photons and neutrons using the
particle velocity β . Photons will naturally be very close to β = 1. If the resolution of the
neutral particle is well understood it can be used to develop a covariance matrix for that
particle for use in a kinematic fitting procedure. The covariance matrix for photons and
neutrons in the EC is quite different; however, the standard (target frame) CLAS variables
can be used for both.
13.1 Neutron EC Covariance Matrix
The neutron covariance matrix is developed using the neutron resolutions in p, θ , and
φ as outlined in Section 11.5. Because the full covariance matrix has already been done
for the charged particles it is only the neutron EC submatrix that is required to purse this
technique. The neutron subcovariance matrix can be expressed in the same coordinates;
however, the neutron momentum is weakly correlated with the other measured variables
leading to a matrix of the form,
Vni =
V pp
i V pθ
i V pφ
i
V pθ
i V θθi V φθ
i
V pφ
i V φθ
i V φφ
i
. (13.1)
The momentum resolution for the the neutron depends on how the neutron momentum
was achieved. The term V pp is strictly depends on the velocity β of the neutron and the
calibration of the EC with regard to the EC time of flight (EC time). For many
photoproduction experiments with CLAS, the electromagnetic calorimeter is calibrated
using fast pions. This leads to poor time resolution for neutrals. For neutrons this can
make a big difference in momentum resolution. There is usually a correction to neutron
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path and EC time of flight to get accurate momentum reconstruction. The inner and outer
layers can have different neutron momentum resolutions and must also be taken into
account.
The position and angular resolution for neutrons have dynamic dependencies on
momentum that can be studied using a prominent channel for a particular data set, as well
as Monte Carlo. Studying the various measured variable resolutions for all EC sector at
various θ , φ , and p dependence leads to functional relations that can be used as error
estimates to fill in the angular elements of the subcovariance matrix.
Once the covariance matrix is constructed and tested for all particle types, the
kinematic fitting procedure can be used.
13.1.1 Diagonal Terms
There is often dynamic dependence on the various measured parameters as well a
particle dependence for each resolution term. The residual quantities δvi in Vii should be
accurately depicted for all kinematic ranges and experimental configuration changes.
When investigating resolutions it is necessary to break up the study into all variables and
as many kinematic ranges as possible and analyze the effects on
δv = vtrue− vmeas,
which is the difference in the measured quantity vmeas and the true vtrue, known quantity
for that variable. With simulations it is relatively easy to extract the vtrue from what is
generated. Experimentally it is necessary to use the other known variables to extract the
vtrue. One example is to use a missing 4-vector with all other decay products detected. In
the case of neutrons it is possible to use a kinematic fit to the missing neutron using the
other charged decay products in the drift chambers that already have well established
resolution parameters. This gives a true value for the position and momentum of the
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neutron that can be compared to the reconstructed values from the Electromagnetic
Calorimeter.
The residuals over various ranges and variables for each δvi can then be sliced up and
binned for fitting with a Gaussian. These fit results are then used to develop trends that
map out and express each component as functionals with a designated four-vector going in
and the error estimate coming out.
13.1.2 Off-Diagonal Terms
In order to construct the full covariance matrix for any particle it is necessary get all
of the correlations between the measured variables represented in the off-diagonal terms.
One method to obtain these correlations is to find the Pearson product-moment correlation
which is a measure of dependence between any two observed quantities. The population
correlation coefficient between two observed data values vi and v j and with standard
deviations σi and σ j is defined as,
ρi j =Vi j
σiσ j=
Vi j√ViiVj j
=〈(vi− vi)(v j− v j)〉
σiσ j,
for finite standard deviations and non-zero vi and v j such that the off-diagonal terms to the
covariance matrix become,
Vi j = ρi jσiσ j,
for ρi j where (i , j). It is a corollary of the Cauchy-Schwarz inequality that the |ρ| cannot
exceed 1. The correlation coefficient is also symmetric such that ρi j = ρ ji.
For a correlation of +1 there is a ideal positive or increasing linear relationship
showing a slope of 1 between the two variables, and for a correlation of -1 there is a ideal
decreasing or negative linear relationship. I linear relation of some slope between the two
indicates the degree of linear dependence between the measured parameters. Naturally a
value close to zero indicates a small correlation. Zero correlation implies ideal
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independence between the measured parameters; however, the Pearson correlation
coefficient can only detect linear dependencies between two variables.
For a set of n measurements of variables x and y, the Pearson correlation can be
obtained from the sample correlation coefficient. The correlation coefficient of the
measured variables are,
ρxy =1
(n−1)σxσy
n
∑l=1
(xl− x)(yl− y),
where x and y are the sample means for each measured variable, and σx and σy are the
corresponding standard deviations. This value is then used to acquire Vxy for the
covariance matrix.
As an example, assume that for the neutron in the EC the φ resolutions were found to
be dependent on θ and momentum in the target frame. The θ resolution parameters are a
function of θ as well so that
σ2i j = ρi jσiσ j = Vi j,
or
σ2θφ = ρθφ σφ (θ , p)σθ (θ) = Vθφ .
The resulting value would then fill two off-diagonal terms Vθφ and Vφθ in the covariance
matrix.
13.2 Neutron Fitting
For the neutron, the development of the neutron covariance matrix is the main
challenge for kinematic fitting. There are other issues of concern, such as detection
efficiency and misidentifying photons as neutrons, but once the neutron resolution
functionals are found for a data set, kinematic fitting can make a big difference in analysis
without introducing extra systematic uncertainty. As an example using simulations,
consider the reaction γ p→ K0Σ+→ π+π−π+n. After identifying the K0, it is possible to
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fit the decay products in a (4-C) fit with nothing missing, Figure 13.1 shows the
confidence level distribution and pull distributions are shown in Figure 13.4.
To demonstrate some adjustments in measured values, a (4-C) kinematic fitting with
a neutron is used and an invariant mass is obtained after the correction to the 4-vectors are
made. Clearly, a nice improvement is seen in Figure 13.2. It is also possible to make a
(5-C) fit to utilize an additional constraint on the π+n to have the invariant mass of the Σ+
making the distribution essentially a delta function. This can improve the missing mass off
the Σ+.
Figure 13.1: Confidence level distribution for a (4-C) kinematic fit π+π−π+n.
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Figure 13.2: Left:Showing the (4-C) kinematic fit to get the invariant mass of π+n, Right:improvement in missing mass off the Σ+ (both in GeV).
Figure 13.3: Left:Showing the (5-C) kinematic fit to the invariant mass of π+n, Right:improvement to the missing mass off the Σ+ (both in GeV).
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Figure 13.4: Pull distributions for a (4-C) kinematic kit π+π−π+n.
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14 SIMULATIONS
As in Part 1, the relative contribution of different reaction channels have been studied
using Monte Carlo simulations. The experimental photon energy distribution for the
subset of g11a at 4.0186 GeV was again used to determine the energies of the incident
photons in the simulation. The weighting of each channel and its contribution to the
variation in the final result is discussed in Section 15.3.
The event generator FSGEN was used with a variable t-dependence such that a
channel with a kaon is generated uniformly in the center-of-mass frame in φ with a
t-dependent distribution in θcm according to P(t) ∝ e2.0t . The real photon energy range
setting used was 1.0-4.2 GeV. Gaussian distributions in x and y with σ = 0.5 cm were
used to approximate the beam width in the target. The target length of the g11a target at
40 cm from (30 cm,-10 cm) was again used to generate events uniformly along the length
of the target along the beam axis. The generated events were fed into GSIM to simulate
the CLAS detector in the Monte Carlo.
This analysis relies on an understanding of the contributing leakage of background
channels into the π0 and radiative signal counts used in the ratio calculation. For example
π0 leakage from a background channel such as γ p→ ωN∗→ π+π−π0nπ+ will make the
ratio of Σ∗+→ Σ+γ to Σ∗+→ Σ+π0 smaller then it should be. The Monte Carlo of various
possible contribution has been used to study the possible background leakage at various
stages of the analysis. The acceptance of each possible background have been used to
estimate the counts that survive all cuts and contribute in the final signal counts.
One likely background reaction is γ p→ ωN(1440) followed by N(1440)→ nπ+
decay. The ω decays primarily to π+π−π0 followed by π0→ 2γ . The full reaction
γ p→ ωN(1440) then has the same final state as the K0Σ∗+→ Σ+π0 and should be
carefully considered. The N(1440) has a relatively large decay width at 250-450 MeV
[60] which implies that almost any cut around the π+n at the invariant mass of the Σ+ can
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still have contributions from the N(1440) in this spectrum. Figure 14.1 shows the
spectrum of the mass off the invariant mass of the π+n. A cut of ±0.05 from the N∗ mass
is used in each case. A cut near a mass of 1440 MeV is shown in the left plot, giving a
clear ω(782) peak. Next, a cut near the mass of the Σ+ is shown in the right plot,
demonstrating that the ω(782) peak is still present.
Figure 14.1: The missing mass off the N∗ using a ±0.05 invariant mass cut on the π+ncombination at a mass of 1440 MeV (left) demonstrating a clear ω(782) peak; (right) at amass of the Σ+ demonstrating that the ω(782) peak is still present.
Another source of possible background it the reaction γ p→ m∗N∗ where m∗ is any
meson that can decay to ρπ0, and the N∗→ Nπ+ provides the detected pion. Similarly,
γ p→ ρN∗ where the N∗ decay to nπ+π0 is also a possible contaminant. In addition to the
kinematic constraints previously mentioned, these backgrounds cannot contribute for low
W . For testing purposes of these types of reaction, the channel ρN(1520) is considered.
The ρ(770) has a width of Γ = 150.3 MeV and decays almost 100% to ππ so it is
possible to leak under the K0 invariant mass cut.
The K∗ is also checked. The constraints on the K0 combined with the missing mass
constraint off the K0 to be the mass of the Σ∗+ should minimize any K∗ contribution.
However, because the reaction γ p→ K∗0Σ+ has the same possible final states that are
being analyzed, it should be investigated.
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Based on the possible final state decay products, the reactions γ p→ ηnπ+,
γ p→ K0Σ0π+, and γ p→ K0Σ+ are also considered. While these backgrounds cannot be
dismissed, careful comparison of the mass spectra from the data and the Monte Carlo
mass spectrum provide an indication of what channels are relevant. The Monte Carlo
channels generated and studied are listed in Table 14.1.
Table 14.1: The set of Monte Carlo channels and amount of events generated for theacceptance studies.
Reaction Generated Generated Events
K0Σ∗+→ K0Σ+γ 4×106
K0Σ∗+→ K0Σ+π0 4×106
K∗0Σ+→ K0Σ+γ 4×106
K∗0Σ+→ K0Σ+π0 4×106
ρN(1520)→ π+π−π0π+n 4×106
ωN(1440)→ π+π−π0nπ+ 4×106
ωN(1440)→ π+π−γnπ+ 4×106
ηnπ+→ π+π−π0π0nπ+ 4×106
γ p→ K0Σ0π+→ π+π−nπ0γπ+ 4×106
γ p→ K0Σ+→ π+π−nπ+ 4×106
To tune the Σ∗+ Monte Carlo, an approximate differential cross section for the
reaction γ p→ K0Σ∗+→ K0Σ+π0 was generated. A 1/Eγ photon energy distribution was
used in the generator. The acceptance for the Σ∗→ Σ+π0 channel was determined by the
ratio of accepted events to thrown events in each Eγ bin. The data and the Σ∗+→ Σ+π0
Monte Carlo were cut on the Y ∗ mass range of 1.34-1.43 GeV, the Σ+ invariant mass was
cut at ±0.005 GeV around the PDG value of the Σ+ mass, and a cut around the missing
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mass squared of 0.017-0.022 GeV2, in order to isolate the Σ∗+→ Σ+π0 channel in the
data. The yield was determined by the ratio of the raw Σ+π0 events to the number of
incident photons in each Eγ bin, so as to normalize with the Bremsstrahlung spectrum.
Corrections were made for each bin with the newly obtained acceptances. The density of
the target was assumed to be constant, and no background subtraction was performed. The
cross sections are used to modify the data similar to the discussion in Method 1 of Part 1.
From the adjusted Monte Carlo, acceptance corrections are found for bins in the kaon
cosine center-of-mass angle and the approximated differential cross sections are used to
adjust the Σ∗+ generator. Each corresponding bin was filled according to the distributions
of the data. Each angle bin is broken into Eγ bins and represented accordingly in the new
event weighting scheme of the generator. After these modifications were made, the
resulting Monte Carlo was compared with the data, using the momentum distributions for
the kaon, pion, and proton tracks (as well as the kaon lab frame angle distribution), and
found to be reasonable.
The same differential cross sections for the reaction γ p→ K0Σ∗+→ K+Σ+π0 was
also used to generate the radiative channels. A 1 GeV2 t-dependence was used to produce
the other Monte Carlo background channels.
The simulations for γ p→ K0Σ∗+→ K0Σ+γ and γ p→ K+Σ∗+→ K0Σ+π0 were
produced and studied. Figure 14.2 shows the γ p→ K0Σ∗+→ K0Σ+γ channel distributions
for the π+1 -π− invariant mass, the missing mass off the π
+1 -π− combination, the missing
energy from all detected particle, and the missing mass squared of all detected particles.
Figure 14.3 shows the same for the γ p→ K+Σ∗+→ K0Σ+π0 channel. For comparison the
same distributions for the ωN(1440)→ π+π−π0nπ+, ρN(1520)→ π+π−π0π+n,
ηnπ+→ π+π−π0π0nπ+, and K∗0Σ+→ K0Σ+π0 channels are shown in Figure 14.4,
14.5, 14.6, and 14.7 respectively. The background channels are shown without any cuts
applied but with the selection of π+1 and π
−2 made as described previously for the K0.
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As expected many of the background channel lead to a similar total missing mass
squared spectrum as for the γ p→ K0Σ∗+→ K0Σ+π0 channel. The most important thing
to notice from the distributions in Figure 14.2-14.7 is that the missing mass off of the π+2 n
combination (middle right plot) in Figure 14.3 compares well with the distribution of the
data in Figure 12.5. Had it looked more like the ω distribution in Figure 14.4 (middle
right), then this would be indicative that the events selected do not come from the channel
of interest. This would be cause for concern even if the acceptance for the ω (or other
background) is very small. The conclusion that can be drawn by this study of the Monte
Carlo is that the required channels have indeed been isolated as required in the data. Any
background from the γ p→ K0Σ∗+→ K0Σ+γ and γ p→ K0Σ∗+→ K0Σ+π0 channels
appear to be only minor contributions.
To calculate the acceptance of the signal and background reactions, the extraction
method used to resolve the radiative and π0 channels will be discussed next.
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Figure 14.2: The γ p→K0Σ∗+→K0Σ+γ Monte Carlo distibutions for the π+1 -π− invariant
mass, the missing mass off the π+1 -π− combination, the missing energy from all detected
particle, and the missing mass squared of all detected particles.
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Figure 14.3: The γ p → K0Σ∗+ → K0Σ+π0 Monte Carlo distibutions for the π+1 -π−
invarinat mass, the missing mass off the π+1 -π− combination, the missng energy from all
detected particle, and the missing mass squared of all detected particles.
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Figure 14.4: The ωN(1440)→ π+π−π0nπ+ Monte Carlo distributions for the π+1 -π−
invarinat mass, the missing mass off the π+1 -π− combination, the missing energy from all
detected particle, and the missing mass squared of all detected particles, no cuts yet applied.
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Figure 14.5: The ρN(1520) → π+π−π0π+n Monte Carlo distibutions for the π+1 -π−
invariant mass, the missing mass off the π+1 -π− combination, the missing energy from
all detected particle, and the missing mass squared of all detected particles, no cuts yetapplied.
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Figure 14.6: The ηnπ+ → π+π−π0π0nπ+ Monte Carlo distibutions for the π+1 -π−
invarinat mass, the missing mass off the π+1 -π− combination, the missng energy from all
detected particle, and the missing mass squared of all detected particles, no cuts yet applied.
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Figure 14.7: The K∗0Σ+ → K0Σ+π0 Monte Carlo distributions for the π+1 -π− invariant
mass, the missing mass off the π+1 -π− combination, the missing energy from all detected
particle, and the missing mass squared of all detected particles, no cuts yet applied.
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15 EXTRACTION METHOD
The missing energy boosted in the frame of the Σ∗+ is analyzed to help separate the
γ p→ K0Σ∗+→ K0Σ+γ signal and the γ p→ K+Σ∗+→ K0Σ+π0 background. The
missing energy of the Σ∗+ should be the difference in mass energy between the Σ∗+ and
Σ+ (∼ 0.193 GeV). The two step kinematic fitting procedure of Part 1 is not used in
attempt to maximize statistics. A missing energy restriction can help reduce the
overwhelming amount of π0 background before applying a kinematic fit to the radiative
signal. The intial energy restriction for the radiative signal used is a cut a 0.01 GeV below
the expected missing energy of 0.193 GeV.
Similar to Part 1, a kinematic fit is used to fit each track of all detected particles to a
particular missing particle hypothesis. This is again done by using the undetected particle
mass in the constraint equation. The (1-C) kinematic fit is use to kinematically fit at the
final stage of analysis. In the attempt to separate the contributions of the Σ∗+ radiative
decay and the decay to Σ+π0, the events were fit using different hypotheses for the
topology:
γ p→ π+π−π+n(π0) (1-C)
γ p→ π+π−π+n(γ) (1-C).
The constraint equations are
F =
Ebeam +Mp−Eπ+−Eπ−−Eπ+−En−Ex
~pbeam−~pπ+−~pπ−−~pπ+−~px
=~0. (15.1)
~Px and Ex represent the missing momentum and energy of the undetected π0 or γ .
15.1 Acceptance
The π0 leakage into the γ channel is still the dominant correction to the radiative
branching ratio. To properly calculate the ratio, the leakage into the π0 region from the γ
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channel is also used. The same style of notation is use from Part 1 of this thesis. Taking
just these two channels into consideration, the number of true counts can be represented
as N(Σγ) for the Σ∗+→ Σγ channel and N(Σπ) for the Σ∗+→ Σ+π0 channel. The
acceptance under the Σ∗→ Σ+γ hypothesis can be written as Aγ(X), with the subscript
showing the hypothesis type and the actual channel of Monte Carlo input to obtain the
acceptance value is indicated in the parentheses. For the calculated acceptance for the
Σ∗+→ Σγ channel under the Σ∗+→ Σ+γ hypothesis the acceptance is Aγ(Σγ), and for the
Σ∗+→ Σ+π0 hypothesis it is Aπ(Σ+γ). It is now possible to express the measured values
for each channel nγ and nπ as
nγ = Aγ(Σγ)N(Σγ)+Aγ(Σπ)N(Σπ) (15.2)
nπ = Aπ(Σπ)N(Σπ)+Aπ(Σγ)N(Σγ). (15.3)
The desired branching ratio of the radiative channel to the π0 channel using the true
counts is then R = N(Σγ)/N(Σπ). Solving for R to get the branching ratio expressed in
terms of measured values and acceptances,
R =nγAπ(Σπ)−nπAγ(Σπ)nπAγ(Σγ)−nγAπ(Σγ)
. (15.4)
Equation 15.4 is under the assumption that there are no further background contributions.
The formula for the branching ratio to take into account background from the ωN(1440)
can be expressed as
R =∆nγAΣ
π(Σπ)−∆nπAΣγ (Σπ)
∆nπAΣγ (Σγ)−∆nγAΣ
π(Σγ),
(15.5)
∆nπ = nπ −Nπ(ω → π+
π−
π0)−Nπ(ω → π
+π−
γ) (15.6)
∆nγ = nγ −Nγ(ω → π+
π−
γ)−Nγ(ω → π+
π−
π0). (15.7)
243
The nγ (nπ ) terms come directly from the yield of the kinematic fits and represent the
measured number of photon (pion) candidates. In the notation used, lower case n
represents the measured counts, while upper case N represents the acceptance corrected or
derived quantities. The Nγ,π terms are corrections needed for the leakage from the ωN∗
channel (this background is used as an example). The notation utilized is such that the
pion (photon) channel identifications are denoted AΣπ(Σ+π−) (AΣ
γ (Σ+π−)) so that AΣγ (Σπ)
denotes the relative leakage of the Σπ channel into the Σ+γ extraction and AΣπ(Σγ) denotes
the relative leakage of the Σ+γ channel into the Σ+π extraction.
Each Monte Carlo channel listed in Table 14.1 is run through the analysis with the
same cuts as used for the data. These cuts for the extraction of the radiative and π0 signal
are listed is Table 15.1. The cuts are listed in the order implemented. The first cut, (1), is
the confidence level cut taken on the π+1 and π− kinematic fit to a K0 invariant mass with
total missing mass off the K0 of the Σ∗+. The second cut, (2), lists the invariant mass cut
used on the π+-n combination around the value of the Σ+. The energy restriction used
only for the radiative signal is number (3). Then number (4) and (5) list the final
confidence level cuts used from the kinematic fit to the missing π0 and radiative
hypothesis. The second column lists whether the cut was applied to just one channel or
both.
The acceptances are found for each channel in Table 14.1 with the set of cuts listed in
Table 15.1. Table 15.2 lists all channels taken into consideration and the value of the
acceptance for the Pπ0(χ2) > 10% and Pγ(χ2) > 10%. The table lists two columns sorted
by hypothesis Aγ , Aπ . The uncertainty is statistical only.
244
Table 15.1: The cuts used to extract the final radiative and π0 counts. (See text for details.)
Cut Used (Applied)
(1)PK0(χ2) > 0.1% (both)
(2)|M(π+2 n)−MΣ+|< 0.02 (both)
(3)Ex(γ) < 0.183 (γ)
(4)Pπ0(χ2) > 10% (π0)
(5)Pγ(χ2) > 10% (γ)
Table 15.2: Acceptances (in units of 10−3) for the channels used in the calculation of thebranching ratios. All the cuts used to obtain the acceptance values are listed in Table 15.1.The uncertainties are statistical only. The two columns contain the acceptance for eachhypothesis Aγ , Aπ . In some cases the values are rounded up to 0.0001.
Reaction Aπ Aγ
K0Σ∗+→ K0Σ+γ 0.01130±0.0020 1.5510±0.0140
K0Σ∗+→ K0Σ+π0 1.3530±0.0120 0.0337±0.0021
K∗0Σ+→ K0Σ+γ 0.0055±0.0008 0.0009±0.0005
K∗0Σ+→ K0Σ+π0 0.0001±0.0000 0.0002±0.0000
ρN(1520)→ π+π−π0π+n 0.0001±0.0000 0.0001±0.0000
ωN(1440)→ π+π−π0nπ+ 0.0035±0.0009 0.0008±0.0003
ωN(1440)→ π+π−γnπ+ 0.0001±0.0000 0.0001±0.0000
ηnπ+→ π+π−π0π0nπ+ 0.0001±0.0000 0.0001±0.0000
γ p→ K0Σ0π+→ π+π−nπ0γπ+ 0.0001±0.0010 0.0001±0.0000
γ p→ K0Σ+→ π+π−nπ+ 0.0001±0.0000 0.0001±0.0000
245
The acceptance values indicate that very little background gets in to the final
radiative and π0 counts. Had there been a structure seen is Figure 12.5 indicating some
background present it would be necessary to obtain a measure of the surviving counts
from that channel and calculate the contribution to the ratio. This is true even for the small
values of the acceptances due to the sensitive nature of the signal extraction. Based on the
simulation distributions in Figures 14.2-14.7 in comparison to that of the data in Figure
12.1-12.5, no background will be taken into account in the initial calculation of the
branching ratio. However it is possible to study the effect of significant background as part
of the systematic studies (next section). Due to the fact that the ωN(1440) is very difficult
to differentiate from the π0 channel, there could be a large amount of systematic
uncertainty.
To proceed with the background calculation, consider the background of the
ωN(1440). In order to obtain an estimate for the ωN(1440) counts present in the final
singal extraction a relationship between the number of counts from the K0Σ∗+→ K0Σ+π0
to the number of ωN(1440)→ π+π−π0nπ+ present in the peak seen in the full missing
mass squared in the final stage of the analysis before taking the confidence level cuts for
each of the radiative and π0 hypothesis (the mass of the Σ+ could also be used). For this
example calculation, it is assumed that the radiative and all other contribution to the
missing mass squared peak are negligible. The relationship between the π0 counts from
the Σ∗0 and the ω is then
nω = f nΣ∗. (15.8)
Because this calculation of background is used only in a test, the value of f is specified for
a particular systematic study, see Section 15.3 for details.
246
The total number of events from ωN(1440)→ π+π−π0nπ+ present (N(ω)) at the
final stage of the analysis is calculated as,
N(ω) =nω
R(ω∗→ π+π−π0)Aω(π+π−π0). (15.9)
Here R(ω → π+π−π0) is the probability that the ω will decay to the π+π−π0 and
Aω(π+π−π0) is the probability that this decay channel will be observed after all the
applied cuts. An estimate of the number of counts in the π0 peak coming from some
decay mode of the ω using Eq. 15.9, can be calculated;
Nπ(ω) = R(ω → π+
π−
π0)Aω
π (X)N(ω) =Aω
π (X)nω
Aωγπ(π+π−π0)
. (15.10)
The notation here Aγπ again denote the counts that survive all other cuts but do not
satisfy either the γ or π0 hypothesis in the final confidence level cuts and still fit in the π0
missing mass squared peak.
To look at the contribution form the ωN(1440)→ π+π−γnπ+ channel an estimate
can be obtained from,
Nπ(ω) = R(ω → π+
π−
π0)Aω
π (π+π−
γ)N(ω)R(ω → π+
π−
γ) =
Aωπ (π+π−γ)R(ω → π+π−γ)nω
Aωγπ(π+π−π0)
(15.11)
where R(ω → π+π−γ) is the branching ratio for the radiative decay of the ω with a value
of 3.6×10−3. It is now possible to express all other associated ω corrections used in a
systematic study for a given value of f , along with the acceptance terms for that particular
channel. For example the corrections for the γ channel can be written as
Nγ(ω → ηγ) =Aω
γ (ηγ)R(ω → ηγ)nω
Aωγπ(π+π−π0)
, (15.12)
247
Nγ(ω → π+
π−
π0π
0) =Aω
γ (π+π−π0π0)R(ω → π+π−π0π0)nω
Aωγπ(π+π−π0)
, (15.13)
Nγ(ω → π+
π−
γ) =Aω
γ (π+π−γ)R(ω → π+π−γ)nω
Aωγπ(π+π−π0)
,
Nγ(ω → π+
π−
π0) =
Aωγ (π+π−π0)nω
Aωγπ(π+π−π0)
. (15.14)
For the π0 channel they take the form,
Nπ(ω → ηγ) =Aω
π (ηγ)R(ω → ηγ)nω
Aωγπ(π+π−π0)
, (15.15)
Nπ(ω → π+
π−
π0π
0) =Aω
π (π+π−π0π0)R(ω → π+π−π0π0)nω
Aωγπ(π+π−π0)
,
Nπ(ω → π+
π−
γ) =Aω
π (π+π−γ)R(ω → π+π−γ)nω
Aωγπ(π+π−π0)
,
Nπ(ω → π+
π−
π0) =
Aωπ (π+π−π0)nω
Aωγπ(π+π−π0)
. (15.16)
where R is used for the corresponding branching ratio in each case. The value R(ω → ηγ)
is listed as 4.9×10−4 and the value of R(ω → π+π−π0π0) is listed as < 2% [60].
The same sort of calculation can be used to check the variation in the ratio for the K∗
contribution and the ρN(1520). The possible contributions these channels is discussed in
Section 15.3.
248
15.2 Results
To calculate the branching ratio Eq. 15.4 is employed. All terms that take into
account any channel other than the π0 and radiative signal are for the time being ignored.
The acceptance values are taken from Table 15.2. The raw values obtained out of the final
kinematic fit are nγ = 63 and nπ = 1798.
The ratio of the K0Σ∗+→ K0Σ+γ channel to the K0Σ∗+→ K0Σ+π0 is then,
R =nγAπ(Σπ)−nπAγ(Σπ)nπAγ(Σγ)−nγAπ(Σγ)
. (15.17)
Plugging in values for the numerator,
nγAπ(Σπ)−nπAγ(Σπ) = (63)(1.353)− (1798)(0.0337). (15.18)
The denominator is then,
nπAγ(Σγ)−nγAπ(Σγ) = (1798)(1.551)− (63)(0.0113). (15.19)
This leads to the final value of R to be,
RΣ∗+→Σ+γ
Σ∗+→Σ+π0 =nγAπ(Σπ)−nπAγ(Σπ)nπAγ(Σγ)−nγAπ(Σγ)
= 0.88402±0.39%. (15.20)
The uncertainty is statistical only propagated from δnγ = √nγ and δnπ =√
nπ .
The results for the extracted values after the acceptance corrections are listed in Table
15.3. The raw counts for the radiative and π0 extraction using Pγ = Pπ0 = 10% and the
cuts from Table 15.1 are also present in Table 15.3. After adjusting for acceptance the
true signal counts become ∆nγ = 24.65 and ∆nπ = 2787.99.
To calculate the width from the branching ratio achieved here one uses the full width
of the Σ∗+ which is ΓFull = 35.8±0.8 MeV with the the branching ratio that the radiative
signal is being normalized to, which is the R(Σ∗+→ Σ+π0) = 5.85±0.75%. The partial
width calculation is then
249
Table 15.3: Breakdown of statistics for each term in Eq. 15.4 for the Σ(γ) and Σ(π0)hypothesis. Each counts for each hypothesis is subtracted accordingly as shown in Eq. 15.4.Th raw count are taken directly from the kinematic fit to use in the final ratio calculation.The uncertainties listed are statistical only.
Hypothesis Σ(γ) Σ(π0)
Raw counts 63±7.94 1798±42.40
Σ(1385)→ Σ+γ 85.24±10.77 60.59± 3.87
Σ(1385)→ Σ+π 2788.70±70.42 0.712± 0.155
Adjusted counts 24.65 2787.99
ΓΣ∗+→Σ+γ = RΣ∗+→Σ+γ
Σ∗+→Σ+π0R(Σ∗+→ Σ+
π0)ΓFull = 18.5±8.6 keV. (15.21)
Again only statistical uncertainty is presented. The issue to determine is exactly how
reliable are the presented numbers. As with the previous methods used in Part 1, some
systematic variation in the ratio based on the choice of confidence level is expected.
Unlike the previous method, there is a cut introduced on the extraction of the radiative
signal to restrict the missing energy. The choice to used an energy restriction comes from
the attempt to maximize statistics rather then using the two step kinematic fit. However a
simple energy restriction can not be expected to filter out that much of the π0 background
due to the fact that the radiative signal is so small compared to the π0. This variation and
all other systematic studied are considered in next section.
250
15.3 Systematic Studies
The value of each cut is varied to study the effect on the final acceptance corrected
ratio. For each variation the new acceptance terms in Equation 15.4 are recalculated with
the corresponding Monte Carlo. Each major systematic uncertainty contribution is
collected together in a table at the end of this section. Each term is numbered as it is
discussed to tie the discussion to the corresponding term listed in Table 15.5.
Several ∆β cut variations are checked starting with ∆β < 0.022 for all charged
particles leading to a ratio of 1.11±0.44%. There is also a check at ∆β < 0.1 giving a
ratio of 0.84±0.39%. The ∆β selected uses a ±1 ns timing cut, while keeping ∆β < 0.025
for both the π+ with only a loose cut of ∆β < 0.1 for the π−. This variation is presented
at the top of Table 15.5 as number (1).
The first kinematic fit used in this analysis is applied only to the π+ and π−
associated with the K0. To investigate the systematic variation in the ratio due to the
confidence level cut applied the ratio signals are re-extracted and new acceptance terms
are calculated over a set of confidence level cuts.
251
Figure 15.1: The variation in the ratio due to the confidence level cut from the kinematicfit of π
+1 -π− combination to the invariant mass of K0 with missing mass of Σ∗0.
252
It is again difficult to obtain an estimate of the systematic uncertainty in the Monte
Carlo’s capacity to match the data. The method used in Part 1 is again employed by
checking the effects of various tuning parameters on the final ratio. The systematic effects
of the Monte Carlo are studied by making adjustments to the generator, producing a new
simulation and then re-evaluating the acceptance terms. It is possible to check the
variation in the ratio (based on the new acceptances) by modifying only the generator to
match the production cross section seen in the photon energy distribution from the data
and then checking various exponential t-dependences. Prior to using the approximated
differential cross section, the best data matching t-slope value in the Monte Carlo is found
to be ∼ 1.8 GeV2. To study the generator’s effect on the ratio, a t-slope in the range of
∼±20% the ideal matching value of 1.8 GeV2 is selected to test the effects on the
resulting acceptance corrected ratio. The change in t-slope is varied simultaneously for the
radiative and π0 channels. The acceptances for both channels are recalculated and the
resulting ratio for various t-slopes is shown in Figure 15.2. The branching ratio using a
t-slope of 1.8 GeV2 is0.891%, and using the differential cross sections to match the data,
the branching ratio is 0.884%. In the final ratio reported, the modification in the generator
used to match the differential cross sections seen in the data is used. The t-slope is only
used as a way to study the variation in the ratio. Only the flat region in Figure 15.2 is used
in the systematic uncertainty contribution. The values are between 0.76-0.99. The
variation is listed as number (2) in the table.
253
Figure 15.2: Variation in the acceptance corrected branching ratio for various t-slope settingin the generator.
254
To demonstrate the possible effect of background on the ratio at values of f from
Equation 15.8 can be used to test the variation on the ratio for a certain fraction of the final
missing mass squared distribution. For this study the value of f is taken to be 10% which
implies for the case of the ωN(1440) that the number of count present in the missing mass
squared peak of the π0 are 10% from ω → π+π−π0. The choice of f is based on the
estimation of the ω counts seen in Figure 14.1 that survive all the cuts except the
kinematic fitting cuts to the missing π0 or radiative signal. After calculating the resulting
effects for each decay mode for the ω , the contribution is shown in Table 15.4.
Table 15.4: The resulting contribution to the ratio for a f value of 10%. Some values arerounded up. Uncertainty is statistical only.
Contribution Counts
Nγ(ω → π+π−π0) 0.001±0.0001
Nγ(ω → π+π−γ) 0.004±0.0002
Nγ(ω → π+π−π0π0) 0.0001±0.0001
Nγ(ω → ηγ) 0.001±0.0001
Nπ(ω → π+π−π0) 20.31±4.89
Nπ(ω → π+π−γ) 1.42±0.51
Nπ(ω → π+π−π0π0) 0.11±0.027
Nπ(ω → ηγ) 0.002±0.0001
Clearly very few counts will contribute to the ratio however due to the sensitive
nature of the ratio still some change can be seen. The ρN(1520) and K∗0Σ+ are
determined to be negligible due to the lack of structure seen in Figure 12.2 and 12.3 in the
missing mass off the π+-n. The acceptances for all other possible background
255
contribution are too small to make an impact. The resulting ratio including the ω
contribution (with f =10%) is 0.923±0.39.
An additional method of checking the effects of background come from cutting
directly on the peaks that can be identified as background. For example using the missing
mass off all detected particles and the π+ and π− from the K0 it is possible to make a
three particle invariant mass. A cut can then be applied around the mass of ω and the
analysis procedure continued. By using a cut of ±0.01 around the PDG value of the ω in
the three particle invariant mass spectrum a new ratio is found to be R = 0.90±42.
Because the variation in method using f =10% is larger, its value is used in the systematic
uncertainty calculation. This values is listed in Table 15.5 as number (3).
To look at the systematic dependency on the choice of the confidence level cuts in
both the radiative and π0 hypothesis the ratio is recalculated from the resulting raw counts
in each case with the new acceptance terms for a set of equal cuts such that
Pπ(χ2) = Pγ(χ2). A range of confidence level cuts from 0.5%-30% is used. Figure 15.3
show the ratio at each tested confidence level.
The variation in confidence level is also independently tested for each channel. The
confidence level cut for the π0 hypothesis is kept at 10% while the confidence level cut for
the radiative hypothesis is varied. The resulting ratio is shown in Figure 15.4.
Finally the variation in the confidence level for the radiative hypothesis is kept at
10% while the confidence level cut for the π0 hypothesis is varied. The resulting ratio is
shown if Figure 15.5.
256
Figure 15.3: Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts are equal for each hypothesis suchthat Pγ(χ2) = Pπ(χ2).
257
Figure 15.4: Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts for the π0 hypothesis is kept at10% while the radiative hypothesis confidence level cut, Pγ(χ2), is varied.
258
Figure 15.5: Variation in the ratio for a range of confidence level cuts on the final extractionof radiative and π0 signal. Here the confidence level cuts for the radiative hypothesis is keptat 10% while the π0 hypothesis confidence level cut, Pπ0(χ2), is varied..
259
The change in the ratio over the variation in Pπ0(χ2) is a little bit more stable. This is
expected because the challenge of this analysis is to isolate the radiative signal not the π0.
The largest change is seen in Figure 15.3 over P(χ2) showing a variation in the ratio from
starting a 1.62% however once ratio come down to the P(χ2)∼ 0.90% range and the
numbers stabilize, for this reason only the points in the linear fit are used in the systematic
uncertainty contribution. The values the ratio including all confidence level plots range
from 0.99%-0.79% and are listed in Table 15.5 as number (4).
The missing energy cut on the radiative candidates is also studied. This cut is used to
reduce the amount of π0 background going into the final kinematic fit used to resolve the
radiative signal without greatly reducing statistics. A set of cuts is chosen to remove the
majority of the π0 events to prepare for the final kinematic fit. The range chosen runs from
0.16GeV-0.19GeV. This range is based on simulation studies that suggest that ∼ 0.01
lower than the ∼ 0.193 GeV centroid (0.18 GeV)gives the optimal signal to background
ratio. Again the acceptance for each cut is determined and the ratio is recalculated. Figure
15.6 shows the resulting ratio for each cut.
Clearly the cut is not as stable as the previous studied systematics, however to obtain
a range of uncertainty in the ratio from this cut only the point shown in Figure 15.6 within
the linear fit are used. Two points to the left and two point to the right of the 0.18 GeV cut
used in the final ratio are taken as the systematic range. The resulting variation is from
0.69%-1.22%. These numbers are the largest contributors to the systematic uncertainty
and listed in the table as (5).
There is also the variation from the cut on the π+2 -n combination. This can be studied
by taking various cut in |M(π+2 n)−MΣ+|< Mcut and again analyzing the change in the
ratio. Figure 15.7 show the variation over a range in the cut value from 0.005-0.06 GeV.
Again only the range in the linear fit are used in the systematic uncertainty calculation.
260
This rage is chosen qualitatively to be the most stable leading to a variation from
0.66%-1.15%. This variation is listed the Table 15.5 as number (6).
Figure 15.6: Variation in the ratio for a range of missing energy cuts on the final extractionof the radiative signal. Here the cuts are only applied to the radiative hypothesis.
261
Figure 15.7: The variation in ratio from the cut on the Σ+ invariant mass.
262
Table 15.5 shows a summary of the systematic studies and the higher and lower range
of the ratio based on the variations mentioned for each type of uncertainty.
Table 15.5: Ranges of systematic variation in resulting ratio in (%) showing L(Low)-contribution and H(High)-contribution and rang in each case.
Source Low Range Low Contribution High range High Contribution
(1)∆β 0.840±0.39 -0.044 1.110± 0.440 +0.226
(2)t-slope 0.760±0.39 -0.124 0.99±0.39 +0.106
(3)Background 0.884±0.39 -0.000 0.920± 0.39 +0.036
(4)P(χ2) 0.790±0.39 -0.094 0.990± 0.33 +0.106
(5)Ex 0.690±0.45 -0.194 1.22± 0.48 +0.336
(6)MΣ+ 0.884±0.39 -0.000 1.150± 0.66 +0.266
Total Uncertainty -0.253 +0.508
To calculate the final systematic uncertainty the difference in the ratio R = 0.884%
and the high range of the ratio for each case in Table 9.3 is added in quadrature to obtain a
value for the uncertainty of 0.508 greater than the ratio. The lower systematic uncertainty
is based on the difference between the ratio R = 0.884% and the low range of the ratio for
each case, resulting in a value of 0.253 less than the ratio.
263
15.4 Conclusion
The increase in the statistical uncertainty in ratio compared to Part 1 is due to the
detection of an additional particle, and the requirement of using the EC. The increase in
systematic uncertainty is due mostly to the method used in extracting the radiative signal
out from under the π0. Using the missing energy restriction can only help this filtration in
a very mild way. The branching ratio reported is 0.884±0.39(stat)+0.51−0.25(sys)%.
264
16 OVERALL RESULTS
From the U-spin SU(3) multiplet representation the prediction for the ratio of the
∆→ nγ partial width to the Σ∗0→ Λγ partial width from Section 2.2 is,
Γ(∆0→ nγ)Γ(Σ∗0→ Λγ)
=(
Mn
M∆
)(MΛ
MΣ∗0
)−1( qn
qΛ
)3 43
= 1.56,
leading to the U-spin prediction of the partial width of the electromagnetic decay using the
wdith of the ∆0→ nγ decay of,
1.56−1×Γ(∆0→ nγ) = 1.56−1×660±60 = 423±38 keV.
Using the results from method-2 of Part 1 the ratio of the Σ∗0→ Λ+γ partial width to
the Σ∗0→ Λπ0 partial width is,
RΛγ
Λπ=
Γ[Σ0(1385)→ Λγ]Γ[Σ0(1385)→ Λπ0]
= 1.42±0.12(stat)+0.11−0.07(sys)%
The width for the branching ratio achieved for Part 1 comes from the full width of the Σ∗0
which is Γ(Σ∗0)Full = 36.0±5 MeV with the the branching ratio that the radiative signal
is being normalized to, which is the R(Σ∗0→ Λπ0) = 87.0±1.5% [60]. The partial width
calculation is then
ΓΣ∗+→Σ+γ = RΣ∗0→Λγ
Σ∗0→Λπ0R(Σ∗0→ Λπ0)Γ(Σ∗0)Full = 444.74±72.87(stat)+71.3
−66.0(sys) keV.
(16.1)
This can be compared to the previously published work on this channel which yielded a
ratio of 1.53±0.39(stat)+0.15−0.17% [73]. Again calculating a width from this ratio gives,
Γ = 479±120(stat)+81−100(sys) keV.
The final result from Part 2 with systematic uncertainties is
RΣ∗+→Σ+γ
Σ∗+→Σ+π0 =nγAπ(Σπ)−nπAγ(Σπ)nπAγ(Σγ)−nγAπ(Σγ)
= 0.88402±0.39+0.51−0.25(sys)%, (16.2)
265
To calculate the partial width from the branching ratio achieved in Part 2 the full
width of the Σ∗+ which is ΓFull = 35.8±0.8 MeV with the the branching ratio that the
radiative signal is being normalized to, which is the R(Σ∗+→ Σ+π0) = 5.85±0.75%.
The partial width calculation including systematic uncertainties leads to,
ΓΣ∗+→Σ+γ = RΣ∗+→Σ+γ
Σ∗+→Σ+π0R(Σ∗→ Σ+
π0)ΓFull = 18.51±8.58(stat)+11.04
−5.81 (sys) keV. (16.3)
Table 16.1 shows the previous model predictions along with the U-spin prediction
and the final results from this analysis in each case. Statistical and systematic
uncertainties are combined in the present partial widths.
Table 16.1: Comparison of theorectical model predictions for the radiative decay widthswith the present results, all in keV.
Model Σ(1385)0→ Λγ Σ+(1385)→ Σ+γ
NRQM 273 104
RCQM 267
χCQM 265 105
MIT Bag 152 117
Soliton 243 91
Skyrme 157-209 47
Algebraic model 221.3 140.7
U-spin 423±38 223±20
Present Results 445±102 18.5±14
266
Interestingly enough, the partial width of the Σ∗0→ Λγ is still larger than any
prediction listed except for the U-spin prediction. In contrast the partial width of the
Σ∗+→ Σ+γ is smaller than all predictions.
The U-spin prediction for the Σ∗0→ Λγ partial width is well validated by the
experimental results here. In this case U-spin symmetry is confirmed within the
experimental uncertainties. However, the study in Part 2 suggests that within the
experimental uncertainties the Σ∗+ EM decay partial width does not agree with U-spin
symmetry and the other model predictions. It is important to note that the U-spin
prediction for the Σ∗+ EM decay partial width ignores the effects of the interference of the
isovector and isoscaler components of the photon. If the isoscaler component interfered
destructively the resulting prediction could indeed be much smaller.
The results in Ref. [41] reveal that the meson cloud effect can contribute significantly
(∼ 40%) to the overall electromagnetic decay width of the ∆→ Nγ . This puts the
prediction from the model at about 80% of the experimental measurement. As stated
previously it has not yet been determined from a theoretical standpoint if the meson cloud
effects contribute and if so to what degree for the radiative decay of the Σ∗0 and Σ∗+. This
may be the reason for such a difference in the predictions seen from experiment and why
the deviation from the experimental result does not indicate a similar trend for each of the
Σ∗0 and Σ∗+ results. Because the U-spin prediction for Σ∗0 EM decay width uses
empirical information from the ∆ EM decay, contributions from phenomena like the
meson cloud effect should be inherent. The agreement between experiment and the U-spin
prediction, and the disagreement with the other models listed in Table 16.1, suggests that
meson cloud effects are important for calculations of Σ∗0→ Λγ .
Perhaps this work can prompt more encompassing calculations that are necessary to
probe the structure of the baryon resonances and motivate consideration of the meson
cloud contributions for electromagnetic decay predictions.
267
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273
APPENDIX A: RATIO CALCULATION DETAILS
A.1 Full Ratio Calculation
In this section the branching ratio and uncertainty is calculated for Method-2 using
nγ = 635 and nπ = 13950. The values from Table 8.7 are used for each acceptance term in
the following calculations.
The ratio is calculated using the expression in Eq. 15.5 to take all backgrounds into
account such that,
R =∆nγ
(AΣ
π(Λπ)+ RΣπΛπ
2 AΣπ(Σπ)
)−∆nπ
(AΣ
γ (Λπ)+ RΣπΛπ
2 AΣγ (Σπ)
)∆nπAΣ
γ (Λγ)−∆nγAΣπ(Λγ)
,
where the adjusted counts considering the background are,
∆nπ = nπ −Nπ(Λ∗→ Σ+
π−)−Nπ(Λ∗→ Σ
0π
0)−Nπ(Λ∗→ Σ0γ)
−Nπ(Λ∗→ Λγ)−Nπ(K∗→ Kπ0),
∆nγ = nγ −Nγ(Λ∗→ Σ+
π−)−Nγ(Λ∗→ Σ
0π
0)−Nγ(Λ∗→ Σ0γ)
−Nγ(Λ∗→ Λγ)−Nγ(K∗→ Kγ).
The radiative background term derived from the Λ(1405) counts is, using Γγ(Λ∗) = 27±8
keV and Γtot(Λ∗) = 50 MeV,
Nγ(Λ∗→ Λγ) =AΛ
γ (Λγ)R(Λ∗→ Λγ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 3.0409.
(A.1)
274
To carry out an example error calculation for this term,
δN2γ (Λ∗→ Λγ) =
(RnΛδAγ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)
)2
+
(AγRδnΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)
)2
+
(AγnΛδR
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)
)2
+
(AγRnΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)δAΛ
γπ(Σ0π
0)
)2
+
(AγRnΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)δAΛ
γπ(Σ+π−)
)2
= 0.592.
Where R(Λ∗→ Λγ) = 5.4×10−4±1.62×10−4 is from Ref. [74]. This leads to the
statistical uncertainty in Nγ(Λ∗→ Λγ) of 0.769. Each set of counts and uncertainties is
calculated accordingly,
Nγ(Λ∗→ Σ0γ) =
AΛγ (Σ0γ)R(Λ∗→ Σ0γ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 0.1297±0.0442,
Nγ(Λ∗→ Σ0π
0) =AΛ
γ (Σ0π0)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 3.413±0.355,
Nγ(Λ∗→ Σ+
π−) =
AΛγ (Σ+π−)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 4.437±1.087.
Where R(Λ∗→ Σγ) = 2−4±8.05×10−5 is from Ref. [74]. For the π0 channel the
background terms derived from the Λ(1405) counts are,
Nπ(Λ∗→ Λγ) =AΛ
π (Λγ)R(Λ∗→ Λγ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 0.002±0.0005,
Nπ(Λ∗→ Σ0γ) =
AΛπ (Σ0γ)R(Λ∗→ Σ0γ)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 0.116±0.04,
Nπ(Λ∗→ Σ0π
0) =AΛ
π (Σ0π0)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 168.943±11.649,
Nπ(Λ∗→ Σ+
π−) =
AΛπ (Σ+π−)nΛ
AΛγπ(Σ0π0)+AΛ
γπ(Σ+π−)= 98.977±7.797.
275
The background terms derived from the K∗ counts subtracted from the radiative signal
come from an estimate of the total K∗ present (see Table 12).
N(K∗+) =n(K∗+→ K+π0)
R(K∗+→ K+π0)A(K∗+→ K+π0)
= 1287/0.66666(0.01)
= 1.93×105±0.06×105,
where the term A(K∗+→ K+π0) = 0.010±0.0003 is the acceptance for the K∗+→ K+π0
using the extrapolation method to achieve the K∗+ counts. The counts 1287 come from
Table 8.9. To find the counts to the radiative contribution, using
R(K∗→ Kγ) = 9.9×10−4,
Nγ(K∗+→ K+γ) = R(K∗+→ K+
γ)Aγ(K∗+→ K+π
0)N(K∗+)
= 0.03±0.0015.
The background counts from the K∗ to the π0 contribution is
Nπ(K∗+→ K+π
0) =Aπ(K∗+→ K+π0)n(K∗+→ K+π0)
A(K∗+→ K+π0)
= 27.41±1.71.
All of these terms are subtracted out of the initial nγ and nπ to obtain
∆nγ = 623.92±25.23 and ∆nπ = 13654.56±118.95. The term that takes the Σ∗→ Σ+π−
counts into consideration is calculated as
RΣπΛπ
2AΣ
π(Σπ) = (0.135±0.011
2)0.161×10−3±0.01×10−3
= 1.087×10−5±8.888×10−6
276
then added to the other dominant term
AΣπ(Λπ)+
RΣπΛπ
2AΣ
π(Σπ) = 1.421×10−3±0.0278×10−3 +1.087×10−5±8.88×10−6
= 1.432×10−3±2.91×10−5
and
AΣγ (Λπ)+
RΣπΛπ
2AΣ
γ (Σπ) = 0.0321×10−3±0.002×10−3 +1.715×10−7±7.68×10−8
= 3.227×10−5±2.0×10−6,
which can then be used in the numerator to calculate R,
R = 623.92(1.432×10−3)−13654.56(0.03227×10−3)13654.56(2.335×10−3)−623.92(0.0184×10−3) = 0.0142. (A.2)
A.2 Statistical Uncertainty Calculation
To calculate the statistical uncertainty, the ratio derivation is differentiated with
respect to every respective contributing variable to obtain,(δRR
)2
=((
Aπ(Λγ)nπAγ(Λγ)−nγAπ(Λγ)
+Aπ(Λπ)
nγAπ(Λπ)−nπAγ(Λπ)
)δ∆nγ
)2
+((
Aγ(Λγ)nπAγ(Λγ)−nγAπ(Λγ)
+Aγ(Λπ)
nγAπ(Λπ)−nπAγ(Λπ)
)δ∆nπ
)2
+(
nγ
nγAπ(Λπ)−nπAγ(Λπ)δAπ(Λπ)
)2
+(
nπ
nγAπ(Λπ)−nπAγ(Λπ)δAγ(Λπ)
)2
+(
nπ
nπAγ(Λγ)−nγAπ(Λγ)δAγ(Λγ)
)2
+(
nγ
nπAγ(Λγ)−nγAπ(Λγ)δAπ(Λγ)
)2
,
where the terms containing the acceptance statistical uncertainties turn out to be small.
The dominating contributions come from the δ∆nπ and δ∆nγ terms. To evaluate, use the
277
previously given terms in Eq. A.2 to fill in the acceptance and denominators,(∆RR
)2
=((
Aπ(Λγ)31.88
+Aπ(Λπ)0.443
)δ∆nγ
)2
+((
Aγ(Λγ)31.88
+Aγ(Λπ)0.443
)δ∆nπ
)2
=((
0.0184×10−3
31.88+
1.432×10−3
0.443
)25.23
)2
+((
2.33×10−3
31.88+
0.03227×10−3
0.443
)118.95
)2
,
resulting in the final fractional uncertainty of,
δRR
= 0.08228
which means δR = 0.08228×0.0142 = 0.00117.
278
APPENDIX B: BOS CLAS DATA STRUCTURE
The data structure management of CLAS is a dynamical storage bank knows as BOS
banks. These data structures contain information on raw events, detector status, geometry
and calibration constants, reconstructed event information, tracking information, cluster
reconstruction information and in essence all required CLAS detector information needed
for physics analysis. The recorded units of information are contained in each bank which
is identified by a name consisting of up to four characters to identify the data structure
associate with it.
The communication between detector modules and the acquisition of the CLAS data
stream uses only the BOS bank data structures. Each bank structure type is constructed
using a readable DDL definition. The DDL definition for several BOS banks mentioned in
this analysis are listed here.
279
PART:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE PART ! create write display delete ! Hit Based Tracking Result bank
!
!CLAS COORDINATE SYSTEM USDED.
!
!ATTributes:
!———–
!COL ATT-name FMT Min Max ! Comments
!
1 pid I 0 10 ! particle id (GEANT)
2 x F -100. 100. ! vector3 t vert; Vertex position x,y,z
3 y F -100. 100. ! y
4 z F -500. 500. ! z
5 E F 0. 16. ! vector4 t p; Energy
6 px F -16. 16. ! momentum x,y,z
7 py F -16. 16. ! py
8 pz F -16. 16. ! pz
9 q F -16. 16. ! charge
0 trkid I -16. 16. ! index to TBID bank, counting from 1
11 qpid F -100.0 100.0 ! quality factor for the pid
12 qtrk F -100. 100.0 ! quality factor for the trk
13 flags I 0 10000 ! set of flags defining track (ie, BEAM)
!
280
TAGR:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE TAGR ! create write display delete ! Tagger result bank
!
! ATTributes:
! ———–
!COL ATT-name FMT Min Max ! Comments
!
1 ERG F 0. 10. ! Energy of the photon in GeV
2 TTAG F -20. 200. ! Time of the photon has reconstructed in the Tagger
3 TPHO F -20. 200. ! Time of the photon after RF correction
4 STAT I 0 4096 ! Status ( 7 or 15 are Good)
5 T id I 1 121 ! T counter Id
6 E id I 1 767 ! E counter Id
!
281
MVRT:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE MVRT ! create write display delete ! vertex Result bank
!
! THE DETECTOR COORDINATE SYSTEM IS USED.
!
! ATTributes:
! ———–
!COL ATT-name FMT Min Max ! Comments
!
1 v id I -1000 1000 ! info about track ids
2 ntrk F -100. 100. ! number of tracks used to make vertex
3 x F -1000. 1000. ! x vector3 t vertx,y,z
4 y F -1000. 1000. ! y
5 z F -1000. 1000. ! z
6 chi2 F -1000. 1000. ! chi2
7 cxx F -1000. 1000. ! Covariance matrix array element
8 cxy F -1000. 1000. ! Covariance matrix array element
9 cxz F -1000. 1000. ! Covariance matrix array element
10 cyy F -1000. 1000. ! Covariance matrix array element
11 cyz F -1000. 1000. ! Covariance matrix array element
12 czz F -1000. 1000. ! Covariance matrix array element
13 stat I -1000. 1000. ! status integer, not used yet
!
282
! note v id is based upon the track id used to make the
! vertex. v id = (summed over all tracks used) 2(tber id of track(1-10))
! + 1 if beamline info used
283
ECHB:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE ECHB ! create write display delete ! Forward calorimeter result bank !
! ATTributes:
! ———– !COL ATT-name FMT Min Max ! Comments
!
1 Sect I 0 0xFFFF ! Sector number & Layer number
2 E hit F 0.0 6.0 ! energy found
3 dE hit F 0.0 6.0 ! error on the energy found
4 t hit F 0.0 9999.0 ! time found
5 dt hit F 0.0 9999.0 ! error time found
6 i hit F 0.0 9999.0 ! sector rectangular coordinate
7 j hit F 0.0 9999.0 ! sector rectangular coordinate
8 di hit F 0.0 9999.0 ! sector rectangular coordinate error,
9 dj hit F 0.0 9999.0 ! sector rectangular coordinate error,
10 x hit F 0.0 9999.0 ! lab coordinate,
11 y hit F 0.0 9999.0 ! lab coordinate,
12 z hit F 0.0 9999.0 ! lab coordinate,
13 dx hit F 0.0 9999.0 ! lab coordinate error,
14 dy hit F 0.0 9999.0 ! lab coordinate error,
15 dz hit F 0.0 9999.0 ! lab coordinate error,
16 u2 hit F 0.0 9999.0 ! second moment of u hit pattern
17 v2 hit F 0.0 9999.0 ! second moment of v hit pattern
18 w2 hit F 0.0 9999.0 ! second moment of w hit pattern
284
19 u3 hit F -9999.0 9999.0 ! third moment of u hit pattern
20 v3 hit F -9999.0 9999.0 ! third moment of v hit pattern
21 w3 hit F -9999.0 9999.0 ! third moment of w hit pattern
22 u4 hit F -9999.0 9999.0 ! forth moment of u hit pattern
23 v4 hit F -9999.0 9999.0 ! forth moment of v hit pattern
24 w4 hit F -9999.0 9999.0 ! forth moment of w hit pattern
25 centr U F 0.0 9999.0 ! peak position on U axis
26 centr V F 0.0 9999.0 ! peak position on V axis
27 centr W F 0.0 9999.0 ! peak position on W axis
28 path U F 0.0 9999.0 ! path length from hit position to U axis
29 path V F 0.0 9999.0 ! path length from hit position to V axis
30 path W F 0.0 9999.0 ! path length from hit position to W axis
32 Nstrp V I 0 36 ! Number of V strips in the hit
31 Nstrp U I 0 36 ! Number of U strips in the hit
32 Nstrp V I 0 36 ! Number of V strips in the hit
33 Nstrp W I 0 36 ! Number of W strips in the hit
34 MatchID1 I 0 30 ! Id of matched hit in the layer1
35 CH21 F 0. 999. ! Quality measure of matching with layer1
36 MatchID2 I 0 30 ! Id of matched hit in the layer2
37 CH22 F 0. 999. ! Quality measure of matching with layer2
38 istat I 0 0xFFFF ! Number of hits & hit ID
!!
! For matching if current layer is WHOLE then layer1=INNER and layer2=OUTER
! if current layer is INNER then layer1=WHOLE and layer2=OUTER
! if current layer is OUTER then layer1=WHOLE and layer2=INNER
285
TBER:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE TBER ! create write display delete ! Time Based Tracking ERror bank
! record no=0
! ! Fit parameter and Covariance matrix: (Cij)
! ! Track# = row# (cf. TBTR bank)
! note these are in the sda tracking coordinate system
! (x=beamline, y=radially outward, z=parallel to axial wires)
! ATTributes:
! ———–
!COL ATT-name FMT Min Max ! Comments
!
1 q over p F 0. 100. ! q/p
2 lambda F -10. 10. ! dip angle (pi/2 - theta)
3 phi F -60. 60. ! phi
4 d0 F -100. 100. ! min.distance from (x=0,y=0,z=?) [cm]
5 z0 F -100. 100. ! z position of starting point [cm]
6 c11 F -10. 10. ! element C1,1
7 c12 F -10. 10. ! element C1,2
8 c13 F -10. 10. ! element C1,3
9 c14 F -10. 10. ! element C1,4
10 c15 F -10. 10. ! element C1,5
11 c22 F -10. 10. ! element C2,2
12 c23 F -10. 10. ! element C2,3
286
13 c24 F -10. 10. ! element C2,4
14 c25 F -10. 10. ! element C2,5
15 c33 F -10. 10. ! element C3,3
16 c34 F -10. 10. ! element C3,4
17 c35 F -10. 10. ! element C3,5
18 c44 F -10. 10. ! element C4,4
19 c45 F -10. 10. ! element C4,5
20 c55 F -10. 10. ! element C5,5
21 chi2 F 0. 50. ! Chisquare for this Track
22 layinfo1 I 0. 0. ! layerhit info
23 layinfo2 I 0. 0. ! layerhit info§or&track#in hber
! the layer hit info is stored in the following way
! for layinfo1= sum over each layer used in track(layers 1-30) Of 2(layer#-1)
! for layinfo2 = sum of 2(layer#-31) for (layers 31-36)
! + 256 * track# in sector+2562*track# in hber
! + 2563 * sector
287
MCTK:
!********************************************************************
! BANKname BANKtype ! Comments
TABLE MCTK ! create write display delete ! GSIM Monte Carlo track bank
! ! ATTributes:
! ———–
!COL ATT-name FMT Min Max ! Comments
! 1 cx F -1. 1. ! x dir cosine at track origin
2 cy F -1. 1. ! y dir cosine
3 cz F -1. 1. ! z dir cosine
4 pmom F 0. 20. ! momentum
5 mass F 0. 10. ! mass
6 charge F -1. 1. ! charge
7 id I -5000 5000 ! track Particle Data Group id
8 flag I 0 0xFFFF ! track flag
9 beg vtx I 0 65536 ! beginning vertex number
10 end vtx I 0 65536 ! ending vertex number
11 parent I 0 65536 ! parent track
288
MCVX
!********************************************************************
! BANKname BANKtype ! Comments
TABLE MCVX ! create write display delete ! GSIM Monte Carlo vertex parameters
!
! ATTributes:
! ———–
!COL ATT-name FMT Min Max ! Comments
!
1 x F -1000. 2000. ! x of vertex
2 y F -1000. 2000. ! y
3 z F -1000. 2000. ! z
4 tof F 0.0 999999. ! secs of flight
5 flag I 0 65536 ! vertex flag
!
