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20 2.2. Kaon production in the Regge limit)2dσ/dt (µb/GeV10 110 2101310 4+p(γ,K )Λ112345 GeV8 GeV11 GeV16 GeV+ 0p(γ,K )Σ0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.02t (GeV )Figure 2.3 – Differential cross section as a function of the momentum transfer |t| at four photon LABenergies E γ = 5, 8, 11 and 16 GeV. For the p(γ, K + )Λ channel (left panel), the Regge-2 model is shown.The p(γ, K + )Σ 0 results (right panel) are obtained with the Regge-3 (solid line) and Regge-4 (dashed line)models. Data from Ref. [113].Photonbeam asymmetry Σ1.51.00.50.00.51.01.51.00.50.0+ 0.5+ 0p(γ,K )Λp(γ,K )Σ1.00.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.02t (GeV )Figure 2.4 – Photon-beam asymmetry as a function of the momentum transfer |t| at photon LAB energyE γ = 16 GeV. For p(γ, K + )Λ results (left panel) are the Regge-2 model. The p(γ, K + )Σ 0 results (rightpanel) are obtained with the Regge-3 (solid line) and Regge-4 (dashed line) models. Data from Ref. [114].As a consequence, the amplitude corresponding to K (∗)+ exchange in the t-channel effectivelyincorporates the transfer of an entire trajectory. When considering the exchange of K + (494) andK ∗+ (892) trajectories, the Regge model for p(γ, K + )Y has a mere three parametersg K + Y p , and G v,tK ∗+ = κ K ∗+ K + × gv,t K ∗+ Y p , (2.45)with g K + Y p, g v K ∗+ Y p and gt K ∗+ Y p the coupling constants at the strong-interaction vertex and κ K ∗+ K +the K ∗+ (892)’s transition magnetic moment (see Paragraph D.3.3).A crucial constraint for the kaon-production amplitude is gauge invariance. It is well-known that thet-channel Born diagram by itself does not conserve electric charge. In Ref. [38], an elegant recipeto correct for this was outlined. Adding the electric part of a Reggeized s-channel Born diagramensures that the amplitude is gauge invariant. Thus, the transition current operator for high-energykaon production in the Regge limit readsĴ K+ (494) (892)Regge+ ĴK∗+ Regge+ ĴBorn-s,elecFeynman× P K+ (494)Regge× ( t − m 2 )K . (2.46)+At sufficiently high energies (E γ 4 GeV), a limited amount of p(γ, K + )Y data points are available.For the K + Λ channel a total of 72 data points exist, comprising 56 differential-cross-sectiondata points [113], 9 photon-beam asymmetries [114], and 7 recoil asymmetries [115]. Even fewerdata is available for K + Σ 0 production: 48 differential cross sections [113] and 9 photon-beam
Chapter 2. The Regge-plus-resonance formalism 21Recoil asymmetry P1.00.50.00.51.0E γ= 5 GeV0.0 0.2 0.4 0.6 0.8 1.02t (GeV )Figure 2.5 – Hyperon-recoil asymmetry for the p(γ, K + )Λ channel as function of the momentum transfer|t| at photon LAB energy E γ = 5 GeV. Data from Ref. [115].asymmetries [114]. In Refs. [39, 40], these data sets have been adopted to constrain the three freeparameters (2.45) of the Regge model, as well as the phases of the exchanged trajectories. Becauseof the meagre database, a unique and optimal model could not be established and several modelvariants attain a comparable χ 2 . After carrying through the full RPR strategy (see Section 2.3) andmaking use of photo- and electroproduction data in the resonance region, the number of possibleRegge models can be significantly reduced. For the K + Λ amplitude, a single Regge model with tworotating trajectories, coined Regge-2, emerged. It is encouraging that a later Bayesian analysis of theRegge model in the high-energy domain was able to single out this particular model using high-energydata alone [56]. The p(γ, K + )Σ 0 data was found to be compatible with a rotating phase for theK + (494) and a constant phase for the K ∗+ (892) trajectory. The sign of the tensor coupling constantremained ambiguous, however. The K + Σ 0 -production models are labelled Regge-3 and Regge-4.The coupling constants for the Regge models that are employed in this work are summarised inAppendix I.The Regge models are compared to the available high-energy data in Figures 2.3, 2.4 and 2.5. Thedifferential cross section for both Λ and Σ 0 production are nicely reproduced. From Figure 2.3,one notices that both Regge models for the K + Σ 0 channel give comparable results at low |t|. Thephoton-beam asymmetries of Figure 2.4 have a striking feature which is perfectly reproduced by theRegge model. The asymmetry rises quickly to +1 as a function of −t. This behaviour is reminiscentof the dominant exchange of a natural-parity state [116], which we identify as the K ∗+ (892) trajectory.For the p(γ, K + )Λ reaction, a set of recoil-polarisation asymmetries at forward kaon angles areavailable. The sign and shape of the angular dependence can be reproduced in the model. The sizeof the asymmetry, on the other hand, is considerably underestimated.2.3 Kaon production in the resonance regionThe Regge model’s amplitude can be interpreted as the asymptotic form of the full amplitude forlarge s and small |t|. Owing to the t-channel dominance and the absence of a prevailing resonance,the Reggeized background has been observed to account for the gross features of the kaon productiondata in the resonance region [39, 40, 67]. Near threshold, the energy dependence of the measureddifferential cross sections exhibits structures which hint at the presence of resonances. These areincorporated by supplementing the background with a number of resonant s-channel diagrams.This approach was coined Regge-plus-resonance (RPR) and has also been applied to double-pion
Page 1: FACULTEIT WETENSCHAPPENRegge-plus-r
Page 6: viPrefaceUniversity, the Hercules F
Page 9 and 10: ContentsixD.3.2 Effective Lagrangia
Page 11 and 12: CHAPTER 1IntroductionConventional w
Page 13 and 14: Chapter 1. Introduction 3σ (µb)γ
Page 15 and 16: Chapter 1. Introduction 5meson-prod
Page 17 and 18: Chapter 1. Introduction 7+ +Figure
Page 19 and 20: Chapter 1. Introduction 9are set ag
Page 21 and 22: CHAPTER 2The Regge-plus-resonance f
Page 23 and 24: Chapter 2. The Regge-plus-resonance
Page 29: Chapter 2. The Regge-plus-resonance
Page 41 and 42: CHAPTER 3Kaon production in data-po
Page 43 and 44: Chapter 3. Kaon production in data-
Page 65 and 66: CHAPTER 4Radiative kaon captureAfte
Page 67 and 68: Chapter 4. Radiative kaon capture 5
Page 69 and 70: Chapter 4. Radiative kaon capture 5
Page 71 and 72: CHAPTER 5Formalism for electromagne
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Chapter 5. Formalism for electromag
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CHAPTER 6Results for kaon productio
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Chapter 6. Results for kaon product
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CHAPTER 7Conclusions and outlookMot
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Chapter 7. Conclusions and outlook
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APPENDIX ANotations and conventions
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Appendix A. Notations and conventio
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Appendix A. Notations and conventio
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APPENDIX BBiquaternionsAnd here the
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Appendix B. Biquaternions 125The bi
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APPENDIX CHelicity spinorsIn Sectio
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Appendix C. Helicity spinors 129whe
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APPENDIX DFeynman rulesIf I could e
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Appendix D. Feynman rules 133D.3.1P
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Appendix D. Feynman rules 135Born s
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APPENDIX ELorentz-invariant cross s
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Appendix E. Lorentz-invariant cross
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APPENDIX FDensity matrixScattering
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Appendix F. Density matrix 143The s
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Appendix F. Density matrix 145When
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APPENDIX GParametrisations of the d
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Appendix G. Parametrisations of the
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Appendix G. Parametrisations of the
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APPENDIX HConnecting (non-)relativi
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Chapter H. Connecting (non-)relativ
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APPENDIX IParameters of the Regge-p
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Chapter I. Parameters of the Regge-
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APPENDIX JExperimental dataFacts ar
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Chapter J. Experimental data 163Tab
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APPENDIX KHyperon-nucleon interacti
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Appendix K. Hyperon-nucleon interac
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Appendix K. Hyperon-nucleon interac
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SamenvattingNederlands is geen taal
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Samenvatting 173namelijk ondubbelzi
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Samenvatting 175de subtiele structu
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Samenvatting 177Productie van neutr
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Samenvatting 179off-shell-extrapola
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List of publicationsPart of the res
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BibliographyIf you steal from one a
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Bibliography 185[27] S. Janssen, J.
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Bibliography 187[58] D. G. Ireland,
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Bibliography 189[93] J. Laget, Pent
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Bibliography 191[127] M. Dugger et
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Bibliography 193http://www.jlab.org
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Bibliography 195[190] M. Galassi, J
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Bibliography 197[223] R. H. Dalitz,
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NomenclatureI strive to be brief, a
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Nomenclature 201r(⃗ω) Rotation t
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