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Non-Leptonic Decays of If-Mesons Within the Chiral Quark Model

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Non-Leptonic Decays of If-Mesons Within the Chiral Quark Model NO9800028 NEI-NO--889 Non-leptonic Decays of if-mesons within the Chiral Quark Model Audun Elias Bergan Thesis submitted for the Degree of Doctor Scientiarum Department of Physics University of Oslo November 1996 Acknowledgements During the work with this thesis I have been very grateful for the help and support from my supervisor Professor Jan Olav Eeg. I also want to thank him for reading the manuscript and giving valuable comments. I thank both employees and students at the theory group for a nice time and I will especially mention Susanne "Maradona" Viefers and Tor Haugset who I shared office with. They have always contributed to a positive and enjoyable atmosphere in our office which I highly appreciate. This attitude was also characteristic for Jan Melsom who shared my interest in the Chiral Quark Model and with whom I had many valuable discussions. I also want to thank Harek Haugerud and Mark Burgess for a very good management of the computer network. I am cordially grateful to the Department of Physics at the University of Oslo for given me many short-time contracts as an associate professor. The Norwegian Research Council and the Nansen Fund and its Associated Funds are also gratefully acknowledged for a 5 months short-time position and a scholarship, respectively. Furthermore, I am indebted to the theory group in Oslo, the Norwegian Research Council/Professor Hallstein H0gasen and NORDITA for financial support of my par- ticipation at various meetings, schools and conferences. I will also thank my relatives and especially my parents, Signe and Kjell Bergan, for their support during my studies. Last but not least I will thank my wife, Monika, for her encouragement, patience, love and care during the work with this thesis. Preface This thesis has been submitted for the degree of Doctor Scientiarum and is mainly based upon four papers • A. E. Bergan and J. 0. Eeg, K - K Mixing in a Low Energy Effective QCD, Zeitschrift fur Physik C 61, 511-516 (1994) A. E. Bergan and J. O. Eeg, Momentum Dependence of the Penguin Interaction, Accepted for publication in Zeitschrift fur Physik C e-Print Archive:hep-ph/9606245 • A. E. Bergan, Non-factorizable Contribution to BK of order (G3), Physics Letters B 387 (1), 191-194 (1996) e-Print Archive:hep-ph/9608249 • A. E. Bergan and J. 0. Eeg The Self-penguin Contribution to K -¥ 2n Accepted for publication in Physics Letters B e-Print Archive:hep-ph/9609262 In the papers made together with my supervisor, I have worked through all the calculations. In addition I have done some calculations with a fermion propagator in an external field. This yielded some momentum dependent functions that has been used in the calculation of gluon condensate corrections. Some of the results in table 5.3 have not been seen elsewhere. However, they have not been published. I have also done some unpublished calculations about the effects of a condensed penguin. Contents Introduction 8 1.1 Motivation 8 1.2 Introduction to the Standard Model 9 1.3 Hadronic Interactions and the A/ = 1/2 Rule 13 1.4 External, Internal and Hidden Symmetries 15 Non-leptonic Decays 20 2.1 Introduction 20 2.2 K Decays in Light of CPT Invariance 21 2.3 Relations between K —> 2x Amplitudes 23 2.4 Historical Aspects of the A/ = 1/2 rule 25 2.5 Measurable CP-violating Quantities 26 Effective Lagrangian at Quark Level 30 3.1 The Standard Approach to K —> 2ir Transitions 30 3.2 The K° - ~K° Transition 35 3.3 Renormalization 37 Chiral Perturbation Theory (xPT) 41 4.1 Effective Low Energy Field Theories 41 4.2 \PT 44 4.2.1 An Example of the FKW Theorem in xPT 49 The Chiral Quark Model (xQM) 51 5.1 XQM 51 5.1.1 Relations between /*, M, A and (qq) 55 5.2 Vacuum Saturation Approximation 58 5.3 Construction of Weak XPT by using the xQM 58 5.4 Condensates 59 5.4.1 The Point Gauge/Fock-Schwinger Gauge 60 5.4.2 The Fermion Propagator in an External Field 61 5.4.3 The Quark Condensate 65 5.5 Regularization Schemes 65 5.5.1 Sharp Cut-off Regularization 66 5.5.2 Pauli-Villars Regularization 66 5.5.3 Proper Time Regularization 67 5.5.4 Dimensional Regularization 67 6 Applications 70 6.1 The K°-~K° Transition 70 6.1.1 Non-local Effects from the Siamese Penguin Diagram 70 6.1.2 Non-factorizable Condensate Effects to BK 75 6.2 K -*• 2n Transitions 77 6.2.1 Bosonization of Q6 77 6.2.2 Effects of a Momentum Dependent Penguin Coefficient .... 80 6.2.3 Non-diagonal Self-energy 5 —> d Effects 85 6.2.4 Penguin Condensation 95 7 Resume 99 8 Appendix 101 8.1 Clebsch-Gordan Coefficients Applied on a Member of the 27-plet ... 101 8.2 Technical Details of Dimensional Analysis of K -¥ w 102 List of Tables 1.1 Table of current and constituent quark masses 14 1.2 Transformation properties of Dirac bilinears under C, P and T. ... 17 1.3 Transformation properties of different quantities under C, P and T transformations 18 4.1 The renormalized coupling constants Li(Mp). L\\ and L12 are not directly accessible to experiment 46 5.1 These are the exceptions where a\ is not equal to unity. See also text for explanation 53 5.2 These are the exceptions where ai is not equal to unity. See also text for explanation 54 5.3 Functions denned in text 64 6.1 Values of g(Q&)s and g{Qs)p, for constant and momentum dependent penguin coefficient respectively, calculated for different values of A and M. The corresponding modified values obtained when compensating for wrong values obtained for fn and (qq) are also given 84 6.2 Coefficients related to the self penguin diagram. Each term includes (not written) integration over the coupled Feynman parameters y (from 0 to 1 — z) and z (from 0 to 1). A is defined as A — m2 + 2\ _L -,2 . 2 2 m + q\z{\ - z) and ( = M& - z(M& - m ) + q Ez{\ - z) 90 List of Figures 2.1 Left: Basic diagram for s —> d transition. Right: Approximated four quark interaction point 21 2.2 The box diagrams contributing to the AS = 2 transition 22 3.1 The penguin interaction 31 3.2 Two typical cases of the /^-function. Figure a) shows an ultraviolet stable fixed point go as s —¥ oo. Figure b) shows an infrared stable fixed point go as 5 —> 0 38 4.1 The three stage model 43 4.2 Contributions in the process K° -» n+n~. An example of the FKW theorem. The constant term drops out by considering the tadpole term. 49 5.1 Typical diagrams generated by the ALX Lagrangian. There exists also vertices with three, four etc. mesons interacting with the quarks. ... 54 5.2 Diagrams contributing to /j0'. The small black circles denotes the vertex of the axial vector 57 5.3 Contributions to the fermion propagator. The nabla symbols denote contributions from terms in the expansion of Aa(x) which contain co- variant derivatives 62 6.1 Radiative corrections contributing to the AS = 2 transition 71 6.2 Diagrammatic representation of the Siamese penguin diagram in the XQM 73 6.3 Three diagrams contributing to the K ->• n transition. The first two diagrams are baptized as the "eight" and "keyhole" diagram, respectively. 79 6.4 A typical self penguin diagram 87 6.5 Self energy diagrams for K —)• TT transition 93 6.6 A typical diagram for K —} 2n with a selfpenguin insertion 94 6.7 Two diagrams with two-meson vertex for K —> 2n with a selfpenguin insertion 94 6.8 The condensed penguin diagram 95 6.9 Condensate contributions in K -4 IT transitions 97 6.10 The eye diagram 98 Chapter 1 Introduction 1.1 Motivation The aim of elementary particle physics is to discover new fundamental laws of nature and to classify and develop a profound theoretical framework that arranges particles in simple patterns and describes their interactions in a unified way. The Standard Model of elementary particle physics fits this prescription and it has withstood all experimental tests so far. Quantum chromodynamics, QCD, is a part of the Stan- dard Model which describes interactions among quarks. It contains roughly two parts which are essentially different. The high energy part which can be described by a per- turbative framework and in principle does not represent any problem while in practice is very tedious to calculate. The second part concerns low energy effects which can not be calculated in Standard Model QCD due to a strong coupling constant. How- ever, phenomenological models or ansatzes based on appropriate symmetries can be used to mimic the effects of low energy QCD. One of the frontiers of todays particle physics is attached to effective models of QCD. Even though there has not been observed any deviation from the Standard Model there are some sectors that still need to be investigated more deeply due to seemingly improper descriptions which probably are due to their complexity or maybe more interesting, new physics. One of the most interesting areas in this concern is non- leptonic decays which describes interactions containing only hadronic states. An example of such processes, comprising merely mesons, are K —>• 2ir decays. These processes have puzzled physicists in many years. They exhibit unexpected ratios between different decays. This is referred to as the A7 = 1/2 rule which refers to a change of isospin in the initial and final states.
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