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A Relativistic One Pion Exchange Model of Proton-Neutron Electron-Positron Pair Production

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A Relativistic One Pion Exchange Model of Proton-Neutron Electron-Positron Pair Production Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-1973 A Relativistic One Pion Exchange Model of Proton-Neutron Electron-Positron Pair Production William A. Peterson Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Physics Commons Recommended Citation Peterson, William A., "A Relativistic One Pion Exchange Model of Proton-Neutron Electron-Positron Pair Production" (1973). All Graduate Theses and Dissertations. 3674. https://digitalcommons.usu.edu/etd/3674 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. ii ACKNOWLEDGMENTS The author sincerely appreciates the advice and encouragement of Dr. Jack E. Chatelain who suggested this problem and guided its progress. The author would also like to extend his deepest gratitude to Dr. V. G. Lind and Dr. Ackele y Miller for their support during the years of graduate school. William A. Pete r son iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS • ii LIST OF TABLES • v LIST OF FIGURES. vi ABSTRACT. ix Chapter I. INTRODUCTION. • . • • II. FORMULATION OF THE PROBLEM. 7 Green's Function Treatment of the Dirac Equation. • • • . • . • • • 7 Application of Feynman Graph Rules to Pair Production in Neutron- Proton Collisions 11 III. DIFFERENTIAL CROSS SECTION. 29 Symmetric Coplanar Case 29 Frequency Distributions 44 BIBLIOGRAPHY. 72 APPENDIXES •• 75 Appendix A. Notation and Definitions 76 Appendix B. Expressi on for Cross Section. 79 Appendix C. Trace Theorems and "Y Identities 84 iv TABLE OF CONTENTS (Continued) Page Appendix D. Phase Space Distributions for Neutron- Proton Electron- P ositron Pair Production . • . • • 86 Appendix E. Computer Program PHASE . 111 VITA. 136 v LIST OF TABLES Figure Page I. Glossary of symbols ... ...•.•.•.. 1 2 vi LIST OF FIGURES Figure Page l. Typical Feynman diagrams for nucleon-nucleon electron-positron pair production. • • • 5 2. Vertices for (a) spinor electrodynamics, (b) elec­ trodynamics of spin-zero boson, and (c) meson- nucleon scattering. • • • • • • • • • • • • • 15 3. Feynman graphs for process considered in this paper ••••••.••••• 17 4. Symmetric al geometry for cross section 30 5. Theorectical curves of differential scattering cross section for the symmetric geometry of Figure 4 • 34 6. Angular distribution of final proton in laboratory system with the directi~'S of incident proton with matrix for weights< 10 . • . • • • • • • . 48 7. Angular distribution of electron in laboratory with the direction §f the incident proton with matrix for weights < I o- . 50 B. Angular distribution in final proton with the direc­ tion of the incident proton in center of momentum system with matrix for weights < I o-B. 52 9. Angular distribution of electron with the direction of the incident proton in center of momentum sys- tem with matrix for weights < 1 o-B . 54 1 o. Angular distribution between pair in lg-bot·atory system with matrix for weights < 1 o- . 56 11. Angular distribution between pair in center of _ 8 momentum system with matrix for weights < 1 0 • 58 2 2 12. A = (p + p ) distribution of gamma ray with matrix for w;ights < 1 o-8 • . • . • . 60 vii LIST OF FIGURES (Continued) Figure Page 2 2 13. '-"A =-(nf- n . ) d1stn• •b ut10n. o f ;rO w1t• h matr1x• f or -8 1 < 10 •..•.•.•••••••••••• 62 14. Angular distribution between final proton-neutron in the laboratory system with matrix for weights < 1 o-8 ••••••••..•••.••••• 64 15. Angular distribution between final proton-neutron in center of momentum system with matrix for weights< l o-8 . • • • • • • • • • • • • • • • 66 16. Momentum distribution of electron in laboratory system with matrix for weights < 1 o-8 68 17. Momentum distribution of electron in center of_ 8 momentum system with matrix for weights < 10 • 70 18. Angular distribution of final proton in laboratory with the direction of incident proton without matrix. 87 19. Angular distribution of electron in laboratory with the direction of the incident proton without matrix • 89 20. Angular distribution of final proton with the direc­ tion of the incident proton in center of momentum system without matrix • • • • • • • • • • • • 21. Angular distribution of electron with the direction of the incident proton in center of momentum sys­ tem without matrix. • • . • . • • • • . 93 22. Angular distribution between pair in laboratory system without matrix • . • , . 95 23. Angular distribution betwe en pair in center of momentum system without matrix •. .• 97 2 2 24. !::,. = (p_+ p+) distribution of gamma ray without matrix . ................ .. 99 viii LIST OF FIGURES (Continued) Figure Page 25. '-"A2 = - ( nf- ni )2 dtstn" "b uhon . o f .,.0 wtt• h out matrtx• l 01 26. Angular distribution between final proton-neutron in the laboratory system without matrix . • . l 03 27. Angular distribution between final proton-neutron in center of momentum system without matrix l 05 28. Momentum distribution of electron in laboratory system without matrix • . • . • . • l 07 29. Moment um distribution of electron in center of momentum system without matrix • . l 09 ix ABSTRACT A Relativistic One Pion Exchange Model of Proton- Neutron Electron- Positron Pair Production by William A. Peterson, Doctor of Philosophy Utah State University, 1973 Major Professor: Dr. Jack E. Chatelain Department: Physics Proton-neutron electron-positron pair production c ross sections are calculated in the framework of the ps e udoscalar one- pion exchange model in a fully relativistic manner, A computer program has been developed to evaluate invariants and Dirac traces for a given data point. The c ross sections for symmetric coplanar events for labora- tory kinetic energies of l 0 to 250 MeV were calculated for pair angles Frequency distributions were also calculated, at a laboratory energy of 200 MeV, using a random number generator to select data points. The frequency distributions are illustrated by curves. It was noted that the inclusion of heavier bosons will not significantly improve the results at laboratory energies less than 200 MeV. ( 145 pages ) CHAPTER I INTRODUCTION The early investigations of scattering and absorption of gamma­ rays showed that the interaction with heavy elements exceeded that predicted by the Compton and photoelectric effect alon e . In 1932 Anderson observed the ejection of electron-positron pairs from lead foil subjected to c osmic ray irradiation; this experiment provided evidence for the Oppenheimer and Plesset (1933) interpetration of pair production on the basis of Dirac 1s electron theory. The first studies of the problem of electron- positron pair pro­ duction were theoretical. Some of the first papers were those of Furry and Carlson (1933) and of Heitler and Nordheim (Heitler, 1960). In these papers a cross section was calculated by the use of the Born­ Molter perturbation theory for relativistic particles . They found that for cosmic-ray energies the production of pairs is as important fo r electrons as it is for gamma-rays . Another early calculation was done by Bhabha (193 5 ). The Bhabha calcu lation uses the Weizsacker­ Williams approximation to calculate the cross- sec tion for pair pro­ du ction by an e lectron scattering off a fixed electromagnetic field (the trident process). In the Weizsacker-Williams method the incident particle is placed in a rest frame by the Lorentz transformation. 2 The scattering p roduced by the nucleus is treated as if its field of virtual photons resemble a flux of photons distributed over a frequency spectrum. Racah (1937) also did a calculation using the 1 93 0 version of pe rturbation theory. His r esults did not include exchange of iden- tical particles in the final state. Experimental evidence for pair production by an electron scattering off a fixed electromagnetic field has been consid e red by a n umber of authors and a summary of some of these attempts has been discussed by Johnson (1965 ). In 1 935 Yukawa first proposed the meson theory of nuclear fo r c es . Since then, a large number of theoretical as well as experimental in- vestigations have b een carried out to i n crease ou r k n owled ge a bout the nature of the nucleon- nucleon interaction. One way to study the basic problem is to investigate nucleon- nucleon e lectron-positron p a ir production. Although this process is more complicated than elastic nucleon- nucleon scattering , it y ields / more information than can be obtained from the study of the e lastic case. In addition i t i s worthwhile to study the process of pair pro- duction partly for its own sake , i.e. , as inte r esting phenomena in itself which must be compared with our theoretical knowledge of ele- mentary processes, and partly b ecause it can be present aa part of the "background" when other processes are studi ed. Only a thoro u gh understanding will permit us to eliminate this background. Also high- energy electron-positron pairs probe the small-distance b e havior of 3 quantum electrodynamics (Bjorken and Drell, 1 959 ). In particular, it offers the possibility of exploring the photon propagator at small distances . Petiau (1951) and Havas (1 952) studied the c r eation of pairs by collision of particles . They treated the interactions in te r ms of virtual exchange particles with mass and estab lished the matrix ele­ ments for the process to lowest order in perturbation theory. Their papers have been reviewed thoroughly by Parker (1968). Parker calculated a differential cross-section fo r proton- proton electron­ positron pair production. He considered only electromagnetic inter­ actions or photon exchange. Wheeler (1968) made a calculation for neutron - proton electron -posit ron pau production u sing on! y neutral one pion exchange between nucleons . This excluded the possibility of the effect of charge exchange.
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