Positronium Hyperfine Splitting Corrections Using Non-Relativistic QED Seyyed Moharnmad Zebarjad Centre for High Energy Physics Department of Physics, McGil1 University Montréal, Québec Novernber 1997 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulWment of the requirements of the degree of Doctor of Philosophy @ Seyyed Moh~unmadZebarjad, 1997 National Library Bibliothèque nationale of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Weiiington Street 395. nie Wellington OttawaON K1A ON4 Ottawa ON K 1A ON4 Canaâa Canada Your fïb Votre reiemnw Our fî& Notre reIPrmce The author has granted a non- L'auteur a accordé une Licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of ths thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. 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Contents Abstract vii Résumé viii Acknowledgement s x Statement of Original Contributions xi 1 Motivation and Outline of this Thesis 1 2 Introduction to NRQED 7 2.1 NRQED Lagrangian. 7 2.2 Matching at Leading Order. 10 2.2.1 Two-Fermi Operators . 11 2.2.2 Four-Fermi Operators . 13 2.3 NRQED Propagators. 15 2.4 NRQED Power Counting . 18 2.4.1 Soft Photons . 19 2.4.2 Ultra-Soft Photons . 21 3 Leading-Order Bound State Energy Shifi. 26 3.1 Positronium Energy Shift to Order a'. 26 3.2 Scalar-Fermion and Scalar-Scalar Bound State Energy Shift to Order cu4. 32 4 Next-to-Leading Order Bound State Energy Shift. 35 4.1 Matching at Next-to-Leading Order for TweFermi Operators 37 4.1.1 Lamb Shift in Hydrogen Atom. 41 4.1.2 Lamb Shift in the Scalar-Scalar Bound State. 47 4.2 Matching at Next-to-Leading Order for Four-Fermi Operators. 51 4.2.1 Hyperfine Splitting in Positronium to Order as. 58 5 Next-to-Next-Leading Order Bound State Energy Shift. 64 5.1 Derivative Interaction . 66 5.2 Revisiting One-Loop Matching . 67 5.3 Matching at NNLO for Four-Fermi Operators Coming from Propagator Corrections. 71 5.3.1 Hyperfine Splitting in Positronium to Order (r6 Coming from Propagator Correction. 73 5.4 Matching at NNLO for Four-Fermi Operators Coming from Vertex Corrections. 76 5.4.1 Hyperfine Splitting in Positronium to Order asComing F'rorn Vertex Correction. 83 6 Discussion 90 A Table of Integrds(1) 94 B Table of Integrals(2) 99 List of Figures NRQED Vertices . The matchings which determine cl), t$) and cg). The graphs on the LHS represent QED Feynman des, while those on the right are for NRQED. (0) (0) (0) (01 and The mîtching which determines c, , o, , c, , c4 . .. Separation of a transverse photon into a soft, instantaneous contribution (represented by a vertical line) and an ultra-soft propagator (represented by the broken wavy line). The two non-covariant perturbation theory diagrams which correspond to a single covariant diagram using tirne-ordered propagator. Photon propagators. Generic NRQED diagrams in the presence of soft photons only. Two non-covariant perturbation theory diagrams correspond- ing to a transverse photon exchange. Generic bound state potentials. The lowest order diagrams contributing to the fermion-fermion bound state energy. The matchings which determine $) and CL).. One-loop matching for the NRQED vertices. Diagrams contributing to order cr4 of hydrogen atom in the non-recoil limit. (a): Order a5contribution from the Uehling potential. (b): Order a5from ultra-soft photon(any number of Coulomb lines in the intermediate state are understood.) . One-loop matching for the scalar non-relativistic QED vertices. C.. Diagram contributing to order cr4 of scalar-scalar bound state in the non-recoii limit. 50 The graphs which give the one-photon annihilation contribu- tiontoc,(1) . 52 Diagrams which contribute to two-photon annihilation contri- butions to c5(1) . 54 The one-ioop t-channel matching diagrams. 55 The NRQED graphs which contribute to the hyperfine struc- ture at order ma4. 59 The general form of matching for the annihilation diagrams in a triplet state up to two loops. 65 The matching that determines c, (2) (1-7am). 65 The rnatching above the threshold which determines cr) and c,..................................(0) 66 One-loop matching required for two-loop calculation. 68 The rnatching which determines [~f)(l-~ann)lpc. 72 Double annihilation diagram with one Coulomb line inside. 73 The whole double annihilation bound state diagrams. 75 The rnatching which determines [~y)(l-~ann)lv, . 77 Double annihilation diagram with one Coulomb line outside. 82 Al1 the bound state diagrams which contribute to as except the dianrams with the Double Annihilation interaction. 84 List of Tables 5.1 Summary of the theoretical calculations to the HFS . Only the references with the first correct calculations are given ...... 88 A.1 Three dimensional integrais ................... 95 A.2 Three dimensional integrals (continued) ............. 95 A.3 One dimensional integrals .................... 96 A.4 Second type of one dimensional integals ............ 97 A.5 Third type of one dimensional integrals ............. 98 Abstract We use Nonrelativistic Quantum Electrodynamics (NRQED) as an effective field theory to calculate the single photon annihilation contribution to the positronium hyperîme splitting (HFS)in its ground state at order 0(or6).Our andyticd result completes the hyperfine splitting cdculation to order meab. Using NRQED, we derive the Lamb shift of a scalar-scalar bound state and 0(a5)hyperfine splitting of positronium for a general excited state. Using the same technique, we also rederive the Lamb shift of Hydrogen atom and 0(a5)HFS of positronium in its ground state. Résumé Nous utilisons Électro-~~nami~ueQuantique Non-Relativistique (NRQED) comme théorie des champs effective afin de calculer la contribution de l'anihi- lation d'un seul photon au découplage hyperfin du positronium dans son état de base à l'ordre O(&). Ce résultat analytique complète les calcuis du découplage hyperfin à l'ordre m,a4. En utilisant toujours NRQED, nous dérivons le décalage de Lamb de l'état scalaire-scalaire lié ainsi que le découplage hyperfin du positronium pour un état excité général n et 1 à l'ordre C(a5). Finalement, nous utilisons la technique NRQED pour redériver le décalage de Lamb de l'atome d'hydrogène ainsi que le découplage hyperfin du positronium à son état de base. vii Acknowledgement 1 would like to thank my supervisor, Professor Clifford Burgess for his sci- entific and financial support. 1 have learned a lot from the courses 1 took with him and 1 am sure this work would have been impossible without his help. 1 would like to thank him for his patience and wish him the best in his research and his personal life. 1 was definitely lucky to have met Dr. Patrick Labelle who introduced me the wonderful world of NRQED. He shared with me his invaluable expe- riences and knowledge of NRQED. 1 am sure, it would be very difficult, if not impossible, for me to have learnt al1 this material by myself. This last year of my studies during which 1 have worked with him has been the best time 1 have ever had in my life. 1 cannot thank him enough for his kindness and patience and 1 am certain he will one day make it to the top. I also thank Professors R.C. Myers and C.S. Lam for the informative courses 1 took with them. I would like to highly appreciate the financial support of the Ministry of Culture and Higher Education of Iran, without which this work would not have been feasible. 1 also thank Dr. Reza Hoseini, Dean of Higher Education Advisory, who has been taking care of al1 students affairs. 1 should also thank Dr. A.H. Hoang for his collaboration to part of the hy- perfine splitting calculation and also for his excellent suggestions concerning chapter(5) of this thesis. My thanks also to Martin Kamela and Chris Roderick who helped me with the proofread of the manuscript, and Stéphane Legault for translating the abstract into French. No doubt the smooth running of business in Our physics Department would not be possible without the skills of secretaries, Mrs. Paula Domingues, Joanne Longo, Diane Koziol, Elizabeth Sharon, and the expertise of the com- puter system managers, Mr. Paul Mercure and Juan Gallego, aU of whom 1 wish to thank. 1 am geatly indebted to my wife, Roza Aleyasin, and my daughters, Fatemeh and Marzieh, whose patience helped me to bear dl the difnculties of these years. They invested five of the best years of their lives in my thesis without cornplaining about the extraordinary weather of Montreal or nagging about being away fiom our relatives. 1 hope that one day 1 can make it up to them. Special thanks to my parent, Khadijeh Semati, S. Mohesn Zebarjad, for their constant encouragment. I am deeply greatful to my fnend Dr. Omid Hamidi-Ravari with whom 1 had many unforgettable times in Physics Department and am eagerly waiting for his special ceremony in Kerman City. It would not be fair to end this acknowledgment without mentioning the name of my friends with whom we had so much fun in Montreal: Dr.