UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO PION PRODUCTION REACTIONS IN NUCLEON-NUCLEON COLLISIONS Verónica de Ataíde Malafaia Lopes dos Santos (Licenciada) Dissertação para obtenção do Grau de Doutor em Física Orientador: Doutora Maria Teresa Haderer de La Peña Stadler arXiv:nucl-th/0602026v1 9 Feb 2006 Júri: Presidente: Reitor da Universidade Técnica de Lisboa Vogais: Doutor Dan-Olof Riska Doutor Jorge Manuel Rodrigues Crispim Romão Doutora Maria Paula Frazão Bordalo e Sá Doutora Maria Teresa Haderer de La Peña Stadler Doutora Ana Maria Formigal de Arriaga Doutor Jirka Adam Janeiro 2006 Resumo A produ¸c˜ao de pi˜oes em colis˜oes nucle˜ao-nucle˜ao, junto ao limiar, tem sido um desafio nas ´ultimas d´ecadas. A reac¸c˜ao pp ppπ0 em particular, ´emuito sens´ıvel a mecanismos → de curto alcance porque a conserva¸c˜ao do isospin suprime o termo de troca de pi˜oes que de outra forma seria dominante. Assim, tem sido muito dif´ıcil estabelecer a relativa importˆancia dos v´arios processos de reac¸c˜ao. Ap´os rever o estado-da-arte da teoria, abordamos a validade da distorted-wave Born approximation (DWBA), atrav´es da sua liga¸c˜ao aos diagramas da teoria das perturba¸c˜oes ordenadas no tempo (TOPT). Analisamos igualmente as escolhas poss´ıveis para a energia do pi˜ao trocado, inevit´aveis no formalismo n˜ao-relativista subjacente a DWBA. O operador de re-dispers˜ao resultante de TOPT ´ecomparado com o obtido pela abor- dagem mais simples de matriz-S, que tem sido usada eficazmente abaixo do limiar. A t´ecnica de matriz-S, tendo reproduzido os resultados de TOPT para a re-dispers˜ao em pp ppπ0 ´eaplicada `aprodu¸c˜ao de pi˜oes neutros e carregados, descrevendo-se com → sucesso as sec¸c˜oes eficazes nestes diferentes canais. Os principais mecanismos de produ¸c˜ao e ondas parciais correspondentes a momento angular mais elevado s˜ao inclu´ıdos. Final- mente, discutimos o efeito na sec¸c˜ao eficaz das aproxima¸c˜oes usuais para a energia do pi˜ao trocado. PALAVRAS-CHAVE: Produ¸c˜ao de pi˜oes, prescri¸c˜oes de energia, matriz-S, DWBA, teoria das perturba¸c˜oes ordenadas no tempo, re-dispers˜ao. i Abstract Understanding pion production in nucleon-nucleon collisions near threshold has been a challenge for the last decades. In particular, the reaction pp ppπ0 is highly sensitive to → short-range mechanisms, because isospin conservation suppresses the otherwise dominant pion exchange term. However, the relative importance of the various reaction processes has been very difficult to establish. After reviewing the state-of-the-art of the theoretical approaches, we address the va- lidity of the distorted-wave Born approximation (DWBA) through its link to the time- ordered perturbation theory (TOPT) diagrams. As the energy of the exchanged pion is not determined unambiguously within the non-relativistic formalism underlying DWBA, we analyse several options to determine which one is closer to TOPT. The S-matrix technique, successfully used below threshold, is shown to reproduce the results of TOPT for the re-scattering mechanism in π0 production. It is afterwards applied to full calculations of both charged and neutral pion production reactions, the cross sections of which are described successfully. The main production mechanisms and partial waves corresponding to high angular momentum are included in the calculations. Finally we discuss the effect on the cross section of the frequent prescriptions for the energy of the exchanged pion. KEYWORDS: Pion production, energy-prescriptions, S-matrix, DWBA, time-ordered perturbation the- ory, re-scattering. iii Acknowledgments I am deeply grateful to my advisor, Professor Teresa Pe˜na, who first motivated me to study Nuclear Physics. I wish to thank her for her constant support, encouragement, and extensive advice throughout my Ph.D., and for all the important remarks in the writing of this manuscript. I am also very grateful for the possibility to establish valuable international collaborations and for participating in international conferences. I wish to thank my collaborators, Professor Jiri Adam and Professor Charlotte Elster, for their advice. I cannot forget their kind hospitality during my stays in Reˇzandˇ in Athens, OH. My gratitude goes to CFTP (Centro de F´ısica Te´orica de Part´ıculas), and all its members, for the excellent working conditions and stimulating environment during the preparation of my Ph.D. I also wish to thank CFIF (Centro de F´ısica das Interac¸c˜oes Fundamentais) for hosting me during the first two years of my investigations, before CFTP was created in 2004. The Physics Department secretaries, Sandra Oliveira and F´atima Casquilho, offered constant and valuable help, and I thank them for all their kindness and assistance. I am indebted to FCT (Funda¸c˜ao para a Ciˆencia e Tecnologia) for the financial support of my research and participation in several international conferences. Regarding computer support, I would like to thank Doctor Juan Aguilar-Saavedra, for helping me to install Linux, and Doctor Ricardo Gonz´alez Felipe and Doctor Filipe Joaquim for their precious assistance in many computer-crisis. I am also grateful to Eduardo Lopes, especially for his readily help concerning the cluster of computers. I am indebted to Professor Alfred Stadler for his valuable comments during my in- vestigations. I wish to thank Doctor Zoltan Batiz, my office colleague during 2002, and Doctor Gilberto Ramalho, for their suggestions. v I would like to express my gratitude to my teachers during my graduate studies, Professor Pedro Bicudo, Professor L´ıdia Ferreira, Professor Jorge Rom˜ao and Professor V´ıtor Vieira, for their encouragement. My gratitude goes to my colleagues and friends at Instituto Superior T´ecnico, Ana Margarida Teixeira, Bruno Nobre, David Costa, Filipe Joaquim, Gon¸calo Marques, Juan Aguilar-Saavedra, Miguel Lopes, Nuno Santos and Ricardo Gonz´alez Felipe, for their friendship and immense support, and for creating an excellent working environment. I will never forget their help and advice, as well as the many stimulating discussions we had. I wish to thank my friends outside Instituto Superior T´ecnico, Patr´ıcia, Rita and Ana Filipa, for our weekly lunches and for their incessant encouragement, and my movie- companions Miguel and Inˆes. I am very grateful to my dear friends Ana Filipa and Ana Margarida for our leisure trips together and for their precious suggestions in the writing of this thesis. Without the continuous support of my family I could not have completed this work. I wish to thank my grandparents for all their kindness and unceasing support. I am deeply grateful to my parents, for having always supported my options, and for their unconditional love and constant encouragement. Finally, I wish to thank Gon¸calo for his endless patience and unfailing love during so many years. vi Este trabalho foi financiado pela Funda¸c˜ao para a Ciˆencia e Tecnologia, sob o contrato SFRD/BD/4876/2001. This work was supported by Funda¸c˜ao para a Ciˆencia e Tecnologia, under the grant SFRD/BD/4876/2001. vii Contents Preface ....................................... 1 1 Meson production close to threshold 7 1.1 What physics can we learn from meson production reactions? . ..... 8 1.2 Specific aspects of hadronic meson production close to threshold . 11 1.2.1 Rapidlyvaryingphase-space . 11 1.2.2 Initial and final relative momenta . 12 1.2.3 Remarks on the production operator and nucleonic distortions ... 13 1.3 Theoretical considerations on NN NNx reactions ............ 14 → 1.4 Thecrosssection ................................ 15 2 State of the art of theoretical models for pion production 19 2.1 DWBAintheMesonexchangeapproach . 21 2.1.1 Problems of the earlier calculations . 21 2.1.2 The first quantitative understandings . 26 2.2 Coupled-channel phenomenological calculations . ... 32 2.2.1 TheHannovermodel .......................... 32 2.2.2 TheJ¨ulichmodel ............................ 33 2.3 Chiralperturbationtheory .......................... 33 2.3.1 Powercountingfortheimpulseterm . 37 2.3.2 Why π0 isproblematic ......................... 38 2.3.3 Charged pion production in χPT ................... 39 2.3.4 The Vlow k approach .......................... 40 − 2.4 Energyprescriptionfortheexchangedpion . ... 40 ix Table of Contents 3 From Field Theory to DWBA 43 3.1 Extractionoftheeffective productionoperator . .... 46 3.1.1 Final-state interaction diagram . 46 3.1.2 Initial-state interaction diagram . 51 3.2 StretchedBoxesvs.DWBA .......................... 55 3.3 The logarithmic singularity in the pion propagator (ISI) . .. 58 3.4 Conclusions ................................... 59 4 The S-matrix approach for quantum mechanical calculations 61 4.1 The pion re-scattering operator in the S-matrix technique . ...... 62 4.1.1 Factorisation of the effective re-scattering operator . .... 63 4.1.2 TheS-matrixtechnique ........................ 67 4.2 Energy prescriptions for the exchanged pion . ... 70 4.2.1 Expansion of the effective pion propagator . 80 4.3 Conclusions ................................... 84 5 Charged and neutral pion production reactions 89 5.1 Pionproductionoperators ........................... 91 5.1.1 Themechanismsandtheiroperators . 91 5.1.2 Nucleon-nucleon potentials at intermediate energies . 93 5.2 Calculational details . 95 5.2.1 Three-bodykinematics . 95 5.2.2 The role of the different isospin channels . 96 5.2.3 Partialwaveanalysis .......................... 98 5.2.4 Selection rules for NN NNx ....................101 → 5.3 Convergenceofthepartialwaveseries.