Supersymmetric Dark Matter – aspects of sfermion coannihilations Mia Schelke Stockholm University Department of Physics 2004 Thesis for the degree of doctor in philosophy in theoretical physics Department of Physics Stockholm University Sweden c Mia Schelke 2004 ISBN 91-7265-909-2 pp i–viii,1–157 Printed by Universitetsservice US-AB, Stockholm, 2004 Abstract There is very strong evidence that ordinary matter in the Universe is outweighed by almost ten times as much so-called dark matter. Dark matter does neither emit nor absorb light and we do not know what it is. One of the theoretically favoured candidates is a so-called neutralino from the supersymmetric extension of the Standard Model of particle physics. A theoretical calculation of the expected cosmic neutralino density must include the so-called coannihilations. Coannihilations are particle processes in the early Universe with any two supersymmetric particles in the initial state and any two Standard Model particles in the final state. In this thesis we discuss the importance of these processes for the calculation of the relic density. We will go through some details in the calculation of coannihilations with one or two so-called sfermions in the initial state. This includes a discussion of Feynman diagrams with clashing arrows, a calculation of colour factors and a discussion of ghosts in non-Abelian field theory. Supersymmetric models contain a large number of free parameters on which the masses and couplings depend. The requirement, that the pre- dicted density of cosmic neutralinos must agree with the density observed for the unknown dark matter, will constrain the parameters. Other con- straints come from experiments which are not related to cosmology. For instance, the supersymmetric loop contribution to the rare b sγ decay → should agree with the measured branching fraction. The principles of the calculation of the rare decay are discussed in this thesis. Also on-going and planned searches for cosmic neutralinos can constrain the parame- ters. In one of the accompanying papers in the thesis we compare the detection prospects for several current and future searches for neutralino dark matter. i Accompanying papers I J. Edsj¨o, M. Schelke, P. Ullio and P. Gondolo, JCAP 0304 (2003) 001. Accurate relic densities with neutralino, chargino and sfermion coannihilations in mSUGRA. [hep-ph/0301106]. II J. Edsj¨o, M. Schelke and P. Ullio, Draft, May 6, 2004. Direct versus indirect detection in mSUGRA with self-consistent halo models. III P. Gondolo, J. Edsj¨o, P. Ullio, L. Bergstr¨om, M. Schelke and E.A. Baltz, Draft, May 10, 2004. DarkSUSY – A numerical package for supersymmetric dark matter calculations. Paper I is reproduced with permission from JCAP [http://jcap.sissa.it], IOP Publishing Ltd. iii Acknowledgements First of all I want to thank my two supervisors, Lars Bergstr¨om and Joakim Edsj¨o. Lars Bergstr¨om for welcoming me at Fysikum already when I was an undergraduate at the Niels Bohr Institute. Thanks for encouraging me to present my work at conferences and to help finding means for travel expenses. Joakim Edsj¨o, thanks for support in the last busy months. I also appreciate the conversations we had at many different occasions. I would like to thank the whole DarkSUSY group, Ted Baltz, Lars Bergstr¨om, Joakim Edsj¨o, Paolo Gondolo and Piero Ullio for sharing the dedicated interest in supersymmetric dark matter and always aiming at the highest possible accuracy in the calculations. Piero Ullio, thanks for instructive visits at SISSA, Trieste. Thanks to you and Paola for hospitality and delicious italian dinners. Thanks Martin Eriksson, Christofer Gunnarsson and Michael Gustafs- son for many pleasant hours in our study group. Many other colleagues also deserve a special thanks. Among others Rahman Amanullah, Johan Br¨annlund, Gast´on Folatelli, Gabriele Garavini, Martin Goliath, Boris Gudiksen, Stefan Hofmann, S¨oren Holst, Edvard M¨ortsell and Serena Nobili. Thanks to the secretariat for readiness to help. Special thanks to Britta Schmidt. Thanks also to my friends and family in Stockholm, Denmark and elsewhere. A warm thanks to my parents. v Contents Abstract i Accompanying papers iii Acknowledgements v 1 Overview 1 1.1 Dark matter density from cosmology . 2 1.2 Supersymmetry........................ 4 1.2.1 Supersymmetric particles . 5 1.2.2 Softbreaking ..................... 8 1.2.3 R-parityandtheLSP . .. .. .. .. 9 1.2.4 mSUGRA....................... 10 1.3 Paper III and the DarkSUSY computer package . 12 1.3.1 My contributions to DarkSUSY and Paper III . 13 1.4 Paper I on coannihilations and density . 14 1.4.1 My contributions to Paper I ............ 17 1.5 Paper II on detection rates . 18 1.5.1 My contributions to Paper II ........... 20 2 Relic neutralino density and coannihilations 23 2.1 The Boltzmann equation . 24 2.2 Thermalaveraging . .. .. .. .. .. .. 34 2.3 Solving the Boltzmann equation . 44 2.4 Putting together the numerical solution . 46 0 3 Sfermion–χ /χ± coannihilations 51 3.1 Modifiedcrossing....................... 51 3.2 The gauge boson plus fermion final state . 57 vii 3.2.1 The s process..................... 57 3.2.2 The modified t process – including discussion about clashing arrows . 78 3.3 The Higgs boson plus fermion final state . 92 3.3.1 The s process..................... 92 3.3.2 The modified t process................ 103 4 Sfermions coannihilating into two bosons 111 4.1 The two gauge boson final state . 111 4.2 Colourfactors......................... 115 4.3 Thetwogluonfinalstate . 122 4.3.1 The optical theorem . 124 4.3.2 Calculation with ghosts . 128 4.3.3 Sum of physical polarization vectors . 129 4.3.4 Explicit polarization vectors . 131 5 The rare b sγ decay 135 → 5.1 The Standard Model contribution . 136 5.1.1 The operator product expansion . 137 5.1.2 The RGE of the Wilson coefficients . 139 5.1.3 Thebranchingratio . 144 5.1.4 Thenumericalresult . 146 5.2 Supersymmetric corrections . 147 5.2.1 Implementation and results for the full model . 150 6 Summary 153 Bibliography 155 Chapter 1 Overview The subject of this thesis falls within the framework of astroparticle physics. Astroparticle physics considers particle physics phenomena of importance for cosmology. It combines the very small with the very large by considering the interplay between elementary particles and the gen- eral history and large scale structure and kinematics of the Universe. This thesis deals with the so-called supersymmetric dark matter. Super- symmetric particles are elementary particles predicted by the theory of supersymmetry for which we still do not have any experimental evidence. Dark matter is one of the biggest puzzles of modern cosmology. Dark matter does neither emit nor absorb light, but there are other strong in- dications of its existence. It is unknown what this dark matter is, but one of the theoretically favoured candidates is a population of a super- symmetric particle. Our interest here will be the theoretical aspects of the phenomenology of supersymmetric dark matter. If supersymmetric dark matter exists, it must be a relic from the big bang. We have to understand the particle processes in the early Universe in order to make a theoretical prediction of the present amount of the supersymmetric par- ticles. These cosmic relics will still today occasionally interact with each other or with ordinary matter. It is possible to make theoretical predic- tions of how sensitive astroparticle detectors should be to find evidence for these interactions. We will mention some of the cosmological evidence for dark matter in section 1.1 and we will give an introduction to supersymmetry in section 1.2. In the remaining part of the chapter we give a summary of some of the aspects of supersymmetric dark matter which are covered by the the- 1 2 Chapter 1. Overview sis and the accompanying papers. In section 1.3 we briefly discuss Paper III which describes the DarkSUSY computer package, that we have devel- oped for the numerical calculations of theoretical predictions. In sections 1.4 we review Paper I on the relic density of supersymmetric dark mat- ter. It is here calculated within the supersymmetric model called minimal supergravity and the calculation includes all so-called coannihilation pro- cesses in the early Universe. In section 1.5 we discuss the work done in Paper II. This paper compare the possibilities for different astroparticle detectors to detect the cosmic relics of a minimal supergravity model. The discussion of the three papers will also contain references to the rest of the thesis. In this thesis we will go through some of the aspects of the papers in more detail. 1.1 Dark matter density from cosmology For many years there has been compelling evidence that luminous mat- ter only makes up a small fraction of all matter in the Universe. The observed rotation curves of galaxies were compared with the rotation in- ferred from Kepler’s third law and the estimated mass of the luminous galaxy. Around ten times as much mass was needed to explain the obser- vations. This missing mass was called dark matter, but the studies of the rotation curves did not give any hint of the nature of the dark matter. The picture of a large component of missing mass has been confirmed in the recent years by various cosmological observations. This has even revealed that the main part of the dark matter cannot be ordinary matter. Also, most of the missing mass is cold dark matter. The word cold means that these particles were non-relativistic at the time where structures began to form in the Universe. Even more surprisingly, cosmological observations have revealed that an even larger part of the energy in the Universe is something which has a negative pressure.