On Antihyperon-Hyperon Production in Antiproton-Proton Collisions with the PANDA Experiment A Thesis submitted for the degree of Master of Science in Engineering Physics Catarina E. Sahlberg Department of Nuclear and Particle Physics Uppsala University March 2007 2 Abstract The PANDA project is an international particle physics collaboration, aimed at investigating unsolved questions regarding the strong interaction. This will be done through the construction of a state-of-the-art particle detector, to allow detection of particles produced in antiproton-proton annihilation in experiments planned to be preformed at the future FAIR research centre in Darmstadt, Germany. The aim of this work is to contribute to the development of a software for simulations of reactions in the PANDA experiment. An event generator for the reaction pp → ΛΛ → pπ+pπ− was created, with regard to spin observables and target properties. Experimental information for the differential cross sec- tion of the pp → ΛΛ reaction, Λ/Λ-polarisation and Λ-Λ spin correlation was considered. ii Contents 1 Introduction 1 2 Theoretical Background 3 2.1 Introduction . 3 2.2 Matter Particles . 3 2.2.1 Quarks . 3 2.2.2 Leptons . 4 2.3 Forces and Force Carriers . 4 2.4 Hadrons . 6 2.4.1 The Quark Model . 6 2.4.2 Hadrons within the Quark Model . 6 2.4.3 Exotic Hadrons . 9 2.5 Open Questions . 9 2.5.1 Confinement . 9 2.5.2 The Origin of Mass . 9 2.6 Symmetries . 10 2.6.1 Parity . 10 2.6.2 Charge conjugation . 10 2.6.3 Time Reversal . 11 2.6.4 G-parity . 11 2.6.5 Broken Symmetries . 11 2.7 Note on the Units . 12 3 The PANDA Project 13 3.1 Introduction . 13 3.2 Physical Motivation . 14 3.3 FAIR . 14 3.4 Detector . 15 3.4.1 Interaction Region . 15 3.4.2 Target Spectrometer . 16 3.4.3 Forward Spectrometer . 17 3.5 Software . 18 4 The pp → ΛΛ → pπ+pπ− Reaction 19 4.1 The pp System . 19 4.2 The pp → ΛΛ Reaction . 20 4.2.1 Coordinate System . 21 4.2.2 Production Kinematics . 22 iii 4.2.3 Spin Observables . 24 4.2.4 Symmetries . 27 4.2.5 Cross Section . 28 4.3 Decay of Λ . 30 4.3.1 The Λ → pπ− Decay Channel . 31 4.3.2 Angular Distribution . 32 5 Simulations 35 5.1 Introduction . 35 5.2 Event Generation . 35 5.2.1 Extended Target . 36 5.2.2 Decay Vertices . 37 5.2.3 Differential Cross Section . 41 5.2.4 Polarisation . 42 5.2.5 Spin Correlations . 48 5.3 Reconstruction . 51 5.3.1 Extended Target . 51 5.3.2 Momentum of Λ from Opening Angles . 51 5.3.3 Production Angle of Λ.................... 56 6 Conclusion and Outlook 61 6.1 Summary and Conclusion . 61 6.2 Outlook . 62 A Statistics 69 A.1 The Method of Moments . 69 A.2 Weighting . 70 A.3 Random Number Generation . 70 A.3.1 Transformation Method . 70 A.3.2 Rejection Method . 71 B Relativistic Kinematics 73 B.1 Four-vectors . 73 B.2 Reference Frames . 74 B.3 Lorentz Transformation . 74 B.4 Mandelstam Variables . 75 B.4.1 Invariant Mass . 76 B.4.2 Four-momentum Transfer . 76 C Momentum in two-body decay 79 iv Chapter 1 Introduction In 1947, during an experiment studying the interactions of cosmic rays, G. Rochester and C. Butler discovered a new type of particle. The particle had a surprisingly long life time, approximately 13 orders of magnitude longer than what had been expected. This property of the particle along with the fact that it decayed via the weak interaction although being produced through the strong interaction, puzzled scientists. In the following years Rochester and Butler found other particles which showed similar such strange properties. These particle were assigned a property dubbed strangeness, and the particles were later to be referred to as ’strange particles’. [1] The discovery of the strange particles caused great excitement at the time, since they indicated the existence of a new form of matter which was completely unexpected at the time.[2] With this and other contemporary discoveries, the notion of a ’particle zoo’ was created, which referred to the multitude of new particles that were being discovered.[1] In 1964 Gell-Mann postulated the exis- tence of quarks to organize these particles, which lay the foundation of modern particle physics. Today, particle physics is one of the most active and expanding fields of physics. It sets out to explain the universal principles that govern even every- day phenomena by studies of the most elementary levels of the universe. The goal of particle physics is to gain understanding of the building blocks of matter and the forces between these that makes them stay together. To address ques- tions regarding these issues, particle physicists seek to create experiments that might show properties of the elementary interactions, by isolating and identify- ing reactions between elementary particles. Although experiments studying naturally occurring particles for instance in cosmic rays, similar to the experiments of Rochester and Butler, are still performed, most of the particle physics experiments today are made in large accelerator facilities. Here a beam of particles is created, and then sent to collide with another beam of particles or a slab of some material. The reactions between the beam particles and the colliding particles are then carefully detected and analysed. All of these experiments rely on sophisticated detectors that employ a range of advanced technologies to measure and record particle properties. In Germany, a new particle accelerator facility called FAIR (Facility for Antiproton and Ion Research) is being built that will be able to produce beams of antiprotons with higher intensities and energy resolutions than ever before. 1 It will be suited for a number of experiments, of which the PANDA experiment is one of the most prominent. The PANDA project is focused on developing a state-of-the-art particle detector to be used in the accelerator to study the properties of the force that enables the production of strange particles, the strong force. It will make use of a beam of antiprotons accelerated to high momenta, colliding with an internal target of protons. The aim of this Diploma thesis is to make a contribution to the development of a computer framework for simulations of the reactions thought to take place at the PANDA detector. This work has focused on the production of the light- est antihyperon-hyperon pair decaying to a proton-pion and an antiproton-pion pair, i.e. the reaction pp → ΛΛ → pπ+pπ− – where the Λ particle happens to be the second of the strange particles discovered by Rochester and Butler. The work includes the construction of an event generator for this reaction, with particles produced according to distributions based on experimentally deter- mined differential cross sections, polarisations and spin correlations. Regard has also been taken to the properties of the two main target types envisioned for PANDA. Although the work is limited to the discussion of Λ particle pro- duction and decay, the methods presented here should be possible to implement on other production and decay channels as well. This report starts with giving a brief introduction to particle physics in Chapter 2, to make the reader up to date with the theoretical background of the PANDA project. This chapter also treats some of the unresolved questions that explains the importance of the project. The project itself is described in the following chapter. Chapter 4 discusses the theory specific to the simulations that have been investigated and the work with the simulations is described in Chapter 5. The report is finished with a short conclusion and outlook. In the appendices some awkward but relevant theory and derivations are presented. 2 Chapter 2 Theoretical Background This chapter discusses the general theoretical background for the work. It gives a brief introduction to the Standard Model of particle physics (Section 2.1), and thereafter a description of the different parts of this theory: fundamental particles (Section 2.2), force carriers (Section 2.3) and non-elementary particles (Section 2.4). This is completed with a discussion of some of the complications with the Standard Model and some remaining question within the field of hadron physics (Section 2.5). The chapter is ended with a discussion of symmetries in quantum mechanics (Section 2.6) and a note on the units used in this work (Section 2.7). 2.1 Introduction The so-called Standard Model of Particles and Forces is a quantum field theory that describes all the current knowledge about particle physics. It describes the fundamental particles, of which all matter is composed, and the interaction be- tween these. The Standard Model includes 12 fundamental matter particles and their antiparticles, 12 force carrying particles that are responsible for the inter- action between the matter particles, as well as a number of thus far unobserved particles that has been predicted based on the theory. 2.2 Matter Particles The fundamental particles that make up the matter of the world can be orga- nized in two groups, the quarks and the leptons. These are both fermions1 of spin 1/2 and, as far as we know point-like. 2.2.1 Quarks There are six known quarks, ordered in three different categories, or generations, depending on their mass and charge properties (see Table 2.1). The first gen- eration of quarks consists of the light up (u) and down (d) quarks, the second generation of the strange (s) and the charm (c) quark, and the third generation of the heavy bottom (b) and top (t) quarks.