On Antihyperon-Hyperon Production in Antiproton-Proton Collisions with the PANDA Experiment

Home , Antiproton, Hypercharge, Hyperon, Particle physics, Standard Model, Up quark

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 section 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 existence 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 questions 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 lightest 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 determined 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 production 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 between these. The Standard Model includes 12 fundamental matter particles and their antiparticles, 12 force carrying particles that are responsible for the interaction 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 organized 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 generation 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.

1Particles of half-integer spin.

3 Generation Name Symbol Charge Mass Flavour Anti- e [MeV/c2] particle Up u +2/3 1.5 – 3.0 I = +1/2 u 1 3 Down d –1/3 3.0 – 7.0 I3 = −1/2 d Strange s –1/3 95 ± 25 S = –1 s 2 Charm c +2/3 1250 ± 90 C = 1 c Bottom b –1/3 4200 ± 70 B = –1 b 3 e Top t +2/3 171400 ± 2100 T = 1 t

Table 2.1: The properties of the six quarks.[3] All quarks carry baryon number 1/3 and spin 1/2. The anti-quarks have the same mass as their respective quark, but opposite ﬂavour, electric charge and baryon number.

All quarks carry a property called flavour (see column 6 of Table 2.1). The flavour of the u and d is the isospin (I), the up quark carry half a unit of positive third component isospin while the down quark carry half a unit of negative third component isospin. The s quark carry one unit of negative strangeness (S), the c quark one unit of charm (C), the b quark one unit of negative bottomness or beauty (Be) and the t quark one unit of topness or truth (T ). Quarks and antiquarks cannot be observed individually due to the property of the strong interaction and are always confined in particles called hadrons. The quarks are fermions and as such they have to obey the Pauli exclusion principle. But since they all have the same spin, in order to be distinguishable they carry an additional quantum number, called colour charge. Quarks carry either red, blue or green colour, while antiquarks carry antired, antiblue or antigreen colour. The only objects that can be observed are colourless, which arises from the fact that a colour together with its corresponding anticolour, or all three colours together, produce colour neutral objects.

2.2.2 Leptons There are six known leptons, and in the same way as with the quarks, they can be organized in three generations, as is indicated in Table 2.2. The electron (e) and the electron neutrino (νe) make up the first generation, the muon (µ) and the muon neutrino (νµ) the second, and the tau (τ) and the tau neutrino (ντ ) the third. The leptons carry a so-called lepton number, distinct for each generation. The total lepton number in a system is always conserved in any reaction. All visible matter in the universe is composed of the quarks and leptons of the first generation. Particles composed of quarks and leptons of higher generations are short lived and decays to particles composed of first generation fundamental particles. [4]

2.3 Forces and Force Carriers

There are four fundamental forces, from which all other forces can be derived: the strong, the weak, the electromagnetic and the gravitational. The ﬁrst three are included in the Standard Model, although eﬀorts – so far unsuccessful –

4 Generation Name Symbol Charge Mass Anti- e [MeV/c2] particle Electron e− –1 0.511 e+ 1 −6 Electron neutrino νe 0 < 2 · 10 νe Muon µ− –1 106.5 µ− 2 Muon neutrino νµ 0 < 0.19 νµ Tau τ − –1 1777 τ − 3 Tau neutrino ντ 0 < 0.018 ντ

Table 2.2: The properties of the six leptons.[3] The anti-leptons have the same mass as their respective lepton, but opposite lepton number and electric charge. have been made to include the last as well. But since the gravitational force is signiﬁcantly weaker than the others on the level of particle physics, it can be neglected. Each force within the Standard Model is mediated by gauge bosons, and couple to a certain property of the particles on which it acts. The properties of the gauge bosons are listed in Table 2.3. The notation J PC refers to the quantum numbers of the gauge bosons where J is the total spin, P is parity and C is C-parity. Parity and C-parity will be discussed in greater detail in Section 2.6.

Force Name Symbol Charge Mass J PC e [GeV/c2] Electromagnetic Photon γ 0 0 1−− Z boson Z0 0 91.16 ± 0.03 1−− Weak W boson W ± ±1 80.6 ± 0.4 1−− Strong Gluon g 0 0 1−

Table 2.3: The properties of the four gauge bosons.[3]

The electromagnetic interaction is mediated by the exchange of massless photons and acts on the electrical charge. It has infinite range and is practically the only force within the Standard Model that we notice in our daily life. The electromagnetic force is responsible for keeping the electrons and nucleons of the atom together, as well as keeping together the different atoms in a molecule. The phenomena involving the electromagnetic interaction is described in the theory of Quantum Electrodynamics (QED). [5] The weak interaction is mediated by the heavy bosons Z0, W + and W −, and acts on the flavour of particles. It has a relative short range, about 10−15 m, and is the only force that can violate the conservation of flavour in an interaction. Therefore it plays a major role in the process of a nuclear decay, for example β− decay in which a neutron is converted into a proton together with an emission of an electron and an electron neutrino. [5] The strong interaction is carried by massless particles called gluons and cou- ples to the colour charge of the particles. The strong force has a smaller range than the weak force, only about 10−18 m and is the force that creates the interaction between quarks. It is also responsible for binding the nucleons together inside the nucleus. The theory describing the strong interaction is called Quan- tum Chromodynamics (QCD). Unlike the one photon of the electromagnetic

5 interaction, and the three W and Z bosons of the weak interaction, the Stan- dard model includes eight independent gluons for the strong interaction. And unlike the photon, which is electrically neutral, the gluons themselves carry colour charge, which means that they can interact among themselves. It is this property that causes the quarks to be conﬁned within the hadrons. Gluons are thought to simultaneously carry both colour and anticolour. [5]

2.4 Hadrons

Particles composed of quarks and gluons are called hadrons, defined by their interactions via the strong force. The hadrons can be either baryons, defined as particles of baryon number B = 1, or mesons, defined as particles of baryon number B = 0, where the baryon number of a particle is the sum of the baryon number of its constituents. Hadrons are seen as containing both so-called valence quarks – and possibly also valence gluons – and a sea made up of virtual gluons and antiquark-quark pairs. The valence quarks (and gluons) give the hadron its characteristic quantum numbers and also its dynamical properties regarding mechanisms in decay and particle production. Hence, when discussing the constituents of a particle, it is the valence quarks that are referred to. The quark-gluon sea has no effect on the quantum numbers, but determines other properties of the hadron, such as the electric charge distribution and magnetic moment within the particle. [6]

2.4.1 The Quark Model The quark model is a mean of structuring the hadrons, based on their quantum numbers. The quark model predates the theory of QCD, and was developed by Gell-Mann in order to classify the multitude of particles that was being discovered in the 50’s and 60’s. It was soon understood that the observed particles could not all be elementary particles. Gell-Mann (together with Nishijima) be- gun by classifying the hadrons, and went on to postulate the existence of the quarks as the particles composing the hadrons. The quark model describes essentially all hadrons that have been observed thus far. It is, however, far too simple to include all hadrons that can be predicted by the theory of QCD. Particles that cannot be described by the quark model are called exotic hadrons. The quark model organizes the hadrons in a structure based on their I, C and Y quantum numbers, where I is the particle isospin, C is the charm, and Y is the hypercharge. The hypercharge is deﬁned as Y ≡ S + C + Be + T + B, where S is strangeness, C is charm, Be is bottomness, T is topness and B is baryon number. The hadrons reﬂect the properties of their constituents, and thus the quantum numbers of a hadron can be found by considering the quantum numbers of its valence quarks. Figure 2.1 shows the structure of the quark model.

2.4.2 Hadrons within the Quark Model Hadrons that can be classiﬁed according to the quark model are normally referred to as ’ordinary’ hadrons, as opposed to the ’exotic’ ones that are found outside of this model. The hadrons within the quark model are either baryons

6 (a) (b)

Figure 2.1: Quark model structuring of the hadrons showing the baryon octet (a) and decuplet (b) as well as the meson pseudoscalar (c) and vector (d) 20- plets.[3] The axes of the coordinate system are I, C and Y , where I is the particle isospin, C is the charm, and Y is the hypercharge.

7 consisting of three quarks (qqq) or mesons consisting of a quark and an antiquark (qq). Baryons are fermions, while mesons are bosons.

Name Symbol Quark Mass J PC Mean life Anti- structure [MeV/c2] [s] particle Proton p uud 938.3 1/2+ > 6.6 · 1036 p Neutron n ddu 939.6 1/2+ 885.7 ± 0.8 n Delta ∆ udd 1.232 3/2+ 6 · 10−24 ∆ Lambda Λ uds 1.115 1/2+ 2.6 · 10−10 Λ Sigma Σ uds 1.197 1/2+ 1.5 · 10−10 Σ Xi Ξ uss 1.315 1/2+ 2.9 · 10−10 Ξ

Table 2.4: The properties of some baryons.[3] All baryons have baryon number 1, and all antibaryons have baryon number -1.

A list of some important ordinary baryons and their properties is presented in Table 2.4. The only stable baryon is the proton, which has a mean life that by far exceeds the estimated age of the universe, which is currently approximated to 13.7 billion years [7]. Protons and neutrons form the nucleus of an atom, and are therefore referred to as nucleons. Bound neutrons are stable, since it is energetically impossible for them to decay within a stable nucleus. Baryons that have non-zero strangeness, but zero charm, bottomness and topness are called hyperons. The lightest of the hyperons is the Λ hyperon. The hyperons are relatively long lived. All decay via the weak interaction – apart for the Σ0 that decay electromagnetically – directly or through a series of decays to a nucleon and one or more mesons. The hyperons do not normally form bound states, but can occur in short lived so-called hypernuclei. Some important mesons and their properties are presented in Table 2.5. Note that the neutral pion, π0, is its own antiparticle. The only relatively stable meson are the charged pions and the kaons, which means that they are the only mesons that can be detected before they decay. In certain models of the strong interaction, the interaction is mediated by mesons through a so-called meson exchange.

Name Symbol Quark Mass J PC Mean life Anti- structure [MeV/c2] [s] particle (uu−dd) π0 √ 135.0 0−+ 0.84 · 10−16 π0 Pion 2 π+ ud 139.6 0−+ 2.6 · 10−8 π− K+ us 493.7 0−+ 1.2 · 10−8 K− Kaon 0 −+ −8 0 K(L) ds 497.7 0 5.2 · 10 K D+ cd 1869 0−+ 1.1 · 10−12 D− D meson 0 D0 cu 1864 0−+ 4.1 · 10−13 D

Table 2.5: The properties of some mesons.[3].

8 2.4.3 Exotic Hadrons The theory of QCD predicts the ordinary hadrons, but it does not rule out the possible existence of other, more complex, types of particles, provided these are colour neutral. However, the existence of such hypothetical particles, referred to as exotic hadrons, remains a subject of controversy. Exotic hadrons could be consisting of just quarks, such as the tetraquarks (qqqq) or the pentaquarks (qqqqq). It is also possible that gluons could form compounds, either on their own – creating glueballs (gg) – or in combination with quarks – forming so-called hybrids (qqg-mesons or qqqg-baryons). [2] The exotic hadrons can be of two different categories. The first type are those with quantum numbers that cannot be fitted into the schematic system of the quark model. This could be either particles with anomalous flavour or charge, or particles with anomalous spin-parity quantum numbers. The detection of particles with such forbidden quantum numbers would thus indicate the existence of exotic hadrons. So far, the only available candidates for particles within this category are mesons with spin-parity J PC = 1−+, although these results are not conclusive. [6] The second type are those exotic hadrons that have coinciding quantum numbers with other ordinary hadrons, but with a different valence structure, resulting in anomalous dynamical properties. These are called cryptoexotic, or hidden-exotic hadrons. All of the serious candidates for exotic particles are found within this category, for example baryons with hidden strangeness (qqqss) and mesons with hidden charm (qqcc). [6]

2.5 Open Questions

Apart from wanting to find the particles predicted by QCD discussed in the previous section, research in the field of hadron physics is made to answer some of the existing unsolved questions. These include the question of confinement of the quarks in hadrons, and the question regarding the origin of the mass of the hadrons.

2.5.1 Confinement Unlike all other forces, which grows weaker with distance, the impact of the strong force is small at close distances, and grows stronger if the distance is increased. This is a part of the explanation to the question of confinement of the quarks. If trying to separate the constituents of a meson – that is trying to separate the quark from the antiquark – the gluonic field of the strong force would eventually get so large that a new quark and anti-quark would be formed, in between the separated pair. Consequently, two antiquark-quark pairs would be formed, replacing the one pair that existed in the beginning. To explain why this mechanism occurs is one of the remaining tasks of hadron physics. [8]

2.5.2 The Origin of Mass From Tables 2.1 and 2.4, it is evident that the mass of a nucleon (proton or neutron) is signiﬁcantly larger than the sum of the masses of its three constituent

9 valence quarks. The rest of the nucleon mass has to be attributed to the kinetic energy of the quarks and to the energy of the interactions between these. Analogously, only a part of the spin of the nucleon can be attributed to the valence quarks, but must be explained by other means. These eﬀects of mass generation are not described by the standard model, but there is hope that new experiments will shed some light on this issue. [9]

2.6 Symmetries

The concept of symmetry is important in quantum mechanics, and particularly in particle physics. The standard model has three related symmetries of the matter universe, namely: parity (P ), which is the reflection of space; charge conjugation (C), which is the reflection to the antimatter universe; and time reversal (T ), which is the reflection of time.

2.6.1 Parity The operation of parity reverses the momentum of a particle, but conserves the direction of its spin. Consequently it changes the handedness of a system, turning a right handed system into a left handed and vice versa. The operation of parity can be seen as turning a system into its mirror image, for which the same physical laws are assumed to be valid as for the original system. If the spatial part of the wave function for a system is symmetric under the parity operation, it is said to have even parity, in particle physics denoted P = +. If it on the other hand is antisymmetric under the operation, the state is said to have odd parity, denoted P = −. The parity of a composite system is given by the parity of its constituents, according to

L P = P1P2(−1) , (2.1) where Pi denotes the parity of the constituents, and L is the orbital angular between the constituents. [2]

2.6.2 Charge conjugation The operation of charge conjugation turns a particle into its antiparticle, con- serving the direction of the spin. The operation can be seen as taking a system in the matter world and turning into its image in the antimatter world, where the same physical laws are assumed to be applicable as in the matter world. The eigenvalues of the charge conjugation operator are called the C-parity of the system. However, it is only a very few particles that have wave functions that are eigenstates to the charge conjugation operator; only the truly neutral particles such as γ and π0 will have an associated C-parity. This is because such particles are their own antiparticles. The C-parity of a composite system made up of a particle-antiparticle pair is given by the C-parity of its constituents, but has diﬀerent formulae depending on if the particles are fermions or bosons. For bosons it is given by

C = (−1)L, (2.2)

10 where L is the orbital angular momentum of the composite system. For fermions, on the other hand, it is given by

C = (−1)L+S, (2.3) where L is the angular momentum of the composite system, and S is the spin angular momentum of the system. [2]

2.6.3 Time Reversal The operation of time reversal changes the direction of time, but keeps all another quantities conserved. This symmetry is rather counter intuitive, and in fact, the overall universe does not seem to be symmetric under the change of the direction of time. In this larger picture, the notion of time is closely intertwined with the idea of entropy, giving a distinct notion of ’past’ and ’future’ based on the increase or decrease of the entropy. In particle physics, however, the universe is seen on a much smaller scale, where global quantities such as entropy has no real meaning. When viewing the world from this scale, local properties show a ﬁne symmetry under the operation of time reversal.

2.6.4 G-parity G-parity is not a symmetry as such, but combination of charge conjugation and a rotation. It is given by the C operation in addition to a rotation of the angle π around one of the axis of the isospin space of a particle. As with the charge conjugation, the G operation will only have eigenvalues for neutral systems. The G-parity of a system of a boson and antiboson pair is given by G = (−1)S+I , (2.4) where S is the spin angular momentum of the composite system and I is the isospin. For a fermion-antifermion pair the G-parity is given by

G = (−1)L+S+I , (2.5) where S is the spin angular momentum of the composite system, L is the orbital angular momentum and I is its isospin. [2]

2.6.5 Broken Symmetries The parity and the charge conjugation is conserved in both the strong and electromagnetic interaction, but is broken by the weak interaction. Since the strong interaction also conserves the isospin, this means that also the G-parity is conserved in this interaction. It is not, however, conserved in the weak and in the electromagnetic interactions, since these do not conserve isospin. The standard model predicts that if applying all three symmetry operators at the same time, the result would always show symmetry. This phenomenon is called CPT invariance, and has so far proved to be true. There are, however, systems that show a broken symmetry under the combined C and P operations. This also means, under the assumption of CPT invariance, that the T symmetry must be broken, and that the system thus shows a preference for one direction of time. This violation of CP -symmetry,

11 and thereby of T -symmetry as well, has so far only been observed in the weak interaction. The breaking of the CP -symmetry might give an explanation to why the world is made up of only matter and not equal parts of matter and antimatter, as is predicted by the theory of the Big Bang.

2.7 Note on the Units

In most fields of physics, it is often convenient to use a system of units appropriate to that specific field. Particle physics is no exception, and has thus adopted a system of so-called natural units. The system is chosen so that the two fundamental constants of quantum mechanics, the reduced Planck’s constant, ~, and the speed of light in vacuum, c, are set to unity. These two constants would in conventional SI-units be given by h ≡ = 1.055 · 10−34 Js = 6.582 · 10−22 MeVs ~ 2π c = 2.998 · 108 ms−1.

In the system where ~ = c = 1, ~ can be seen as one unit of action, and c as one unit of speed, and with the addition of one unit of energy as 1 eV, the system of units is completely deﬁned. [10] By adopting the system of natural units, ~ and c can be omitted in formulas, which leads to considerable simpliﬁcations. The dimension of all quantities will also have some power of energy; mass (m), momentum (mc) and energy (mc2) are expressed in MeV, and both time (~/mc) and length (~/mc2) are described in units of MeV−1. It is, however, easy to convert a quantity back to practical units by using the conversion factor

−13 ~c = 1.973 · 10 MeVm. In the remainder of this work, these natural units will be used in calculations, although some quantities will be given in SI-units.

12 Chapter 3

The PANDA Project

This chapter will address the PANDA project, including a short introduction (Section 3.1), physical motivations to the experiments (Section 3.2) and a description of the facility where the experiment will take place (Section 3.3). The detector itself will be described in Section 3.4 and the computational framework of the experiment in Section 3.5.

3.1 Introduction

The PANDA (antiProton ANihilations DArmstadt) project is an international collaboration, involving more than 300 researchers, at 40 different institutions in 15 countries worldwide.[11] The project started a few years back, and will continue for many years to come. The aim of the project is to study the properties of the strong interaction. This will be done by accelerating antiprotons to large speeds and letting them collide with protons, and observing the outcome of these collisions. In order to do this, two things are required: Firstly, an advanced accelerator facility to obtain the required energies and intensities of the antiprotons and secondly, a sophisticated detector system to be able to the detect the produced particles. To meet the first requirement, the PANDA experiment will take place at the Gesellschaft für Schwerionenforschung (GSI) in the German city of Darmstadt, where a new accelerator facility called FAIR (Facility for Antiproton and Ion Research) is currently being built. FAIR will be able to provide the experiment with antiprotons accelerated to the necessary energies and intensities, and thus it is envisioned that predicted particles never seen before will able to be detected. The second requirement will be met by the construction of a state-of-the-art detector. This detector is the heart of the PANDA project. It is currently in its research and development phase, and is planned to start taking data at the High Energy Storage Ring (HESR) at FAIR in 2012.[12] The hope is that experiments like PANDA will be of decisive importance for developing an understanding of the properties of the strong force, and also both confirm earlier predictions made from QCD and at the same time generate observations that can serve as an input to the development of the theory.

13 3.2 Physical Motivation

The physics program for the PANDA experiment includes many diﬀerent topics all related to the properties of the strong interaction. As a start, experiments such as precision spectroscopy of charmonium, the search for exotic objects such as hybrids and glueballs, the study of the properties of charmed hadrons and γ-ray spectroscopy of hypernuclei are foreseen.[9] Charmonium, for example the J/ψ meson, is a bound state of a charmed quark and antiquark pair, cc. It was named in analogy with positronium, the bound state of an electron and a positron. Since charmonium has a net zero charm, it is often said that its states contain hidden charm. The charm quarks are relatively massive, making their motion almost non-relativistic and the potential they move in almost static.[4] This gives the charmonium a positronium like spectrum, with energy levels described by the potential between the charmed quark and antiquark. Studies of the charmonium spectrum would therefore give information about properties of the interaction between the quarks. The physics program studying charmonium states would include precise measurement of mass, width and decay branches of all states through spectroscopy.[13] Hybrids and glueballs are discussed in Section 2.4.3. Here, the aim of the PANDA project is to establish the predictions from QCD regarding these, using high statistics measurement and advanced spin-parity analysis.[13] Such studies of heavier quarks would give insights in the gluon interaction responsible for the generation of a part of the hadron masses. Hypernuclei are nuclei that contain not only nucleons, but also one or more hyperons. Precision γ-ray spectroscopy of hypernuclei will gain knowledge about their structure and the nature of the interaction between nucleons and hyperons as well as between hyperons and other hyperons.[13] Further along the project, other subjects of study are envisioned. These will include the search for CP -violation in the strange and charmed regions, i.e. in the decays of D mesons or in the ΛΛ system, as well as spectroscopy of D meson decay in the search for rare leptonic and hadronic decay. [13]

3.3 FAIR

The new international research facility FAIR will be a major upgrade of the current GSI facility. The construction of the new parts is planned to start this year (2007), with the ﬁrst experiments taking place in 2012, and the whole construction being completed by 2015. The costs for this building project are estimated at approximately 1.2 billion euro, 65 percent of which is paid by the German government. It is estimated that four diﬀerent experiments will be able to run simultaneously at the facility. [14] The outlines for the existing GSI and the upcoming are shown in Figure 3.1, where the existing GSI facility with its linear accelerator UNILAC is shown to the left; and the upcoming FAIR facility to the right with the double ring synchrotron SIS 100/200 and the High-Energy Storage Ring (HESR). The new double ring synchrotron accelerator, which has a circumference of about 1.1 km, will use the current GSI facility as an injector. The SIS 100/200 will produce high energy protons, which will be used to create an antiproton beam. The antiprotons will be collected and stochastically cooled in the CR

14 Figure 3.1: The existing GSI facility to the left (shaded area) and the new FAIR facility to the right. [12] and RESR rings, and then injected into the 574 m long HESR. The HESR is envisioned to be able to store 1 · 1011 antiprotons of momenta from about 1.5 up to 15 GeV at a time. [15]

3.4 Detector

The PANDA detector is designed to be a versatile system, able to measure both electromagnetic and hadronic ﬁnal states. The event rate, i.e. the number of particle reactions per second, is estimated to 2 · 107 per second. The goal is to make the detector cover nearly the full solid angle. To manage this, the detector is composed of two parts: a cylindrical target spectrometer and an extensive forward spectrometer. [15] Figure 3.2 is showing a cross section view of the PANDA detector in the horizontal xz-plane. The coordinate system of the detector is given by the accelerated beam going along the z-axis, and the target beam travelling in the negative y-direction. Although the design, location and properties of most of the detector subssys- tem are decided, there are still some unsettled questions. In some parts more than one solution is possible, and the question to answer is which one of these that would be optimal, in that it could fulﬁll all the requirements while at the same time keeping the costs at a minimum.

3.4.1 Interaction Region The detector in the PANDA experiment makes use of an internal target of protons. Various target options are being considered, although the two most

15 Figure 3.2: The PANDA detector in the xz-plane, showing the antiproton beam coming in from the left, and the proton target going into the page. [16] prominent are a cluster jet target and a pellet target. The cluster jet target equipment produces a jet of ultra-dense hydrogen that is sent through the antiproton beam. The pellet target consists of a stream of small pellets made from frozen hydrogen that is sent through the beam. Both these targets are constructed in such a way so that the horizontally incoming beam will hit the target particles, arriving to the interaction region in a vertical stream. The interaction between the beam and target will then occur in a volume that depends on the beam and target widths, centred around the origo of the detector system. The pellet target can be somewhat cumbersome to use, and the cluster jet is thus favourable from a practical point of view. However, the cluster jet has not yet managed to deliver the luminosity desired for PANDA, although eﬀorts to attain this are still being made. Furthermore, the cluster jet creates larger interaction region, which complicates the reconstruction of events, in particular when dealing with produced particle of short life times. Currently, a solution is foreseen where both target options would be possible to use in the detector, depending on the requirements of the speciﬁc experiment. [4]

3.4.2 Target Spectrometer The target spectrometer (TS) is a detector system with cylindrical symmetry that detects particle emitted at relatively large angles. The planned outline is shown in Figure 3.2, and the diﬀerent components, from the interaction region and outwards are the following [13]:

MVD A micro-vertex detector (MVD) for detection of charge particles is directly located around the interaction region. The MVD is arranged in a barrel structure with ﬁve layers together with four additional layers in the

16 forward direction. The MVD barrel consists of pixel detectors while the forward layers are made up of microstrip detectors. STT/TPC The next layer will be either a Straw Tube Tracker (STT) or a Time Projection Chamber (TPC). The STT, consisting of self-supporting straws in double layers, is considered being a safe fall-back solution to the technically more challenging TPC. MDC There are two multi-wire drift chambers (MDC) positioned in the forward direction from the STT/TPC, to detect particles emitted at small forward angles. Their function is similar to the STT. TOF In the layer following the STT/TPC, as well as behind the second MDC, Time-Of-Flight counters (TOF) will be placed to measure the flight time of the produced particles. One option is to use fast scintillating materials in thin strips to be read by photomultiplier tubes. [4] DIRC Outside of the cylindrical TOF there is a Detector of Internally Reflected Cherenkovˇ light (DIRC), which is a type of Ring Imaging Cherenkovˇ Counter (RICH). The DIRC is composed of quarts rods, in which the Cherenkovˇ light is internally reflected to photon detectors at the edges. A second DIRC, made of quartz radiators, will be placed after the two MDC:s. EMC The next component is the Electromagnetic Calorimeter (EMC), made up of both a barrel part and a forwards and backwards layer. The EMC will likely be made of crystals of P bW O4, which is a scintillating material that gives fast signals and has fair resolution. MUO All these components are all surrounded by a solenoid coil and iron yoke. The solenoid will yield a magnetic field of approximately 2 T. The iron yoke stops all produced particles, with the exception of muons. The last component of the TS, and the one furthest away from the interaction region, is therefore a set of muon counters (MUO).

3.4.3 Forward Spectrometer The forward spectrometer (FS) is designed to detect particles emitted at relatively small angles, approximately at angles below 5o in the vertical direction and below 10o in the horisontal direction, as well as give additional information to that of the TS system for particles emitted at angles below 22o. The outline is shown in Figure 3.2, and the components are the following, from the edge of the TS and downstream [13]:

MDC A row of six vertically placed MDC:s is envisaged from the very edge of the TS and continuing more than half the length of the FS. These will be similar to those of the TS, and will detect the charged particles that are emitted at small angles. TOF Forward TOF:s will be placed behind the last MDC, as well as on both sides of the row of MDC:s, to detect and identify forward going particles with a moderate momentum. They will be made of plastic scintillating strips coupled to photomultiplier tubes.

17 RICH After the TOF:s, a Ring Imaging Cherenkovˇ Counter (RICH) will be used. It is designed to compensate for the uncovered space of the DIRC in the forward region of the TS. The RICH will probably be made of some type of aerogel, connected to photon detectors. EMC Because of the size of the forward EMC, a less expensive alternative to the high-performance target EMC is sought. The solutions considered are either a lead-glass or a so-called Shashlyk EMC, both being about one order of magnitude cheaper than the target EMC, while at the same time only decreasing the resolution by a factor of two. HC A Hadron Calorimeter (HC) is placed right next to the EMC, consisting of steel and scintillator plates arranged in two layers. It will measure the energies of hadrons as well as energy losses of muons, MUO The outermost component of the FS will, similar to the TS, be a set of moun counters.

3.5 Software

The antiproton-proton annihilations in the PANDA experiment have been simulated using so-called Monte Carlo methods1 in software speciﬁc to this experiment. Such simulations are made to imitate the response of the proposed detector to get input for further improvements regarding materials and geometry and to make sure that the goals of PANDA are met. It is also important as a test bench for the development of the reconstruction and analysis software to be used for the experimental data from the actual experiment. [15] The PANDA framework is a complete simulation system, written in C++. All parts are not completely implemented yet, but the basic structure is. It consists of four major components: event generation, detector simulation, reconstruction and analysis. It uses the latest version of the CERN platform GEANT (GEometry ANd Tracking), Geant42, in the particle propagation and classes from the CERN analysis program ROOT3 are used in the event generation. Both these systems enable the handling and analyzing of large amounts of data in eﬃcient ways.

1Monte Carlo methods are computational algorithms that are based on random numbers. They are often used when simulating the properties of physical systems. 2Geant4 is an object-oriented software toolkit that uses Monte Carlo methods to simulate particle propagations in material.[17] 3ROOT is an object-oriented software, developed at CERN and designed for particle physics data analysis.[18]

18 Chapter 4

The pp → ΛΛ → pπ+pπ− Reaction

This chapter discusses some theoretical aspects for the reaction that is in focus of this work, the pp → ΛΛ → pπ+pπ− reaction. It starts with treating the properties of the pp system (Section 4.1). Then the ΛΛ production in the pp annihilation (Section 4.2) is discussed, and the chapter ﬁnishes with treating the decay of the Λ hyperon (Section 4.3).

4.1 The pp System

The collision of an antiproton beam with a beam of protons can result in several different reactions. At a beam energy of 2 GeV, about 40 percent of the collisions results in elastic scattering.[19] The rest is referred to as antiproton-proton inelastic scattering and annihilations. The latter will be considered here. To find out which particles are allowed in the final state from the pp annihilation process, the quantum numbers of the initial state needs to be considered. The quantum numbers of the proton and the antiproton are given in Table 4.1.

Quantum number Symbol Proton Antiproton Electric charge Q +1 –1 Baryon number B +1 –1 Total spin J 1/2 1/2 Isospin I 1/2 1/2 Third component isospin I3 +1/2 –1/2 Parity P +1 –1

Table 4.1: The quantum numbers of the proton and anitproton.

The quantum numbers of the antiproton and proton are combined to establish the total quantum numbers of the initial pp system. The baryon number and charge are scalars and as such simply additive, while other quantum numbers are slightly more complicated to handle. The parity of the composed system L is given by (see Section 2.6.1) P = PpPp(−1) , where L is the orbital angular

19 momentum of the system, and the spin and isospin can be found by treating theses quantities as vectors. Therefore the total quantum numbers of the initial pp system are:

• Electric charge, Q = 0;

• Baryon number, B = 0;

• Isospin, (I,I3) = (1, 0) or (I,I3) = (0, 0); • Spin, S = 0 or S = 1

• Parity, P = (−1)L+1

In addition, the system also has zero strangeness and charm. Furthermore, the charge conjugation of the system is given by (cf. Section 2.6.2) C = (−1)L+S and the G-parity (cf. Section 2.6.4) by G = (−1)L+S+I . All these quantum numbers puts constraints on the possible ﬁnal state particles that can occur in the pp annihilation. There is, however, a multitude of possible particles that can be produced in the antiproton-proton annihilation. Examples are two body processes such as pp → YY , where Y denotes a hyperon or pp → mm, where m denotes a meson or three body processes, such as pp → Y Y π.[19] The channels of interest here are the hyperon decay channels, namely pp → ΛΛ, pp → ΛΣ0, pp → Σ0Λ, pp → Σ+Σ+, pp → Σ−Σ−, pp → Ξ−Ξ−, pp → Ξ0Ξ0 and pp → Ω−Ω−. All of the produced particles here decay via the weak interaction, with the exception of the Σ0 that decays electromagnetically to Λγ. The Λ and the Σ± all decay to Nπ, where N denotes a nucleon, whilst the other decay in one or more steps to a Λ particle and one or more pions. Thus, the most straightforward approach to study the hyperon production channels would be to start with the ΛΛ channel. This is also the only hyperon channel were high quality experimental data on relevant quantities such as spin observables are available.[20] Consequently, the subject of the following sections will be a more detailed discussion of the pp → ΛΛ reaction.

4.2 The pp → ΛΛ Reaction

The reaction pp → ΛΛ takes place via the strong interaction, which conserves parity and charge conjugation as well as ﬂavour. The most interesting feature of the reaction is the process where strange quarks are created. There are two ways to look at the reaction, as depicted in Figure 4.1. Figure 4.1(a) shows the reaction in a so-called quarkline diagram. Here the proton and the Λ hyperon are viewed as composed of a diquark and a quark. The diquark has the same ud quantum numbers with isospin and spin zero for both the proton and the Λ and is indicated by the shaded areas in Figure 4.1(a). In this view, the important process is the annihilation of the uu quark pair and the production of an ss pair, whilst the diquarks of the proton and antiprotons are merely spectators to this process. The observables for the pp → ΛΛ reaction should thus indicate properties of the underlying uu → ss process. [21] An alternative way to view the pp → ΛΛ reaction is through meson exchange, illustrated in Figure 4.1(b). In this model, a K+ meson (consisting of a u and

20 (a) (b)

Figure 4.1: Two diﬀerent ways to view the reaction pp → ΛΛ: (a) A quarkline diagram; (b) A meson exchange diagram. [20] an s quark), is exchanged and thus creating the strangeness of the ﬁnal state. [21]

4.2.1 Coordinate System

The pp → ΛΛ reaction has two initial and two ﬁnal state particles, and thus only two truly independent momentum vectors, namely the initial momentum vector of the beam antiprotons ~pi and the momentum vector of one of the produced Λ particles ~pf . The plane formed by these two vectors is called the production plane, and is unique for each event. Using this information, a coordinate system for each of the Λ and Λ particles in each event, can be created. This is usually done in the Centre-of-Mass (CM) frame of the reaction, letting the z-axis of the coordinate system be in the direction of the Λ particle, i.e. in the direction of the vector ~pf . The y-direction is then taken as the direction of the normal of production plane, which will mean that the z- and y-directions are orthogonal. The x-direction is then chosen orthogonal to both the z- and the y-direction, and in such a way that the constructed coordinate system is right handed. This is illustrated in Figure 4.2, where θ∗ is the production angle of the Λ particle in the centre of mass of the reaction.

Figure 4.2: Coordinate system for the reaction pp → ΛΛ as it is constructed in the CM frame of the reaction.[22]

21 More formally, the coordinate systems are constructed according to ~p ~p × ~p z = Λ , y = i f and x = y × z , (4.1) |~pΛ| |~pi × ~pf | and ~p ~p × ~p z = Λ , y = i f and x = y × z , (4.2) |~pΛ| |~pi × ~pf | where {x, y, z} denotes the coordinate system of the Λ rest frame and {x, y, z} denotes the Λ rest frame coordinate system.

4.2.2 Production Kinematics The kinematics of the reaction pp → ΛΛ will be treated using relativistic kinematics. This ensures, at all times, a correct kinematical treatment. Thus, four- vectors are used to describe the properties of the particles involved. A summary of some, for this discussion, important aspects of relativistic kinematics is given in Appendix B.

Threshold momentum The beam momentum that corresponds to the minimum amount required for a speciﬁc reaction to take place is called the threshold momentum of the reaction. At threshold, the particles will be produced with zero relative momentum, along the direction of the antiproton beam. As the beam momentum increases the kinetic energy of ΛΛ increases, and the particles will be produced with increasing angles. Taking data from Table 2.4 the minimum total energy for producing a ΛΛ pair can be obtained as

Emin = mΛ + mΛ = 2 · 1.1157 GeV = 2.2304 GeV, (4.3) which corresponds to an antiproton beam energy in laboratory of

1 2 2 2 Ep = (Emin − mp − mp) = 1.713 GeV, (4.4) 2mp or a threshold momentum of

q 2 2 pp = Ep − mp = 1.435 GeV. (4.5) Thus, the antiprotons in the incoming beam must have a momentum larger than 1.435 GeV for the ΛΛ reaction to occur.

Invariant Mass The invariant mass squared of a system of two-particle scattering, is given by (B.12) as 2 2 s = m1 + m2 − 2p1p2 (4.6) In the laboratory frame of reference, where an incoming antiproton beam in- teracts with a proton target at rest, the invariant mass squared can be reduced to q 2 2 2 2 2 s = mp + mp + 2mpEp = 2mp + 2mp |~pp| + mp . (4.7)

22 In the CM frame of reference, the invariant mass can be expressed according to (B.14) as ∗ 2 s = (Etot) , (4.8) ∗ where Etot is the the total energy in the CM frame. In this case, the total energy in the CM frame can either be expressed as the sum of the energy of the antiproton/proton or of the Λ/Λ,

∗ ∗ 2 ∗ 2 ∗ 2 2 s = (Ep + Ep ) = (2Ep ) = 4( ~pp + mp) , (4.9) ∗ ∗ 2 ∗ 2 ∗ 2 2 s = (EΛ + EΛ) = (2EΛ) = 4( ~pΛ + mΛ) . (4.10) √ From this, it is easy to see that the invariant mass, s, at the threshold momentum of the pp → ΛΛ reaction, will be simply the sum of the rest masses of the hyperons √ s = 2.23GeV. (4.11)

Four-Momentum Transfer The four-momentum transfer squared of a general two-particle scattering process is given by (B.19). For the case of the pp → ΛΛ reaction this can be written as

2 2 t = mp + mΛ − 2pppΛ, (4.12) where the last term can be expressed as

2pppΛ = 2(EpEΛ − ~pp~pΛ). (4.13)

The four-momentum transfer is always negative, due to the fact that the masses of the proton and hyperon are small compared to the product of their four- momenta. It can sometimes be advantageous to express the four-momentum transfer in terms of the invariant mass of the system. This can be done by considering the process in the CM frame and using the expressions for the invariant mass squared given in Equations (4.9) and (4.10) (see the derivation in Section B.4.2). This gives the ﬁnal expression of the four-momentum transfer squared as 1 1q t = m2 + m2 − s + (s − 4m2)(s − 4m2 ) cos θ∗ . (4.14) p Λ 2 2 p Λ The smallest absolute value of the four-momentum transfer squared occurs when cos θ∗ = 1, i.e. when the CM-scattering angle is zero and the Λ particle is produced along the direction of the incoming antiproton. It is often convenient to consider the so-called reduced four-momentum transfer squared [19],

0 ~∗ ~∗ ∗ t = t − t0 = 2 p p p Λ (cos θ − 1) , (4.15)

∗ where t0 denotes the four-momentum transfer squared at θ = 0. This can also be written in terms of the invariant mass, using Equations (4.9) and (4.10), as 1q t0 = (s − 4m2)(s − 4m2 )(cos θ∗ − 1) . (4.16) 2 p Λ It is evident that also the reduced four-momentum transfer squared is a quantity that is always negative.

23 Excess Energy The excess energy in a reaction is given by √ X = s − mf , (4.17) where mf denotes the mass of the ﬁnal state particle. The excess energy for the pp → ΛΛ is given by √ = s − 2mΛ. (4.18) This quantity represents the total kinetic energy that is available for the ΛΛ in the pp CM system. In the case of an incoming beam hitting an almost ﬁxed target, as is the case here, the expression for s in (4.7) can be used, to obtain r q 2 2 2 = 2mp + 2mp |~pp| + mp − 2mΛ. (4.19)

Thus it is possible to determine the excess energy from merely the masses of the proton and Λ and the beam momentum.

4.2.3 Spin Observables The polarisation, P~ , of the hyperon describes the direction of its spin, J. It is defined as [21] hJi P~ = , (4.20) J where hJi is the expectation value of the spin operator J. The spin correlation coefficients Cij describe the correlation between the spin projection in the i-direction of the antihyperon with the spin projection in the j-direction of the hyperon. It is defined as [21]

hJiJji Cij = , (4.21) JΛJΛ where hJiJji is the expectation value of the operator JiJj and JΛ and JΛ are the spin of the particles.

Available Data

Data for the polarisation in the y-direction, Py, of the reaction is shown in Figure 4.3 for excess energies between 0.57 and 3.95 MeV, i.e. very close to the threshold of the reaction. Figure 4.4 shows the y-component of the polarisation for beam momenta of 1.642 GeV and 1.918 GeV. There is only one complete set of data for the spin correlation coeﬃcients.[24] This can be found at an antiproton beam momentum of 1.637 GeV. Figure 4.5 shows this experimental data for the four independent spin correlation parameters, as functions of cos θ∗ (cf. the following section). To facilitate the use of these experimental results, they have been approximated with polynomials up to degree 5, according to which [24]

∗ 0 1 ∗ 2 2 ∗ 3 3 ∗ Cij(cos θ ) ≈ Aij + Aij cos θ + Aij cos θ + Aij cos θ 4 4 ∗ 5 5 ∗ (4.22) + Aij cos θ + Aij cos θ

24 Figure 4.3: Hyperon polarisation in the reaction pp → ΛΛ for beam energies just over threshold as functions of cos θ∗.[23] The solid lines represents ﬁts to Legendre polynomials.

Figure 4.4: Hyperon polarisation in the reaction pp → ΛΛ for beam energies of 1.642 GeV (open circles) and 1.918 GeV (closed circles) as functions of cos θ∗.[22]

25 n where the coeﬃcients Aij are given by

0 0 0 0 Axx = 0.43484 Ayy = 0.51782 Azz = −0.7148 Axz = 0.56455 1 1 1 1 Axx = 1.0498 Ayy = 0.54426 Azz = −1.2273 Axz = 0.63164 2 2 2 2 Axx = −2.532 Ayy = −0.47872 Azz = 0.86644 Axz = −0.9645 3 3 3 3 Axx = −1.7598 Ayy = −0.48097 Azz = 4.5709 Axz = −0.64014 4 4 4 4 Axx = 2.5511 Ayy = 0.0 Azz = −0.68308 Axz = 0.0 5 5 5 5 Axx = 0.0 Ayy = 0.0 Azz = −3.6507 Axz = 0.0 (4.23) n n and by symmetry Axz = Azx, as is discussed in the next section.

(a) (b)

Figure 4.5: Experimental data of the non-zero spin correlation coeﬃcients at beam energies of 1.637 GeV.[24] The dashed lines indicates the functions that the polynomials in Equation (4.22) have been ﬁtted to.

26 4.2.4 Symmetries The fact that the strong interaction conserves parity (P ) and charge conjugation (C) reduces the number of independent spin observables. Also, the fact that there is a rotational symmetry in θ∗ = 0 and π in the production plane, imposes further constraints on the set of independent spin observables. The ΛΛ system is self-conjugate, meaning that under the operation of charge conjugation it is simply transformed back into the original state. This implies that the polarisation of the Λ produced at the CM angle θ must be equal to the polarisation of the Λ produced at the CM angle π − θ. Since the production angle of the reaction θ∗ is defined as the opening CM angle of Λ, then the polarisation of the hyperon must be equal to that of the antihyperon for each value of θ∗. [21] As a consequence, the spin correlation coefficient Cij of the reaction must be equal to the spin correlation coefficient Cji of the charge-conjugate reaction. Again, since the ΛΛ system is self-conjugate, this implies that

Cij = Cji (4.24) for every i and j. Since the strong interaction conserves parity, the observables must be unaf- fected by a parity transformation performed on them. If the initial state system pp is unpolarised and parity is conserved, then the ﬁnal state hyperons cannot have any polarisation in the production plane of the reaction. The production plane is simultaneously the xz-plane of the hyperon and the xz-plane of the antihyperon (cf. Section 4.2.1). This means that

Px = Pz = Px = Pz = 0 (4.25) and the only non-zero polarisation can be in the direction perpendicular to the production plane, that is in the y = y direction. Hence, the only non-zero spin polarisation observables are Py and Py. Furthermore, the reasoning above puts further constraints on the spin correlation coeﬃcients, since all spin correlation between the direction along the normal of the production plane (y/y) and those directions within the production plane (x/x and z/z) becomes zero

Cxy = Cyx = Czy = Cyz = 0 . (4.26)

At the production angles 0 and π, the direction of the produced hyperon coincide with the direction of the initial baryon, thus it is not possible to determine a unique production plane. The lack of a normal direction means that also the polarisation in the y/y-direction must be zero, since this direction will be ill-deﬁned. It also results in a symmetry between the x/x and y/y-directions, such that

Cxx = −Cyy (4.27) and also

Cxz = Czx = 0 (4.28) for θ∗ = 0 and θ∗ = π. [21]

27 4.2.5 Cross Section The cross section for a reaction can be viewed as the effective scattering area per target particle, as it is seen by the impinging particles of the beam. It is defined as the reaction rate per unit of incident flux per target particle, and is normally measured in units of barns (1 barn = 10−28 m2). It is given by

N σ = , (4.29) R L where N is the number of events of the whole experiment, L is the luminosity, i.e. the beam flux per target particle, and the integral is over the whole running of the experiment. [21] The differential cross section dσ/dΩ of the reaction, describes the cross section per unit solid angle. It is a measure of the probability to observe a scattered particle per unit solid angle. This quantity is not invariant under Lorentz transformation, and it must thus be given in a specific reference frame. The most commonly used is the CM of the reaction.

Available Data Up to beam momenta of 2 GeV, there is plenty of high quality experimental data available for the differential cross section from the PS185 experiment at LEAR, CERN. These data can be used to find approximate functions for the differential cross section in the range from the threshold momenta up to 2 GeV, which was the maximum energy at LEAR where the PS185 experiment took place. The experimental data of the differential cross section dσ/dΩ of the pp → ΛΛ reaction at antiproton beam momenta of 1.771 are shown in Figure 4.6.

Figure 4.6: Diﬀerential cross section in CM of the reaction pp → ΛΛ as a function of cos θ∗ for a beam momentum of 1.771 GeV.[25]

The data from the PS185 experiment of the diﬀerential cross section in CM have been ﬁtted to Legendre polynomials up to degree eight as functions f of

28 the production angle θ∗, according to

∗ ∗ ∗ ∗ f(cos θ ) ≈A0(P0(cos θ ) + A1P1(cos θ ) + A2P2(cos θ )+ ∗ ∗ (4.30) + A3P3(cos θ ) + ... + A8P8(cos θ )), where the Legendre polynomial Pn(x) of degree n is given by 1 dn P (x) = ((x2 − 1)n). (4.31) n 2nn! dxn

The coeﬃcients An are dependent on the energy used, and have been ﬁtted with polynomials of the excess energy up to degree three,

2 3 An() = an,0 + an,1 + an,2 + an,3 , (4.32) where the coeﬃcients an,m are given experimentally by [24]

−3 −4 −7 a0,0 = −3.12 · 10 a0,1 = 0.0970 a0,2 = −4.79 · 10 a0,3 = 8.39 · 10 a1,0 = 0.394 a1,1 = 0.275 a1,2 = 0.0 a1,3 = 0.0 −5 a2,0 = 0.626 a2,1 = 0.0103 a2,2 = −2.06 · 10 a2,3 = 0.0 −5 a3,0 = 0.174 a3,1 = 0.0105 a3,2 = −2.22 · 10 a3,3 = 0.0 −3 −5 a4,0 = 0.098 a4,1 = 2.33 · 10 a4,2 = 2.38 · 10 a4,3 = 0.0 −4 −7 a5,0 = −0.854 a5,1 = 0.0262 a5,2 = −2.33 · 10 a5,3 = 7.33 · 10 −5 a6,0 = 0.615 a6,1 = −0.0113 a6,2 = 6.02 · 10 a6,3 = 0.0 −3 a7,0 = −0.537 a7,1 = 4.39 · 10 a7,2 = 0.0 a7,3 = 0.0 a8,0 = 0.1 a8,1 = 0.0 a8,2 = 0.0 a8,3 = 0.0 (4.33) Thus, in the energy range between threshold and 2 GeV, the differential cross section in CM can be approximated by dσ (cos θ∗) = f(cos θ∗), (4.34) d(cos θ∗) with f(cos θ∗) given as in Equation (4.30). For higher energies, the available data for the differential cross section of the reaction pp → ΛΛ are more scarce. The experimental results of the differential cross section dσ/dt0 in CM as a function of the reduced four momentum transfer squared t0 at 6.0 GeV are shown in Figure 4.7. It can be approximated with the formula [26] dσ 0 0 = aebt + cedt (4.35) dt0 where t0 is given by (4.15), and the coefficients have been experimentally determined to be [27]:

a = 24.2 ± 2.2 (4.36a) b = 10.1 ± 0.6 (4.36b) c = 16.5 ± 2.1 (4.36c) d = 3.0 ± 0.3 (4.36d)

The relation between dσ/dt0 and dσ/d(cos θ∗) is given by dσ dσ dt0 = · . (4.37) d(cos θ∗) dt0 d(cos θ∗)

29 Thus, by diﬀerentiating (4.15) and using (4.35), the diﬀerential cross section dσ/d(cos θ∗) at 6.0 GeV can be found to be

dσ dσ ∗ ∗ ∗ ∗ bt0 dt0 = · 2 ~p ~p = 2 ~p ~p · (ae + ce ) , (4.38) d(cos θ∗) dt0 p Λ p Λ and, using the expression of the reduced four momentum transfer in (4.15), as a function of cos θ∗ according to

dσ ∗ ∗ ∗ 2b|~p∗||~p∗ |(cos θ∗−1) 2d|~p∗||~p∗ |(cos θ∗−1) (cos θ ) = 2 ~p ~p · (ae p Λ + ce p Λ ), d(cos θ∗) p Λ (4.39) where the coeﬃcients a, b, c and d are given by (4.36).

Figure 4.7: Experimental data on the diﬀerential cross section at 6.0 GeV.[26] The dashed line indicates the ﬁrst term and the solid line the second term of the expression in Equation (4.35).

4.3 Decay of Λ

The Λ particle is the lightest of the hyperons, and therefore it must decay via the ﬂavour changing weak interaction.[20] There are two main decay channel for Λ(Λ); in most cases it decays to either a proton and a charged pion pπ− (pπ+) or to a neutron and a neutral pion nπ0 (nπ0). The former channel is the most common, with a branching ratio1 of around 64 percent (0.641 ± 0.005), while the latter has a 36 percent (0.359 ± 0.005) branching ratio [3].

1The ratio between the decay rates of an individual decay channel and the total decay rate

30 The charged decay is much easier to detect and evaluate than the uncharged decay, since charged particles will leave tracks in the detector. Consequently, it is this decay channel that will be the subject of the following discussion.

4.3.1 The Λ → pπ− Decay Channel It is convenient to begin the discussion of the pπ− decay channel by considering aspect of two of the most important properties of this decay, namely the decay asymmetry parameter and the expected decay point.

Decay Asymmetry Parameter Decays that take place via the weak interaction does not have to conserve parity. Therefore there can be an asymmetry in the distribution of decay particles with respect to the direction of the spin of the decaying particle. A measure of this asymmetry is the decay asymmetry parameter α. The parameter is speciﬁc for each decay channel and is a measure of the probability for the decay proton to be emitted along the spin direction of the decaying hyperon. [20] The value of α has been experimentally determined to be 0.642±0.013 [3] for the Λ decay, meaning that the proton has a 64 percent probability of decaying in the direction of the hyperon spin. The decay asymmetry parameter for Λ is denoted α and is α = −α if parity and charge conjugation (CP ) is conserved (see Section 2.6). A measure of this CP -invariance is the quantity A given by α + α A = (4.40) α − α which obviously will be zero if CP is conserved. The current value of this quantity cited by the Particle Data Group is 0.012 ± 0.021 [3], although lower values have been observed [20].

Decay Point The decay point of the hyperon is mostly referred to as a second vertex of the pp annihilation, as opposed to the ﬁrst vertex coming from the interaction between the antiproton beam with the protons. The probability density function of particle decay is given by [4]

m − m x P (x) = e |~p|τ , (4.41) |~p|τ where m is the particle mass, ~p is the three-momentum and τ is the mean lifetime of the particle. The expectation value of the decay point is then given by Z ∞ Z ∞ m − m x |~p|τ xˆ = xP (x)dx = x e |~p|τ dx = = γβτ, (4.42) 0 0 |~p|τ m where γβ is the relativistic factor indicating the speed of the decaying particle relative to the laboratory rest frame. Thus, the quantity τc will give an indication of the mean distance to the second vertex. From Table 2.4 the average lifetime of the Λ particle is found to

31 be approximately 0.26 nanoseconds (2.631 ± 0.020 · 10−10 s [3]), corresponding to a value of τc of approximately 7.9 cm. This shows that the average value of the distance to the decay point will be relatively large for Λ – provided that β is not too small – and will mean that the second vertices in the pp → ΛΛ → pπ+pπ− reaction for the most part will be well separated from the ﬁrst interaction vertex, thus greatly facilitating the analysis of experimental data. [20]

4.3.2 Angular Distribution The distribution of decay particles in the pp → ΛΛ → pπ+pπ− reaction is not isotropic, but has an asymmetry that is dependent on the spin direction of the hyperon and antihyperon. The angular distribution of decay nucleons in the hyperon decay can be related to the spin observables, according to [20]

 3 3  IΛΛ X X I (θ, φ, kˆ , kˆ ) = 0 ααχ P BP T k k , (4.43) pp p p 64π3  klµν k l 1,µ 2,ν  µ,ν=0 k,l=0

B T where Pk and Pl are the beam and target polarisations, k1,µ and k2,ν are the direction of the respective decay particle, and χklµν is the generalised notation for a spin observable. If considering an unpolarised beam and target – as is the case here – the normalised angular distribution of decay particles becomes reduced to [19]  1 X X I(θ , θ ) = 1 + α P Λ cos θ + α P Λ cos θ i j 16π2  i i j j i j  (4.44) X +αα Cij cos θi cos θj i,j

Λ Λ where Pi is the polarisation of Λ in the i-direction and Pj is the polarisation of Λ in the j-direction. Also, θi is the emission angle of the antiproton with respect to the i-direction and θj is the emission angle of the proton with respect to the j-direction. Here the index i refers to the Λ coordinate system {x, y, z} and j refers to the Λ coordinate system {x, y, z}.

Hyperon Polarisation If considering only the polarisation and taking the symmetry conditions that gives Px = Pz = 0 into account, the distribution of decay protons from the Λ decay can be expressed as 1 I(θ ) = (1 + αP cos θ ), (4.45) y 4π y y that is, it is only the polarisation perpendicular to the production plane that aﬀects the distribution. Consequently, the distribution of decay nucleons with respect to the θy-angle is anisotropic, while the distribution with respect to the θx and θz-angles will be isotropic.

32 From these distribution functions, it is possible to use statistical methods to obtain the expectation value of the polarisation of Λ/Λ as a function of the measured values of the directional cosines. The method used here is called the Method of Moments and is described in Appendix A.1. The distribution of the produced particles can be expressed using (4.44), relating it to the polarisation Pj(Pi) of the Λ(Λ) particle as

I(θj) ∝ 1 + αPj cos θj. (4.46)

Thus, by substituting cos θj for x and normalising, the following probability density function is obtained 1 f(x|P ) = (1 + αP x). (4.47) j 2 j

The observations are here the xn:s, ranging from -1 to +1, and the wanted parameter is the polarisation Pj. By choosing the function g in the easiest possible way, simply g(x) = x, the ﬁrst moment of g becomes (from Equation (A.1)) Z Z 1 1 2 αPj E(g(x)) = g(x)f(x|Pj)dx = (x + αPjx )dx = . (4.48) Ω 2 −1 3 Combining this result with (A.2) gives αP x = (g(x)) =γ ˆ(P ) ≈ E(g(x)) = j . (4.49) j 3 And thus an estimate of the polarisation in the j-direction is given by

N 3 3 3 1 X P = x = cos θ = cos θ . (4.50) j α α j α N jn n=1

The variance for Pj can be obtained using A.5 and is given by N 1 1 9 X 2 V (P ) = (cos θ 2 − cos θ ). (4.51) j N N − 1 α2 jn j n=1 Correspondingly for Λ

N 3 1 X P = cos θ (4.52) i α N in n=1 where cos θin denotes the i-directional cosine of the antiproton in the Λ rest frame coordinate system, for the nth event generated, and N is the total number of events.

Antihyperon-Hyperon Spin Correlations If considering the spin correlation between the antihyperon and hyperon, the normalised angular distribution of the decay particles is given by [19] 1 X I(θ , θ ) = (1 + αα C cos θ cos θ ) (4.53) i j 16π2 ij i j i,j

33 The symmetry conditions of Section 4.2.4 gives that the summation only have to be made over the five non-zero spin correlation coefficients Cij. These spin correlations coefficients Cij can each be estimated in the same way as the polarisation using the method of moments. The angular distribution of decay particles with respect to spin correlation can be expressed

I(θi, θj) ∝ 1 + ααCij cos θi cos θj. (4.54)

Considering the set of observables cos θin cos θjn and conducting the same derivation as for the polarisation, an estimate of the spin correlation coeﬃcient Cij, can then be found to be

N 9 1 X C = cos θ cos θ , (4.55) ij αα N in jn n=1 with a variance given by ! 9 2 1 P (cos2 θ cos2 θ ) P cos θ cos θ 2 V (C ) = n in jn − n in jn . ij αα N − 1 N N (4.56)

34 Chapter 5

Simulations

This chapter treats the work for simulations of the pp → ΛΛ reaction that have been preformed. The software of the PANDA experiment is effective, but it handles so many different components, that it becomes awkward when being used from a normal computer to run a large number of events. Also, there are some flaws and, as of today, still some unresolved issues as to what certain parts of it will ultimately look like. As a consequence, the PANDA software has not been used in this work. Instead, a computational framework of the PS185 experiment has worked as the back bone of the simulations. This chapter gives a brief introduction to the PS185 experiment and the framework used there (Section 5.1), and continues with discussing the alterna- tions that have been made to this code (Section 5.2) in order to make it work as the event generator of the PANDA simulations. The chapter also treats a reconstruction code, and the results obtained there (Section 5.3).

5.1 Introduction

The PS185 project started in 1981, and took data from 1984 until 1996 at the LEAR Antiproton Ring at CERN in Switzerland. It was created to study antihyperon-hyperon production, and thus it has a software that is advantageous for simulating the pp → ΛΛ reaction. The maximum momentum of the LEAR antiproton beam of 2 GeV limited the studies to single strangeness hyperons.

5.2 Event Generation

The used PS185 framework is a complete software using Monte Carlo methods, to simulate the detector system. Since the detector setup for the PS185 is different to that of PANDA, only the part of the program that generates the events was kept. To better suit the task here, changes – small as well as large – were made to most parts of the code. The four major changes were the following: the simulation of the interaction region was adopted to resemble the situation at PANDA; the generation of hyperons was adopted to reflect the experimental data of the differential cross section; the generation of the decay particles from the hyperon decay was

35 adopted to both simulate the experimental data on polarisation and spin correlation. These changes will all be discussed in detail in the following. The reaction between the antiproton beam and the proton target takes place in the interaction point, located at some distance from the origo of the detector coordinate system. The exact location of the interaction point is determined by the type of target used, since the two foreseen target types have different widths and require different widths of the beam. This will result in different distributions of interaction points. The method for generating the extended interaction region is described in Section 5.2.1. The event generation uses subroutines in the CERN program library, CERN- LIB. The ΛΛ pairs are produced with a call to the CERN routine FOWL. This routine returns the four-momenta of the hyperons in the ΛΛ Centre-of-Mass on a statistical basis. The distribution of the produced hyperons returned by the FOWL routine will therefore be isotropic, whilst the actual distribution of Λ in CM is strongly forward peaking, as is indicated by the differential cross section (cf. Section 4.2.5). This difference can be corrected by assigning weighting factors to the events depending on how well these fit the experimental data of the differential cross section, as is described in Section 5.2.3. The four-momentum of the hyperons are then transformed into the laboratory frame, and propagated in space until their simulated decay. The decay particles are produced back-to-back in the hyperon rest frame. The polarisation of the hyperons and the spin correlation between them determine the angular distribution of the produced protons and pions (cf. Section 4.3.2). The generation of the decay particles are thus changed to fit the distribution determined by the experimental data available. An alternative method is to assign weighting factors, as for the angular distribution of the hyperons. Both these approaches have been used and are described in Sections 5.2.4 and 5.2.5. Each one of these adaptations of the event generation to fit the experimental results will be done separately. This has been done for facilitating the evaluation of the results obtained.

5.2.1 Extended Target The two main options for the target in the PANDA experiment, the pellet target and the cluster jet target, were discussed in Section 3.4. There is, presently, a development of the pellet target that would allow individual pellets to be tracked in order to determine their position with very high accuracy. This would signiﬁcantly reduce the uncertainty of the volume of the interaction region. It is not, however, yet known whether this is a workable solution for the experiment. Therefore, in these simulations, only the two standard targets will be considered: the cluster jet target and the untracked pellet target.

Method

When determining the spread of the interaction region for the diﬀerent targets, the extension of both the beam and the target need to be considered. Both the cluster jet and the pellet target beams can be assumed to be uniformly distributed, while the antiproton beam will have a Gaussian distribution in the plane perpendicular to its direction of motion. [4]

36 The width of the antiproton beam is dependent on the requirements of the used target. A cluster jet target would need a beam rms width of σx = σy = 0.1 mm and would itself have a diameter of 15 mm. The untracked pellet target would have a smaller diameter of just 2 mm1, but would require a wider beam of σx = σy = 1 mm. [4] Furthermore, the antiproton beam must be limited in the directions perpendicular to its direction of motion. In these simulations, the maximum radius of the beam have been assumed to be 10 mm.[24] This would result in a interaction volume of approximately 4 mm3 for the cluster jet target and approximately 25 mm3 for the pellet target. The extended targets are created by generating two random numbers from a double Gaussian distribution with the widths σx and σy given by 0.1 mm or 1.0 mm depending on the target. Theses two numbers are then used as the respective x- and y-components of the position vector of the first vertex. In order to limit the beam in the xy-plane, all x- and y-components that do not fulfill the condition 2 2 x + y ≤ RB, (5.1) where RB denotes the maximum radius of the beam, are discarded. Since the target is assumed to be uniformly distributed, the z-component of the position vector of the first vertex is generated by

z = r · 2RT , (5.2) where r is a uniform random number on the interval [0, 1] and RT is the maximum radius of the target beam. Since the target beam cross section in the xz-plane is circular, the z-component also need to fulﬁll the condition

2 2 x + z ≤ RT . (5.3)

The z-values that do not meet this condition are discarded and the event generation is redone.

Results

Figure 5.1 shows the distribution of production vertices in the xy- and xz-planes using the pellet target. Figure 5.2 shows the distribution of production vertices in the xy- and xz-planes using the cluster jet target.

5.2.2 Decay Vertices

Since both the hyperons are neutral, they do not leave tracks in the detector system, and it is only from their decay products that the path of the hyperons can be reconstructed. The extension of the target will be important in the reconstruction of the event using the location of the decay points.

1If using tracked pellets it would be possible to determine the target location within a diameter of 0.1 mm.

37 (a) (b)

Figure 5.1: Interaction region using the untracked pellet target, seen in the (a) xy-plane and (b) xz-plane in the detector coordinate system.

(a) (b)

Figure 5.2: Interaction region using the cluster jet target, seen in the (a) xy- plane and (b) xz-plane in the detector coordinate system.

38 Method The decay vertices are simulated using the expression for the distribution probability given in Equation (4.41), according to which

m − m x P (x) = e |~p|τ , (5.4) |~p|τ where x is the distance from the interaction vertex, and P (x) is the probability of decay at that position. Using the transformation method described in Section A.3.1, a random number χ corresponding to the probability density function can be found be taking a uniformly distributed random number r ∈ [0, 1] and letting Z χ Z χ m − m x r = P (x)dx = e |~p|τ dx. (5.5) −∞ 0 |~p|τ This integral can be calculated to be Z χ m − m x − m χ e |~p|τ dx = −e |~p|τ + 1, (5.6) 0 |~p|τ resulting in − m χ r = 1 − e |~p|τ . (5.7) Solving this equation for χ gives

|~p|τ χ = − ln(1 − r) . (5.8) m Since r is a uniform random number on [0,1], so is 1 − r. Thus, by taking a uniformly distributed random number r0 ∈ [0, 1], the distance to the decay point can be simulated as |~p|τ x = − ln(r0) . (5.9) m

Results Considering an antiproton beam momentum of 1.5 GeV, which is the smallest beam momentum in the PANDA experiment and just above threshold of the reaction, the decay points will be distributed according to the histogram of Figure 5.3 when using this method. Figure 5.4 shows the distribution of decay vertices using the two different targets at a beam momentum of 1.5 GeV. When using the pellet target about two percent of the decay vertices will be located in the actual interaction region, compared to approximately four percent when using the cluster jet target. This is surprising, since the interaction volume is smaller for the cluster jet target than for the pellet target. The presence of an interaction region that is extended as opposed to an ideal interaction point in origo, poses some difficulties when reconstructing the kinematics of the event. The two different target types yield different extensions of the interaction area and thus also different accuracies. It will, of course, be very hard to reconstruct the events where the second vertices are situated within the interaction region.

39 Figure 5.3: Distribution of Λ decay points at a beam momentum of 1.5 GeV.

(a) (b)

Figure 5.4: Decay vertices of the reaction using (a) the pellet target and (b) the cluster jet target for an antiproton beam momentum of 1.5 GeV.

40 As shown in [4], this has severe consequences for the evaluation of the experimental result when dealing with produced particles with short lifetimes. Although the Λ particle has a relatively long lifetime (cf. Table 2.4), there are still some events that will be close to impossible to reconstruct if the decay will occur within the interaction region. This is discussed further in Section 5.3.

5.2.3 Differential Cross Section The angular distribution of produced hyperons in the antiproton-proton annihilation is described by the differential cross section dσ/d(cos θ∗) as a function of the excess energy. In order to obtain a realistic distribution of the produced particles, the event generation will use the experimental data of the differential cross sections that are available.

Method The experimental data of the diﬀerential cross section of pp → ΛΛ is of high quality for low beam momenta, but more scarce at higher momenta. For momenta below 2 GeV, the data from the PS185 experiment can be used. Here, an angular distribution of the produced particles can be obtained, using Legendre polynomials, for every given value of the excess energy , as is discussed in Section 4.2.5. The Legendre polynomials can be calculated using the recursion formula

(n + 1)Pn+1 = (2n + 1)xPn − nPn−1 (5.10) and the fact that the ﬁrst two Legendre polynomials are given by

P0(x) = 1 and P1(x) = x. (5.11)

The differential cross section in CM for the reaction at this particular beam momentum can be obtained by calculating the excess energy (Equation (4.19)), and using the experimentally determined coefficients in (4.33). For beam momenta of 2 GeV and up to 15 GeV – which is the maximum beam momenta of FAIR – the experimental data from [27] is used. This is based on measurements taken at 6.0 GeV, and here it is a fair approximation to assume that the differential cross section will have a similar behaviour for the momenta in the interval from 2 GeV up to 15 GeV, which is the maximum momentum of the PANDA experiment.[26] The angular distribution of the produced hyperons particles can be obtained from the expression of the differential cross section in CM for the pp → ΛΛ reaction given by either Equation (4.34) for beam momenta below 2 GeV or by Equation (4.39) for momenta above 2 GeV. The normalised form of these expressions corresponds to the probability distribution function of cos θ∗ for the Λ particle. Since the hyperons are generated in an external routine, it is not possible to change the generation of these according to the desired distribution. Thus, here a routine assigns a weighting factor to each event according to the normalised probability distribution. This weight is propagated along with the particle and is used for example when histogramming a quantity.

41 Results Figure 5.5 shows the simulated angular distribution in CM of produced Λ particles at a beam momentum of 1.771 GeV (histogram). This corresponds well to the experimental results of the diﬀerential cross section in Figure 4.6, indicated as data point.

Figure 5.5: Normalised angular distribution of simulated Λ particles at 1.771 GeV as a function of cos θ∗ (histogram). The data points are the experimental data shown in Figure 4.6.

The distribution of produced Λ particles at 6.0 GeV beam momentum using 100,000 generated events is shown in Figure 5.6. Figure 5.6(a) shows the angular distribution of produced Λ as a function of cos θ∗, while Figure 5.6(b) shows the normalised distribution as a function of the negative reduced four-momentum transfer squared, −t0 up to t0 = −1.8 GeV. The latter distribution corresponds well to the data of the diﬀerential cross section presented in Section 4.2.5, as is indicated by the solid lines representing the exponential ﬁts equal to those in Figure 4.7.

5.2.4 Polarisation In the original event generation, the particles from the decay of the hyperons were generated isotropically. However, as shown in Section 4.3.2, the distribution of the decay particles is not isotropic but has a distribution that is depending on the polarisation of the decaying hyperon.

Method There are two possible ways to simulate the experimental distributions of the decay product particles. One method is to generate them according to this distribution. In most situations this is the more favourable approach, although it might not be in all cases. In the second method, the particles are generated

42 (a) (b)

Figure 5.6: Distribution of produced Λ particles at 6.0 GeV as a function of (a) the antihyperon production angle in CM, cos θ∗, and (b) the negative reduced four-momentum transfer squared, −t0. The solid lines in (b) represents the exponential terms in the expression of Equation (4.35). Note the logarithmic scale on the y-axes. isotropically and each event is assigned a weighting factor that depends on how well it corresponds to the desired distribution. When evaluating the result from such a simulation, the events are filled into the histogram bins with their appropriate weight. Hence, the resulting histogram would show the desired distribution and not the generated – isotropic – distribution. In order to change the event generation to fit the distribution given in (4.45) the value of the polarisation needs to be known. Since the value of θ∗ is known for a specific event, the polarisation can be calculated as well. The experimental data shows that the polarisation is highly dependent on the momentum of the beam. For the extremes of the angular span, i.e. for θ∗ = 0 and θ∗ = π it is zero, as is expected from the predicted symmetry of the observables (cf. Section 4.2.4). For reasons of simplicity, it is thus reasonable to assume that the polarisation of the hyperons can be illustrated by a sine function depending on the opening angle in CM

∗ Py = Py = sin θ . (5.12) The probability density distribution as a function of the y-directional cosine for the decay protons can be obtained using (4.47) and the range of the directional cosines, here denoted with the variable x, as 1 (1 + αP x) if − 1 ≤ x ≤ 1 P (x) = 2 y , 0 otherwise which is obviously already normalised. The distribution of produced protons as a function of its y-directional cosine would then have a distribution as shown in Figure 5.7. The slope of the curve is given by the value of the polarisation for that speciﬁc value of θ∗. The directional

43 cosines of the proton along the y-axis can then be obtained by generating a random number on the interval [-1,1] according to this distribution. This can be done using the method described in A.3.1.

Figure 5.7: Distribution function of the decay particles as a function of cos θy, for the CM angle θ∗ = 0 .

The ﬁrst step in this method is taking a uniformly distributed random number, r ∈ [0, 1], and letting this be equal to the integral of the probability density function over the interval from −∞ to χ, where χ is the sought random number. Z χ 1 Z χ r = P (x)dx = (1 + αPyx)dx (5.13) −∞ 2 −1 The integral on the right hand side of (5.13) is easy to evaluate

Z χ 2 αPy αPyχ (1 + αPyx)dx = 1 − + χ + , (5.14) −1 2 2 which gives (5.13) the following expression

1 αP αP χ2 r = 1 − y + χ + y . (5.15) 2 2 2

Solving (5.15) for χ, s 1 1 1 χ = − ± + αPy − 2 + 4r . (5.16) αPy αPy αPy gives an expression for ﬁnding a random number according to the distribution given in (4.45) from a uniformly distributed random number, r, on the interval [0,1]. Since (5.16) has two distinct solutions for each value of r it is also necessary to evaluate which solution that should be used. By evaluating the two diﬀerent solutions, it can be seen that if the quantity αPy is positive then it will be the solution with the negative square root that should be used. If, on the other hand, the quantity is negative, then it will be the solution with the positive square root that should be used. Here, since α = 0.64 and α = −0.64, while Py = Py > 0, the negative solution will be used for the directional cosines of the protons, and the positive solution for the antiprotons.

44 Since symmetry considerations implies that the polarisation in the x- and z-directions are zero, the directional cosine of the proton in these directions should be uniformly distributed, thus adding up to zero. So, by knowing the y-directional cosine to be χ, it is possible to create a unit vector in the hyperon rest frame. The y-component will be given by χ and the x- and z-components make up a two-dimensional vector of length, l, where p l = 1 − χ2 (5.17) and where the two components, denoted c1 and c2, are uniformly distributed. The two components are found by generating a random number s on the interval [0,1] and letting the ﬁrst component c1 = l · s. This gives the second component p 2 2 to be c2 = l − c1. This method generates a three-dimensional unit vector, which after being multiplied with a factor describing the three-momentum of the proton, can be used as the spatial part of the four-momentum of the proton in CM of the hyperon. The other decay product, the pion, will then simply be generated by taking the negative of the three-momentum of the proton, since the total momentum in CM is zero. Using the other method, producing the decay particles uniformly and assigning weighting factors for each event, the starting point is still the distribution function of Equation (4.45), which can be evaluated for each value of the angle θ∗. The distribution function will give a diﬀerent value for each value of the directional cosine cos θyn of the produced proton/antiproton for the event n. This value will be assigned as the weighting factor wn to the event. The weighting factor will be propagated along with the particle, to be used when calculating quantities involving the directional cosines.

Results Regardless of the method used, the accuracy of the method can be evaluated by calculating the expectation value of the polarisation from the generated values of the directional cosines. For the case of the particles being directly generated according to the distribution, the polarisation can be reconstructed using Equation (4.50)

N 3 1 X P = cos θ (5.18) i α N in n=1 where i denotes the component of the directional cosine. The standard deviation of this expectation value is given by s 1 1 9 X 2 σ = (cos2 θ − cos θ ). (5.19) i N N − 1 α2 in i n If the decay particles of the hyperons have been generated according to a uniform distribution and the observations of cos θi have been weighted to compensate for this, the weights needs to be considered in the evaluation of the polarisation. This is discussed in Appendix A.2, giving the estimate of polarisation in the i-direction as P 3 n wn cos θin Pi = P . (5.20) α n wn

45 The standard deviations of the these reconstructed polarisation estimates are v u P 2 P 2! u 1 9 n wncos θin n wn cos θjn σi = t 2 P − P . (5.21) N − 1 α n wn n wn

When using the method of particles generated according to the distribution and considering the whole set of observed directional cosines and using 1,000,000 simulated events, the reconstructed polarisation is obtained as

Px = −0.0007 ± 0.0027 Py = 0.7879 ± 0.0026 Pz = 0.0012 ± 0.0027 (5.22) for Λ and

Px = −0.0039 ± 0.0027 Py = 0.7904 ± 0.0026 Pz = −0.0008 ± 0.0027 (5.23) for Λ. If instead using the method of weighting factors, the reconstructed values of the polarisation for 1,000,000 simulated events become

Px = −0.0027 ± 0.0027 Py = 0.7848 ± 0.0026 Pz = 0.0013 ± 0.0027 (5.24) for Λ and

Px = −0.0006 ± 0.0027 Py = 0.7846 ± 0.0026 Pz = −0.0031 ± 0.0027 (5.25) for Λ. This corresponds well to the average value of the polarisation function that was used here, since the expectation value of the polarisation in the y-direction should correspond to the arithmetic mean of sin θ∗ on the interval [0, π], given by2 sin θ∗ ≈ 0.7858 (5.26) while the expectation value of the polarisation in the x- and z-directions should be equal to zero. Since the standard deviations are of the same order as the difference between the calculated expectation values and the assumed average of the polarisation these discrepancies can be assumed to be caused by statistical errors. The polarisation can be obtained as functions of the antihyperon production angle by putting the directional cosines in bins depending on the value of θ∗. This result is shown in Figure 5.8 for 1,000,000 events using the method of directly generating the events according to the distribution. As is shown in the figures, the calculated polarisation in the y-direction of Λ corresponds well to the sine function that was used to illustrate the polarisation. The polarisation in the x- and z-direction is very close to zero, in accordance with the discussion of symmetries in Section 4.2.4. The fluctuations shown can be explained with statistical variances. Using fewer events, for example 10,000 events, the expectation values of Λ becomes

Px = 0.022 ± 0.027 Py = 0.7321 ± 0.026 Pz = 0.020 ± 0.027 (5.27)

2The mean of sin θ∗ is not, as might be expected, equal to 0.64, which is the average of sin(x) for x uniformly distributed on the interval [0, π]. This is caused by the fact that the hyperons are not uniformly distributed with respect to θ∗, but rather with respect to cos θ∗.

46 (a)

(b) (c)

Figure 5.8: Reconstructed polarisation components as functions of the Λ production angle θ∗. The deviations of the reconstructed values are of the order of 0.01 and are thus within the indicated points.

47 Distributing the directional cosines in bins depending on the θ∗ angle of the event, the plot in Figure 5.9 is obtained, showing the polarisation of Λ in the y-direction for 10,000 generated events. Clearly, the polarisation can not be

Figure 5.9: Reconstructed polarisation Py for as a function of the production angle, using 10,000 generated events. reconstructed in the same accurate way when using fewer events. There is one feature of the simulated polarisation that is not evident. Con- sidering the results in Equations (5.22) to (5.25), it is clear that the standard deviation of the expectation value in the y-direction is different to those in the x- and z-direction, for both the hyperon and the antihyperon. This property persists regardless of the method used, and regardless of how good statistics that are being used. It is not a significant difference, and it has no impact on the analysis, but it is peculiar that this phenomenon occurs, since the standard deviation is expected to be equal for all directions. It could possibly be an effect of numerical inaccuracies.

5.2.5 Spin Correlations The angular distribution of decay particles is also dependent on the spin correlation, as is discussed in Section 4.3.2. The generation of decay particles can be adapted to better correspond to the experimental data of the spin correlation coeﬃcients.

Method As with polarisation, it would be advantageous if the distribution based on the experimentally determined spin correlation coeﬃcients could be obtained both through direct generation and by assigning weights. The weighting factors can be created in much the same way as for the polarisation. However, when it comes to directly generating events according to the distribution, the case of the spin correlation is slightly more complicated than that of the polarisation.

48 In order to evaluate the spin correlation of the Λ and Λ particles, the decays Λ → pπ+ and Λ → pπ− must be considered simultaneously. The generation of the decay particles is made for one of them at a time, and it is therefore no easy way to directly generate the produced particles according to the desired distribution. Thus, the event generation will be done through a method of ’brute force’ instead, where the generated particles that does not fit the distribution will simply be rejected and the generation redone until a satisfactory particle is produced. This is a far less efficient than the method used for the polarisation, but it will still represent a relatively small decrease in the total efficiency of the generation program. Out of the spin correlations, it is only the Cxx, Cyy, Czz and Cxy = Cyx that are non-zero. Equation (4.53) can be used to describe the distribution function of the particles produced in the Λ/Λ decay giving the distribution function for each of the Cij:s as 1 I(θ , θ ) = (1 + ααC cos θ cos θ ) . (5.28) i j 4π ij i j Thus, there are five different distributions, one for each of the non-zero spin correlation coefficients, and the generation of the decay particles needs to be adopted to fit each and everyone of these. For a specific event the θ∗ is known, and thereby also the approximate value of the non-zero spin correlation coefficients as given by the parameterisation in (4.22). The distribution of decay particles can then be calculated according to (5.28). The polynomials of (4.22) do not, however, fit the symmetry conditions that Cxz = Czx = 0 and Cxx = −Cyy for the extremes of the opening angle of the antihyperon, cos θ∗ = 1 and cos θ∗ = −1 (cf. Section 4.2.4). They are also only verified at a beam momentum of 1.637 GeV, and thus not, a priori, valid at other momenta. However, for the purpose of this work, it will be assumed that the coefficients at other momenta will have a similar appearance and that these parameterisations are valid. The method used for the production of decay particles according to the desired distribution is similar to the statistical method for generating random numbers by rejection as described in Section A.3.2. For each of the non-zero spin correlation coefficients, a random number t in the interval [0, fmax] is generated, where fmax is the maximum value of the distribution function. This is done by letting 1 t = r · f = r · (1 ± ααC ), (5.29) max 4π ij where r is a uniformly distributed random number on the interval [0,1] and the sign is positive if the value of Cij is negative, and negative if Cij is positive. If t is larger than the calculated distribution for the specific Cij, then the produced event is discarded and a new generation is made. However, if t is less than the distribution, the produced particle is kept and the next random number t is generated to test the next spin correlation coefficient. In the end, a decay particle has been generated that fits all the five anisotropic distributions. In the case of producing the decay particles isotropically, the starting point is the distribution function of each of the non-zero spin correlation coefficients of Equation (5.28). The distributions can be independently evaluated for each value of the angle θ∗, using the fitted values for the experimental data (Equation

49 (4.22)). The value of the distribution function for the particular values of the directional cosines will be assigned as the weighting factor wn to the event n. Each event will thus be assigned ﬁve diﬀerent weighting factors, that are multiplied to obtain a total weighting factor of the event. This total weighting factor will be propagated along with the particle, and used when quantities are calculated that use the directional cosines.

Results As with the polarisation, the expectation value of the spin correlation coeﬃcients can be calculated in order to evaluate the result of the generation. The expectation value for each spin correlation coeﬃcient is given by (4.55)

N 9 1 X C = cos θ cos θ . (5.30) ij αα N in jn n=1 The standard deviation, σ, of the coefficient is given by q σ(Cij) = V (Cij), (5.31) where V denotes the variance, given by ! 9 2 1 P (cos θ cos θ )2 P cos θ cos θ 2 V (C ) = n in jn − n in jn . ij αα N − 1 N N (5.32) If the events are weighted to fit the distribution, the average of the spin correlation coefficients cannot be calculated as above, since the weighting factors must be taking into account (cf. Appendix A.2). Instead they are given by P 9 n cos θin cos θjnwn Cij = P (5.33) αα n wn with a variance of 2 P 2 P 2! 9 1 n (cos θin cos θjn) wn n cos θin cos θjnwn V (Cij) = P − P αα N − 1 n wn n wn (5.34) The average values for the spin correlation coefficients can be obtained by summing over all directional cosines according to Equations (5.30) and (5.33). These values are listed below for a simulation of 1,000,000 events at a beam energy of 1.637 GeV, using the method of weighting factors.

Cxx = 0.1246 ± 0.0073 Cyx = −0.0048 ± 0.0073 Czx = 0.2367 ± 0.0073 Cxy = 0.0077 ± 0.0073 Cyy = 0.3620 ± 0.0073 Czy = 0.0009 ± 0.0073 Cxz = 0.2500 ± 0.0073 Cyz = 0.0090 ± 0.0073 Czz = −0.5537 ± 0.0073 (5.35) The corresponding values using the rejection method are

Cxx = 0.1001 ± 0.0073 Cyx = −0.0017 ± 0.0073 Czx = 0.2577 ± 0.0073 Cxy = −0.00037 ± 0.0073 Cyy = 0.3605 ± 0.0073 Czy = 0.0044 ± 0.0073 Cxz = 0.2465 ± 0.0073 Cyz = −0.0067 ± 0.0073 Czz = −0.5776 ± 0.0073 (5.36)

50 These values should be compared to the corresponding average values of the functions for the spin correlation coeﬃcients given in (4.22), given by

Cxx = 0.1199 Cyx = 0.0 Czx = 0.2431 Cxy = 0.0 Cyy = 0.3562 Czy = 0.0 (5.37) Cxz = 0.2431 Cyz = 0.0 Czz = −0.5559

The correspondence between these and the reconstructed values from the event generation is fairly good for both methods. Also, here there is no significant discrepancy of standard deviation in one direction similar to the one found for the polarisation. The value of the spin correlation coefficients as a function of the antihyperon opening angle can be obtained by filling the product if directional cosines ∗ cos θi cos θj, for each event in bins of θ . This result is shown in Figure 5.10 using 1,000,000 events at a beam momentum of 1.637 GeV. The plots in Figure 5.10 correspond well to the initial plots of Figure 4.5. Thus, the spin correlation coefficients can be simulated and reproduced with good accuracy using high statistics.

5.3 Reconstruction

For the reconstruction of the generated events it is here assumed that the detection of the particles can happen in an ideal way, something that will not occur in practice. It is also assumed that the particle identification is always correct. These are huge simplifications, but they are legitimate for the properties studied in this work. The work with the reconstruction was concentrated upon trying to reproduce the momentum of the Λ, as well as its generated production angle in CM, i.e. the angle denoted θ∗. The aspects of interest here was to investigate how the extension of the interaction area would effect the reconstruction of this angle, and what the difference between the two different target options would be.

5.3.1 Extended Target When generating events using the extended target, the vertex of the decay will be located at some distance from the nominal interaction point. In the reconstruction, however, it is not possible to determine where the production took place, and it must therefore be assumed that it occurred in the origo of the coordinate system. By this assumption, an uncertainty in the calculation of the opening angle of Λ is unavoidable. By assuming that the decay vertex is accurately determined, and assuming that the production vertex is located in origo, an estimation of the opening angle of the emitted Λ particles can be made. As a ﬁrst step towards this, the momentum of the Λ is to be determined, assuming that the momenta of the detected decay particles are not known.

5.3.2 Momentum of Λ from Opening Angles Figure 5.11 shows a schematic view of the Λ decay in the laboratory frame of reference. The angle ϕ is the angle between the momentum vectors of the

51 (a) (b)

(e)

Figure 5.10: Reconstructed spin correlation coeﬃcients at simulated beam momenta of 1.637 GeV (histograms), using 1,000,000 generated events. The data points indicate the experimental data of Figure 4.5 and the dashed lines are the ﬁt to these values.

52 antiproton and pion, while the angles α and β are the opening angles of the proton and pion, respectively.

π+

ϕ Λ β

Figure 5.11: Schematic view of the reaction Λ → pπ+ in lab.

It is not possible to reconstruct the momentum of the Λ from just the angle ϕ. This can be understood by considering Figure 5.12, which shows a simulation of the angle ϕ as a function of the momentum of the Λ particle. Clearly, there is not a unique correspondence between a certain momentum and a certain angle ϕ.

Figure 5.12: Simulation of the opening angle of the decay particles in the Λ → pπ+ reaction, as a function of the momentum of the Λ particle.

However, if the angles of the decay particles with respect to the direction of the Λ particle are known, i.e. if both α and β are known, it is possible to calculate the momentum of the Λ particle.

Method If the directions of the decay particles are known, the decay point can be determined as the intersection of these. By assuming that the production point is located in origo, the direction of the decaying hyperon can be determined as

53 well. This gives an approximate value of the opening angles of the two decay particles. The momentum can be derived from the opening angles of the decay particles using the conservation of energy and momentum of the reaction. A detailed derivation is given in Appendix C and the result is that the modulus of the momentum of the decaying Λ particle, |~pΛ|, is found to be v u s u L L 2 M |~p | = t ± − , (5.38) Λ K K K where K, L and M in the case of decay to a proton and a pion are given by

sin β 4 sin α 4 sin β 4 K =1 + + − 2 − sin (α + β) sin (α + β) sin (α + β) sin α 4 sin β 2 sin α 2 −2 − 2 (5.39) sin (α + β) sin (α + β) sin (α + β) sin β 2 sin α 2 L =m2 + m2 + m2 − (m2 + m2) − c sin (α + β) a sin (α + β) b a b sin β 2 sin α 2 − (m2 + m2) − (m2 + m2) (5.40) sin (α + β) b c sin (α + β) c a 4 4 4 2 2 2 2 2 2 M =mΛ + mp + mπ − 2mΛmp − 2mΛmπ − 2mpmπ, (5.41) where mΛ, mp and mπ are the Λ, proton and pion rest mass respectively, and α and β are the angles between the antiproton and the pion with respect to the direction of the Λ in the laboratory frame of reference (see Figure 5.11). The relation between K, L and M is, in most cases of this reconstruction, s L L 2 M < − , (5.42) K K K which means that, in the very majority of events, the only valid solution in (5.38) is the one using the positive square root, v u s u L L 2 M |~p | = t + − . (5.43) Λ K K K

Results Figure 5.13 shows the relative error in the reconstructed momenta as a function of the generated momenta for an ideal target, i.e. a target where all production occurs in the nominal interaction point, at 6.0 GeV beam momentum. It seems that the errors are increasing with increasing momenta, but that the overall errors are small. If considering a antiproton beam of 6.0 GeV, 98 percent of the reconstructed events have an error that is less than one percent of the momentum. Thus, this method for reconstructing the momentum of the hyperon can be considered extremely accurate if the opening angles of the decay can be determined with high precision.

54 Figure 5.13: Box plot of the error in the reconstruction of the hyperon momentum from the opening angles (in percent of the generated momentum) as a function of the momentum for an ideal target.

When instead using the two targets of PANDA, the extension of the interaction area will complicate the determination of the opening angles of the decay particles. Figure 5.14 shows the relative error in the reconstructed momenta as a function of the generated momenta for the two target types at a beam momentum of 6.0 GeV for 100,000 generated events. Table 5.1 shows the percentage of the reconstructed events that have an error in the reconstruction of the hyperon momentum that is less than ten and 30 percent, respectively, for several beam momenta. Clearly, the momentum is more accurately determined at lower beam momenta than at higher. Also, the cluster jet target seems to be more accurate than the pellet target at low beam momenta, while the situation is reversed at higher momenta.

Beam momentum Target Error less than [GeV] type 10% 30% Pellet 55.9% 79.4% 1.5 Cluster 66.1% 86.6% Pellet 50.0% 70.6% 6.0 Cluster 42.7% 63.8% Pellet 47.8% 67.5% 15.0 Cluster 45.4% 64.6%

Table 5.1: Percent of the reconstructed events that have an error less than ten and 30 percent, respectively, in the reconstruction of the Λ momentum, for the two diﬀerent targets at diﬀerent beam momenta.

The direction of the decaying hyperon is more accurately determined at higher momenta than at lower, since the decay points will be located further

55 (a) (b)

Figure 5.14: Box plots of the error in the reconstructed value of the hyperon momentum (in percent of the generated momentum) as a function of the momentum for (a) the pellet target and (b) the cluster jet target. from the production region. On the other hand, the opening angles α and β are rapidly decreasing with increasing beam momenta, which means that an error of a certain magnitude would have more impact at higher beam momenta than at lower. For the cluster jet target the reconstruction of the decay particle opening angles is significantly more accurate at lower beam momenta than at higher. For the pellet target, there is no significant difference between different beam momenta. This explains why the best results for the antihyperon momentum reconstruction when using an extended interaction region are given at low beam momenta, despite the inaccuracy in the determination of the hyperon direction.

5.3.3 Production Angle of Λ

When reconstructing the production angle of Λ, it is assumed that the decay point of the antihyperon can be determined precisely. Also, the production vertex is assumed to be located in the nominal interaction point.

Method

The modulus of the momentum of the antihyperon can be assumed to somehow be known beforehand, alternatively it can be taken from the calculation described in the previous section. The direction and modulus of the momentum is then used to construct the four-momentum of the antihyperon in the laboratory frame. This is then transformed back to the centre of mass frame using the general Lorentz transform, given in Equation (B.7). The production angle θ∗ can be obtained by taking the scalar product of the CM four-momentum and the beam direction, which is in the z-direction of the detector system.

56 Results To evaluate the result, the difference between the reconstructed and the generated value of the production angle was determined, and plotted versus the generated θ∗ value, as well as versus the distance of the decay point from the nominal interaction point. This was done both for the case of the modulus of the hyperon momentum being calculated as above, and for the case of the momentum somehow being known beforehand. If assuming that the momentum is exactly determined, the Λ production angle in CM can be estimated fairly well. Figure 5.15 shows the difference between the reconstructed value of θ∗ and the actual value as a function of the distance from the decay vertex to the nominal interaction point for the two different targets at a beam momentum of 6.0 GeV for 100,000 events. As expected, the difference between the reconstructed and the actual value of θ∗ decreases with increasing distance from the interaction region. The pellet target also seems to be significantly more accurate than the cluster jet.

(a) (b)

Figure 5.15: Discrepancies in the reconstructed value of the production angle θ∗ as a function of the distance from the decay vertex to the origo for (a) the pellet target and (b) the cluster jet target.

Figure 5.16 shows the diﬀerence between the reconstructed value of θ∗ and the generated value as a function of the latter at a beam momentum of 6.0 GeV for 100,000 generated events. The pellet target seems to have slightly more events with small errors than the cluster jet target. Table 5.2 shows the percentage of the reconstructed events that have an error in the reconstruction of θ∗ that is less than two, four and ten degrees, respectively, for several beam momenta. Clearly, the reconstruction is more accurate at higher momenta, and the pellet target gives higher accuracies than the cluster jet, as can be expected. If instead using the reconstructed momenta, the inaccuracy in the determination of this will aﬀect the accurateness of the determination of θ∗. Table 5.3 shows the percentage of the reconstructed events that have an error in the reconstruction of θ∗ that is less than two, four, ten and 20 degrees, respectively,

57 (a) (b)

Figure 5.16: Box diagrams of the discrepancies in the reconstructed value of the production angle θ∗ as a function of θ∗ at a beam momentum of 6.0 GeV for (a) the pellet target and (b) the cluster jet target.

Beam momentum Target Error less than [GeV] type 2o 4o 10o Pellet 59.6% 74.8% 89.0% 1.5 Cluster 56.8% 73.3% 87.5% Pellet 89.7% 94.7% 98.0% 6.0 Cluster 75.9% 86.7% 94.4% Pellet 93.0% 96.4% 98.7% 15.0 Cluster 85.9% 92.6% 96.9%

Table 5.2: Percent of the reconstructed events that have an error less than two, four and ten degrees, respectively, in the reconstruction of the Λ production angle, for the two diﬀerent targets at diﬀerent beam momenta when the hyperon momenta is accurately determined.

58 for several beam momenta when the hyperon momentum is determined as in the previous section.

Beam momentum Target Error less than [GeV] type 2o 4o 10o 20o Pellet 14.2% 24.8% 44.5% 61.7% 1.5 Cluster 22.2% 43.2% 53.3% 68.5% Pellet 49.5% 62.6% 76.5% 83.1% 6.0 Cluster 41.7% 54.0% 69.3% 77.8% Pellet 60.9% 71.3% 79.3% 82.8% 15.0 Cluster 56.8% 67.1% 76.4% 81.0%

Table 5.3: Percent of the reconstructed events that have an error less than two, four, ten and 20 degrees, respectively, in the reconstruction of the Λ production angle, for the two diﬀerent targets at diﬀerent beam momenta.

Clearly, the diﬀerence between the cluster jet and the pellet target is not as evident in Table 5.3 as in Table 5.2. The cluster jet target seems to give more accurate results than the pellet target at lower beam momenta, while the pellet target seems to be more accurate than the cluster jet at moderate and high beam momenta.

59 60 Chapter 6

Conclusion and Outlook

6.1 Summary and Conclusion

The aim of this Diploma thesis project was to create a event generator for the simulation of the pp → ΛΛ → pπ+pπ− reaction, with special regard taken to the available experimental data for differential cross section of the pp → ΛΛ reaction, as well as for hyperon polarisation and antihyperon-hyperon spin correlation. The simulated interaction region have been made to fit the two target types that are envisioned for the PANDA experiment, the untracked pellet target and the cluster jet target. When doing simulations using the cluster jet target, more of the hyperon decay points were situated in the actual interaction region than when using the pellet target. This cause greater difficulties when trying to reconstruct these events if using the cluster jet target than if using the pellet target. The generated events have been reconstructed, assuming that the direction of the decay particles in the laboratory frame can be determined. From this, the momentum of the antihyperon was reconstructed, with higher accuracy at lower beam momenta than at higher. The cluster jet target gave more accurate results than the pellet target at lower beam momenta, while the pellet target gave better results at higher momenta. The CM production angle of the antihyperon has also been reconstructed, with varying results. The reconstruction of the angle was more accurate at higher beam momenta than at lower beam momenta. If using the reconstructed momenta of the antihyperon, the production angle could not be reconstructed very accurately. Here, the cluster jet target gave higher accuracy than the pellet target at lower beam momenta, while the situation was the reverse at higher beam momenta. If instead assuming that the antihyperon momenta could be precisely determined otherwise, the production angle could be reconstructed more accurately, with the pellet target giving better accuracies for all beam momenta. The distribution of hyperons has been adapted to fit the distribution indicated by the experimental data for the differential cross section. This was done by assigning a weighting factor to each event depending on how well it fitted the distribution. If using good statistics, the distribution of simulated particles

61 corresponds well to the desired distribution. The event generation of decay particles has been adapted to fit angular distributions determined by the hyperon polarisation and the antihyperon-hyperon spin correlation. This was done using two different methods, either changing the actual generation of the particles to fit the distribution, or by assigning each event with a weighting factor depending on how well it fitted the distribution. The generation of decay particles was changed in different ways for the polarisation and the spin correlation: For the polarisation the decay particles were generated directly according to the angular distribution; for the spin correlation the particles not fitting the distribution were rejected. If enough events were generated, the initial polarisation and spin correlation coefficients can then be reconstructed with very satisfying accuracy. Both methods used here seemed to generate a similarly accurate result when using high statistics.

6.2 Outlook

This work is a small step towards the implementation of an event generator for hyperons based on experimental data on the differential cross section, polarisations and spin correlations in the PANDA framework. A lot of work to further complete it can be envisioned. First and foremost, once the software of PANDA is fairly established, the routines described in this work could be implemented there. There are, however, difficulties regarding the propagation of the weights through this program. Since the PANDA frame work uses Geant4, some of the implementations made in this work, might have to be included there. Also, the indication that the simulation need to include at least 100,000 events for the spin observables to give obtain solid results, which may make the simulations time demanding. It would need to be determined which of the methods used to generate the accurate angular distribution of the decay particles that is the easiest to use in the PANDA software. The method using weighting factors has, as mentioned above, the disadvantage of the weights needing to be propagated along with the event. The advantage of this method is that it is far simpler to implement than the method of directly generating the decay particles according to distributions. Here, the best option might be the method of rejecting the events that do not fit the distribution. The advantage with this method is that it is much simpler to implement compared to the method of directly generating the decay particles according to the distribution, while at the same time not having the problem of the propagating weights. The only disadvantage with the rejection method is that it increases the event generating time. This extra time of the event generation is, however, insignificant in comparison with the time required for the particle propagation, detector simulation of particle detection and the analysis of the results. In this work all three criteria for the particle distributions – differential cross section, polarisation and spin correlation – have been considered separately. One would, however, like all three distribution conditions to be fulfilled at the same time, and this is something that should be done in any further work on this event generator. Furthermore, the function used as the polarisation should be alternated to

62 resemble the available experimental data. This would mean generating different polarisation functions for different values of the excess energy, similar to the case of the differential cross section at low energies. The functions used as the spin correlation parameters should also be constructed in such a way as to fit available experimental data for different energies. The fits also needs to be changed in order to take the symmetry conditions of the parameters into account.

63 64 Acknowledgments

First and foremost, I would like to thank my supervisor on this project, pro- fessor Tord Johansson at the Department of Nuclear and Particle Physics at the Uppsala University. It was through his inspiring lectures that I first got interested in this field, and by offering me the chance to do this Diploma thesis project at the department I was able to further explore that interest. Tord has throughout the project shown encouragement and support, pushing me to find my own solutions and learn how to work independently. I would also like to thank all the staff at the Department of Nuclear and Particle Physics for creating such a wonderful work environment. It has been a truly good experience to work alongside so many talented, funny and interesting people. The coffee breaks have often been the best part of my day while being here at the department, and I would like to thank Karen, Henrik, Lotta, Markus, Henrik, Fredrik, Bengt, Annica, Anni, Karin, Agnes and all the others for making it that way. I would also like to especially thank Karen for being such a good office room mate and for always being there whenever I needed someone to talk to. Furthermore, I am very grateful to all the people – both at the department and elsewhere – that on their own accord have helped me with little bits and pieces here and there, involving everything from getting past computer errors to understanding the unwritten code of the academic world. It might all have seemed like insignificant details and trivial obstacles at the time, but in the end the help I got made all the difference. Among these, I would like to mention Markus who helped me with every single LATEX bug I could possibly come up with and Henrik who gave me invaluable feedback on my report. Finally, I would, as always, like to thank my mum and my boyfriend. Despite lacking even the slightest interest in physics, they have both tried to understand my fascination and listened to my endless rambling about thing that neither of them will ever understand. For this, I am always grateful.

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67 [16] A. Sokolov. PANDA@FAIR – A novel detector for frontier physics, 2006. Talk given at the XVIII International Baldin Seminar On High Energy Physics Problems. [17] Geant4 Home Page. URL: http://geant4.cern.ch. [18] ROOT System Home Page. URL: http://root.cern.ch. [19] N. H. Hamann. Stange particle physics at LEAR. Technical report, Uni- versity of Freiburg, 1991. [20] T. Johansson. Antibaryon-baryon production in antiproton-proton collisions. In Proceedings of the International School of Physics Enrico Fermi, Course CLVIII, Hadron Physics, 2004. [21] R. Tayloe. A Measurement of the pp → ΛΛ and pp → ΣΛ + cc. Reactions at 1.726 GeV/c. PhD thesis, University of Illinois, 1995. [22] P. D. Barnes et al. Observables in high-statistics measurement of the reaction pp → ΛΛ. Phys. Rev. C, 54(4):1877–1886, 1996. [23] P. D. Barnes et al. Observables in high-statistics measurement of the reaction pp → ΛΛ. Phys. Rev. C, 54(4):1877–1886, 1996. [24] T. Johansson. Private communication. [25] P. D. Barnes et al. Measurement of the pp → ΛΛ and pp → Σ0Λ + c.c. reactions at 1.726 GeV/c and 1.771 GeV/c. Phys. Rev. C, 54(6):2831–2842, 1996. [26] S. Pomp et al. String model description of polarisation and angular distributions in pp → ΛΛ at low energies. Eur. Phys J. A, 15:517–522, 2002. [27] H. Becker et al. Measurement of the reactions pp → ΛΛ, pp → ΣΛ0 and pp → ΛΛ (missing mass) at 6 Gev. Nuclear Physics B, 141(48):48–64, 1978. [28] A. G. Frodesen et al. Probability and Statistics in Particle Physics. Uni- versitetsforlaget, 1979. [29] P. R. Bevington et al. Data reduction and error analysis for the physical sciences. WCB/McGraw-Hill, 2003. [30] M. E. Peskin. An Introduction to Quantum Field Theory. Addison-Wesley, 1995.

68 Appendix A

Statistics

In this appendix certain statistical methods used in the discussion in Chapters 4 and 5 are presented.

A.1 The Method of Moments

The statistical Method of Moments (MM) is a method widely used in particle physics experiments. The method is fairly simple, and can be used to determine, for example, the polarisation from a sample of experimental data, as is shown in Section 4.3.2. If a certain distribution is described by a probability density function f(x|θ), where the parameter θ is unknown, but a set of observations x1, x2,..., xn is available, the moments method can give an estimate of the value of the parameter.[28] In the general one-parameter case1, we will consider a function of the parameter, γ(θ), and identify this function as the ﬁrst moment, or estimate, of another function g(x) Z γ(θ) ≡ E(g(x)) = g(x)f(x|θ)dx. (A.1) Ω A reasonable estimate of the function γ(θ) is the arithmetic mean of the function g(x) over the whole set of observations xi, n 1 X γˆ(θ) = g(x) = g(x ). (A.2) n i i=1 The variance of the estimator is 2 n ! 1 X 1 V (ˆγ) = V g(x ) = V (g(x)), (A.3) n i n i=1 where the variance of the function g(x) can be described as

n 1 X V (g(x)) ≈ (g(x ) − g(x))2. (A.4) n − 1 i i=1 1The MM can be used for a set of parameters as well, but here we will only consider the simplest case of just one parameter.

69 Thus the variance of the estimate of the function γ(θ) is

n 1 X V (ˆγ) ≈ (g(x ) − g(x))2 n(n − 1) i i=1 (A.5)  n n !2 1 X 1 X = (g(x ))2 − g(x ) . n(n − 1)  i n i  i=1 i=1

Thus, from (A.2) and (A.5), the estimate and variance of a function of the wanted parameter θ are obtained.

A.2 Weighting

When dealing with a set of observations where some are more important than others, a useful technique is to weigh each observation with a weighting factor wn when comparing the set. The same can be used when simulating the production of particles of some known distribution. It is often easier to generate events according to a uniform distribution and then, to compensate for this inaccuracy, each event is given a weight according to their actual distribution. There is, however, disadvantages with this technique. Not only do the weights have to be propagated through the whole simulation, but the calculation of the mean and variance of some quantity associated with the event are also complicated when the weight of the events have to be taken into account. If the set of observations xn have been given weighting factors wn, then the weighted mean of the observations is [29] P n wnxn x = P , (A.6) n wn and the weighted average variance of the observations is

P 2 2 N n wnxn 2 σ = P − x . (A.7) N − 1 n wn A.3 Random Number Generation

To generate random numbers from a probability distribution, two methods described in [29] were used: the transformation method and the rejection method.

A.3.1 Transformation Method This method is easy to apply, providing that the distribution function is inte- gratable over the whole real axis and that the resulting function is invertible2. It works by ﬁnding the top end point of the interval, at which the value of the integrand is the same as some uniformly distributed random number. To obtain random numbers distributed according to some probability density distribution P (x), the transformation method starts with ﬁnding a uniformly

2Although not in a strict mathematical sense, since the inverse of a function here can have more than one solution.

70 distributed random number r on [0,1], and letting the probability at a point χ be equal to this number r, that is Z χ r = P (x)dx (A.8) −∞ Calculating the integral and solving the obtained expression for χ will give an expression for the χ as a function of the generated random number r. With this method, a set of random numbers χi will be obtained distributed according to the desired distribution.

A.3.2 Rejection Method The rejection method is far less eﬃcient than the transformation method, but while the latter can be somewhat hard to apply if the probability distribution for example is complicated to integrate, the former is almost always possible to use. To obtain random numbers on the interval [a, b], distributed according to some probability distribution P (x) = f(x), the rejection method starts with ﬁnding a uniformly distributed random number x0 on [a, b]. This can be done by using x0 = r(b − a) + a (A.9) where r is some uniformly distributed number on [0, 1]. Secondly, the random number y0 is found, uniformly distributed between 0 and the maximum value of the distribution fmax on the interval [a, b]. This can be found using 0 y = r · fmax (A.10) where r denotes yet again some uniformly distributed number on [0, 1]. Now, if y0 is less than the value of the probability function in x0, the generated random number (that is x0) is kept, but if y0 > f(x0) then x0 is discarded and the procedure is redone. 0 With this procedure repeated the result will be a set of random numbers xi distributed according to the probability function P (x).

71 72 Appendix B

Relativistic Kinematics

When dealing with high energy particles it is often necessary to use relativistic kinematics to describe the motion of the involved particles. In this appendix a review of relativistic kinematics is presented, with focus on the aspects that are relevant to the discussion in Chapters 4 and 5.

B.1 Four-vectors

In relativistic kinematics all quantities are described as four-vectors, which can be seen as generalisations of the classical three-vectors to four-dimensional space-time. This has to do with the fact that it is not just the measurement of space that is relative of the observer, but also the measurement of time. The diﬀerence between the three-vector of classical mechanics and the relativistical four-vector, is thus a fourth component that somehow indicates the observed measurement of time. A four-vector is normally denoted vµ where µ = 0, 1, 2, 3, where the ﬁrst index refers to the time-component of the vector and three later indices refer to the spacial part, i.e. to the classical three vector. The inner product of two four-vectors u and v is   v0 µ −v1 u · v = u vµ = (u0, u1, u2, u3)   , (B.1) −v2 −v3

µ where u is a so-called covariant four-vector and vµ is contravariant, and using Einstein summation convention1. The most commonly used four-vector in the ﬁeld of particle physics is the four-momentum. The covariant four-momentum of a particle is pµ = (E, −~p), where ~p is the classical three momentum vector and E is the total energy. The four-vectors vary depending on how they are looked at, that is they are diﬀerent depending on what reference frame is being used. Some quantities are however invariant. For an individual particle, the norm of a four-vector is

1Einstein summation convention is a notation that implies that whenever the same index appears once as a subscript and once as a superscript in the same term, summation should be carried out over all possible combinations of the components the index refers to.

73 always constant, no matter what reference frame. An example of this is the rest mass, m, of a particle, deﬁned as

2 2 µ 2 2 m = p = pµp = E − |~p| . (B.2)

Thus, from (B.2) an expression for the energy of a relativistic particle can be obtained as a function of the particle rest mass and momentum q E = |~p|2 − m2 . (B.3)

B.2 Reference Frames

The two most useful reference frames when dealing with any sort of scattering is the frame of the laboratory system (L), and the frame of the Centre-of-Mass (CM) of the reaction. The advantage of viewing the system in the L frame is that it treats the reaction the way we would normally see it, and when dealing with a ﬁxed target experiment as is the case for PANDA, the calculations can simplify a great deal. Viewing the system from the CM frame is also advantageous in that it facilitates the calculations, since the CM frame is deﬁned as the reference frame where the momenta adds up to zero.

B.3 Lorentz Transformation

In order to get from one reference frame to another, a Lorentz transformation, or a Lorentz boost, is applied to the four vector in the original frame, and the four vector in the desired frame is obtained. Considering the four-momentum p of a particle in the rest frame S of some observer O, and another frame S0 that is moving in the x-direction relative to S, the four-momentum p0 as is would be measured by an observer O0 at rest in S0 can be obtained as 0µ µ ν p = Λν p , (B.4) where Λ is the transformation matrix. In the simplest case, the direction of the boost coincide with the direction of one of the axes of the coordinate system used. If the boost is in the, say, x-direction, the transformation matrix is given by

 γ −γβ 0 0 µ −γβ γ 0 0 Λ =   (B.5) ν  0 0 1 0 0 0 0 1

Here, β and γ are given by

v 1 β = and γ = , (B.6) c p1 − v2/c2 where v is the speed of the frame S0 relative to the frame S. γ is normally referred to as the Lorentz factor of the frame S0.

74 A general Lorentz boost to the reference frame of a momentum four vector pref is given by

  γ −γβ1 −γβ2 −γβ3 2 (γ−1)β1 (γ−1)β1β2 (γ−1)β1β3 −γβ1 1 + 2 2 2  µ  β β β  Λ = 2 (B.7) ν  (γ−1)β1β2 (γ−1)β2 (γ−1)β2β3  −γβ2 β2 1 + β2 β2   2  (γ−1)β1β3 (γ−1)β2β3 (γ−1)β3 −γβ3 β2 β2 1 + β2 where γ and β are given by

pref,1 pref,2 pref,3 1 β = , , and γ = p (B.8) Eref Eref Eref 1 − β2

B.4 Mandelstam Variables

The scattering process of two particles to two particles can be viewed schemat- ically in Figure B.1, where the four-momenta of the particles are indicated and the white circle representing some kind of scattering process. The initial and ﬁnal state particles do not necessarily have the same mass.

p 1 p3

p p2 4

Figure B.1: Two particle scattering process, indicating the four-momentum of the individual particles.

By introducing the Mandelstam variables the discussion of this kind of process can be facilitated. The Mandelstam variables are Lorentz-invariants, which means that they are independent of the chosen reference frame, and are deﬁned in terms of the ingoing and outgoing four-momenta of the particles according to [30]

2 2 s = (p1 + p2) = (p3 + p4) (B.9) 2 2 t = (p1 − p3) = (p2 − p4) (B.10) 2 2 u = (p1 − p4) = (p2 − p3) (B.11)

75 B.4.1 Invariant Mass The ﬁrst of these variables, s, is known as the invariant mass squared. Using 2 2 the fact from (B.2) that pi = mi for any particle i, s can be simpliﬁed to

2 2 2 s = (p1 + p2) = p1 − 2p1p2 + p2 = 2 2 (B.12) = m1 + m2 − 2p1p2

In the general case, the invariant mass squared is deﬁned as [19]

 2  2 X X s ≡  Ej −  ~pj (B.13) j j where Ej is the total energy of the particle j and pj is the corresponding mo- P mentum. In the CM reference frame, where j ~pj = 0, the general expression for s would become

∗ 2 √ ∗ ∗ ∗ ∗ ∗ s = (E ) ⇔ s = E = E1 + E2 = E3 + E4 (B.14) where E∗ is the the total energy in the CM frame. Thus, the invariant mass is a measure of the energy available for particle production of the system. If the two initial state particles have the same rest mass, mi, the invariant mass is simply √ q ∗ ∗ 2 2 s = 2E1 = 2 |~p1| + m1, (B.15) since the modulus of the momentum in the CM frame is the same for the initial ∗ ∗ state particles, |~p1| = |~p2|. In the same way, if the two ﬁnal state particles have the same rest mass, mf , the invariant mass will reduce to √ q ∗ ∗ 2 2 s = 2E3 = 2 |~p3| + m4, (B.16)

∗ ∗ since |~p3| = |~p4| in the CM frame. Using (B.15) and (B.16), it is then possible to express the modulus of the three-momenta in terms of the invariant mass of the system 1q |~p∗| = |~p∗| = s − 4m2 (B.17) 1 2 2 i and 1q |~p∗| = |~p∗| = s − 4m2 . (B.18) 3 4 2 f

B.4.2 Four-momentum Transfer The second of the Mandelstam variables, t, is known as the four-momentum transfer squared. It can be expressed, using (B.2), as follows

2 2 2 t = (p1 − p3) = p1 − 2p1p3 + p3 = 2 2 = m1 + m3 − 2p1p3 = (B.19) 2 2 = m1 + m3 − 2(E1E3 − ~p1 · ~p3)

76 q and using the fact that E = |~p|2 + m2

q q 2 2 2 2 2 2 t = m1 + m3 − 2( |~p1| + m1 |~p3| + m3 − ~p1 · ~p3) (B.20) The expression of the four-momentum transfer squared can be simpliﬁed if considering the situation in the CM frame of the reaction. This is shown in Figure B.2, where θ∗ denotes the opening angle in CM for the outgoing particle (3). The scalar product of the three-momentum for the incoming particle (1) and the three-momentum of the scattered particle can be written as

∗ ∗ ∗ ∗ ∗ ~p1 · ~p3 = |~p1| |~p3| cos θ , (B.21) since the angle between the two vectors is θ∗. The four-momentum transfer squared can now be written as q q 2 2 ∗ 2 2 ∗ 2 2 ∗ ∗ ∗ t = m1 + m3 − 2( |~p1| + m1 |~p3| + m3 − |~p1| |~p3| cos θ ). (B.22)

p * 3

p * θ* p * 1 2

p * 4

Figure B.2: Two particle scattering process viewed in the centre of mass frame of reference. If the two initial particles have the same rest mass and the two ﬁnal state particles have identical rest masses as well, then using (B.17) and (B.18), the four momentum transfer squared can be expressed in terms of the invariant mass squared 1 1q t = m2 + m2 − s + (s − 4m2)(s − 4m2) cos θ∗ . (B.23) 1 3 2 2 1 3

77 78 Appendix C

Momentum in two-body decay

This appendix treats the kinematics of a two-body decay, and an expression of the momentum of the decaying particle is derived from the scattering angles of the two decaying particles and the masses of the three particles involved in the decay. The reaction of a two-body decay can be described as

c → a + b, i.e. a particle c decays into a particle a and a particle b. The energy and momentum of each particle is described by

2 2 2 Ea = pa + ma (C.1a) 2 2 2 Eb = pb + mb (C.1b) 2 2 2 Ec = pc + mc. (C.1c)

In the decay, both energy and momentum needs to be conserved. The expression for the conservation of momentum is

~pa + ~pb = ~pc. (C.2)

By assumption that the c particle is only moving in the, say, x-direction and the decay takes place in the xy-plane, it is possible to simplify the expression (C.2) by separating it into its components in the x- and y-directions as is shown in Figure C.1. This yields the following relations

| ~pa| sin α − |~pb| sin β = 0 (C.3)

| ~pa| cos α + |~pb| cos β = |~pc| (C.4)

Now, by taking (C.3) and inserting it into (C.4), the following expression is obtained: sin α | ~p | cos α + | ~p | cos β = |~p | (C.5) a a sin β c

79 b

p b p sin b β p p cos c β b β c p cos α a α p sin a α p a x y a

Figure C.1: Schematic view of the two-body decay. which, by simpliﬁcation, becomes:

|~pc| | ~pa| = sin α cos α + sin β sin β = |~p | (C.6) c sin β cos α + sin α cos β sin β = |~p | c sin (α + β)

2 Thus, an expression for pa is obtained as

sin β 2 p2 = p 2 . (C.7) a c sin (α + β)

2 In the same way, a similar expression to (C.7) can be obtained for pb

sin α 2 p2 = p 2 . (C.8) b c sin (α + β)

The conservation of energy in the decay, gives the following expression:

Ea + Eb = Ec, (C.9) which is equivalent to

2 2 (Ea + Eb) = Ec (C.10) ⇔ 2 2 2 Ea + 2EaEb + Eb = Ec (C.11) ⇔ 2 2 2 2EaEb = Ec − Ea − Eb (C.12)

80 Now, taking the square of each side

2 2 2 2 2 (Ec − Ea − Eb ) = (2EaEb) (C.13) ⇔ 4 4 4 2 2 2 2 2 2 2 2 Ec − Ea − Eb − 2Ec Ea − 2Ec Eb + 2EaEb = 4EaEb (C.14) ⇔ 4 4 4 2 2 2 2 2 2 Ec − Ea − Eb − 2Ec Ea − 2Ec Eb − 2EaEb = 0 (C.15)

The left hand side of (C.15) can be expressed using the energy-momentum relations of (C.1).

4 4 4 2 2 2 2 2 2 Ec − Ea−Eb − 2Ec Ea − 2Ec Eb − 2EaEb = 2 2 2 2 2 2 2 2 2 2 2 2 2 = (pc + mc) − (pa + ma) − (pb + mb ) − 2(pc + mc)(pa + ma)− 2 2 2 2 2 2 2 2 − 2(pc + mc)(pb + mb ) − 2(pa + ma)(pb + mb ) 4 4 4 2 2 2 2 2 2 2 2 2 2 = pc + pa + pb − 2pcpa − 2pcpb − 2papb + 2pcmc + 2pama+ 2 2 2 2 2 2 2 2 2 2 + 2pb mb − 2pc(ma + mb ) − 2pa(mb + mc) − 2pb (mc+ 2 4 4 4 2 2 2 2 2 2 + ma) + mc + ma + mb − 2mcma − 2mcmb − 2mamb (C.16)

By making the following substitution to (C.16), a far more manageable expression can be obtained

4 4 4 2 2 2 2 2 2 x = pc + pa + pb − 2pcpa − 2pcpb − 2papb (C.17) 2 2 2 2 2 2 2 2 2 y = 2pcmc + 2pama + 2pb mb − 2pc(ma + mb ) − 2 2 2 2 2 2 − 2pa(mb + mc) − 2pb (mc + ma) (C.18) 4 4 4 2 2 2 2 2 2 z = mc + ma + mb − 2mcma − 2mcmb − 2mamb (C.19)

Equation (C.15) then becomes simply

x + y + z = 0 (C.20)

2 2 Replacing pa and pb with the expressions given by (C.7) and (C.8) respectively, and treating x, y and z separately, we get for x

sin β 4 sin α 4 sin β 4 x = p4 + p 4 + p4 − 2p4 − c c sin (α + β) c sin (α + β) c sin (α + β) sin α 4 sin β 2 sin α 2 − 2p4 − 2p 4 c sin (α + β) c sin (α + β) sin (α + β)

sin β 4 sin α 4 sin β 4 = p4 1 + + − 2 − c sin (α + β) sin (α + β) sin (α + β) ! sin α 4 sin β 2 sin α 2 −2 − 2 sin (α + β) sin (α + β) sin (α + β) 4 = pcK, (C.21)

81 where K is a function only depending on the angles α and β. The variable y is treated in the same way:

sin β 2 sin α 2 y = 2p2m2 + 2p 2 m2 + 2p 2 m2− c c c sin (α + β) a c sin (α + β) b sin β 2 − 2p2(m2 + m2) − 2p2 (m2 + m2) c a b c sin (α + β) b c sin α 2 − 2p 2 (m2 + m2) c sin (α + β) c a

sin β 2 sin α 2 = 2p 2 m2 + m2 + m2 − (m2 + m2)− c c sin (α + β) a sin (α + β) b a b ! sin β 2 sin α 2 − (m2 + m2) − (m2 + m2) sin (α + β) b c sin (α + β) c a 2 = 2pc L, (C.22) where L is a function of the angles α and β as well as the rest masses of the particles ma, mb and mc. The variable z simply becomes

z = m4 + m4 + m4 − 2m2m2 − 2m2m2 − 2m2m2 c a b c a c b a b (C.23) = M, where M is a function only of the rest masses of the particles, ma, mb and mc. So, in the end (C.20) can be written as

4 2 pcK + 2pc L + M = 0, (C.24)

2 which is a quadratic equation of pc , with coeﬃcients that are only dependent of the angles α and β and the rest masses of the particles involved. The momentum of the particle c can thus be found to be v u s u L L 2 M p = t ± − , (C.25) c K K K where K, L and M are given by Equations (C.21), (C.22) and (C.23) respectively.