Monte Carlo simulation study of the e+e- →Lambda Lamba-Bar reaction with the BESIII experiment

UPPSALA UNIVERSITY

DEPARTMENT OF PHYSICS AND ASTRONOMY

BACHELOR OF SCIENCE DEGREE IN PHYSICS, 15 credits

Monte Carlo simulation study of the e ⁺ e ⁻ → Λ¯ Λ reaction with the BESIII

experiment

Author : Niklas FORSSMAN

Supervisor : Karin SCH ¨ONNING, Division of nuclear physics Subject reader : Tord JOHANSSON, Division of nuclear physics

March 17, 2017

Sammanfattning

Studying the reactions where electrons and positrons collide and annihilate so that hadrons can be formed from their energy is an excellent tool when we try to improve our understanding of the standard model. Hadrons are composite quark systems held together by the strong force. By doing precise measurements of the, so called, cross section of the hadron production that was generated during the annihilation one can obtain information about the electromagnetic form factors, GE and GM, which describe the inner electromagnetic structure of hadrons. This will give us a better understanding of the strong force and the standard model. During my bachelor de- gree project I have been using data from the BESIII detector located at the Beijing Electron-Positron Collider (BEPC-II) in China. Uppsala university has several sci- entists working with the BESIII experiment. My task was to do a quality assurance of previous results for the reaction e⁺e⁻ → Λ ¯Λ at a center of momentum energy of 2.396 GeV. During a major part of the project I have been working with Monte Carlo data. Generating the reactions was done with two generators, ConExc and PHSP.

The generators were used for different means. I have analyzed the simulated data to find a method of filtering out the background noise in order to extract a clean signal.

Dr Cui Li at the hadron physics group at Uppsala university have worked with several selection criteria to extract these signals. The total efficiency of Cui Li’s analysis was 14%. For my analysis I also obtained total efficiency of 14%. This gave me confidence that my analysis have been implemented in a correct fashion and that my analysis now can be transferred over to real data. It is also reassuring for Cui Li and the rest of the group that her analysis has been verified by and independently implemented selection algorithm.

Abstract

Att studera vad som h¨ander vid reaktioner d¨ar elektroner och positroner kolliderar och annihilerar s˚a att hadroner kan bildas ur energin kan vara till stor hj¨alp n¨ar vi vill f¨orst˚a standardmodellen och dess krafter, i synnerhet den starka kraften, som kan studeras i s˚adana reaktioner. Genom att utf¨ora precisa m¨atningar av tv¨arsnitt f¨or hadronproduktion f˚ar man fram de elektromagnetiska formfaktorerna GE och GM som beskriver hadronernas inre struktur. Hadroner ¨ar sammansatta system av kvarkar och den starka kraften binder dessa kvarkar.

Under mitt examensarbete har jag anv¨ant mig av data fr˚an detektorn BESIII som finns vid BEPC-II (Beijing Electron-Positron Collider) i Kina. Uppsala universitet har flera forskare som jobbar med BESIII experimentet. M˚alet var att kvalitetss¨akra den tidigare analys som gjorts f¨or reaktionen e⁺e⁻→ Λ ¯Λ vid 2.396 GeV. Jag b¨orjade med att g¨ora Monte Carlo-simuleringar. Reaktionerna har genererats med tv˚a olika generatorer, ConExc och PHSP. Dessa generatorer har anv¨ants till olika ¨andam˚al. De genererade partiklarnas f¨ard genom detektorn har sedan simulerats. D˚a bildas data av samma typ som dem man f˚ar fr˚an experiment. Jag har analyserat dessa simulerade data f¨or att hitta en metod som kan filtrera bort bakgrundsst¨orningar samtidigt som intressanta data sparas. Kriterier utarbetade av Dr. Cui Li har anv¨ants f¨or att skapa denna metod. Min algortim gav en total effektivitet p˚a 14%, vilket st¨ammer bra med den tidigare algoritmen som Cui Li skapade, ¨aven d¨ar var effektiviteten 14%. Detta ger f¨ortroende f¨or min algortim och den st¨arker ¨aven Cui Lis resultat.

Contents

1 Introduction 1

1.1 The Standard Model . . . 1

1.2 The strong force . . . 1

1.3 Hadrons . . . 2

1.3.1 Quantum numbers . . . 3

1.3.2 Mesons . . . 3

1.3.3 Baryons . . . 4

1.3.4 Hyperons . . . 5

2 Formalism 5 2.1 Relativistic two-body kinematics . . . 5

2.2 Cross section . . . 7

2.3 Form factors . . . 8

3 The BES III experiment 9 3.1 BEPC-II . . . 10

3.2 BESIII detector . . . 10

4 The e⁺e⁻ → Λ ¯Λ reaction 12 5 Software tools 13 5.1 BOSS . . . 13

5.2 Software environment . . . 13

6 Monte Carlo simulations 14 6.1 Generators . . . 14

6.2 Particle interaction with the detector . . . 14

6.3 Digitization . . . 15

6.4 Reconstruction . . . 16

6.5 Analysis . . . 16

7 Analysis 17 7.1 Event selection . . . 17

7.1.1 Pre-selection . . . 17

7.1.2 Final selection . . . 17

7.2 Total efficiency . . . 24

7.3 Efficiency and correction . . . 24

8 Results and discussion 26

9 Summary 27

10 Outlook 28

11 Collaborations 28

12 Acknowledgments 29

13 References 30

1 Introduction

This project serve as a pilot project for bachelor students working with the BESIII software at the hadron physics group at Uppsala university. The goal is that the virtual machine setup by Michael Papenbrock at Uppsala university can be used for future bachelor projects and other short time projects. In this project, the vir- tual machine have been used to study hadron production in electron-positron colli- sions. By studying electron-positron annihilation with the subsequent production of a hadron-antihadron pair we obtain valuable information about the inner electromag- netic structure of hadrons. Hadrons are tied together by the strong force and studying the internal structure of hadrons improves our understanding of the strong force that binds them together.

1.1 The Standard Model

The Standard Model (SM) is a theory of the fundamental particles in nature and how they interact. The standard model has been successful in predicting particles and other features of the microscopic world. In the standard model the fundamental particles are quarks, leptons and gauge bosons. The leptons can be organized in three generations: electrons, muons and tauons, with their respective neutrinos. The Standard Model also comprise the quarks (denoted q) and antiquarks(denoted ¯q).

All quarks have an antiquark with the same mass but opposite charge. The quarks can also be organized into three generations, with the up(u) and down(d) quarks constituting the first generation, the charm(c) and strange (s) the second and top(t) and bottom(b) the third. The elementary particles of the SM are summarized in figure 1, where quarks are particles in purple, leptons in green and gauge bosons in red.

There are four fundamental forces in nature.The strong force, the weak force, the electromagnetic force and gravity. The SM presently incorporates the first three of these forces but not yet gravity. In the SM the forces are mediated by the gauge bosons, also referred to as the force carriers. There are four gauge bosons in the SM seen to the right in figure 1 : photon, gluon, Z - and W -boson. The force carrier for the electromagnetic force is the photon while the gluon is the force carrier for the stronge force. The Z- and W-bosons are force carriers of the weak force. A recent confirmation of the SM is the Higgs’ boson found at the Large Hadron Collider(LHC) at CERN [1].

1.2 The strong force

The strong force holds particles together in the nucleus of an atom and it also holds the quarks together in composite systems, such as the proton or neutron. The range of the strong force is 10⁻¹⁵m [2] and the theory of strong interaction is called Quantum ChromoDynamics(QCD) [2]. To classify particles constituted by quarks a quantity named color charge is introduced. The colors are called red, blue and green. While the electromagnetic force acts on electrically charged objects (like quarks and electrons), the strong force acts on particles carrying color charge (like quarks and gluons). The fact that gluons carry color charge leads to gluons coupling with other gluons, while the photons, which are neutrally charged, in the electromagnetic case will not couple to other photons. As a consequence, the strong interaction becomes stronger with a

Figure 1: The current standard model with quarks, leptons and gauge bosons. Mass charge and spin is displayed in the figure. [3]

larger separation between two colored objects, in contrast with the electromagnetic case. The strength of a force is parameterized in terms of its coupling constant. For the strong force this is denoted αs and for the electromagnetic case it is denoted αEM. When the coupling constant αs becomes large, it becomes impossible to calculate experimental observables analytically and theories are therefore difficult to verify.

This is the cost for the strong interaction, if you try to separate a quark and an antiquark, they will create another quark-antiquark pair that sticks to the first pair since it is more energy efficient. Inside the nucleus the quarks move freely since the strong force is very weak at small distances. This effect is called asymptotic freedom.

The other side of the same feature of the strong force is so called confinement. It states that no quarks can be observed as free isolated objects. One feature of the strong interaction is the mass generation. When bound into composite systems, the quarks are only responsible for a fraction of the total mass of the particles they constitute, one example is the three quarks that constitute the proton(uud) only make up 1% of the total mass while the strong force generated the other 99%.

1.3 Hadrons

Hadrons are composite systems of quarks. While the quarks carry color, the hadrons are color neutral in the same way as atoms are electrically neutral through consisting of electrically charged entities. The major part of the observed hadrons can be described either as baryons, i.e three-quark systems (denoted qqq) or as mesons, i.e a quark- antiquark system(denoted q ¯q).

1.3.1 Quantum numbers

Baryons and mesons are organized into multiplets according to their properties. The main properties, or quantum numbers, that describe the baryons and mesons are spin, parity, charge and isospin.

Spin: Particles and systems of particles have spin which, slightly simplified, is a quantum mechanics analogue of intrinsic angular momentum. The spin of leptons, quarks and gauge bosons are displayed in figure 1. Quarks and leptons are fermions, i.e they have ¹₂ spin, while the bosons have spin 1. The spin (denoted S) of a ¹₂ fermion, e.g a quark, is either up(↑) or down(↓). The total spin (denoted J) of a system constituted by quarks is described in the following equation:

J = L + S (1)

where L is the orbital angular momentum of the system [2].

Parity: The parity operator revert the spatial coordinates of a system. The par- ity (denoted P) of a system of particles M, comprising particle a and b, is described in the following equation:

PM = PaPb(−1)^L (2)

where P_a and P_bare the internal parities of particle a and b and L the relative orbital angular momentum of a and b[2].

Isospin: Hadrons containing the light u and d quarks, can be organized in ”fam- ilies” with similar mass. This is because the mass difference of the u and the d quark is very small. Within a family or an isospin multiplet, particles have the same spin and parity but with different charge. They also have the same total isospin I. However, the z projection of the isospin (denoted I3), differs within a multiplet. The isospin of a system of particles is defined as:

I3≡ Q − Y /2 (3)

where Q is the electric charge and Y is the hypercharge of a particle defined as:

Y ≡ B + S + C + ˜B + T (4)

B is the baryon number of the system (B=1/3 for quarks and B=-1/3 for antiquarks, while B=0 for mesons). S, C, ˜B and T depend on the quark flavor of the system and denoted strangeness (S), charm (C), bottom ( ˜B) and top (T), respectively [2].

1.3.2 Mesons

Mesons can be organized by their quantum numbers J and P, J^P. The representation of the pseudoscalar nonet (J^P = 0⁻[2]) is displayed in figure 2a and the vector nonet (J^P = 1⁻[2]) is displayed in figure 2b. The nonets comprise mesons with the same quantum numbers J^P, but different charge, isospin and strangeness. Properties of the lightest mesons, the pions, are displayed in table 1.

(a) Meson pseudoscalar nonet, spin 0 [4] (b) Meson vector nonet, spin 1 [5]

Figure 2: Two meson nonets, with spin 0 and spin 1, displaying particles with similar mass. Q represents charge. The Y-axis represent strangeness(S) of the particles and the X-axis represent the isospin(I₃).

Table 1: Properties of the lightest mesons, showing quark constituents, mass and lifetime of the charged, and neutral pions. [6].

Particle Quarks Mass[MeV] Lifetime [s]

π⁻, π⁺ d¯u, u ¯d 1.3950718 · 10²± 3.5 · 10⁻⁵ (2.6033 ± 5.0 · 10⁻⁴) · 10⁻⁸ π⁰ u¯u + d ¯d 1.349766 · 10²± 6.0 · 10⁻⁴ (8.52 ± 1.8 · 10⁻¹) · 10⁻¹⁷

1.3.3 Baryons

Baryons can be organized according to total spin and parity, J^P. The representation of the baryon octet (J^P = ¹₂⁺[2]) is displayed in figure 3a and the baron decouplet (J^P = ³₂⁺[2]) is displayed in figure 3b. The baryon decouplet and octet display baryons with similar quantum numbers but different strangeness.

(a) Baryon octet, spin ¹₂[7]

(b) Baryon decouplet, spin ³₂[8]

Figure 3: The organization of baryons into a octet of spin ¹₂ and an decouplet with spin ³₂. The mass of the baryons is displayed to the left in figure (a) and to the right in figure (b). the Y-axis represent strangeness (S) and the X-axis represent isospin (I₃).

1.3.4 Hyperons

Hyperons are baryons where at least one of the light u- and d quarks of the nucleon is replaced by a heavier one. Hyperons have a larger mass then protons and neutrons and they are unstable particles. The lightest hyperon is the Lambda (Λ) hyperon.

The properties of the Λ-hyperon and the properties of the lightest baryons, the proton and the neutron, are displayed in table 2.

Table 2: Properties of the lightest baryons, showing quark constituents, mass and lifetime of the protons and neutrons. *The proton is considered stable due to a mean life time of around 10³¹ to 10³³ years [6].

Particle Quarks Mass[MeV] Lifetime [s]

p uud 9.38272046 · 10²± 2.1 · 10⁻⁵ stable*

n udd 9.39565379 · 10²± 2.1 · 10⁻⁵ 8.803 · 10²± 1.1

Λ uds 1.115683 · 10³± 6.0 · 10³ (2.632 ± 2.0 · 10⁻²) · 10⁻¹⁰

Due to the Λ-hyperon being neutral it is very difficult to measure the Λ-hyperon directly. In particle physics experiments, one therefore often reconstructs the Λ- hyperon from its charged decay products. The major decay modes for the Λ-hyperon is displayed in table 3, the relative probability of the final state is normally referred to as branching ratio (BR) and is displayed in table 3.

Table 3: Branching ratio with the decay modes for the Λ-hyperon [6].

Decay modes Relative decay probability (BR) Λ → pπ⁻ (63.9 ± 0.5)%

Λ → nπ⁰ (35.8 ± 0.5)%

Λ → nγ (1.75 ± 0.15) · 10⁻³% Λ → pπ⁻γ (8.4 ± 1.4) · 10⁻⁴% Λ → pe⁻ν¯e (8.32 ± 0.14) · 10⁻⁴% Λ → pµ⁻ν¯µ (1.57 ± 0.35) · 10⁻⁴%

2 Formalism

2.1 Relativistic two-body kinematics

In a given particle reaction, 1 + 2 → 3 + 4 + 5 + .., where the numbers denote a parti- cle, and the time between the initial and final state is sufficiently long on a quantum mechanics scale, kinematic constraints have to be fulfilled. This means that energy and momentum needs to be conserved from the initial state to the final state. Of- ten, particle reactions are considered in different reference frames, for example the lab frame or the center-of-momentum frame. Different quantities, in a given frame, like energy and momentum, need to be related to the corresponding quantities in a different frame. Relativistic kinematics is the formalism which is used and in the following I will go through some basic definitions and relations which I use in my work.

The Center-Of-Momentum(CMS) system is in the rest system of the interacting particles (1 + 2, or 3+4+5..), which means that the total momentum in the CMS system is equal to zero.

N

X

i=1

¯

pi= 0 (5)

where N is the number of particles in the system. The CMS frame is the preferred frame for numerical calculations, for its simplicity. For example, the following two- body decay 1 + 2 → 3 + 4 would in the CMS system have momentum:

p₁+ p₂= p₃+ p₄= 0 (6)

where p₁is the notation of a three-vector for particle 1.

Euclidean three-vectors can describe, e.g. momentum in 3D space, while the four- vectors describe, e.g. four-momenta in space-time (4D space). The contravariant [9]

four-momentum of a particle is:

p^µ= (p⁰, p¹, p², p³) (7) and the covariant vector [9], with the (+ - - -) metric, is displayed in the following equation:

pµ= (+p0, −p1, −p2, −p3) (8) The dot product of p_µp^ν is defined as:

pµp^µ = +p²₀− p²₁− p²₂− p²₃ (9) By definition, p⁰ correspond to the time coordinate while p^1,2,3 correspond to the spatial coordinates, in such a way that in a four-momentum vector, p⁰ = E and p = (p¹, p², p³). This means that we can write p^µ in terms of p⁰ and p as displayed in equation 10:

p^µ = (E, p) (10)

For the two-body reaction 1 + 2 → 3 + 4 where the four momenta are denoted:

p₁= (p⁰₁, p¹₁, p²₁, p³₁) (11) and correspondingly for p2, p3, p4. The CMS energy (denoted s) can be determined from the four-momentum vectors as displayed in equation 12:

√s = |p₁+ p₂| = |p3+ p₄| = q

(p₃+ p₄)_µ(p₃+ p₄)^µ (12) Consider a two-body decay where 1 → 2 + 3. The momentum of particle 1 can be calculated from the momentum of particles 2 and 3.

p1= p2+ p3= (p⁰₂, p¹₂, p²₂, p³₂) + (p⁰₃, p¹₃, p²₃, p³₃) = (p⁰₂+ p⁰₃, p¹₂+ p¹₃, p²₂+ p²₃, p³₂+ p³₂) = (p⁰_tot, p¹_tot, p²_tot, p³_tot) = ptot

(13) From this the square of the invariant mass (denoted M²) can be defined, this is displayed in equation 14:

M²= ptotµp^µ_tot= (p⁰_tot)²− (p¹_tot)²− (p²_tot)²− (p³_tot)²= E_tot² − |p_tot|² (14) In my work, I study Λ → pπ⁻ and ¯Λ → ¯pπ⁺ decays. The mass of the Λ( ¯Λ) can then be calculated from the invariant mass of the proton, p(¯p) and pion, π⁻(π⁺):

M_Λ = M (pπ⁻) = q

E_tot² − |p_tot|²=q

(E_p+ E_π−)²− (p_p+ p_π−)² (15)

2.2 Cross section

The cross section is one of the most important observables in scattering experiments.

To perform scattering experiments, one can either let a beam of particles bombard a stationary target, or let two beams collide. Figure 4 shows an incident beam and figure 5 display an incident beam of particles to the left being scattered of a target into the solid angle element, dΩ. The probability that a cross section element dσ, of the beam is scattered into a solid angle element dΩ, is denoted the differential cross section and is written as _dΩ^dσ.

Figure 4: Scattering of an incoming beam to the left. [10]

Figure 5: From the left a beam of particles, denoted with lines and arrows, are scattered of a target and detected in the detector that covers a solid angle dΩ [10]

The beam intensity I0, denotes the number of particles per unit area and unit time. The number of particles scattered into a cone with a solid angle, dΩ per unit time is displayed in equation 16:

I(θ, φ)dΩ (16)

where θ is the polar angle and φ is the azimuthal angle. The differential cross section is then defined by equation 17:

dσ

dΩ =I(θ, φ) I0

(17) Theoretically, the differential cross section is expressed in terms of the scattering amplitude f (θ, φ) which is displayed in the following equation:

dσ

dΩ= |f (θ, φ)|² (18)

The scattering amplitude describes the physical processes in the scattering and incor- porates the relevant interactions (for example, the strong and electromagnetic forces) and the structure of particles involved. The total cross section is obtained by inte- grating over the full solid angle:

σ = Z dσ

dΩdΩ = Z

|f (θ, φ)|²dΩ (19)

2.3 Form factors

The cross section of reactions involving scattering of virtual photons, γ^∗, producing composite systems such as hadrons, can be parameterized in terms of ElectroMagnetic Form Factors (EMFF). Scattering involving virtual photons, γ^∗, can for example be elastic electron-nucleon scattering(e⁻N → e⁻N ) where the electromagnetic force is carried by a virtual photon, see figure 6a. It can also be electron-positron annihilation into a virtual photon with the subsequent production of a hadron-antihadron pair (e⁺e⁻→ γ^∗→ h¯h), see figure 6b.

The EMFFs are fundamental observables of QCD and describe the structure of hadrons, i.e they show the deviation from a point-like structure. In a certain refer- ence frame, the electric FF, GE and the magnetic FF, GM are related to the charge density and magnetization density respectively. Form factors can be studied in the Time-Like (TL) or Space-Like(SL) regions. The momentum transfer carried by the virtual photon (denoted q²) of a reaction defines the nature of the form factors. For elastic scattering experiments, such as electron-nucleon scattering of a nucleon, the momentum transfer is negative, q²< 0. The form factors are then SL and the EMFFs are real numbers. The Uppsala hadron physics group is interested in measuring the EMFF’s of hyperons. There are no experiments where a hyperon is used as a targets.

This is because the short life-times of hyperons make them unsuitable as station- ary target. Hyperons are instead investigated by the collision of an electron and a positron which annihilate and create an intermediate virtual photon which sub- sequently produces hadrons, for example a hyperon-antihyperon pair. This means that for hyperons, the TL region is experimentally accessible, by the process where e⁺e⁻→ Y ¯Y . The momentum transfer carried by the virtual photon will be positive, q²> 0, and the form factors GE and GM will be complex numbers.

(a) SL reaction of e⁻N → e⁻N⁰ [11]

(b) TL reaction of e⁺e⁻ → N ¯N [11], where the nucleon N could be exchanged for a hadron (denoted h).

Figure 6: The SL 6a and TL 6b reactions. q² is the momentum transfer for the reaction and for hyperons the TL reaction is used. The virtual photon is denoted with the curly line.

The cross section, parameterized in terms of form factors for a reaction, e⁺e⁻ → B ¯B where B is any spin 1/2 baryon, is given by:

dσ(q², θ_B)

dΩ = α²_sβC

4q² [(1 + cos²θ_B)|G_M(q²)|²+1

τsin²θ_B|GE(q²)|²] (20) There B denotes a baryon, αs the strong coupling constant, τ = ^4M_q2^B2 and q² the momentum transferred squared. Furthermore β =p1 − 1/τ , θ is the angle between the beam of the reaction and the outgoing baryon and C is the Coulomb factor [12]

(C=1 for neutral hadrons). In equation 21 the total cross section can be calculated from the form factors.

σ(q²) = 4πα²_sβC

3q² [|GM(q²)|²+ 1

2τ|GE(q²)|²] (21) Only a very limited amount of measurements have been performed on the hyperon form factors [13] [14]. The Uppsala group has therefore, in collaboration with insti- tutes from Mainz in Germany, Torino and Frascati in Italy and USTC Hefei in China, written a proposal to the BES III experiment for collecting new, unprecedented data sample for precision measurement of hadron form factors. The Uppsala group was responsible for the hyperon part. This proposal was approved in 2014 and the data was collected during 2014 and 2015. These data have the potantial to give us a greater understanding of how the strange quark affects the structure of the nucleus.

3 The BES III experiment

The only currently running facility in the world where time-like hyperon form factors can be studied is the BESIII experiment at the BEPC-II storage ring. In the following an introduction to the BESIII detector and BEPC-II will be given.

3.1 BEPC-II

The Beijing Electron-Positron Collider (BEPC-II) is a particle collider located in Beijing, China. It is a double storage ring collider which accelerates electrons and positrons to energies between 1 and 2.3 GeV, which means that the total CMS energy of the collision is between 2 and 4.6 GeV [15]. The electrons and positrons are accelerated in a linear accelerator before being injected into the rings. The BEPC-II has a circumference of 237.5m [15] and at the collision point the particles will collide with a total crossing angle of 22 mrad [15]. In figure 7, the electron ring is shown in red and the positron ring in blue. The green arrow points to the collision point where the BESIII detector is located.

Figure 7: The BEPC-II storage ring, the green arrow points towards the BESIII detector which also is the collision point. Each beam have a crossing angle of 11 mrad giving a total crossing angle of 22 mrad. [16]

3.2 BESIII detector

The BESIII (BEijing Spectrometer III) detector is a multi-purpose detector that cov- ers a solid angle of almost 4π [15]. It consists of several sub-detectors: the Main Drift Chamber (MDC), the ElectroMagnetic Calorimeter (EMC), the Time Of Flight sys- tem(TOF) and the muon chamber (MUC). Together, these sub-detectors will provide the information needed to determine the types of particle we register in the collisions.

An overview of the detector is displayed in figure 8. It is built in several consecutive layers, where the MDC is closest to the interaction point. The other layers come in

the following order: TOF, EMC and MUC. The values of the cosine of the lab polare angle (cos θ) displayed to the right indicate the limits of angular coverage of each detector.

Figure 8: An overview of the BESIII detector[17] an integrated part of the BEPC-II collider. The MDC, TOF and the EMC can be seen in the central part of the detector while the muon chamber is surrounding it.

The MDC is used to measure the momentum of charged particles. A magnetic field of 1 Tesla [18] is applied which bends the trajectories of the charged particle.

This means that the momentum and charge of a particle can be extracted from the curvature of its trajectory in the magnetic field. The bending radius of charged particles in a magnetic field is displayed in equation 22:

ρ = p

qB (22)

where ρ is the bending radius, q is the charge of the particle, B is the magnetic field and p is the momentum of the particle. The MDC is filled with a helium-based gas, the gas will ionize from the traversing particles and the electrons from the ionization process are collected on wires in the MDC. The collected charges will give a signal in the wire which is read out by the electronics.

The TOF system is constituted by plastic scintillators, i.e a material which pro- duces visible light when traversed by a charged particle. The TOF detector consists of two end-caps and a barrel with time, ∆t, resolution of 110ps and 80ps respectively [18]. The TOF system measure the time it takes from the instant when a particle is produced, which is given by the time of the beam-beam collision, and the time it

traverses in the TOF detector. From ∆t, the velocity of the particle is calculated.

Combining the information of the momentum from the MDC and the velocity from the TOF, the mass of a particle, and thus its type, can be determined.

The EMC consist of cesium iodine(CsI) crystals activated by Thallium[12] and measure the energies and directions of mainly photons, electrons and positrons.

The MUC is the sub-detector in the outer layer of the detector which will identify muons.

4 The e

e

→ Λ ¯ Λ reaction

The reaction investigated for this project is e⁺e⁻ → γ^∗ → Λ ¯Λ at a CMS energy of 2.396 GeV. The electron-positron pair will annihilate into a virtual photon which subsequently produces a Λ ¯Λ pair. In figure 9 the reaction is displayed with the coordinate system used in this work. The y(ˆn) axis is defined by the normal of the production plane, which is spanned by the incoming e⁺ or e⁻ and the outgoing Λ or ¯Λ. The z(ˆl) axis is directed along the momentum of Λ or ¯Λ. Finally the x( ˆm) axis is defined by the cross product of the z- and y-axis. The angle θΛ, which in the

Figure 9: The reaction e⁺e⁻ → Λ ¯Λ in a coordinate system [12].

figure is denoted with θ, is the angle between the incident electron-positron beam and the outgoing Λ-hyperon. The most common decay channel is Λ → pπ⁻ for Λ and Λ → ¯¯ pπ⁺ for ¯Λ, and the BR for these decays are (63.9 ± 0.5)%, see table 3 in section 1.3.4. In this work we only consider the Λ → pπ⁻ and ¯Λ → ¯pπ⁺ decay channels.

Since the Λ decay changes flavor it can only occur through weak interaction. That means that the life-time of a Λ-hyperon is relatively long and can traverse parts of the MDC before decaying. The charged pions also decay weakly and have an even longer life-time. On a particle physics time-scale they can therefore be considered stable since they will live long enough to be stopped by the detector before decaying. Thus they will traverse the MDC. The differential cross section of e⁺e⁻ → Λ ¯Λ is given by equation 23:

dσ

dΩ =α²_sβC

4q² [|G_M(q²)|²(1 + cos²θ) + 1

τ|GE(q²)|²(sin²θ)] (23) Experimentally, the cross section is given by equation 24:

σ = N_signal

L(1 + δ)Br(Λ → pπ⁻)Br( ¯Λ → ¯pπ⁺) (24)

where Nsignal is the number of signals in the detector, L is the luminosity,  is the efficiency of the detector, (1 + δ) is a so-called radiative correction factor which takes in consideration the spread of the beam energy and the reaction where a photon may be radiated before the e⁺e⁻ collision and Br(Λ → pπ⁻)Br( ¯Λ → ¯pπ⁺) is the branching ratio of the Λ- and ¯Λ-hyperon, see table 3 in section 1.3.4. In particle physics experiment, statistical significance is very important. Minimizing statistical uncertainty in σ means maximizing the number of observed events, Nsignal.

From equation 24, we see that the number of signals is proportional to the cross section and efficiency of the detector. Therefore a high cross section combined with a high efficiency is preferred.

The focus of the BESIII group in Uppsala is currently the e⁺e⁻ → Λ ¯Λ reaction at √

s = 2.396 GeV, where a large amount of data have been collected for precision measurement of Λ form factors. The choice of√

s energy is motivated by a measure- ment by the BaBar collaboration [13], where a large cross section was observed at

√s < 2.5 GeV. On the other hand, simulation studies [19] show that for√ s < 2.3 GeV, the detection efficiency decreases rapidly. The region around 2.4 GeV should therefore be optimal in terms of number of events.

5 Software tools

With today’s powerful and versatile detectors and high luminosity in particle colliders the data samples are very large. Therefore the demand for software that can handle and analyze all of these data is very high.

5.1 BOSS

The software that have been used in this project is the BESIII Offline Software Sys- tem(BOSS). It was developed as a tool for processing and analyzing data from reac- tions in BES III. The software was constructed with the object oriented programming language C++. The BOSS analysis chain consists of five steps: event generation, simulation, digitization, reconstruction/calibration and analysis[4], for further expla- nation see section 6. For this project the BOSS version 6.6.5.p01 have been used, which provides all tools needed for the steps above. The output from BOSS was fur- ther analyzed using the ROOT [20] framework. Data samples can be stored in ROOT trees, for which an analysis can be applied on.

5.2 Software environment

A virtual machine was used for this project and it is an emulator that can integrate another operating system onto a computer. A scientific operating system is required by BOSS, therefore Scientific Linux 6.7 have been used during this project. The advantage of using a virtual machine is that it is portable and can be used as a common starting environment for students working with projects related to BOSS. The virtual machine provides a BOSS installation, a selection algorithm, the simulation scripts and development tools. As a programming environment the Eclipse IDE for C++

developers [21] have been used. Eclipse is a powerful tool for programming, as it will display errors if the code can not be compiled. Eclipse is also useful for finding bugs in the code.

6 Monte Carlo simulations

The e⁺e⁻ → Λ ¯Λ reaction is the focus of interest of the Uppsala BESIII group. Dr.

Cui Li, post doc in the group, has developed a number of data selection criteria for this channel. I have implemented the same criteria independently in order to verify the results obtained by Cui Li. In particular, I have looked into the detector and reconstruction efficiency of e⁺e⁻ → Λ ¯Λ as a function of the scattering angle for the Λ-hyperon. This is achieved by Monte Carlo (MC) simulations. MC methods utilize randomized sampling to generate numerical values. In particle and hadron physics, the reaction of interest is generated using a MC generator. The produced particles are then propagated through a virtual description of the detector and their interaction with the detector material are simulated. These produced MC data can then be reconstructed and analyzed in the same way as real experimental data. By studying the MC simulated data we can learn how to filter out background while optimizing the yield of the relevant signal events. Furthermore knowing what the generated ”true”

data, i.e MC-data looks like we then also know what to look for in the real data. We can also optimize the analysis methods and verify their correctness. MC data are thus necessary in order to understand the experimental data.

6.1 Generators

Particle generators provide the reactions to be simulated and analyzed. They uti- lize a given initial state as input and produce a final state. The parameters of the initial state, e.g. momenta, energies, beam inclination (see section 3.1 on BEPC-II) and particle types, constrain the parameters of the final state. The output from a generator is typically final state particles with corresponding four momenta and, if the generated particles are unstable also the decay products and decay vertex. The simplest generator is a so-called phase-space generator (in BOSS and in the following referred to as PHSP). In the PHSP generator, the constraints on the final state from the initial state is purely kinematic. The PHSP generator is simple, unbiased and give isotropic distributions of e.g. the angles of the produced particles. Figure 10 show the distribution of the CMS polar angle cos θΛ of the Λ-hyperon produced in a phase space generated e⁺e⁻ → Λ ¯Λ reaction. The isotropic distribution imply that the generated particles are evenly distributed in the detector. The PHSP generator is therefore suitable for studying the characteristics of the detector.

In this work, an additional generator has been used as a cross-check of the analysis, namely the CONEXC generator. It is based on the PHOKHARA [22] generator which does not only take kinematic constraints in consideration but also the dynamics.

The events are generated according to the form factor parameterization presented in section 2.3. It also includes radiative correction, e.g. when the incident e⁺e⁻radiates a photon before colliding, this is displayed in figure 11. In the ConExc generator, the user defines the desired value of the ratio of the electromagnetic form factor, GE and GM, which determines the angular distribution of the outgoing Λ-hyperons. Figure 12 display the CMS polar angle distribution when the ratio of the electromagnetic form factors are set to 1.

6.2 Particle interaction with the detector

The generated particles will be propagated through a virtual detector which was constructed with the GEANT4 [23] software. By taking both active detector material

θΛ

cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

0 500 1000 1500 2000 2500

Figure 10: The true distribution of detected events with respect to the cos θΛ-angle for data generated with the PHSP generator, the y-axis represent the number of events and the x-axis represent the angular distribution of the Λ-hyperon.

Figure 11: A reaction where a photon is radiated from the electrons before the electron-positron annihilation [24]

and passive (mechanical support) detector material of the real detector into account, the description of the virtual detector will be as realistic as possible. The particles will traverse in the virtual detector and the information from each sub-detector will be stored as so called MC-points. Information about energy loss is also stored.

6.3 Digitization

The MC-points gathered from the traversing of particles in the virtual detector will be converted into so-called MC-hits. The MC-hits will be converted to signals in the electronics of the individual detectors, such as a wire in the MDC or a crystal in the EMC. After digitization the output will have the same format as data from real

θΛ

cos

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Events

0 500 1000 1500 2000 2500

Figure 12: The true distribution of detected events with respect to the cos θΛ-angle for data generated with the ConExc generator where the the ratio, |GE|/|GM| of the electromagnetic form factors was set to 1.

experiments. Furthermore, the energy losses will be converted into pulse heights.

6.4 Reconstruction

The BESIII collaboration have a reconstruction package that contains several different algorithms that will take the MC hits from the previous step and combine them into tracks. The pulse heights will be converted to energy using calibration techniques. In BOSS, particle identification probabilities are also calculated in this step. The output from the reconstruction is charged and neutral track candidates.

6.5 Analysis

When analyzing simulated or experimental data, the user defines what criteria the particles that have been reconstructed need to fulfill in order to be kept for further processing. The analysis provides a way to investigate the properties of particles by histograms of interesting quantities, e.g. momenta, energies invariant masses and angles. If the goal of the analysis is to study a specific decay there need to be selection criterias implemented that will reject events that come from a different decay while keeping those which corresponds to the decay channel of interest. To reduce the execution time of my analysis for the large data samples that can be provided by the MC generator, a pre-selection of criteria can be implemented. The goal is to run the pre-selection one time on the generated MC data, thus reducing the overall time for running the analysis.

7 Analysis

7.1 Event selection

A set of event selection criteria has been implemented in order to distinguish good e⁺e⁻→ Λ ¯Λ candidate events from background. The reason is that real experimental data is dominated by events which do not come from the e⁺e⁻ → Λ ¯Λ reaction.

The analysis of the e⁺e⁻→ Λ ¯Λ reaction is currently undergoing within the Uppsala hadron physics group and this project provides a cross-check of the current results. I have implemented the event selection criteria that were developed by Dr. Cui Li. For my analysis the pre-selection will remove events that definitely does not belong to the reaction under investigation. The pre-selection criteria and the final event selection criteria will be introduced below.

7.1.1 Pre-selection

Four charged tracks: For the e⁺e⁻ → Λ ¯Λ reaction with the subsequent decay of Λ → pπ⁻ and ¯Λ → ¯pπ⁺, there will be four charged particles in the final state that can leave a track in the detector. Therefore the pre-selection will select events with at least four charged tracks. The reason why not exactly four charged tracks are required is because there are situations where a e⁺e⁻ → Λ ¯Λ signal can give rise to more then four tracks, e.g in the case of overlapping events or tracks from a secondary interaction with the detector.

Particle identification: In the pre-selection, the identity of the charged tracks was checked, based on energy loss in the detectors and time-of-flight. At least one particle each of type p, ¯p, π⁻, π⁺was required.

Combinations of protons and pions: The protons and pions with opposite charge was combined together and a vertex fit of the oppositely charged tracks was done. If the vertex fit was successful the proton- and pion pairs was considered a Λ-candidate.

∆E difference: The pre-selection look for the difference between the energy of the Λ-candidate and the CMS energy for one of the beams, meaning half of the total CMS energy. This difference, denoted ∆E, was ordered for all the Λ-candidates, where the Λ-candidate with the smallest ∆E was considered the ”best”. The ”best” Λ-candidate was then saved.

Λ, ¯Λ separation: The pre-selection will separate the Λ and ¯Λ, this was performed by a check of the charge of the protons. Positively charged protons lead to a Λ, and negatively charged protons lead to a ¯Λ.

Store Λ, ¯Λ-candidates: The final step of the pre-selection is to store events that correspond to exactly one Λ-candidate and one ¯Λ-candidate, for further analysis.

7.1.2 Final selection

In order to further investigate the Λ- and ¯Λ-candidates a final selection has been implemented. The criteria implemented in the final selection is introduced below:

Momentum of pions and protons: Consider the Λ → pπ⁻ decay. The Λ- hyperon mass is 1.115 GeV and the proton mass is 0.938 GeV, see table 2. The pion mass is much smaller, 0.139 GeV, see table 1. In order to conserve momentum in the Λ → pπ⁻ decay, the Λ-hyperon will give most of its energy to the proton. This means that the proton from a Λ decay will have much larger energy and momentum then the pion from the Λ decay. Therefore, I have implemented a separation criteria of protons

and pions in addition to the previous TOF and ^dE_dx separation. The requirement for the pions is that momentum for each pion has to be less then 200 MeV/c, (p(π⁻) < 200 MeV/c, p(π⁺) < 200 MeV/c), and the requirement for protons is to have a minimum of 200 MeV/c each, (p(p) > 200 MeV/c, p(¯p) > 200 MeV/c). The ”true” momentum from undistorted MC simulations for the protons and pions are displayed in figure 13 along with the reconstructed momentum. In figure 14, the distributions are shown

p / (GeV/c)

0 0.1 0.2 0.3 0.4 0.5 0.6

-1 /(5 MeV/c)dpdN

0 200 400 600 800 1000

p p π-

π+

p / (GeV/c)

0 0.1 0.2 0.3 0.4 0.5 0.6

-1 /(5 MeV/c)dpdN

0 50 100 150 200 250 300

p p π-

π+

Figure 13: The representation of the momentum distribution of the pions and protons for, MC truth data in the upper panel and reconstructed data before final event selection in the lower panel. The Y-axis display the number of events for the pion momentum region, which is displayed on the X-axis. The green data points denote π⁻, the purple denote π⁺, the red denote p and the blue denote ¯p.

after the final selection have been applied to the reconstructed data.

p / (GeV/c)

0 0.1 0.2 0.3 0.4 0.5 0.6

-1 /(5 MeV/c) dpdN

0 20 40 60 80 100 120 140 160 180 200

220 p

p-

π π+

Figure 14: The momentum distribution of protons and pions for reconstructed data after the final selection. The Y-axis represent the number of events for over the momentum region, which is displayed in the X-axis. The green data points denote π⁻, the purple denote π⁺, the red denote p and the blue denote ¯p.

Invariant mass: If a proton and a π⁻ ( or ¯p and a π⁺) comes from a Λ decay (or Λ decay) they should have an invariant mass, M(pπ¯ ⁻), M (¯pπ⁺) see equation 15 in section 2.1, close to the Λ mass. Therefore the invariant mass of the pπ⁻ and ¯pπ⁺ combination is calculated and required to be within 3 standard deviations σ from the nominal Λ mass, see table 2 in section 1.3.4: |M (pπ⁻) − 1.115683| < 0.006 GeV,

|M (¯pπ⁺) − 1.115683| < 0.006 GeV. In order to estimate the standard deviation, a Gaussian curve have been fitted to the invariant mass of the Λ-candidate. In figure 15 the invariant mass of the Λ- and ¯Λ-candidates are displayed together with a Gaussian fitted to reconstructed data. The invariant mass of the Λ- and ¯Λ-candidates before the final event selection is displayed in figure 15a. After the final event selection have been applied to the Λ- and the ¯Λ-candidates the invariant mass distribution is seen in figure 15b. The Λ-candidates are denoted in red and the ¯Λ-candidates in blue.

2) / (GeV/c MΛ

1.1 1.105 1.11 1.115 1.12 1.125 1.13

-1)2 /(1 MeV/c ΛdMdN

0 200 400 600 800

1000

Λ

Λ

(a) The invariant mass distribution for the Λ-candidates and the ¯Λ-candidates before the final event selection. A very long tail can be seen for values well above and below the allowed invariant mass of the Λ-hyperon.

2) / (GeV/c MΛ

1.1 1.105 1.11 1.115 1.12 1.125 1.13

-1)2 /(1 MeV/c ΛdMdN

0 100 200 300 400 500 600

700

Λ

Λ

(b) The invariant mass distribution of reconstructed signals after the final event selection.

Figure 15: The invariant mass before and after the final event selection. A fit with a Gaussian curve have been implemented to test the significance. The extracted width from the Gaussian fit was 1.9 MeV, this corresponds will with the expected 3σ deviation. The Y-axis display the number of events over the mass distribution of Λ- candidates, which is displayed in the X-axis. The red data points denote Λ-candidates and the blue denote ¯Λ-candidates, the red line is the Gaussian fit which have been applied to the invariant mass of the Λ-candidate.

Decay length: To determine if the proton- and pion pairs could reconstruct a Λ- candidate a vertex fit is performed. The vertex fit adjusts the track parameter of the proton and the pion in such a way that resulting Λ-candidate momentum has the same direction as the line between the e⁺e⁻ collision point and the Λ decay vertex. From the vertex fit, the decay length is obtained. The decay length is the distance between the collision point and the decay vertex. If the Λ momentum points oppositely to the direction from the collision to the decay point, then the decay length becomes negative in the vertex fit algorithm. Due to the mean life time of the Λ-hyperon being relatively long on an experimental scale, the decay should be well separated from the e⁺e⁻ collision point. To make sure this is the case a requirement on the decay length is set such that: L(Λ) > 0.2 cm,L( ¯Λ) > 0.2cm.

LΛ

-4 -2 0 2 4 6 8 10 12 14

ΛdL

dN

0 100 200 300 400 500

600

Λ

Λ

(a) 1-dimensional view of the decay lengths before the final event selection.

LΛ

-4 -2 0 2 4 6 8 10 12 14

ΛdL

dN

0 100 200 300 400 500

600

Λ

Λ

(b) 1-dimensional view of the decay lengths after the final event selection.

Figure 16: 1-dimensional view of the decay lengths for the Λ- and ¯Λ-candidates. The Y-axis display the number of events for the decay lengths of the Λ-candidates, which is displayed on the X-axis. The Λ-candidates are denoted in red and the ¯Λ in blue

LΛ

-4 -2 0 2 4 6 8 10 12 14

ΛL

-4 -2 0 2 4 6 8 10 12 14

(a) 2-dimensional view of the decay lengths for the reconstructed Λ ¯Λ-candidates before the final event selection.

LΛ

-4 -2 0 2 4 6 8 10 12 14

ΛL

-4 -2 0 2 4 6 8 10 12 14

(b) 2-dimensional view of the decay lengths for the reconstructed Λ ¯Λ-candidates after the final event selection.

Figure 17: The decay lengths for the reconstructed Λ, ¯Λ-candidates before and after the final event selection. The Y-axis represent the decay length of the ¯Λ-candidates and the X-axis represent the decay length of the Λ-candidates.

Angular limits: Since the Λ and the ¯Λ are emitted back-to-back in the CMS system, the angle between them, θ_{Λ ¯Λ} should be close to 180^◦. Taking the detector resolution into account, a requirement of θ_{Λ ¯Λ}> 177^◦ was found optimal.

Momentum of Λ, ¯Λ: In section 2.1 we concluded that three momentum of a particle produced in a two-body reaction 1 + 2 → 3 + 4 is fixed. For a beam with a CMS energy of 2.396 GeV the Λ( ¯Λ) momentum |p_Λ|(|pΛ¯|) is found to be 0.436 GeV/c.

Therefore we require the events to fulfill ||p_Λ| − 0.436| < 0.02 GeV/c for both Λ and Λ. The momentum of the Λ, ¯¯ Λ-candidates are shown in figure 18.

/ (GeV/c) pΛ

0.41 0.42 0.43 0.44 0.45 0.46 0.47

-1 /(2 MeV/c) ΛdpdN

0 100 200 300 400 500 600 700 800 900

Λ Λ

(a) The momentum distribution of reconstructed Λ- and ¯Λ-candidates before the final event selection have been applied.

/ (GeV/c) pΛ

0.41 0.42 0.43 0.44 0.45 0.46 0.47

-1 /(2 MeV/c) ΛdpdN

0 100 200 300 400 500 600

Λ Λ

(b) The momentum distribution of reconstructed Λ- and ¯Λ-candidates after the final event selection have been applied.

Figure 18: The momentum distribution of reconstructed Λ- and ¯Λ-candidates before and after the final event selection. The Y-axis denote the number of events over the momentum distribution, which is displayed on the X-axis. The red data points denote the Λ-candidates and the blue the ¯Λ-candidates.

7.2 Total efficiency

The analysis proceeds as follows: For each event, my algorithm goes through the event selection and checks if the event fulfills them or not. If it does, then the event is classified as a signal. If the event selection criteria is not fulfilled, it is instead classified as background and is filtered out. The events remaining and the total efficiency after each event selection criteria can be seen in table 4. Table 4 reflect the implementation of subsequent criteria. This means that the momentum criteria on pions and proton, which have been applied after the pre-selection does not necessarily reflect the total number of events removed from that criteria, but rather how many events are still available after the pre-selection and the momentum criteria on pions and protons have been implemented.

Criteria Events MC Events remaining (%)

No cuts 20000 100 %

Pre-selection 4846 24.2%

p(p, ¯p, π⁻, π⁺) 4815 24.0 %

|M (pπ⁻, ¯pπ⁺) − 1.115683 · 10³| < 6.0MeV 4128 20.6 %

L(Λ, ¯Λ) > 0.2 3209 16.0 %

θ_{Λ ¯Λ}> 177 2988 14.9 %

|p(Λ, ¯Λ) − 4.36 · 10²| < 20.0 MeV 2787 13.9 %

Table 4: Display of number of events and efficiency after each criteria has been implemented. Expected final efficiency with respect to previous work done by Cui Li was 14%.

7.3 Efficiency and correction

When analyzing real, experimental data, the goal is to extract a ”true” quantity from an observed one. The observed quantity will be distorted by the limit of the detector and the reconstruction efficiency. In order to unfold the ”true” results, the observed quantity needs to be corrected for e.g. efficiency and resolution. Here, we focus on the efficiency. In MC simulations, the undistorted MC truth data correspond to the true events, whereas the reconstructed results corresponds to observed ones. From the ratio between the reconstructed and the MC truth results a correction factor, or a weight, can be obtained. This weight will later be used to correct experimental data.

The process of correcting the reconstructed signals is introduced below:

1. Generated, unfiltered, MCtruth events are stored in a histogram. For the PHSP generator this is displayed in figure 10, see section 6.1.

2. Another histogram is filled with the reconstructed, filtered events. This his- togram should have the same range and have the same number of bins as the MC truth histogram. The reconstructed events without a correction and the MC truth events are displayed in figure 19. The histogram reveals a clear dif- ference between the reconstructed signals in blue and the MC truth signals in red. The difference reflects the effect of the applied selection criteria and the resolution.

3. We calculate the efficiency for each bin. efficiency(bin i) = #Reconstructed events in bin i

#Generated events in bin i

θΛ

cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

0 500 1000 1500 2000 2500 3000

MCTruth Reconstructed

Figure 19: The clear difference in how the reconstructed signals look before the cor- rection, in comparison to the MC truth data which is how we expect the data to look like.

4. A new histogram with the efficiency is constructed.

5. A fourth order polynomial is fitted to the efficiency histogram. Fourth order polynomial : a · x⁴+ b · x³+ c · x²+ d · x + e =efficiency(x), where x is cos θ_Λ. The histogram with efficiency and fourth order polynomial fit are displayed in figure 20.

6. When analyzing reconstructed MC and real data, the efficiency and the corre- sponding weight is calculated in each event. The reconstructed events are then corrected using a weight (denoted W). W = _efficiency¹ . For the MC data we would then see that each event is corrected according to its weight:

Corrected = Reconstructed

efficiency =Reconstructed

Reconstructed Generated

= Generated

This means that the correction of reconstructed signals should be the same as the true MC data. However, since the correction is performed event-by-event while the correction function is obtained in a binned fit, we do not expect exact agreement.

To verify this we create a histogram with the weighted and corrected, reconstructed signals and the MC truth signals. These are displayed in figure 21, where we can see that the correct reconstructed data of the outermost bins -0.9 and 0.9 disagree with the MC truth data. This means that the efficiency correction does not work well in this region. The reason is likely that the efficiency function varies rapidly at extreme angles which makes it difficult to model the correction. It may even go to zero close to |cosθΛ| = 1.

θΛ

cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Efficiency

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Figure 20: The efficiency of the detector fitted with a fourth order polynomial to calculate the total efficiency of the detector.

θΛ

cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Events

0 500 1000 1500 2000 2500 3000

MCTruth

Reconstructed and corrected

Figure 21: The reconstructed signals, with a correction, in blue and the MC truth signals in red. The need for a limit on the angular distribution of the Λ-candidate is shown, due to reconstructed values in the outer regions not being similar to the MC truth signals, which they are expected to be.

8 Results and discussion

The main part of the project was to see if the construction of an independent check of Dr Cui Li’s previous work with the total efficiency from the reaction e⁺e⁻ → Λ ¯Λ