UPPSALA UNIVERSITY

DEPARTMENT OF PHYSICS AND ASTRONOMY

BACHELOR OF SCIENCE DEGREE IN

PHYSICS, 15 credits

Monte Carlo simulation study of the

e+e− → ΛΛ reaction with the BESIII

experiment

Author : Niklas FORSSMAN

Supervisor : Karin SCHONNING, 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 thathadrons can be formed from their energy is an excellent tool when we try to improveour understanding of the standard model. Hadrons are composite quark systemsheld 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 onecan obtain information about the electromagnetic form factors, GE and GM , whichdescribe the inner electromagnetic structure of hadrons. This will give us a betterunderstanding 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 BeijingElectron-Positron Collider (BEPC-II) in China. Uppsala university has several sci-entists working with the BESIII experiment. My task was to do a quality assuranceof previous results for the reaction e+e− → ΛΛ at a center of momentum energy of2.396 GeV. During a major part of the project I have been working with Monte Carlodata. Generating the reactions was done with two generators, ConExc and PHSP.The generators were used for different means. I have analyzed the simulated data tofind 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 severalselection criteria to extract these signals. The total efficiency of Cui Li’s analysis was14%. For my analysis I also obtained total efficiency of 14%. This gave me confidencethat my analysis have been implemented in a correct fashion and that my analysisnow can be transferred over to real data. It is also reassuring for Cui Li and the restof the group that her analysis has been verified by and independently implementedselection algorithm.

Abstract

Att studera vad som hander vid reaktioner dar elektroner och positroner kollideraroch annihilerar sa att hadroner kan bildas ur energin kan vara till stor hjalp nar vivill forsta standardmodellen och dess krafter, i synnerhet den starka kraften, somkan studeras i sadana reaktioner. Genom att utfora precisa matningar av tvarsnittfor hadronproduktion far man fram de elektromagnetiska formfaktorerna GE ochGM som beskriver hadronernas inre struktur. Hadroner ar sammansatta system avkvarkar och den starka kraften binder dessa kvarkar.Under mitt examensarbete har jag anvant mig av data fran detektorn BESIII somfinns vid BEPC-II (Beijing Electron-Positron Collider) i Kina. Uppsala universitethar flera forskare som jobbar med BESIII experimentet. Malet var att kvalitetssakraden tidigare analys som gjorts for reaktionen e+e− → ΛΛ vid 2.396 GeV. Jag borjademed att gora Monte Carlo-simuleringar. Reaktionerna har genererats med tva olikageneratorer, ConExc och PHSP. Dessa generatorer har anvants till olika andamal. Degenererade partiklarnas fard genom detektorn har sedan simulerats. Da bildas dataav samma typ som dem man far fran experiment. Jag har analyserat dessa simuleradedata for att hitta en metod som kan filtrera bort bakgrundsstorningar samtidigt somintressanta data sparas. Kriterier utarbetade av Dr. Cui Li har anvants for att skapadenna metod. Min algortim gav en total effektivitet pa 14%, vilket stammer bra medden tidigare algoritmen som Cui Li skapade, aven dar var effektiviteten 14%. Dettager fortroende for min algortim och den starker aven Cui Lis resultat.

Contents

1 Introduction 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The strong force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 Hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Formalism 52.1 Relativistic two-body kinematics . . . . . . . . . . . . . . . . . . . . . 52.2 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 The BES III experiment 93.1 BEPC-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 BESIII detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 The e+e− → ΛΛ reaction 12

5 Software tools 135.1 BOSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Software environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Monte Carlo simulations 146.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 Particle interaction with the detector . . . . . . . . . . . . . . . . . . . 146.3 Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Analysis 177.1 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.1.1 Pre-selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.1.2 Final selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.2 Total efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.3 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 BESIIIsoftware at the hadron physics group at Uppsala university. The goal is that thevirtual machine setup by Michael Papenbrock at Uppsala university can be usedfor 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 ofa hadron-antihadron pair we obtain valuable information about the inner electromag-netic structure of hadrons. Hadrons are tied together by the strong force and studyingthe internal structure of hadrons improves our understanding of the strong force thatbinds them together.

1.1 The Standard Model

The Standard Model (SM) is a theory of the fundamental particles in nature andhow they interact. The standard model has been successful in predicting particlesand other features of the microscopic world. In the standard model the fundamentalparticles are quarks, leptons and gauge bosons. The leptons can be organized inthree generations: electrons, muons and tauons, with their respective neutrinos. TheStandard Model also comprise the quarks (denoted q) and antiquarks(denoted q).All quarks have an antiquark with the same mass but opposite charge. The quarkscan also be organized into three generations, with the up(u) and down(d) quarksconstituting 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 infigure 1, where quarks are particles in purple, leptons in green and gauge bosons inred.There are four fundamental forces in nature.The strong force, the weak force, theelectromagnetic force and gravity. The SM presently incorporates the first three ofthese forces but not yet gravity. In the SM the forces are mediated by the gaugebosons, also referred to as the force carriers. There are four gauge bosons in the SMseen to the right in figure 1 : photon, gluon, Z - and W -boson. The force carrier forthe electromagnetic force is the photon while the gluon is the force carrier for thestronge force. The Z- and W-bosons are force carriers of the weak force. A recentconfirmation 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 holdsthe quarks together in composite systems, such as the proton or neutron. The range ofthe strong force is 10−15m [2] and the theory of strong interaction is called QuantumChromoDynamics(QCD) [2]. To classify particles constituted by quarks a quantitynamed color charge is introduced. The colors are called red, blue and green. While theelectromagnetic force acts on electrically charged objects (like quarks and electrons),the strong force acts on particles carrying color charge (like quarks and gluons). Thefact that gluons carry color charge leads to gluons coupling with other gluons, whilethe photons, which are neutrally charged, in the electromagnetic case will not coupleto other photons. As a consequence, the strong interaction becomes stronger with a

1

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

larger separation between two colored objects, in contrast with the electromagneticcase. The strength of a force is parameterized in terms of its coupling constant. For thestrong 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 calculateexperimental 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 anantiquark, they will create another quark-antiquark pair that sticks to the first pairsince it is more energy efficient. Inside the nucleus the quarks move freely since thestrong 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 statesthat no quarks can be observed as free isolated objects. One feature of the stronginteraction is the mass generation. When bound into composite systems, the quarksare 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% ofthe 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 hadronsare color neutral in the same way as atoms are electrically neutral through consisting ofelectrically charged entities. The major part of the observed hadrons can be describedeither as baryons, i.e three-quark systems (denoted qqq) or as mesons, i.e a quark-antiquark system(denoted qq).

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1.3.1 Quantum numbers

Baryons and mesons are organized into multiplets according to their properties. Themain 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 aquantum 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 1

2 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 asystem 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 describedin the following equation:

PM = PaPb(−1)L (2)

where Pa and Pb are the internal parities of particle a and b and L the relative orbitalangular 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 quarkis very small. Within a family or an isospin multiplet, particles have the same spin andparity 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 ofa 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 anddenoted 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, JP . The representationof the pseudoscalar nonet (JP = 0−[2]) is displayed in figure 2a and the vector nonet(JP = 1−[2]) is displayed in figure 2b. The nonets comprise mesons with the samequantum numbers JP , but different charge, isospin and strangeness. Properties ofthe lightest mesons, the pions, are displayed in table 1.

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(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 similarmass. Q represents charge. The Y-axis represent strangeness(S) of the particles andthe X-axis represent the isospin(I3).

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

Particle Quarks Mass[MeV] Lifetime [s]

π−, π+ du, ud 1.3950718 · 102 ± 3.5 · 10−5 (2.6033± 5.0 · 10−4) · 10−8

π0 uu+ dd 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, JP . The representation

of the baryon octet (JP = 12

+[2]) is displayed in figure 3a and the baron decouplet

(JP = 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).

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1.3.4 Hyperons

Hyperons are baryons where at least one of the light u- and d quarks of the nucleonis replaced by a heavier one. Hyperons have a larger mass then protons and neutronsand 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 protonand the neutron, are displayed in table 2.

Table 2: Properties of the lightest baryons, showing quark constituents, mass andlifetime of the protons and neutrons. *The proton is considered stable due to a meanlife 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 ± 1.1Λ 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-hyperondirectly. In particle physics experiments, one therefore often reconstructs the Λ-hyperon from its charged decay products. The major decay modes for the Λ-hyperonis displayed in table 3, the relative probability of the final state is normally referredto 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−νe (8.32± 0.14) · 10−4%Λ→ pµ−νµ (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 quantummechanics scale, kinematic constraints have to be fulfilled. This means that energyand 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 thelab 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 ina different frame. Relativistic kinematics is the formalism which is used and in thefollowing I will go through some basic definitions and relations which I use in my work.

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

N∑i=1

pi = 0 (5)

where N is the number of particles in the system. The CMS frame is the preferredframe 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 p1 is 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:

pµ = (p0, p1, p2, p3) (7)

and the covariant vector [9], with the (+ - - -) metric, is displayed in the followingequation:

pµ = (+p0,−p1,−p2,−p3) (8)

The dot product of pµpν is defined as:

pµpµ = +p2

0 − p21 − p2

2 − p23 (9)

By definition, p0 correspond to the time coordinate while p1,2,3 correspond to thespatial coordinates, in such a way that in a four-momentum vector, p0 = E andp = (p1, p2, p3). This means that we can write pµ in terms of p0 and p as displayedin equation 10:

pµ = (E, p) (10)

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

p1 = (p01, p

11, p

21, p

31) (11)

and correspondingly for p2, p3, p4. The CMS energy (denoted s) can be determinedfrom the four-momentum vectors as displayed in equation 12:

√s = |p1 + p2| = |p3 + p4| =

√(p3 + p4)µ(p3 + p4)µ (12)

Consider a two-body decay where 1 → 2 + 3. The momentum of particle 1 can becalculated from the momentum of particles 2 and 3.

p1 = p2 + p3 = (p02, p

12, p

22, p

32) + (p0

3, p13, p

23, p

33) =

(p02 + p0

3, p12 + p1

3, p22 + p2

3, p32 + p3

2) = (p0tot, p

1tot, p

2tot, p

3tot) = ptot

(13)

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

M2 = ptotµpµtot = (p0

tot)2 − (p1

tot)2 − (p2

tot)2 − (p3

tot)2 = E2

tot − |ptot|2 (14)

In my work, I study Λ→ pπ− and Λ→ pπ+ decays. The mass of the Λ(Λ) can thenbe calculated from the invariant mass of the proton, p(p) and pion, π−(π+):

MΛ = M(pπ−) =√E2tot − |ptot|2 =

√(Ep + Eπ−)2 − (pp + pπ−)2 (15)

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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 bombarda stationary target, or let two beams collide. Figure 4 shows an incident beam andfigure 5 display an incident beam of particles to the left being scattered of a targetinto the solid angle element, dΩ. The probability that a cross section element dσ, ofthe beam is scattered into a solid angle element dΩ, is denoted the differential crosssection and is written as dσ

dΩ .

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, arescattered of a target and detected in the detector that covers a solid angle dΩ [10]

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The beam intensity I0, denotes the number of particles per unit area and unittime. The number of particles scattered into a cone with a solid angle, dΩ per unittime 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 sectionis then defined by equation 17:

dσ

dΩ=I(θ, φ)

I0(17)

Theoretically, the differential cross section is expressed in terms of the scatteringamplitude 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 =

∫dσ

dΩdΩ =

∫|f(θ, φ)|2dΩ (19)

2.3 Form factors

The cross section of reactions involving scattering of virtual photons, γ∗, producingcomposite systems such as hadrons, can be parameterized in terms of ElectroMagneticForm Factors (EMFF). Scattering involving virtual photons, γ∗, can for example beelastic electron-nucleon scattering(e−N → e−N) where the electromagnetic force iscarried by a virtual photon, see figure 6a. It can also be electron-positron annihilationinto a virtual photon with the subsequent production of a hadron-antihadron pair(e+e− → γ∗ → hh), see figure 6b.

The EMFFs are fundamental observables of QCD and describe the structure ofhadrons, 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 chargedensity and magnetization density respectively. Form factors can be studied in theTime-Like (TL) or Space-Like(SL) regions. The momentum transfer carried by thevirtual photon (denoted q2) of a reaction defines the nature of the form factors. Forelastic scattering experiments, such as electron-nucleon scattering of a nucleon, themomentum transfer is negative, q2 < 0. The form factors are then SL and the EMFFsare real numbers. The Uppsala hadron physics group is interested in measuring theEMFF’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 anda positron which annihilate and create an intermediate virtual photon which sub-sequently produces hadrons, for example a hyperon-antihyperon pair. This meansthat for hyperons, the TL region is experimentally accessible, by the process wheree+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.

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(a) SL reaction of e−N → e−N ′ [11]

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

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

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

dσ(q2, θB)

dΩ=α2sβC

4q2[(1 + cos2θB)|GM (q2)|2 +

1

τsin2θB |GE(q2)|2] (20)

There B denotes a baryon, αs the strong coupling constant, τ =4M2

B

q2 and q2 the

momentum transferred squared. Furthermore β =√

1− 1/τ , θ is the angle betweenthe 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 calculatedfrom 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 hyperonform 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 datasample for precision measurement of hadron form factors. The Uppsala group wasresponsible for the hyperon part. This proposal was approved in 2014 and the datawas collected during 2014 and 2015. These data have the potantial to give us a greaterunderstanding 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 factorscan be studied is the BESIII experiment at the BEPC-II storage ring. In the followingan introduction to the BESIII detector and BEPC-II will be given.

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3.1 BEPC-II

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

Figure 7: The BEPC-II storage ring, the green arrow points towards the BESIIIdetector which also is the collision point. Each beam have a crossing angle of 11mrad 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 DriftChamber (MDC), the ElectroMagnetic Calorimeter (EMC), the Time Of Flight sys-tem(TOF) and the muon chamber (MUC). Together, these sub-detectors will providethe 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 consecutivelayers, where the MDC is closest to the interaction point. The other layers come in

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

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

The MDC is used to measure the momentum of charged particles. A magneticfield 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 thecurvature of its trajectory in the magnetic field. The bending radius of chargedparticles 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 fieldand 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 ionizationprocess are collected on wires in the MDC. The collected charges will give a signal inthe 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 consistsof 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 particleis produced, which is given by the time of the beam-beam collision, and the time it

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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 fromthe 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] andmeasure 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 identifymuons.

4 The e+e− → ΛΛ reaction

The reaction investigated for this project is e+e− → γ∗ → ΛΛ at a CMS energyof 2.396 GeV. The electron-positron pair will annihilate into a virtual photon whichsubsequently produces a ΛΛ pair. In figure 9 the reaction is displayed with thecoordinate system used in this work. The y(n) axis is defined by the normal of theproduction plane, which is spanned by the incoming e+ or e− and the outgoing Λor Λ. The z(l) axis is directed along the momentum of Λ or Λ. Finally the x(m)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− → ΛΛ in a coordinate system [12].

figure is denoted with θ, is the angle between the incident electron-positron beam andthe outgoing Λ-hyperon. The most common decay channel is Λ → pπ− for Λ andΛ→ pπ+ for Λ, and the BR for these decays are (63.9± 0.5)%, see table 3 in section1.3.4. 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. Thatmeans that the life-time of a Λ-hyperon is relatively long and can traverse parts of theMDC before decaying. The charged pions also decay weakly and have an even longerlife-time. On a particle physics time-scale they can therefore be considered stablesince they will live long enough to be stopped by the detector before decaying. Thusthey will traverse the MDC. The differential cross section of e+e− → ΛΛ is given byequation 23:

dσ

dΩ=α2sβC

4q2[|GM (q2)|2(1 + cos2θ) +

1

τ|GE(q2)|2(sin2θ)] (23)

Experimentally, the cross section is given by equation 24:

σ =Nsignal

Lε(1 + δ)Br(Λ→ pπ−)Br(Λ→ pπ+)(24)

12

where Nsignal is the number of signals in the detector, L is the luminosity, ε is theefficiency of the detector, (1 + δ) is a so-called radiative correction factor which takesin consideration the spread of the beam energy and the reaction where a photonmay be radiated before the e+e− collision and Br(Λ → pπ−)Br(Λ → pπ+) is thebranching ratio of the Λ- and Λ-hyperon, see table 3 in section 1.3.4. In particlephysics experiment, statistical significance is very important. Minimizing statisticaluncertainty in σ means maximizing the number of observed events, Nsignal.

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

The focus of the BESIII group in Uppsala is currently the e+e− → ΛΛ reactionat√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 shouldtherefore be optimal in terms of number of events.

5 Software tools

With today’s powerful and versatile detectors and high luminosity in particle collidersthe data samples are very large. Therefore the demand for software that can handleand 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 programminglanguage 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 ROOTtrees, 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 integrateanother operating system onto a computer. A scientific operating system is requiredby BOSS, therefore Scientific Linux 6.7 have been used during this project. Theadvantage of using a virtual machine is that it is portable and can be used as a commonstarting environment for students working with projects related to BOSS. The virtualmachine provides a BOSS installation, a selection algorithm, the simulation scriptsand 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 willdisplay errors if the code can not be compiled. Eclipse is also useful for finding bugsin the code.

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6 Monte Carlo simulations

The e+e− → ΛΛ 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 forthis channel. I have implemented the same criteria independently in order to verifythe results obtained by Cui Li. In particular, I have looked into the detector andreconstruction efficiency of e+e− → ΛΛ as a function of the scattering angle for theΛ-hyperon. This is achieved by Monte Carlo (MC) simulations. MC methods utilizerandomized sampling to generate numerical values. In particle and hadron physics,the reaction of interest is generated using a MC generator. The produced particles arethen propagated through a virtual description of the detector and their interactionwith the detector material are simulated. These produced MC data can then bereconstructed and analyzed in the same way as real experimental data. By studyingthe MC simulated data we can learn how to filter out background while optimizing theyield 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. Wecan also optimize the analysis methods and verify their correctness. MC data arethus 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 theinitial 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 agenerator is typically final state particles with corresponding four momenta and, ifthe generated particles are unstable also the decay products and decay vertex. Thesimplest generator is a so-called phase-space generator (in BOSS and in the followingreferred to as PHSP). In the PHSP generator, the constraints on the final state fromthe initial state is purely kinematic. The PHSP generator is simple, unbiased andgive isotropic distributions of e.g. the angles of the produced particles. Figure 10show the distribution of the CMS polar angle cos θΛ of the Λ-hyperon produced in aphase space generated e+e− → ΛΛ reaction. The isotropic distribution imply thatthe generated particles are evenly distributed in the detector. The PHSP generatoris 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 whichdoes not only take kinematic constraints in consideration but also the dynamics.The events are generated according to the form factor parameterization presented insection 2.3. It also includes radiative correction, e.g. when the incident e+e− radiatesa photon before colliding, this is displayed in figure 11. In the ConExc generator, theuser defines the desired value of the ratio of the electromagnetic form factor, GE andGM , which determines the angular distribution of the outgoing Λ-hyperons. Figure12 display the CMS polar angle distribution when the ratio of the electromagneticform factors are set to 1.

6.2 Particle interaction with the detector

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

14

Λθcos

­1 ­0.8 ­0.6 ­0.4 ­0.2 0 0.2 0.4 0.6 0.8 1

Eve

nts

0

500

1000

1500

2000

2500

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

Figure 11: A reaction where a photon is radiated from the electrons before theelectron-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 particleswill traverse in the virtual detector and the information from each sub-detector willbe 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 willbe converted into so-called MC-hits. The MC-hits will be converted to signals in theelectronics of the individual detectors, such as a wire in the MDC or a crystal inthe EMC. After digitization the output will have the same format as data from real

15

Λθcos

­0.8 ­0.6 ­0.4 ­0.2 0 0.2 0.4 0.6 0.8

Eve

nts

0

500

1000

1500

2000

2500

Figure 12: The true distribution of detected events with respect to the cos θΛ-anglefor data generated with the ConExc generator where the the ratio, |GE |/|GM | of theelectromagnetic 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 differentalgorithms that will take the MC hits from the previous step and combine them intotracks. The pulse heights will be converted to energy using calibration techniques. InBOSS, particle identification probabilities are also calculated in this step. The outputfrom the reconstruction is charged and neutral track candidates.

6.5 Analysis

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

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7 Analysis

7.1 Event selection

A set of event selection criteria has been implemented in order to distinguish goode+e− → ΛΛ candidate events from background. The reason is that real experimentaldata is dominated by events which do not come from the e+e− → ΛΛ reaction.The analysis of the e+e− → ΛΛ reaction is currently undergoing within the Uppsalahadron physics group and this project provides a cross-check of the current results. Ihave implemented the event selection criteria that were developed by Dr. Cui Li. Formy analysis the pre-selection will remove events that definitely does not belong to thereaction under investigation. The pre-selection criteria and the final event selectioncriteria will be introduced below.

7.1.1 Pre-selection

Four charged tracks: For the e+e− → ΛΛ reaction with the subsequent decay ofΛ → pπ− and Λ → pπ+, there will be four charged particles in the final state thatcan leave a track in the detector. Therefore the pre-selection will select events withat least four charged tracks. The reason why not exactly four charged tracks arerequired is because there are situations where a e+e− → ΛΛ signal can give rise tomore then four tracks, e.g in the case of overlapping events or tracks from a secondaryinteraction with the detector.Particle identification: In the pre-selection, the identity of the charged tracks waschecked, based on energy loss in the detectors and time-of-flight. At least one particleeach of type p, p, π−, π+ was required.Combinations of protons and pions: The protons and pions with opposite chargewas combined together and a vertex fit of the oppositely charged tracks was done. Ifthe 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 theΛ-candidate and the CMS energy for one of the beams, meaning half of the total CMSenergy. This difference, denoted ∆E, was ordered for all the Λ-candidates, where theΛ-candidate with the smallest ∆E was considered the ”best”. The ”best” Λ-candidatewas then saved.Λ, Λ separation: The pre-selection will separate the Λ and Λ, this was performedby a check of the charge of the protons. Positively charged protons lead to a Λ, andnegatively charged protons lead to a Λ.Store Λ, Λ-candidates: The final step of the pre-selection is to store events thatcorrespond 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 beenimplemented. 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 pionmass is much smaller, 0.139 GeV, see table 1. In order to conserve momentum in theΛ→ pπ− decay, the Λ-hyperon will give most of its energy to the proton. This meansthat the proton from a Λ decay will have much larger energy and momentum then thepion from the Λ decay. Therefore, I have implemented a separation criteria of protons

17

and pions in addition to the previous TOF and dEdx separation. The requirement for

the pions is that momentum for each pion has to be less then 200 MeV/c, (p(π−) < 200MeV/c, p(π+) < 200 MeV/c), and the requirement for protons is to have a minimumof 200 MeV/c each, (p(p) > 200 MeV/c, p(p) > 200 MeV/c). The ”true” momentumfrom undistorted MC simulations for the protons and pions are displayed in figure 13along 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)

dp

dN

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)

dp

dN

0

50

100

150

200

250

300

p

p­

π

+π

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

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

18

p / (GeV/c)

0 0.1 0.2 0.3 0.4 0.5 0.6

­1 /(5

MeV

/c)

dp

dN

0

20

40

60

80

100

120

140

160

180

200

220p

p­

π

+π

Figure 14: The momentum distribution of protons and pions for reconstructed dataafter the final selection. The Y-axis represent the number of events for over themomentum region, which is displayed in the X-axis. The green data points denoteπ−, 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 (orΛ decay) they should have an invariant mass, M(pπ−),M(pπ+) see equation 15 insection 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 fromthe 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, aGaussian curve have been fitted to the invariant mass of the Λ-candidate. In figure 15the invariant mass of the Λ- and Λ-candidates are displayed together with a Gaussianfitted to reconstructed data. The invariant mass of the Λ- and Λ-candidates beforethe final event selection is displayed in figure 15a. After the final event selection havebeen applied to the Λ- and the Λ-candidates the invariant mass distribution is seenin figure 15b. The Λ-candidates are denoted in red and the Λ-candidates in blue.

19

)2 / (GeV/cΛ

M

1.1 1.105 1.11 1.115 1.12 1.125 1.13

­1 )2

/(1

Me

V/c

Λd

MdN

0

200

400

600

800

1000 Λ

Λ

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

)2 / (GeV/cΛ

M

1.1 1.105 1.11 1.115 1.12 1.125 1.13

­1 )2

/(1

Me

V/c

Λd

MdN

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 witha Gaussian curve have been implemented to test the significance. The extractedwidth from the Gaussian fit was 1.9 MeV, this corresponds will with the expected 3σdeviation. 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-candidatesand the blue denote Λ-candidates, the red line is the Gaussian fit which have beenapplied to the invariant mass of the Λ-candidate.

20

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 theproton and the pion in such a way that resulting Λ-candidate momentum has the samedirection as the line between the e+e− collision point and the Λ decay vertex. Fromthe vertex fit, the decay length is obtained. The decay length is the distance betweenthe collision point and the decay vertex. If the Λ momentum points oppositely tothe direction from the collision to the decay point, then the decay length becomesnegative in the vertex fit algorithm. Due to the mean life time of the Λ-hyperon beingrelatively long on an experimental scale, the decay should be well separated from thee+e− collision point. To make sure this is the case a requirement on the decay lengthis set such that: L(Λ) > 0.2 cm,L(Λ) > 0.2cm.

ΛL

­4 ­2 0 2 4 6 8 10 12 14

ΛdLdN

0

100

200

300

400

500

600

Λ

Λ

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

ΛL

­4 ­2 0 2 4 6 8 10 12 14

ΛdLdN

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. TheY-axis display the number of events for the decay lengths of the Λ-candidates, whichis displayed on the X-axis. The Λ-candidates are denoted in red and the Λ in blue

21

ΛL

­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 ΛΛ-candidates before thefinal event selection.

ΛL

­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 ΛΛ-candidates after thefinal event selection.

Figure 17: The decay lengths for the reconstructed Λ, Λ-candidates before and afterthe final event selection. The Y-axis represent the decay length of the Λ-candidatesand 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 CMSsystem, the angle between them, θΛΛ should be close to 180. Taking the detectorresolution into account, a requirement of θΛΛ > 177 was found optimal.

22

Momentum of Λ, Λ: In section 2.1 we concluded that three momentum of aparticle produced in a two-body reaction 1 + 2 → 3 + 4 is fixed. For a beam with aCMS energy of 2.396 GeV the Λ(Λ) momentum |pΛ|(|pΛ|) is found to be 0.436 GeV/c.Therefore we require the events to fulfill ||pΛ| − 0.436| < 0.02 GeV/c for both Λ andΛ. The momentum of the Λ, Λ-candidates are shown in figure 18.

/ (GeV/c)Λ

p0.41 0.42 0.43 0.44 0.45 0.46 0.47

­1 /

(2 M

eV

/c)

Λd

pdN

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 eventselection have been applied.

/ (GeV/c)Λ

p0.41 0.42 0.43 0.44 0.45 0.46 0.47

­1 /

(2 M

eV

/c)

Λd

pdN

0

100

200

300

400

500

600

Λ

Λ

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

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

23

7.2 Total efficiency

The analysis proceeds as follows: For each event, my algorithm goes through theevent selection and checks if the event fulfills them or not. If it does, then the event isclassified as a signal. If the event selection criteria is not fulfilled, it is instead classifiedas background and is filtered out. The events remaining and the total efficiency aftereach event selection criteria can be seen in table 4. Table 4 reflect the implementationof 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 totalnumber of events removed from that criteria, but rather how many events are stillavailable after the pre-selection and the momentum criteria on pions and protons havebeen 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 %θΛΛ > 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 beenimplemented. Expected final efficiency with respect to previous work done by Cui Liwas 14%.

7.3 Efficiency and correction

When analyzing real, experimental data, the goal is to extract a ”true” quantity froman observed one. The observed quantity will be distorted by the limit of the detectorand the reconstruction efficiency. In order to unfold the ”true” results, the observedquantity needs to be corrected for e.g. efficiency and resolution. Here, we focus onthe efficiency. In MC simulations, the undistorted MC truth data correspond to thetrue events, whereas the reconstructed results corresponds to observed ones. Fromthe ratio between the reconstructed and the MC truth results a correction factor, or aweight, 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 PHSPgenerator 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 theMC truth histogram. The reconstructed events without a correction and theMC 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 inred. The difference reflects the effect of the applied selection criteria and theresolution.

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

24

Λθcos

­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 looklike.

4. A new histogram with the efficiency is constructed.

5. A fourth order polynomial is fitted to the efficiency histogram. Fourth orderpolynomial : a · x4 + b · x3 + c · x2 + d · x+ e =efficiency(x), where x is cos θΛ.The histogram with efficiency and fourth order polynomial fit are displayed infigure 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 thencorrected using a weight (denoted W). W = 1

efficiency . For the MC data wewould then see that each event is corrected according to its weight:Corrected = Reconstructed

efficiency = ReconstructedReconstructed

Generated

= Generated

This means that the correction of reconstructed signals should be the same as thetrue MC data. However, since the correction is performed event-by-event while thecorrection 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, reconstructedsignals and the MC truth signals. These are displayed in figure 21, where we can seethat the correct reconstructed data of the outermost bins -0.9 and 0.9 disagree withthe MC truth data. This means that the efficiency correction does not work well inthis region. The reason is likely that the efficiency function varies rapidly at extremeangles which makes it difficult to model the correction. It may even go to zero closeto |cosθΛ| = 1.

25

Λθcos

­1 ­0.8 ­0.6 ­0.4 ­0.2 0 0.2 0.4 0.6 0.8 1

Effic

iency

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 tocalculate the total efficiency of the detector.

Λθcos

­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 truthsignals in red. The need for a limit on the angular distribution of the Λ-candidate isshown, due to reconstructed values in the outer regions not being similar to the MCtruth 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 checkof Dr Cui Li’s previous work with the total efficiency from the reaction e+e− → ΛΛ

26

could be reproduced. Using the same selection criteria as Cui Li, I obtained a totalefficiency of 14%, which agrees with Cui Li’s result. These results gave a strongindication that not only my own algorithm is working as intended but also thatCui Li’s algorithm is correct since an independent algorithm gives the same result.Another important result is the correction of reconstructed events. From figure 21we can see how many events are found in different angular regions. It is clear thatthe reconstruction efficiency drops at extreme angles, i.e near |cosθΛ| = 1. To be ableto extract valuable information from real experimental data, we need to correct forevent losses in the reconstruction. I have implemented a method for this and testedits correctness. It works well except at extreme angles. This is because the rapiddecrease towards zero for the efficiency in this region, see figure 20. It is thereforesuggested that the region |cosθΛ| > 0.9 is excluded from the data analysis.

9 Summary

In this project, I have performed simulation studies of the e+e− → ΛΛ reaction at2.396 GeV, with the BES III experiment. By studying this reaction, we access theelectromagnetic form factors, GE and GM . The electromagnetic form factors givevaluable information about the structure of hadrons and how the quarks are bound inhyperons. In other words, this is one way to gain better understanding of the strongforce. The BES III experiment at the BEPC-II storage ring in China have collecteddata at several energy points. However, at 2.396 GeV, a particularly large amountof data was collected for precision measurements. In order to optimize the analysismethods and to verify their correctness, we perform Monte Carlo-simulations. 20000events of the reaction of interest have been generated and the produced particles arepropagated through a virtual description of the real BES III detector. We obtain datawith the same features as real experimental data. The undistorted MC truth datarepresent the ”true” underlying physics, and the reconstructed data represent themeasurements. The latter reflect the limited efficiency and resolution of the detector.When the electrons and positron collide in the real experiment, only a small fractionof the events come from my reaction of interest. I therefore apply a set of selectioncriteria to filter out the events that corresponds to the reaction I am investigating.The event selection checks every event in my data sample and if all the criteria arefulfilled, the event is saved for further analysis. If not, it is removed and labeled asbackground. The efficiency is the number of reconstructed events fulfilling all criteria,divided by the number of generated MC truth events of the reaction of interest. Iobtained a total efficiency, after all event selections had been implemented, of 14%.Previous work with the same event selection criteria was performed at the hadronphysics group at Uppsala university, and the efficiency extracted then, by Dr. CuiLi. Also then, a value of 14% was obtained which agrees with my result. This giveme confidence that my analysis have been implemented in a correct fashion and thatmy analysis now can be transferred over to real data. It is also reassuring for CuiLi and the rest of the group that her analysis has been verified by an independentlyimplemented selection algorithm.

27

10 Outlook

After verifying the analysis method, the next step is to investigate real data. I havealready started to look into the fraction (denoted R) between the electric and themagnetic form factor, R = |GE |/|GM |. R can be extracted from the angular distribu-tion of the Λ-hyperon, by fitting the angular distribution of the efficiency correctionto the function shown in equation 25:

f(cos θ) = A[(1 + cosθ2) +R2

τ(1− cos2θ)] (25)

Where A is a normalization factor [12].Before applying this fit to real data, one must make sure that the method of extractingthe R-value is correct for the generated data. We can verify the method for both theConExc and PHSP generated MC events.

11 Collaborations

As mentioned in section 2.3, a collaboration with several different institutes fromdifferent countries is working with the extraction of hadronic form factors. This leadsto countries working together. So-called ”Big science” projects will allow countriesthat do not necessarily work so well together, to put their differences aside to work fora common goal when it comes to understanding the world around us. Collaborationwith different countries also give the opportunity for both students and others to workand study in a different country.

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12 Acknowledgments

I would like to end my thesis by thanking those who helped me along the way.

I want to thank Tord Johansson for giving me the opportunity to work within thenuclear physics division, and for the final comments of this thesis.

I want to thank my supervisor Karin Schonning for all the help, encouragementand advice that I have been given during the process of this bachelor project. Thecomments I got on my thesis have been invaluable for further, scientific papers.

I want to thank Michael Papenbrock for all the help and discussions regarding theprogramming aspect of the project, and I also want to thank you for your commentson my thesis.

I want to thank Cui Li for being a great person to share the office with, and also forall your assistance during the project.

I also want to thank everyone at the department for being very welcoming.

As a final note I want to thank my family for all the support I have been given duringmy studies and everything else in life!

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13 References

References

[1] http://home.cern/about

[2] B.R Martin and G.Shaw. Particle Physics Wiley, Third edition, 2008.

[3] https://upload.wikimedia.org/wikipedia/commons/thum

b/0/00/Standard_Model_of_Elementary_Particles.svg/2000px

-Standard_Model_of_Elementary_Particles.svg.png

[4] [5]https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Meson_nonet_-_spin_0.sv

g/2000px-Meson_nonet_-_spin_0.svg.png

[5] [6]$https://upload.wikimedia.org/wikipedia/commons/thumb/1/13/Meson_nonet_

-_spin_1.svg/200px-Meson_nonet_-_spin_1.svg.png$

[6] K.A. Olive et al.(Particle Data Group), Chin. Phys. C, 38, 090001 (2014) and2015 update.

[7] https://upload.wikimedia.org/wikipedia/en/9/98/Baryon_octet_w_mass.png

[8] https://upload.wikimedia.org/wikipedia/en/c/c1/Baryon_decuplet_w_mass.png

[9] http://uspas.fnal.gov/materials/12MSU/Conrad_4vector.pdf

[10] http://wiki.physics.fsu.edu/wiki/index.php/Differential_Cross_Section_and_the_Green

[11] http://inspirehep.net/record/1191015/plots?ln=sv

[12] Precision measurement of R values, high mass charmonium states and QCDstudies with BESIII at BEPCII, BESIII Doc-127 Ver.3

[13] http://www-public.slac.stanford.edu/babar/

[14] https://wiki.classe.cornell.edu/CLEO/WebHome

[15] arXiv:0911.4960 [physics.ins-det]

[16] http://www.hep.umn.edu/bes3/MN_BES3_files/BEPC2.jpg

[17] arXiv:0809.1869 [hep-ex]

[18] BESIII Analysis Memo, BAM-00139 March 25,2015

[19] Y. Liang, private communication

[20] https://root.cern.ch/documentation

[21] https://eclipse.org/downloads/packages/eclipse-ide-cc-developers/mars2

[22] http://ific.uv.es/ rodrigo/phokhara/

[23] S. Agostinelli et al, (GEANT4 collaboration), nucl.instrum.Meth. A506,250(2003)

[24] http://www.hep.phy.cam.ac.uk/drw/pix/opalpix.html

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