Monte Carlo Simulation of e
+e
−→ ¯Σ
0Λ/ ¯
Σ
0Σ
0Reaction
Uppsla University
Department Of Physics and Astronomy
Master Thesis
Author : Halimeh Vaheid
Supervisor : Karin Sch¨
onning
Contents
1 Introduction 2
1.1 Aim . . . 2
1.2 The Standard Model . . . 3
1.3 The Strong Interaction . . . 3
1.4 Hadrons . . . 4 1.4.1 Quantum Numbers . . . 4 1.4.2 Mesons . . . 7 1.5 Baryons . . . 8 1.5.1 Hyperon . . . 9 2 Formalism 10 2.1 Cross section . . . 10 2.2 Relativistic Kinematics . . . 11
2.2.1 Electromagnetic Form Factor . . . 13
3 Hyperon Production in e+e− Annihilation 16 3.1 The e+e− → Λ¯Σ0 reaction . . . . 16
3.2 The e+e− → Σ0Σ¯0 reaction . . . . 18
3.3 Previous studies . . . 19
4 The BESIII Experiment 20 4.1 BEPC-II . . . 20
4.1.1 The BES III Detector . . . 21
5 Software Tools 26 5.1 Jupyter . . . 26
5.2.1 Event generation . . . 27
5.2.2 Particle transport and detector response . . . 28
5.2.3 Digitalization . . . 28
5.2.4 Reconstruction . . . 28
5.2.5 Analysis . . . 28
6 Parameter Estimation Using Monte Carlo method 29 6.1 The Method of Moments . . . 29
6.1.1 Extracting the ratio R from an angular distribution by applying MM . . . 30
6.2 The Least Squares Fit . . . 32
6.3 Hit-or-Miss Generator . . . 32
6.4 Simulating angular distributions . . . 33
6.4.1 The Results . . . 33
6.5 Results and Discussion . . . 35
7 Full Simulation Study with BES III 36 7.1 The pre-selection criteria . . . 38
7.2 The e+e− → Λ¯Σ0 Reaction . . . 38 7.2.1 Photon Selection . . . 40 7.2.2 The reconstruction of ¯Σ0 . . . . 45 7.2.3 Selection of e+e−→ Λ¯Σ0 . . . 48 7.2.4 Predicted precision of R . . . 52 7.2.5 Angular Distribution of ¯Σ0 . . . 53 7.3 The e+e− → Σ0Σ¯0 Reaction . . . . 57 7.3.1 Photon selection . . . 59 7.3.2 The ¯Σ0 reconstruction . . . 60 7.3.3 Selection of e+e− → Σ0Σ¯0 . . . . 61
7.3.4 Predicted precision of R for the Σ0 in e+e− → Σ0Σ¯0 reaction . . . . 64
Acknowledgement
First of all, I would like to thank my thesis advisor Karin Sch¨onning of the Hadron Physics Group at Uppsala University for her support of my study and providing the opportunity for this project for me. I must express my profound gratitude to her, not only because of her academic support but also for her concern about other issues.
I would also like to acknowledge Michael Papenbrock of the Hadron Physics Group at Uppsala University, for his technical help and support. Without his help, this project could last a long time.
Sammanfattning
En av de st¨orsta utmaningarna inom modern fysik ¨ar att f¨orst˚a de grundl¨aggande egenskaperna hos hadroner och k¨arnor i termer av QCD. Under det senaste decennierna har ett flertal experimentella och teoretiska metoder utvecklats f¨or att studera hur kvarkar och gluoner innesluts i hadroner. Ett s¨att att studera detta ¨ar att m¨ata hadroners elektromagnetiska struktur. Denna kan kvantiseras i ter-mer av s˚a kallade formfaktorer. Hyperoner ¨ar ett slags hadroner som liknar v˚ara byggstenar protonen och neutronen. Skillnaden ¨ar at en eller flera l¨atta kvarkar bytts mmot en tyngre s¨ar-, charm- eller bottenkvark. BESIII-experimentet i Peking, Kina, ¨ar en av f˚a faciliteter d¨ar hyperonstruktur kan de-taljstuderas. Uppsalas hadronfysikgrupp ¨ar en del av den internationella BESIII-kollaborationen och h˚aller f¨or n¨arvarande p˚a att f¨orbereda en str˚altidsans¨okan f¨or precisionsm¨atningar av den s˚a kallade Sigma-hyperonens struktur. F¨orh˚allandena f¨or en s˚adan m¨atning ¨ar optimala vid en CMS-energi p˚a 2.5 GeV. Detta project syftar till att bidra till denna ans¨okan genom Monte Carlo-simuleringar av reaktionerna e+e−
→ Λ¯Σ0 och e+e−
→ Σ0Σ¯0. S¨arskilt fokus ligger p˚a fotonerna fr˚an
Abstract
A central objective of the field of nuclear physics is understanding the fundamental properties of hadrons and nuclei in terms of QCD. In the last decade, a large range of experimental and theoretical methods have been developed to study the nature of quark confinement and the structure of hadrons which are composites of quarks and gluons. One important way to address some questions of hadron physics is studying the electromagnetic form factors of hadrons. The electric and magnetic form factors are related to the distribution of charge and magnetization in hadrons.
The internal structure of hyperons, which are a subgroup of hadrons, is a topic of interest of particle physicists. The BES III experiment is one of the few current facilities for studying hadron structure. The Uppsala Hadron Physics group, which is a part of the BES III collaboration, is preparing a proposal for data taking for Λ ¯Σ0 transition form factors and Σ0 direct form factors at 2.5 GeV.
Aiming the electromagnetic form factors of Σ hyperons, this work contributes to this proposal by a simulation study of the e+e−
→ Λ¯Σ0 and e+e−
→ Σ0Σ¯0 reactions. The efficiency and resolution
of the electromagnetic calorimeter sub-detector of BES III and kinematic properties of the detected particles are studied in this work. Our final goal is to provide input for the beam time proposal and optimize the future measurement.
In the first chapter the theoretical background including the Standard Model, strong interaction, QCD and hadrons are studied. In the second chapter, some concepts like the formalism of cross section, relativistic kinematics and electromagnetic form factors are briefly presented. The third chapter is dedicated to introducing the e+e− → Λ¯Σ0 and e+e− → Σ0Σ¯0 reactions. The BES III
Chapter 1
Introduction
1.1
Aim
Understanding the four fundamental interactions - gravitational, electromagnetic, weak and strong interactions - has been the main purpose of contemporary physics. In spite of a fact that the Standard Model (SM) has been successful in describing the particles interactions up to energies around 100 GeV and it is accepted as the more fundamental theory of particle physics, it leaves many questions unanswered. These questions are usually categorized into two groups: one is related to possible new particles at unexplored energy scales and the other to non-perturbative phenomena of the strong interaction. It is suggested that new particles and new interactions appear at higher energies, can say 1 TeV, solve some inconsistencies within the SM. Such physics subjects belong to the first category of questions and are addressed at the Large Hadron Collider (LHC) [1]. The non-perturbative effects of the strong interaction are very basic to the field of particle physics and include e.g. the structure of hadrons (for instance, hyperon form factors) and the spectrum of hadronic states. To address these questions we need lower energy facilities with high luminosity. Getting information about the structure of hadrons help us to find the answer for some fundamental questions like:
How are quarks confined into hadrons?
How does the strong interaction generate the rest 99%? Only 1% of nucleon mass is generated by quarks mass.
How is the total spin of nucleons generated? only 1/3 of total spin of nucleons is from sum of the spin of the valance quarks.
1.2
The Standard Model
The Standard Model (SM) is a theory of particle physics which describes the strong, electromagnetic and weak interactions of elementary particles in a quantum field theory framework. The strengths and ranges of fundamental forces are listed in table 1.1. The SM explains the quarks and leptons which are fermions, and the interactions between them which are mediated by gauge bosons. The types of quarks are referred to as flavor of quarks like up and down (u and d ) flavors. The SM contains three generations of fermions (Fig. 1.1). The first generation, which consists of u and d quarks (the lightest quarks), electron and electron neutrino , constitutes most visible matter in the universe. The second and third generations are only observed in very high-energy environments like particle accelerators and cosmic rays. The members of the first generation have lighter mass than corresponding particles from higher generations [2]. The SM also contains the gauge bosons photons (γ), gluons (g), Z0 boson and W± bosons. The elementary fermions interact by exchanging these
gauge bosons. The electromagnetic force is carried by photons while Z and W bosons mediate weak interactions [3, 4]. The gluons are the force carriers of the strong interaction [5]. The SM has been successful in describing and predicting a wide range of experimental observations. One of them was predicting the Higgs boson, H, whose existence used to be the last unverified part of the SM of particle physics. On the 4th July 2012, scientists of ATLAS [6] and CMS [7] at CERN confirmed the observation of a high-mass Higgs-like boson [8, 9]. What is still missing in the SM is a coherent description of the strong interaction at low energies corresponding to long distances.
Interactions Strength (relative to strong) Range(m)
Strong 1 10−15
Electromagnetic 10−2 ∞
Weak 10−6 10−18
Gravitational 10−38 ∞
Table 1.1: Four types of interactions in order of strength. [10].
1.3
The Strong Interaction
R/G/B2/3 1/2 2.3 MeV up
u
R/G/B−1/3 1/2 4.8 MeV downd
−1 1/2 511 keV electrone
1/2 < 2 eV e neutrinoν
e R/G/B2/3 1/2 1.28 GeV charmc
R/G/B−1/3 1/2 95 MeV stranges
−1 1/2 105.7 MeV muonµ
1/2 < 190 keV µ neutrinoν
µ R/G/B2/3 1/2 173.2 GeV topt
R/G/B−1/3 1/2 4.7 GeV bottomb
−1 1/2 1.777 GeV tauτ
1/2 < 18.2 MeV τ neutrinoν
τ ±1 1 80.4 GeVW
± 1 91.2 GeVZ
1 photonγ
colour 1 gluong
0 125.1 GeV HiggsH
Strong Nuclear F orce Electromagnetic F orce W eak Nuclear F orce Charge Mass Spin Quarks Leptons Fermions Bosons Force Carriers Goldstone BosonsFigure 1.1: The fundamental particles of the Standard Model
a quark and an antiquark, the energy of gluon field is enough to create another quark pair. Quarks are always observed in colorless bound states. For example, combining all three different colors/anti-colors (i.e. rbg/¯r¯b¯g) gives colorless states called baryons. This is also true for mesons, combinations of color-anticolor like b¯b. Color is conserved under strong interaction. Since there are three colors, the symmetry group of the strong interaction is SU(3) which is a non-abelian group [10].
Although QCD is a successful theory at short range, it is still an open question why and how quarks are confined into hadrons [12]. The reason is that the non-perturbative nature of the interactions makes an analytical description impossible.
1.4
Hadrons
The aforementioned hadrons are composite particles made up of quarks and possibly gluons which interact via the strong force1. Most observed states are consistent with either baryons (qqq) or mesons (q ¯q) which have three and two valence quarks, respectively, although any colorless combination of quarks like qq ¯q ¯q is allowed according to QCD. These so-called exotic hadrons have been claimed by many experiments recently.
1.4.1
Quantum Numbers
Each hadron has several quantum numbers which refer to its quark contents. The most important quantum numbers are:
1Quarks and hadrons also interact by the weak and electromagnetic interaction, while leptons do not participate in
Spin: is denoted by S and is a quantum number which parametrizes the intrinsic angular momentum of the particle [13]. Half-integer spin particles are called fermions and integer spin particles are called bosons. Quarks are spin 1/2 particles, so mesons can have spin S = 1 or spin S = 0, the spin of baryons can be 3/2 or 1/2. The following formula relates the total angular momentum, J, to S and orbital angular momentum L.
J = L + S (1.1)
Electric charge: According to the Noether theorem (see appendix A) each continuous sym-metry of the Lagrangian leads to a conserved charge. In other words, charges are the generator of a symmetry group of the system. The invariance of the QED Lagrangian under the uni-tary group U(1) transformation leads to a Noether electric current and the integration of this current gives the electric charge Q. Each quark carries a fraction of the unit electric charge (e = 1.6022× 10−19C). For example, Qu = +2/3 e and Qd = −1/3 e (see Fig. 1.1), e.g. the
proton with uud quark content has electric charge Q = (2/3 + 2/3− 1/3) = 1 e. The Q is conserved in all interactions.
Parity: is denoted by P and refers to the behaviour of a particle’s wave function when reversing the sign of the coordinate system, i.e. (x, y, z)→ (−x, −y, −z). If the sign of the wave function of a particle does not change under parity transformation one says that the system is even under parity, whereas a change in sign means that the parity is odd. By convention, the parity of quarks is fixed as below:
Pu = Pd= Ps = Pc= Pd= Pt = 1 (1.2)
and
Pu¯ = Pd¯= Ps¯= P¯c= Pd¯= Pt¯=−1 (1.3)
Therefore, the parity of a meson, M = a¯b, can be calculated using
PM = PaP¯b(−1)L (1.4)
where L is the orbital angular momentum of the quark-antiquark pair. In a similar way the parity of a baryon, B = abc is given by:
PM = PaPb(−1)L12(−1)L3 = (−1)L12L3 (1.5)
where L12 is the orbital angular momentum of a pair of quarks in their center of momentum
frame and L3 is the orbital angular momentum of the third quark with respect to the center of
mass of pre-chosen pair.
Baryon number: is denoted by B and is 1/3 for all quarks and -1/3 for all anti-quarks. Therefore, for baryons (anti-baryons) B = 1(−1) and mesons B = 0.
Strangeness: is denoted S and is a quantum number which quantifies the net number of strange quarks in a particle. By convention, S = −1 for the s quark and S = +1 for ¯s quark. The strangeness of all other quarks is 0. The strangeness of a composite particle with N (s) strange quarks and N (¯s) strange anti-quarks is calculated as:
S = −Ns =−[N(s) − N(¯s)], (1.6)
Hypercharge: One consequence of the approximately equal masses of the u, d, and s quarks (mu = md= ms = mu+m3d+ms) is that the approximate QCD Lagrangian is invariant under SU(3)
flavor transformation. One of the conserved currents from this symmetry is the hypercharge Y. Therefore, Y is conserved in strong interactions. In the absence of c, b and t quarks in a particle state the hypercharge can be written as:
Y = B + S (1.7)
where B is the baryon number.
Isospin: The lightest quarks, u and d, have almost the same mass, which is much smaller than that of the lightest hadrons namely the pions (mu = md << mlightest hadron). If the masses would
be exactly the same, the Lagrangian would be symmetric under SU(2) flavor transformations. The conserved currents from this symmetry are defined as components of the isospin operator, I, or isospin quantum number. One can write the following relation between hypercharge and third component of I in absence c, b and t quarks:
I3 ≡ Q − Y/2 (1.8)
One consequence of the isospin symmetry is that hadrons with u and d quarks can be grouped into families of particles with approximately equal masses. The members of a family have the same spin, strangeness, baryon number, charm and bottom, but differ in their electric charges. While the hypercharge takes the same value for each member of a multiplet, I3 takes different
values:
I3 = I, I − 1, ..., −I. (1.9)
Charge conjugation: is denoted by C. The charge conjugation operator turns the particles into their anti-particles and vice versa, e.g. C(Λ) = ¯Λ. Therefore, the charge conjugation quantum number can be±1 depending on how a particle changes under C. Only those particles which are their own anti-particles, for instance, π0, are eigenstates of C. The C operator changes
the sign of all intrinsic additive quantum numbers like electric charge, lepton number, baryon number etc but not space-time properties like the sign of momentum and mass of the particle. The quantum numbers for different quarks are summarized in table 1.2:
Name Symbol Mass (MeV/c2) Q B S C B˜ T
Up u md≈ 2.2 2/3 1/3 0 0 0 0 Down d mu ≈ 4.7 -1/3 1/3 0 0 0 0 Strange s ms ≈ 86 -1/3 1/3 -1 0 0 0 Charm c mc≈ 1.28 × 103 2/3 1/3 0 1 0 0 Bottom b mb ≈ 4.18 × 103 -1/3 1/3 0 0 -1 0 Top t mt ≈ 173 × 103 2/3 113 0 0 0 1
Table 1.2: Different quantum numbers of individual quarks: electric charge (Q), strangeness (S), charm (C), Bottom (B) and Top (T) [12].
In the table 1.3 you can see some examples of hadrons with associated quantum numbers: Particle Quark Composition Mass (MeV/c2) Mean Life-Time(s) Q S C B˜
Λ uds 1115.7 (2.632± 0.020) × 10−10 0 -1 0 0 Σ0 uds 1192.6 (7.4± 0.7) × 10−20 1 -1 0 0 Λc udc 2286.5 (2.00± 0.06) × 10−13 1 0 1 0 Λb udb 5619.6 (1.470± 0.010) × 10−12 0 0 0 -1 π+ u¯d 139.6 (2.6033± 0.0005) × 10−8 1 0 0 0 K− s¯u 493.7 (1.238± 0.0021) × 10−8 -1 -1 0 0 D− d¯c 1869.6 (1.040± 0.007) × 10−12 -1 0 -1 0
Table 1.3: Examples of hadrons with their quark compositions and the corresponding values of their quantum numbers [12].
In strong and electromagnetic processes, all quantum numbers are conserved. In weak interactions, quark flavors can change if the decay involves charge exchange. As while as we consider only weak interactions parity and charge conjugation are not conserved, however, the combined CP operation is conserved at rest frame of particles. The baryon number and total electric charge are conserved in all interactions [11]. Most hadrons are highly unstable and decay to lighter hadrons within strong interaction with a very short lifetime (of order 10−23s). Those hadrons, which do not decay by strong interactions, have a long life-time on the timescale of order 10−23s. They decay by either electromagnetic or weak interactions. Table 1.4 shows the typical lifetimes of hadrons decaying by the three interactions. The only exception is neutron2 with life-time 103s.
Interaction Lifetime (s) Strong 10−22− 10−24 Electromagnetic 10−16− 10−21 Weak 10−7− 10−13
Table 1.4: Typical lifetime of hadrons decaying by the three interactions [11].
All aforementioned quantum numbers are not associated with spatial properties of the wave func-tion. Therefore, they are are called internal quantum numbers.
1.4.2
Mesons
Mesons are bound states of a quark and an anti-quark and have integer spin, [11]. The lightest mesons are pions, (π+, π− , π0), which form an isospin triplet. All mesons are unstable and decay
eventually to either a lepton and a neutrino or into photons, or into other mesons.
2Weak decays depend on the characteristic energy (Q) which is the kinetic energy released by the decay of the
particle at rest. Q is typically of order 102
− 103MeV. However, for the neutron Q = 0.79 MeV and the lifetime is
The π+, π− , π0 isospin triplet has isospin I = 1 and the different members have I
3 = −1, 0, 1.
Several isospin multiplets can be grouped into spin-parity multiplets. The pion belongs to the lightest one, with spin-parity Jp = 0−. This is the pseudoscalar meson nonet. The members of the so-called
vector meson nonet have spin-parity Jp = 1−. The weight diagrams of both pseudoscalar and vector
meson nonet are shown in Fig.1.2. The η is an isosinglet with I = 0, I3 = 0. The K+, K0 and K−, ¯K0
with I = 1/2 and I3 =±1/2 form two isodoublets with different Y.
Figure 1.2: Weight diagram for (a) the Jp= 0− mesons nonet and (b) the Jp= 1− mesons nonet[11].
The small difference in the mass of the members of an isospin multiplet comes from spontaneously symmetry breaking of the ground state of the quantum field of these particles.
1.5
Baryons
Baryons are combinations of three quarks. Since the sum of an odd number of half-integer spins will always yield a half-integer total spin, all baryons are fermions. The proton and the neutron are the lightest and the most well-known baryons and they are stable. The proton is a combination of two u quarks and one d quark with total electric charge +1. Exchanging one of the u quarks in a proton by a d quark gives a neutron. While the proton is stable (no proton decay has ever been observed) and it has a lower limit on its lifetime that is longer than the age of the universe, other baryons are unstable and decay almost instantaneously (for example, the strange Λ hyperon has a life-time of 10−10s).
The lightest baryon multiplets are a Jp = 1/2+ octet and a Jp = 3/2+ decuplet. The baryon multiplets can also contain isospin singlets. According to the quark model [14, 15], if we assume that the combined space and spin wavefunctions are antisymmetric under interchange of quarks, the parity of three-quark states containing only u, d and s quarks (see eq.1.2) and with zero angular momentum is defined as bellow,
The Fig. 1.3 shows the weight diagrams of the predicted and observed baryon octet and baryon decuplet.
Figure 1.3: Baryon Multiplet.
Weight diagram for (a) the Jp = 1/2+ baryon octet and (b) the Jp = 3/2+ baryon decuplet[11].
1.5.1
Hyperon
Replacing one or more quark(s) of a proton with heavier quark(s) like strange(s) or charm(c) quark results in a hyperon, for example Λ, Σ0, Λ+
c, Ξ and Ω. The ω baryon was predicted from ”the eightfold
way”/SU(3) symmetry and its experimental confirmation was a great success, that laid the ground for the quark model. The properties of hyperons could be derived from the nucleon characteristic provided that the SU(3) flavor symmetry was exact, hence the measured deviations indicate that the SU(3) symmetry is broken. By comparing the properties of hyperons and nucleon one can learn to what extent the SU(3) symmetry is broken. Furthermore, we can use hyperon decays to study the conservation of quantum numbers like isospin in strong interactions. Many hyperons are neutral and can not be detected or recognized directly but only by their decay products. Hyperons typically decay after traveling an experimentally measurable distance which provides a very characteristic signature. The lightest hyperons, Λ and Σ0, are in focus of this work. They have the same quark contents
(see 1.3) but the isospin, I, of Λ is 0 while the particle Σ0 possesses I=1. The Λ particle has an
Chapter 2
Formalism
One approach to study the strong interaction is studying electron-positron annihilation, which results in hadron production. Understanding the reaction cross section of hadrons helps us to under-stand their structure. One can formulate the reaction cross section in terms of Electromagnetic Form Factors (EMFFs) which directly are related to the structure of hadrons.
2.1
Cross section
The reaction cross section is a basic observable in scattering experiments. Conceptually, the cross section is the area which quantifies the probability of interaction between particles. Consider a beam impinging on a target or colliding with another beam. The reaction may be either elastic (no new particles are produced) or inelastic (new particles are produced). Letting the z axis be directed along the beam (or one of the colliding beams), the polar angle θ and the azimuthal angle φ between the direction of the incoming beam and outgoing particle define the direction of particles. The probability of finding a scattered particle from an area element, dσ, within a solid angle element, dΩ = d cos θdφ, denotes the differential cross section, dσdΩ, (Fig. 2.1).
d θ
d σ d Ω
Figure 2.1: A detector which covers the solid angle dΩ detects scattered beam from a target in the center.
The scattering cross section can be also expressed in terms of scattering amplitude, f (θ, φ), which is the probability of a given scattering process [16], as below:
dσ
dΩ =| f(θ, φ) |
By integrating over the solid angle we have: σ = Z dσ dΩdΩ = Z | f(θ, φ) |2dΩ (2.2)
Experimentally, the differential cross section can be obtained from the number of recorded events Nsignal of a reaction like a + b →P Hi, where Hi → ci+ di+ ..., within a given solid angle.
dσ dΩ = I(θ, φ) I0 ∝ dNsignal dΩ 1 L n Q i BR(H → c + d + ...) (2.3)
where the beam intensity I0 is the number of incoming particles per unit area per unit time and
I(θ, φ)dΩ is the number of scattered particles in solid angle dΩ per unit area and unit time. The L stands for luminosity and depends on particle beam parameters, such as beam cross section area at the interaction point and particle flow rate. The denotes efficiency and
n
Q
i
BR(H → a + b + ...) is the product of all decay branching ratios i.e. the percentage of particle decaying into a given state [17].
2.2
Relativistic Kinematics
Consider the reaction 1 + 2 → 10 + 20 + ... . On a sufficiently large time-scale, momentum and energy are conserved.
~ P1+ ~P2+ ... = ~P10 + ~P 0 2+ ... (2.4) and E1+ E2+ ... = E10 + E 0 2+ ... (2.5)
In particle physics experiments, we need high energy particles which are able to probe the substruc-tures and produce new particles. Therefore, particles collide with the speed close to the speed of light. To study the kinematics of high energy particles we have to use relativistic kinematics rather than Galilean kinematics which is commonly used for low speed particles. In relativistic mechanics, we define the contravariant four-momentum in the following way,
pµ = (E, ~p) = (p0, p1, p2, p3) (2.6) Momentum and energy conservation then instead become four-momentum conservation .
p1+ p2+ ... = p01+ p 0
2+ ... (2.7)
where pi and p0i are four-momentum of initial and final particles, respectively.
In tensor language pµ is a contravariant tensor of rank 1 while the covariant four-momentum is defined as:
pµ = (E,−~p) = (p0, p1, p2, p3) = (p0,−p1,−p2,−p3) (2.8)
In relativistic mechanics, the quantities whose value remain unchanged under Lorentz transfor-mation are denoted as Lorentz invariant. The pµpµ is a Lorentz invariant quantity called invariant
mass.
pµpµ = E2− p2 = M2 (2.9)
where M is the rest mass of the particle. For colliding particles or decay products the invariant mass has special name Mandelstum variable ”s” and is defined as bellow:
s2 = (p1+ p2 + ...)2 = (p01+ p 0 2+ ...)
2 = (E
1+ E2+ ...)2− (~p1+ ~p2+ ...)2 (2.10)
In the center of momentum (mass) system (CMS) the total three momentum of initial and final particles are zero
N X i ~ pi = N0 X i ~ p0 i = 0 , ECM S = N0 X i Ei0 (2.11)
N and N0 are the number of initial and final particles, respectively. In the rest frame of each particle the three momentum (~p) of the particle is zero.
Two-Body Decay
Using the conservation of four momentum one can find useful information about particles included in a decay. Consider a two-body decay, a→ b + c. In the rest mass system of the decaying particle we have:
Ma2 = p2a = (pb+ pc)2 = m2b + m 2
c+ 2EbEc− 2|~pb|2 (2.12)
where Mais the mass of initial particle and pa, pb and pcare four momenta of corresponding particles.
The hyperons, which are the focus of this work, decay in a very short time and some of them like Λ are neutral and difficult to detect. One can use the invariant mass of hyperon decay products to find the decaying particle.
Two-Body reaction
Consider a two-body reaction, 1 + 2 → 3+4. Having information about the four momentum of one of final particles and using kinematic principles one can calculate the missing particle. For example, in the CMS one can use the known energy of particle 3 to find energy and mass of the unknown particle: E1+ E2 = ECM S = E3+ E4 =⇒ E4 = ECM S − E3 (2.13) and ~ p1+ ~p2 = ~p3+ ~p4 = 0 =⇒ ~p4 =−~p3 (2.14) and therefore, M4 = q E2 4 − p24 = q (ECM S − E3)2− p23 (2.15)
2.2.1
Electromagnetic Form Factor
An approach to study the structure of hadrons is studying their Electromagnetic Form Factors (EMFFs), which quantify the deviation from the point-like case. Consider the elastic electron-baryon scattering (Fig. 2.2, left panel) e−B → e−B. Here B denotes the baryon. If the dominant process is the exchange of one virtual photon γ∗, the momentum squared q2 transferred to the baryon via γ∗
can be considered as a scale which is probed in the nucleon. With small q2, or large wavelength, one
can only resolve the nucleon as a whole, whereas at large q2 the internal structure is resolved. For very large q2, one can resolve individual, point-like quarks and then we have q2-independent structure
functions instead of q2-dependent FF.
For elastic scattering experiments q2 < 0 and the form factor and the corresponding process are space-like (left panel of Fig. 2.2). In the case of an inelastic reaction like electron-positron annihilation with subsequent productions of baryons, e+e− → B ¯B, we have a time-like form factor and q2 > 0
(right panel of Fig. 2.2).
γ∗, q2 ≤ 0 e− B e− B γ∗, q2 > 0 e− e+ B ¯ BFigure 2.2: Space-Like process (left), Time-Like process (right)
Space-Like Form Factors
Consider the elastic scattering of an electron from a point like spin 1/2 target which contains magnetic moment like nucleon (Fig. 2.2 left panel). The scattering cross section for this reaction assuming one-photon exchange can be expressed by the Rosenbluth formula as below:
dσ dΩ = dσ dΩ M ott G2 E(q2) + q2 4M2G2M(q2) 1 + 4Mq22 + q 2 4M2G 2 M(q 2) tan2 θ 2 (2.16)
The GE(q2) and GM(q2) are electric and magnetic form factor, respectively, and one can write them
in terms of Pauli, F1(q2) and Dirac, F2(q2), form factors.
GE(q2) = F1(q2)−
q2
4M2F2(q
2) (2.17)
GM(q2) = F1(q2) + F2(q2) (2.18)
Figure 2.3: The coordinate system of e+e−
→ Λ¯Λ reaction [20].
In a special frame called Breit frame, which coincides with the center of mass frame for elastic electron scattering ~ p =− ~P = ~q 2 −~p0 = ~P0 = ~q 2 (2.19)
the Fourier transforms of electric and magnetic form factors give charge density and magnetization density. The ~p/ ~P and ~p0/ ~P 0 are the momenta of incoming and outgoing particles, respectively.
ρ(~r) = Z d3r (2π)3e −i~q·~r M E(~q)GE(q 2) (2.20) µ(~r) = Z d3r (2π)3e −i~q·~r M E(~q)GM(q 2 ) (2.21)
In other frames, this is not valid, but it gives an intuitive picture of electric and magnetic form factors [18, 19].
Time-Like Form Factors
Because of the short life time of most hadrons they are not suitable as a target in elastic scattering. The exceptions are proton and anti-proton and neutron. The life time of antineutron has not been measured thus far, although in principal it should be the same as neutron life time. Instead, we can probe their electromagnetic structures in electron-positron annihilation e+e− → γ∗where hadrons can be produced by the subsequent decay of γ∗. Unstable hadrons decay into other hadrons in a very short time. Consider the process e+e− → Y ¯Y in the time like region where the squared four-momentum
transferred via virtual photon with q2 = ECM S2 > 0 and q2 = ECM S2 > 4MY2. If one photon exchange dominates the process (e+e−
→ γ∗
→ Y ¯Y ), as shown in the right panel of Fig. 2.2, the differential cross section is parametrized in terms of EMFFs in the center of mass system [19].
dσ dΩ = α2β 4q2 | GM(q2)|2 (1 + cos2θ) + 1 τ | GE(q 2) |2 sin2θ (2.22) where α = 1371 is the fine structure constant, τ = 4M2
Y/q2 where MY is the mass of hyperon, and
The angular integration of Eq. 2.22 gives us the total cross section [20]: σ(q2) = 4πα 2β 3q2 | GM(q2)|2 + 1 2τ | GE(q 2) |2 (2.23) It is often convenient to consider a linear combination of GE and GM. Therefore the effective form
factor for spin 1/2 baryons can be defined as:
| F |2= 2τ | GM |2 +| GE |2 2τ + 1 = 2τ (2τ + 1) 3q2σ 4πα2β (2.24)
The ratio of the form factors is defined by
R = | GE(q
2)|
| GM(q2)|
(2.25)
and by substituting into Eq. 2.23 and 2.24 one can express the electric and magnetic form factors | GE |2= R2 2τ + 1 2τ + R2 | F | 2 (2.26) | GM |2= 2τ + 1 2τ + R2 | F | 2 (2.27)
In the timelike region the electric and magnetic form factors are complex functions,
GE(q2) =| GE(q2)| eiΦE (2.28)
GM(q2) =| GM(q2)| eiΦM (2.29)
This means that they have a relative phase.
∆Φ = ΦM − ΦE (2.30)
The phase reflects the fact that in the time-like region, intermediate states can be produced before the decay into hadron-anti-hadron pair. Although the incidental beams e+ and e− are unpolarized, the relative phase between the form factors has a polarising effect on the final state [20]. In the annihilation process, e+e−
→ Y ¯Y the produced hyperon Y (antihyperon ¯Y ) decays via weak interaction into a spin 1/2 baryon like nucleon and a pseudoscalar meson. If the decaying hyperon is polarized, one can see a parity-violating asymmetry in the decay products. In a case that the asymmetry parameter α is known for a weak decay, one can get information about the polarization of parent hyperon using the decay angular asymmetry measurements. This decay is called self-analyzing [21, 18]. For example in e+e−
Chapter 3
Hyperon Production in e
+
e
−
Annihilation
The products of electron-positron annihilation depend on the initial kinetic energy of the process. In a case when the initial particles do not have relative velocity, the process terminates to two or three photons. Colliding electron positron at very high energies (100 GeV scale), as was done at CERN [22], gives rise to heavier particles like B mesons, W and Z bosons, and hadrons. The EMFFs are preferably studied at intermediate energies (1-10 GeV). In particle physics, the minimum energy to produce a particle is called the threshold energy which is equal to the rest mass of the particle. The threshold energies of some hyperon pairs are summarized in the table 3.11.
Hyperon pair Threshold energy (GeV)
λ ¯Λ 2.230
Λ ¯Σ0 2.307
Σ0Σ¯0 2.40
Table 3.1: The threshold energy for producing some hyperon pairs.
3.1
The e
+e
−→ Λ¯Σ
0reaction
To produce the second lightest hyperon, Σ0, and a precision measurement of its EMFFs, one needs
to increase the initial CMS energy in electron-positron annihilation to more than twice as large as the mass of Σ0, 1.192 GeV/C2
to be able to produce this particle. By increasing the initial CMS energy to at least 2.5 GeV one will get both e+e−→ Λ¯Σ0 (Fig. 3.1) and e+e−→ Σ0Σ¯0 reactions. We choose
this energy because it is sufficiently low to have a high cross section [23] and sufficiently high for the final state particles to have enough momentum to reach the detector.
1The Uppsala Hadron Physics group acquired a beam time at BES III and performed precision measurements of
Figure 3.1: The e+e−
→ Λ¯Σ0 reaction.
The produced hyperons decay in turn to other hadrons like Λ → p + π− and ¯Σ0 → ¯Λγ → ¯p+ π+γ.
By looking at the lifetime of Λ from table 1.3 one can conclude that Λ is a long-lived particle in experimental scale and it can travel some centimeters before decaying, whereas Σ0 decays immediately after being produced. In the next section, the decay length of both Λ and Σ0 are calculated. The
pion mean life is 2.6× 10−6s and mostly decays to µ+ν
µand the lower limit of mean life of anti-proton
is of an order of 1-10 million years. This means that on an experimental scale, p, ¯p, π± and γ are considered stable. One can use the four-momenta of daughter particles to find the invariant mass of parents. For example for ¯Σ0 we have:
MΣ¯20 = (pΛ¯ + pγ)2 = (pp¯+ pπ+ + pγ)2 (3.1)
By reconstructing ¯Σ0 and having its energy in CMS one can calculate the energy and mass of the missing particle, in this case Λ.
ECM S = 2.5 GeV = EΣ¯0 + EΛ (3.2) MΛ= q E2 Λ− p2Λ = q (2.5− EΣ¯0)2− p2¯ Σ0 (3.3)
Relativistic kinematics can also be used to calculate the decay length of ¯Σ0
LΣ¯0 = τ v = τ c
r 1− 1
γ2 (3.4)
where v is the velocity of ¯Σ0 in the CMS and
γ = 1
p1 − v2/c2 (3.5)
The first component of four momentum can be written in terms of γ and rest mass:
E = mγc2 (3.6)
Substituting γ from Eq.3.6 in Eq. 3.4 gives
The mass of ¯Σ0 is known, so we just need to calculate its energy. In CMS we have
~
pe+ + ~pe− = ~pΣ¯0 + ~pΛ = 0 =⇒| pΣ¯0 |=| pΛ | (3.8)
and
E =pm2c4+ p2c2 (3.9)
Substituting from the last equations into Eq. 3.2 gives: ECM S = 2.5 GeV = q M2 ¯ Σ0c4+ p2Σ¯0c2+ q M2 Λc4+ p2Λc2 = q M2 ¯ Σ0c4+ p2Σ¯0c2+ q M2 Λc4+ p2Σ¯0c2 (3.10) Therefore, pΣ¯0 = 0.481 GeV/c (3.11)
By plugging it in Eq. 3.9 we get
EΣ¯0 = 1.129 GeV (3.12)
Substituting the lifetime, mass and energy of ¯Σ0 in Eq. 3.7 gives
LΣ¯0 = 7.4× 10−20× 3 × 108 r 1− 1.192 2 1.292 ≈ 8.3 × 10 −10 cm (3.13)
In the same way we get for Λ decay length in CMS: LΛ = 2.6× 10−10× 3 × 108
r
1−1.115
2
1.212 ≈ 3.03 × cm (3.14)
Therefore, Λ has a long decay length in experimental scale and should be detectable directly. One can use the kinematics of the two-body decay to find the four-momenta of the final particles, ¯p and π+. In the rest system of ¯Λ where p
¯
Λ= (MΛ¯, 0) we have
pp¯= pΛ¯ − pπ+ =⇒ p2p¯= p2Λ¯ + p2π+ − 2~pΛ¯ · ~pπ+ (3.15)
Using invariant mass relation, p2 = m2, in Eq. 3.15 we have
m2p¯= MΛ2¯ + m2π+− 2MΛ¯Eπ+ =⇒ Eπ+ = M2 ¯ Λ+ m 2 π+ − m2p¯ 2MΛ¯ = 0.171 GeV (3.16)
Substituting Eπ+ and the its mass in Eq. 3.9 give the momentum of π+ in the ¯Λ rest system:
pπ+ =
√
0.1712− 0.1402 ≈ 100 MeV/c (3.17)
3.2
The e
+e
−→ Σ
0Σ
¯
0reaction
In addition to the aforementioned reaction, also the e+e− → Σ0Σ¯0 reaction is possible at E
CM S =
2.5 GeV. In the same way as last section one can find the invariant mass of ¯Σ0 using the four momenta of reconstructed ¯Λ and γ. To find the rest mass of the missing particle again we use the center of mass system of this reaction. In this case the Σ0 and ¯Σ0 have equal masses and carry the same energy:
MΣ¯0 = r E2 CM S 2 − p 2 ¯ Σ0 =⇒ pΣ¯0 = q 1.252− M2 ¯ Σ0 = 0.376 GeV/c (3.19)
In this case the calculation of the decay length is straightforward because the particles energy in the CMS are known.
LΣ0 = τ c s 1− M 2 Σ0 E2 Σ0 ≈ 6.7 × 10−10cm (3.20)
Comparing equations 3.13 and 3.20 shows that both reactions lead to almost the same decay length for Σ0.
3.3
Previous studies
The Uppsala Hadron Physics group has focused on the properties of hyperons for some years. One special case is the simulation and analysis of the e+e−
→ Λ¯Λ reaction performed in BES III. Before I started my project, two bachelor theses by Niklas Forssman [24] and Camilla Tumlin [25] have been performed on the e+e− → Λ¯Λ reaction for the center of mass energy of 2.396 GeV, which resulted in an efficiency of 14% to detect this reaction and the Uppsala Hadron Physics group has successfully proposed and acquired a beam time at BESIII to do precision measurements of the Λ hyperon considering the findings of the mentioned projects. My analysis, including algorithm and selection criteria, is built on their work but is extended in the sense that it encompasses new criteria for photon selection.
In parallel to my work, another bachelor thesis by Davide Marietti [26] has been done on the simulation of e+e−
→ Λ¯Σ0 reaction at 2.5GeV independently. Some main results of all aforementioned
bachelor projects have been summarized in table 3.2.
Name Reaction ECM S(GeV) (%)
Niklas Forssman [24] e+e−
→ Λ¯Λ 2.396 14
Camilla Tumlin [25] e+e−
→ Λ¯Λ 2.239 14
Davide Marietti [26] e+e−→ Λ¯Σ0 2.5 23.5
Table 3.2: The main results of previous study in Uppsala Hadron Physics group
I have also studied the e+e−
Chapter 4
The BESIII Experiment
There are several facilities for physicists to study hadrons like J-PARC in Japan [27], BESIII at BEPC-II in Beijing, China [28], KLOE-2 at DAΦNE in Italy [29] and the future PANDA experiment at FAIR, Germany [30].
In this work, I have performed simulations for the BESIII (Beijing Spectrometer) experiment. The BES III is the main detector of the Beijing Electron Positron Collider (BEPC-II), which operates in the 2 GeV-4.6 GeV energy range. Within this range, both short-distance and long-distance effects can be probed [28].
The Uppsala Hadron Physics group is a part of the BESIII collaboration. A low energy off-resonance energy scan was initiated by a sub collaboration of e.g. Uppsala, Mainz and Hefei to perform precision measurements of Λ hyperon form factors. The data were collected in 2014 and 2015 [24]. The next proposal which is under preparation by this group is focusing on the center of mass energy of 2.5 GeV where an integrated luminosity of 100 pb−1 is suggested. The purpose is to study the e+e−→ Λ¯Σ0/Σ0Σ¯0 reactions and especially Λ ¯Σ0 transition form factors, Σ0 and ¯Σ0 form factors.
4.1
BEPC-II
The BEPC-II (see Fig. 4.1) is a double ring electron-positron collider with a circumference of 240.4. The electrons and positrons are accelerated by electric fields and bent by dipole magnetic fields in two separate rings. When beams acquire the desired energy, they are brought to collision with a crossing angle of 22 mrad in total. The interaction point is denoted IP and is shown in Fig. 4.1. The BEPC consists of the following components [31]:
The injector: An electron-position linear accelerator which is able to accelerates electrons and positrons up to 1.3 GeV/c.
Figure 4.1: A schematic view of BEPC-II [32]. The IP is the interaction point of colliding beams and two supercon-ducting Radio Frequency (RF) are installed to accelerate the beams in the storage ring. In the SR point, there is a Synchrotron Radiation facility. The figure in the middle shows the crossing angle of beams schematically.
maximum speed and collide at the interaction point. There are a lot of different magnets in the storage ring which accelerates the beam particles in the desired direction
BSRF: The Beijing Synchrotron Radiation Facility (BSRF) was constructed in parallel with BEPC and has been opened to use since 1991. After upgrading BEPC to BEPCII, the BSRF runs in 2.5 GeV full-energy injection and 250 mA beam current which is the dedicated syn-chrotron radiation mode. The energy covered by the synchrotron radiation light of BSRF contains a range from vacuum ultraviolet to hard X-ray.
The computer center Contains a lot of state-of-the-art equipment located in the local station of the injector to control the electrical apparatuses from a distance.
4.1.1
The BES III Detector
that bends the trajectories of the charged particles. The BES III is located around the interaction point (IP) of the BEPC storage ring. It has started data taking in 2009. It has four sub-detectors: The Main Drift Chamber (MDC), the Time Of Flight System (TOF), the Electromagnetic Calorimeter (EMC) and the Muon Chamber (MUC), Fig. 4.2.
Figure 4.2: The schematic view of the BES III detector with its sub detectors [34]. The MDC, TOF and EMC are surrounded by the muon chamber
The Main Drift chamber (MDC)
use p = qBr to find the momentum of the charged particle, where q is the charge, r is the curvature radius of the helix and B is the strength of magnetic field. Because of the cylindrical geometry of the MDC, the trajectories of charged particles are helices with axes parallel to the magnetic field. One can use the helix parameter to find the space coordinate of a charged particle [37].
The TOF
This detector has been made of plastic scintillators which produce visible light when traversed by charged particles. The interaction of ionizing particles with matter excites or ionizes a large number of molecules. These molecules in return to their ground states emit photons in the visible or close to visible energy range. To have a good scintillating detector the material should be transparent for the light emitted by the scintillator and the index of refraction of material should be close to 1.5 so that the extraction of light from the matter is easier [35].
The TOF contains a barrel part and two end caps with time resolutions of 80 ps and 110 ps, respectively. This sub-detector measures the time between the instant of producing a particle (the time of beams’ collision) and the time it traverses the detector. From the time of flight, the velocity of the particles can be extracted. Combining this information with the momentum from MDC gives the mass of the charged particle which thus provides a particle ID. In the TOF detector, 448 photomultiplier tubes (PMTs) have been installed. The current produced by the incident light is multiplied by these PMTs by as much as 100 million times. The arrival time of the PMT signals is measured by corresponding 448 channels’ readout electronics [38].
The EMC
The EMC is a scintillating detector made of Thallium activated Cesium Iodide (CsI(Tl)) crystal located inside the Superconducting Solenoid Magnet (SSM). The CsI(Tl) is a very bright scintillator and most of its emission is in the longer wavelength region of the spectrum (close to 550 nm). The EMC has been designed to measure the energy and position of light mass particles like electrons, positrons and photons with energies above 25 MeV and also to provide trigger signals. In the EMC the high energy particles interact with the dense matter and a cascade, so-called electromagnetic shower, is produced. The procedure can start with ionizing one molecule by an incoming particle. The produced electron emits a bremsstrahlung photon and the photon is annihilated to an electron-positron pair. The process continues with the new particles in the same way and results in a cascade of electrons, positrons and photons (see Fig. 4.3). The EMC has good discrimination capability for e/π particles with momentum higher than 200 MeV/c. The EMC has one barrel region and two end caps made of scintillating crystals covering 93% of 4π. There are 6240 CsI(T1) crystals located in 44 rings along the z direction in the barrel, each with 120 crystals, and 6 layers in the end caps, with a different number of crystals. This large number of crystals provides sufficient granularity to prevent showers from overlapping in the EMC. The EMC has a design energy resolution of σE/E = 2.5%
√ E and the position resolution is σ = 0.6 cm/√E at 1 GeV. The barrel region covers the polar angle 33.5◦ < θ < 144.7◦ (| cos θ |< 0.83) and for the end caps the angular coverage is 21.3◦ < θ < 34.5◦ (0.85 <| cos θ |< 0.95), Fig. 4.4. There is a small gap of 50 mm between the barrel and end caps to support the mechanical structures, cables and cooling pipes [39, 33].
In e+e−
→ Λ¯Σ0 and e+e−
one γ and two γ, respectively, as decay products. Therefore, we focus on EMC in our analysis as most important subdetector. Especially, we will use the geometry of the EMC to collect the photons detected in the allowed regions and consider their energies and momenta for further calculations.
Figure 4.3: The schematic view of electromagnetic shower [40].
Figure 4.4: The schematic view of EMC [41].
MUC
Chapter 5
Software Tools
Jupyter and Boss are two software packages which have been used in this work to simulate the hyperon production reactions. In this chapter, these packages are briefly introduced to show their advantages in simulating the particle physics experiments.
5.1
Jupyter
Jupyter Notebook Application [42] is a server-client application that allows editing and running notebook documents via a web browser. ”Jupyter” is an acronym referring to the first target languages of the Jupyter application:
Julia: a high-level programming language for numerical computing [43].
Python: a high-level programming language for different purposes like making games [44]. R: a programming language for statistical computing and graphics [45].
Recent Jupyter notebook technology also supports many other languages such as ROOT. Root is a scientific software framework mainly written in C++ which provides all functionality needed to deal with Big Data processing, visualization and storage and statistical analysis [46]. The Jupyter notebook app can be executed offline on a local desktop or can be installed on a remote server and accessed online. The Jupyter environment can include many documents like live code, plots, interactive widgets, narrative text, equations, images, and video. The Jupyter notebook combines three components [42]:
It has an interactive notebook web application for writing and running codes and writing doc-uments.
The kernel of the notebook is used to run the users’ code in a programming language and return the output to the notebook application.
The Jupyter is commonly used as a training environment for programming since its user is not required to set up environments and compilers. For example, by choosing a special kernel in a programming cell of Jupyter one can use different libraries of the corresponding programming language without adding any heading to the code. Another example is that different pre-written classes of some object-oriented programming languages like C++ are available for the users to play around with and learn from. In this work, I have used the Jupyter platform for running“ROOT C++”.
5.2
BOSS
The BES III Offline Software System (BOSS) is based on C++ which is an object-oriented pro-gramming language on the operation system of Scientific Linux1 CERN (SLC). The software uses
some external libraries such as CERNLIB [47], CLHEP [48], and ROOT. The main parts of data processing and physics analysis software are simulation, calibration, reconstruction, and analysis [49]. The most common modeling technique in experimental particle physics is Monte Carlo (MC) sim-ulation of a reaction. The general idea of MC methods is instead of performing complex calcsim-ulations, to use random number generation to perform a huge number of virtual experiments. Comparing the MC simulation with real experiment helps us to understand the results. The MC simulations are also used to make predictions and preparations for future experiments. The BES III simulation consists of the following steps:
1. Event generation
2. Particle transport and detector response 3. Digitalization
4. Reconstruction 5. Analysis
5.2.1
Event generation
In this step one uses an MC generator to generate events at the interaction point and simulates a particle physics reaction. To get MC samples we should utilize our MC generator with some information about the kinematics and dynamics of the initial and final state particles depending on the type of generator. For e+e− → Y ¯Y reaction the Phase Space (PHSP) generator or CONEXC generator are used. The former utilizes with the types of initial particles and their kinematics which is involved e.g. the four-momenta ant it generates the final state particles, their decay, and 4-momenta based on users defined criteria. The PHSP generator gives an isotropic distribution of particles e.g. the angular distribution of reaction products. The CONEXC generator does not only take the kinematics of particles into account but also their dynamics using a theoretical or phenomenological models. In this work I have used the MC samples generated by Dr. Cui Li, a researcher at Uppsala Hadron Physics group, using the PHSP generator.
1
5.2.2
Particle transport and detector response
In the next step the resulting MC samples propagate through a virtual detector which is simulated by GEANT4 [50] software. To simulate the real detector, GEANT4 takes into account the geometry and the mechanical support of the detector.
All features of traversing virtual particles through virtual detector like multiple scattering, energy loss, bending in the magnetic field are gathered from sub-detectors and stored as MC-points.
5.2.3
Digitalization
The MC-points have infinite precision while real a detector has a finite granularity and resolution meaning real particle hits also have a finite resolution. Therefore, in the last simulation step, the MC-points are converted into MC-hits where their resolution correspond to detector resolution. Also, energy losses are converted to pulse heights. After digitalization, the outputs have the same features and format as data from a real experiment.
5.2.4
Reconstruction
The central task of data processing is reconstruction. The reconstruction software consists of a set of algorithms to combine hits to construct particle tracks. Some other information like the energy loss, dE/dx, in the sub-detectors and the time of flight are used to identify the particles. The particle ID can be determined with a certain probability. The output from this step is stored as charged or neutral tracks.
5.2.5
Analysis
In a real experiment, not all reconstructed particles come from the reaction of interest. In the analysis, we define some algorithms to filter out interesting events for further analysis and reject background events. This is referred to as event selection. BOSS uses the ROOT data analysis framework which stores the information about particle candidates in a tree structure and analyses them. Histogramming the quantities that represent the properties of selected events is a convenient way to examine them.
Chapter 6
Parameter Estimation Using Monte Carlo
method
In this chapter, the results of a simple simulation study are presented which performs using ROOT in the Jupyter framework. To estimate unknown parameters of a theoretical model, the method of moments and least squared fit have been used. One can use the bias of an estimator, which is defined as the difference between the estimated value and its true value, to assess to what extent the applied methods is biased. When the bias of an estimator is zero it is called unbiased.
6.1
The Method of Moments
The method of moments (MM) is a convenient way to extract an experimental observable from data. The idea is finding the expectation value, average or moment of some quantities and relate it to the parameter of interest.
For simplicity we consider the one-parameter case, but the method can be applied to many param-eters. We denote this parameter θ. Consider a data distribution described by a probability density function f (x). The probability to make a series of observations, ¯x = x1, x2, ..., xn, given the underlying
physical parameter θ is f (¯x | θ). The first moment (or expectation value) of another function g(x) can be obtained from the set of observation ¯x in the following way [51]:
γ(θ)≡ E(g(x)) = g(x)f(x | θ)dx (6.1)
In an experimental data sample with n events, the estimator of the function γ(θ) is the arithmetic mean of the function over the full set of observations:
< γ(θ) >= ¯g(x) = 1 n n X i=1 g(xi) (6.2)
The variance of the function g(x) is obtained by:
and the variance of the estimator: V (< γ >) = (1 n) 2V ( n X i=1 g(xi)) = 1 nV (g(x)) (6.4)
combining equations 6.2 and 6.3 with 6.4 gives V (ˆγ) = 1 n(n− 1) n X i=1 g(xi)− 1 n n X i=1 (g(xi) 2 (6.5)
Since the MM is an unbinned method it is convenient to use it in data analysis when the available data sample is small (∼ a few hundred events). One drawback of binning for a small data sample is that one loses information because one can not allow fine binning. It is also a straightforward method to use since averages are easy to calculate [51]. However, since the MM is only sensitive to the lowest moment, the variance becomes larger compared to methods like the least squares method if the number of events is larger.
6.1.1
Extracting the ratio R from an angular distribution by applying
MM
In the reaction e+e−
→ Y ¯Y the angular distribution of the final state particles was parametrized in Eq. 2.22 which can be rewritten as:
dσ d cos θ = N1 | GM | 2 (1 + cos2θ) + 1 τ(1− cos 2θ)R2 (6.6) Given this distribution we want to extract the ratio R.
From a continuous distribution in cos θ the moment is calculated in the following way:
< cos2θ >= Z +1 −1 cos2θ dσ d cos θ = 1 Nnorm Z +1 −1 N1cos2θ (1 + cos2θ)| GM |2 + 1 τ(1− cos 2θ)R2 | GM |2 d cos θ = N1 Nnorm 16 15(| GM | 2 + 1 4τR 2 | GM |2) = N1 Nnorm 4 15 2τ + 1 τ | F | 2 4τ + R2 2τ + R2 (6.7)
In the last equality, I have substituted GM(q2) from Eq.2.27. In reality, each data point is a discrete
value of θ, which is continuously distributed. The Nnorm is the normalization factor given by
Nnorm = Z +1 −1 N1 | GM |2 (1 + cos2θ) + 1 τ(1− cos 2θ)R2 d cos θ = N1 2 3 2τ + 1 τ | F | 2 (6.8)
substituting Eq. 6.8 in Eq. 6.7 gives:
< cos2θ >= 1 5
4τ + R2
Therefore, R = r 4τ 5 < cos 2θ >−2 2− 10 < cos2θ > (6.10) and σ<cos2θ> = r 1 N − 1[< cos 4θ >− < cos2θ >2] = s 1 N − 1 3 35( 6τ + R2 2τ + R2)− 1 25( 4τ + R2 2τ + R2) 2 (6.11)
where I have used
< cos4θ >= Z +1 −1 cos4θ dσ d cos θ = 1 Nnorm Z +1 −1 N1cos2θ (1 + cos2θ)| GM |2 + 1 τ(1− cos 2θ)R2 | GM |2 d cos θ = N1 Nnorm 4 35 2τ + 1 τ | F | 2 6τ + R 2 2τ + R2 (6.12)
The uncertainty in R can be calculated from the uncertainty in < cos2θ > using error propagation. Suppose we want to find the uncertainty of a function of two variables, f (x, y). According to the error propagation: σf (x,y)= s (∂f ∂x) 2(∆x)2 + (∂f ∂y) 2(∆y)2 (6.13)
Therefore, we have the following results for the uncertainty of R:
σR = r ( ∂R ∂ < cos2θ >) 2(σ <cos2θ>)2 = 1 2R 10τ 1− 10 < cos2θ > +25 < cos2θ >2 × s 1 N − 1 3 35( 6τ + R2 2τ + R2)− 1 25( 4τ + R2 2τ + R2) 2 (6.14)
This expression is only valid when the uncertainty in τ , i.e. the uncertainty in the center of mass energy is negligible. When we have a wide energy range for e+e− in center of mass system for example
when applying initial-state radiation (IRS) methods on resonance data, we should also consider the error of τ in our calculations using Eq. 6.13.
6.2
The Least Squares Fit
The method of least squares fit is another way to estimate the ratio R. This method is used when data are binned into a histogram. This means that we divide the entire range of our measured quantity, e.g. θ, into intervals or bins, and count how many events fall into each interval. This result in a histogram. We fit a function to the histogram, like a Gaussian or a Polynomial using the method of least square. In the least square method, the χ2 is defined by:
χ2 = N X i=1 µi− νi σi 2 (6.15)
where N is the number of bins, µi is number of data in each bin, σi is the corresponding error and νi
is number of the fit result in each bin. The χ2 thus measures the sum of the distance squared between
the measure data point and values given by the fit. Dividing by the uncertainty in the bin gives more weight to bins with small uncertainty. The best fit is obtained when χ2 is minimized. The goodness of the fit is quantified by the reduced χ2, given by χ2 divided by the number of degree of freedom,
commonly writen as χ2/ndf . The ndf is the number of bins minus number of parameters. A good
fit, with correctly estimated uncertainties, should have χ2/ndf ≈ 1 [53].
In this work a first order polynomial in cos2θ has been fitted to the histogram according to:
dσ dΩ = N ((1 + R2 τ ) + (1− R2 τ ) cos 2 θ) = p0+ p1cos2θ (6.16)
From this fitted polynomial one can extract R as: R = r τp0− p1 p0+ p1 (6.17) and σR= s (∂R ∂p0 )2(δp 0)2+ ( ∂R ∂p1 )2(δp 1)2 (6.18)
To find the uncertainty in p0 and p1 I have used the uncertainty of the parameters of the fitted
polynomial.
6.3
Hit-or-Miss Generator
Hit-or-miss is a way of generating data within a range of interest using random numbers. For each event, one generated random number is used to calculate the probability Wi/Wmax where Wi
is the weight of that number and Wmax is the maximum weight occurring for randomly generated
numbers. Then one compares the probability with a random number in the range R ∈ (0, 1). If Wi/Wmax > P (R), the event is accepted (”hit”), otherwise it is rejected.
In this project, an angular distribution for ¯Σ0 is generated (see Eq. 6.6) considering the ratio R = 2,
ECM S = 2.5 GeV, MΣ0 = 1.192 GeV/c2and MΛ = 1.115 GeV/c2 as input. We run the programme
6.4
Simulating angular distributions
The algorithm of extracting R for ¯Σ0 starts by generating a user defined number of events for
example 100 in one ”pseudo-experiment”. Each ”pseudo-experiment” has a different seed in the random number generator. Next, The MM is applied to calculate R from < cos2θ > (Eq. 6.10) with an uncertainty σr (see Eq. 6.18). In another step, the number of experiments is fixed to 1000 and
this series of experiment is repeated for a different number of events varying from 100 to 1000 in each experiment. This is in order to study how much measured quantities fluctuate between different identical ”experiments”. The average of all ratios coming from each experiment as well as the average of uncertainties, σR and the variance of the estimator of R,
√
V < R >, are also been calculated for different number of events.
√ V < R > =√< R2 >− < R >2 = s PNexp i=1 r2 Nexp − PNexp i=1 r Nexp 2 (6.19) where Nexp is the number of experiments and r is the calculated EMFFs ratio using the MM for each
experiment (see Eq.6.10).
I have also used the method of least square fit to extract R from fitting the obtained binned histogram with a first-order polynomial according to Eq. 6.16.
6.4.1
The Results
The results of different ”experiments” with different random seeds, each one with 100 events are summarized in tables 6.1 and 6.2 for the e+e−
→ Σ0Σ¯0 and e+e−
→ Λ¯Σ0 reactions, respectively. For
each run the ratio R and the uncertainty are determined using Eqs 6.10 and 6.14, respectively.
Number of Events=100 RM M σR
1st run 1.59 0.45
2nd run 1.76 0.50
3rd run 2.53 0.85
Table 6.1: Three different ”experiment” where the ratio of the electromagnetic form factor of Σ0in the e+e−
→ Σ0Σ¯0
reaction is calculated. The centre of mass energy is 2.5 GeV. The RM M is the ratio calculated by the MM and σR is
the uncertainty.
Number of Events=100 RM M σR
1st run 1.41 0.45
2nd run 1.73 0.57
3rd run 1.53 0.52
Table 6.2: Three different ”experiments” where the ratio of the electromagnetic form factor of ¯Σ0in the e+e−→ Λ¯Σ0
reaction is calculated. The centre of mass energy is 2.5 GeV. The RM M is the ratio calculated by the MM and σR is
The tables 6.3 and 6.4 summarize the results from a series of 1000 experiments. In each series different numbers of events for the reactions e+e−
→ ¯ΛΣ0 and e+e−
→ Σ0Σ¯0 are simulated. The
results show that √V < R > and σR are in good agreement for the same number of events, as
expected. Increasing the number of events reduces the uncertainty in RM M . Furthermore, the
average RM M becomes closer to the input value R = 2.
Number of events for 1000 experiments RM M σR pV (< R >)
100 2.170 1.055 1.105
300 1.991 0.346 0.343
500 2.014 0.269 0.273
1000 2.010 0.188 0.193
Table 6.3: Reconstructed transition R for ΛΣ0 in e+e−
→ Λ¯Σ0 reaction a 2.5 GeV for 1000 experiments. The first
column shows the number of events and the second column shows the averaged RM M. σR is the average uncertainty
in RM M and V < R > is the variance of the estimator of RM M.
Number of events for 1000 experiments RM M σR pV (< R >)
100 2.133 1.278 1.222
300 2.029 0.362 0.374
500 2.018 0.273 0.284
1000 2.000 0.188 0.188
Table 6.4: Reconstructed R for for Σ0in the e+e−
→ Σ0Σ¯0reaction at 2.5 GeV. The first column shows the number
of events, the second column the averaged RM M. σR is the average uncertainty in RM M and V < R > is the variance
of the estimator of RM M.
The values of RM M in tables 6.3 and 6.4 show that the bias of an estimator for the user defined
value R=2 and RM M approaches zero (unbiased case) when the number of events increases.
In the Figs. 6.1 and 6.2 the histograms of Σ0 angular distributions for one individual experiment
with 100 events have been plotted. For each histogram a first order polynomial has been fitted as explained in section 4 (see Eq. 6.16). In both cases, the ratio χ2/ndf is almost close to one which shows a good fit. Using fitted functions the ratio R and corresponding uncertainties are also calculated and the results are listed in table 6.5 for three different runs. In the table 6.5 the results from MM and least square fit are compared.
Number of Events=100 RM M σR Rf it σf it
1st run 1.59 0.45 1.26 0.43
2nd run 1.76 0.50 1.25 0.42
3rd run 2.53 0.85 1.26 0.47
Table 6.5: RM M and Rf it and their corresponding uncertainties for one experiment and 100 events in the e+e− →
Σ0Σ¯0 reaction for E
Figure 6.1: Angular distribution and fitted function for the e+e−
→ Λ¯Σ0reaction
Figure 6.2: Angular distribution and fitted function for the e+e−→ Σ0Σ¯0 reaction
6.5
Results and Discussion
In order to get a good estimate of how close to the ”true” value the result of an experiment is, one needs to check the performance of the estimators. This can be done in a series of ”pseudo-experiments” as in this work. We do not expect the value of R to be close to the ”true” value in every individual experiment, but if our estimator is unbiased, then < R > −Rtrue → 0 when Nexperiment → ∞. The
value of σR does not go to 0 by increasing the number of experiments but when Nevent → ∞. This
is a measure of consistency of the estimator. Therefore, to reach a good precision for R and σR we
want to maximize the number of events. The number of events in an experiment can be increased by a more efficient event selection or by collecting data for a larger time. In the BES III data from the 2014-1015 energy scan sample, we typically have ∼ 100 events per energy point. This is why we have reported the results of fitting for 100 events. From the table 6.5, one concludes that how the result can fluctuate from one experiment to another. Within 1σ errors both RM M and Rf it systematically
