J/ψ Polarization At Mid-rapidity In P+P Collisions At √s = 200 GeV At

STAR

by

Siwei Luo B.S., Northeast Forestry University(China), 2011 M.S., Southern Illinois University(U.S.), 2013

THESIS

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Chicago, 2020

Chicago, Illinois

Defense Committee: Zhenyu Ye, Chair and Advisor Olga Evdokimov David J Hofman Ho-Ung Yee Zebo Tang, University of Science and Technology of China Copyright by

Siwei Luo

2020 ACKNOWLEDGMENT

My study experience at the University of Illinois at Chicago would play the most influential impact to my life and career. It is one of the most leading universities all over the world that provides me the valuable opportunities to get access to the world class educational resources and advanced teaching practice. Because my research is on high energy physics experiment,

I joined the STAR collaboration at Brookhaven National Laboratory in 2014 that have the chance to witness how the top scientists and scholars solve the hardest problems and operate the most advanced laboratory built for the purpose of particle physics research.

PhD program for me is a long marathon. It is a great chance to learn myself, how to be persevered and how to bridge the dream and the reality.

I am grateful to my English teachers Jennifer Taylor, Kimberly Hansen, Vandana Loomba for helping me improve my language and presentation skills. I had many interesting academic and dynamical discussions with classmates on foods, sports and cultures.

Thanks to physics department assistant Melissa A. Mattingly for keeping my files and records updated and valid. I owe much thanks to physics department graduate advisor James

Nell for sharing information of good restaurants, local radio channels, outdoor skating field and must-go places at Chicago.

I am grateful for all the instruction, support and help from UIC high energy research group.

Thanks to my colleagues and friends Dhanush Hangal, Guannan Xie, Zaochen Ye, Xiao Wang,

Rongrong Ma, Shuai Yang, Jinlong Zhang and Zhen Liu for the academic discussion and im-

iii ACKNOWLEDGMENT (Continued) provement. Thank Bingchu Huang teach me how to read datasets and how to get the efficiency of detectors hand by hand. I would like to especially thank Olga Evdokimov, David Hofman and Zhenyu Ye for your kindness and providing me the opportunity to study at UIC. Following

Zhenyu’s approaches and ideas, we have gone through the principles of the calculation from fundamental statistics perspective, overcome difficulties of numerical calculation and achieved the precise measurements of J/ψ polarization parameters. During the research, I also learned that how to present the result with high level of aesthetics.

Siwei Luo

iv TABLE OF CONTENTS

CHAPTER PAGE

1 INTRODUCTION ...... 1 1.1 Standard Model ...... 1 1.2 Heavy flavor quarkonium ...... 6 1.2.1 J/ψ production mechanism ...... 7 1.2.1.1 color evaporation model ...... 8 1.2.1.2 color-singlet model and color-octet model ...... 8 1.2.2 J/ψ polarization ...... 12

2 STAR EXPERIMENT ...... 17 2.1 Relativistic Heavy Ion Collider (RHIC) ...... 17 2.2 Solenoidal Tracker At RHIC (STAR) ...... 21 2.2.1 Time Projection Chamber ...... 22 2.2.2 Vertex Position Detector and Time Of Flight ...... 27 2.2.3 Barrel ElectroMagnetic Calorimeter ...... 31

3 J/ψ SIGNALS RECONSTRUCTION ...... 35 3.1 J/ψ reconstruction ...... 35 3.1.1 Datasets ...... 36 3.1.2 J/ψ reconstruction from data ...... 39 3.1.3 J/ψ decay products cosθ and φ distribution ...... 40 3.1.4 J/ψ Reconstruction Efficiency from Detector Simulation . . . 41

4 EXTRACTION OF J/ψ POLARIZATION ...... 44 4.1 Maximum Likelihood Estimation of J/ψ Polarization Parameters 44 4.2 Bias of the estimator ...... 47 4.3 minimum point position and 1σ contours ...... 55 4.4 Goodness of fit ...... 57

5 SYSTEMATIC UNCERTAINTY ...... 60 5.1 pT Weight function ...... 60 5.2 pT smearing ...... 61 5.3 DCA cut efficiency ...... 63 5.4 nHitsFit cut efficiency ...... 63 5.5 nHitsDedx cut efficiency ...... 63 5.6 p/E cut efficiency ...... 64 5.7 ADC0 cut efficiency ...... 64 5.8 TOF matching efficiency ...... 64

v TABLE OF CONTENTS (Continued)

CHAPTER PAGE

5.9 TOF 1/β cut efficiency ...... 68 5.10 nσe cut efficiency ...... 70 5.11 Rapidity weight function ...... 72 5.12 Systematic uncertainty combination ...... 72

6 RESULTS ON J/ψ POLARIZATION ...... 74

7 SUMMARY AND OUTLOOK ...... 77

CITED LITERATURE ...... 78

VITA ...... 83

vi LIST OF TABLES

TABLE PAGE I Track quality cuts ...... 38 II Electron identification cuts ...... 38 III Triggered electron cuts ...... 38 IV χ2/NDF and the corresponding p-values between data and different model calculations...... 75

vii LIST OF FIGURES

FIGURE PAGE 1 Elementary particles of standard model...... 2 2 Spectrum and transitions of the charmonium family...... 7 3 Top: J/ψ cross section times branching ratio as a function of pT in p+p collisions at √s = 200GeV. Solid circles, open circles and blue squares are the published results from STAR; triangles are the published results for |η| < 0.35 from PHENIX. Bars and boxes are statistical and systematic uncertainties, respectively. The curves are CEM (green), NLO NRQCD A (orange) , CGC + NRQCD (blue), and NLO NRQCD B (magenta) theoretical calculations, respectively. Bottom: ratios of these results with respect to the central value...... 10 4 PHENIX J/ψ pT spectrum measurement at energy √SNN = 200 GeV compared with different model predictions...... 11 5 Definition of the polarization frames: helicity(HX) and Collins-Soper(CS) frames. The y-axis is perpendicular to the production plane, which is defined by the momenta of the two colliding protons, represented by p1 and p2 respectively...... 13 6 The extreme cases that when polarization parameter is equal to +1 and -1 respectively. The probability density of positron decay from the direction is equal to the distance of the corresponding surface from the origin...... 14 7 Allowed regions for the J/ polarization parameters...... 15 8 The complex structure of Relativistic Heavy Ion Collider at Brookhaven National Laboratory...... 18 9 Through collision of Au nuclei, the QGP is generated in the overlap- ping region with an almond shape...... 19 10 Phase diagram and critical point of QGP...... 20 11 The complex of Time Projection Chamber...... 23 12 The characteristics of energy loss distribution for different particles. 25 13 Main parameters for the STAR TPC...... 26 14 The complex of VPD detectors...... 27 15 Two side views of the structure of a MRPC module. The upper (lower) view shows the long (short) edge. The two views are not shown at the same scale...... 28 16 Inverse velocity vs momentum from 2.6 million TOFr+pVPD-triggered events in d+Au collisions...... 30

viii LIST OF FIGURES (Continued)

FIGURE PAGE

17 Left top: TPC dE/dx vs the momentum in d+Au collisions. Left bottom: TPC dE/dx vs the momentum after TOFr PID selection of |1β| < 0.03. Clean electron identification is achieved. Right: dE/dx from TPC after TOFr PID selection (left bottom panel) for 1.0 < p < 1.5GeV/c. 31 18 The geometry and coverage of BEMC...... 33 19 The E/pc distribution ...... 34 20 Invariant mass spectra of dielectron pairs in different pT bins (from left to right: pT = 0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The black markers (blue filled histograms) are the spectra from unlike-sign (like-sign) charge pairs, while the red marker represent those obtained by subtracting the like-sign spectra from the unlike-sign ones. The latter are fit to Double Crystal functions represented by the red solid curves...... 40 21 2D distribution of J/ψ candidates from data as a function of (cosθ, φ) plane in different pT bins (from left to right: pT =0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The top (bottom) row shows the distributions in the HX (CS) frame...... 41 22 J/ψ reconstruction efficiency from embedding as a function of (cosθ, φ) in different pT bins (from left to right: pT =0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The top(bottom) row shows the efficiencies in the HX (CS) frame...... 43 23 The definition of production, polar angle θ and azimuthal angle φ. . 45 24 For instance, the objective function likelihood as a function of (λθ, λφ) at pT [6-8] GeV/c in helicity frame is convex...... 47 25 Distributions of the estimated central value (top row) and statistical uncertainty (bottom row) for λθ (left), λφ (middle) and λθφ (right) from 1000 pseudo experiments in 6 < pT < 8 GeV/c bin in the HX frame. See text for details...... 49 26 Statistical uncertainties for λθ from data (vertical red lines) and 1000 pseudo experiments (black histogram). Top (bottom): HX (Collins- Soper) frame. From left to right: pT = 0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c. 51 27 Same as Figure 26 but for λφ...... 51 28 Same as Figure 26 but for λθφ...... 51 29 Biases in the central value estimation: the x-axis is the input λ value that are used to generate pseudo data, while the y-axis is the mean of the extracted λ values from 1000 pseudo experiments. From top to bottom, λθ, λφ and λθφ in the HX frame, and λθ,λφ and λθφ in the CS frame, respectively. From left to right, pT = 0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c, respectively. The red line is y = x, while the black line is a linear fit y = ax + b to the points...... 53

ix LIST OF FIGURES (Continued)

FIGURE PAGE

30 Biases in the uncertainty estimation: the x-axis is the RMS of the extracted λ values from 1000 sets of pseudo data, while the y-axis is the mean of the extracted λ uncertainty from 1000 pseudo experiments. From top to bottom, λθ,λφ and λθφ in the HX frame, and λθ,λφ and λθφ in the CS frame, respectively. From left to right, pT = 0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c, respectively. The red line is y = x, while the black line is a linear fit y = ax to the points...... 54 31 −lnLmin +0.5 contour (black area) from data as a function of λθ and λφ. Also shown are the λθ and λφ values estimated from data using the MLE...... 56 32 Same as Figure 31 but for λθ and λθφ ...... 56 33 Same as Figure 31 but for λφ and λθφ ...... 56 34 The comparison of theoretical distribution and raw data distribution in φ. The top (bottom) row shows the HX (CS) frame...... 58 35 The comparison of theoretical distribution and raw data distribution in cosθ. The top (bottom) row shows the HX (CS) frame...... 58 36 The comparison of Levy function and Powerlaw function fit to the measured J/ψ production cross section...... 61 37 Electrons TOF matching efficiency in different η range as a function of pT ...... 65 38 Positron TOF matching efficiency in different η range as a function of pT ...... 66 39 Ratio of TOF efficiency over TOF efficiency condition on no EMC matching for electron and pion in p+p MB at 200 GeV. Pion is taken from David’s D∗ 2011 analysis ...... 67 40 The mean of the Gaussian function sampling the 1/β distribution. . 68 41 The sigma of the Gaussian function sampling the 1/β distribution. . 69 42 Top left is the mean of Gaussian function with the fit by POL0. Top right is the mean of Gaussian function with the fit by POL1. Bottom left is the sigma of Gaussian function with fit by POL0. Bottom right is the sigma of Gaussian function with the fit by POL1 ...... 71 43 Systematic uncertainty of λθ, λφ,λθφ and λinv in Helicity and Collins- Soper frame from different sources...... 72 44 Systematic uncertainty combination of λθ, λφ,λθφ and λinv in Helicity and Collins-Soper frame...... 73 45 The J/ψ polarization parameters (from top to bottom: λθ, λφ, λθφ) as a function of pT in the Helicity (left) and Collins-Soper (right) frames. 75 46 The J/ψ polarization parameters (from left to right: λθ, λφ, λθφ) as a function of pT in the Helicity (top) and Collins-Soper (bottom) frames. 76

x LIST OF FIGURES (Continued)

FIGURE PAGE

xi SUMMARY

Experimental measurements of quarkonium cross section and polarization are two main approaches to gain insight into quarkonium production mechanism. Quarkonium polariza- tion refers to anisotropic spatial distribution of lepton decay from quarkonium with regard to quarkonium momentum. Although many theoretical models achieve good consistency with the experimental measurement on quarkonium transverse momentum spectra, they have very differ- ent predictions on quarkonium polarization. Therefore, quarkonium polarization measurements have a strong capability to distinguish different models and portrait the quarkonium production mechanism. In this study, the main detectors in Solenoidal Tracker At RHIC (STAR), time projection chamber, time-of-flight and Barrel Electromagnetic Calorimeter are fully applied for track reconstruction, particle identification and physical quantity measurement. The raw yield of J/ψ as a function of polar and azimuthal angles is calculated from STAR datasets collected in 2012 from p+p collisions at √s = 200GeV, and the efficiency of detectors is obtained by

GEANT3 simulation. A statistical estimator for J/ψ polarization parameters is constructed based on the principle of maximum likelihood estimation. Its properties are studied via Monte

Carlo simulation. It is the first time that the measurements of λφ, λθφ and λinv are presented by STAR experiment. The results don’t suggest prominent J/ψ polarization. Among avail- able theoretical models compared with the measurement, the CGC+NRQCD achieve the best agreement.

xii CHAPTER 1

INTRODUCTION

1.1 Standard Model

What the universe is made of is an ultimate philosophy question. Standard Model is the modern theory describing the properties of fundamental particles and their interactions. The fundamental particles can be divided into two groups: fermions and bosons, as shown in Fig- ure 1. The fermions include quarks and leptons, which are building blocks of visible matter.

The bosons include gluon, photon, W boson and Z boson, which are responsible for interac- tions among particles, and Higgs boson which gives mass to particles through the subtle Higgs mechanism [1]. The spin of a fermion is an odd multiple of half ¯h, and that of a boson is an even multiple of half ¯h. Quarks and leptons have three generations. Up quark, down quark, electron and electron neutrino are in the first generation. Charm quark, strange quark, muon and muon neutrino are in the second. Top quark, bottom quark, tau and tau neutrino are in the third.

1 2

Figure 1: Elementary particles of standard model [2].

The interactions in the Standard Model include electromagnetic, weak and strong forces.

They are described by quantum field theory (QFT) and share the same framework of calculation

[3]. The bosons exchanged in the process of these interactions are gauge bosons, which have a spin of ¯h. Photons participate in the electromagnetic interaction between charged particles. W 3 boson and Z boson are mediated in the weak interaction. Gluons are responsible for the strong interaction between quarks and gluons. The scattering amplitude for the process of exchanging a gauge boson can be calculated with Feynman diagrams [4]. For electromagnetic interaction, weak interaction and strong interaction at high energy, the calculation can be written as power series expansion based on perturbation theory [5], where the dominant contribution is from the lowest order terms. To avoid the divergence of the sequence, all the terms are renormalized [2].

Often, the evolution of a system is calculated by finding the minimum of Lagrangian which is the kinematic energy minus potential energy with respect to general coordinates. Quantum electrodynamics (QED) describes the electromagnetic interaction between charged particles [4].

QED is an Abelian gauge theory with the symmetry group U(1) [6]. The Lagrangian of QED reads 1 L = ψ¯(iγµD − m)ψ − F Fµν, µ 4 µν q X µ where γ are Dirac matrices; ψ a bispinor field of spin-1/2 particles; Dµ = ∂µ + ieAµ + ieBµ is the gauge covariant derivative; m is the mass of charged particle; Aµ is the covariant four- potential of the electromagnetic field; Bµ is the external field imposed by external source;

Fµν = ∂µAν − ∂νAµ is the electromagnetic field tensor [7].

The process of beta decay that a neutron decays into a proton, electron, and electron neu- trino is due to the weak interaction [8]. The observation of CP-violation in weak interaction has improved our understanding of the symmetry of the universe [9]. The electromagnetic interac- tion and weak interaction can be unified into electroweak interaction described by electroweak theory with an SU(2) U(1) group. × 4

Quantum chromodynamics (QCD) is the theory that describes the principles of the strong interaction between quarks and gluons [10]. Analog to the electrical charge of particles in quantum electrodynamics, quarks carry colour charge which is called red, green and blue, respectively. The SU(3) colour symmetry states that the strong interaction is invariant under rotation in color space. QCD is a non-Abelian gauge theory because the SU(3) do not commute

[11].

Two important phenomena, color confinement [12] and asymptotic freedom [13], have been discovered for the strong interaction. Since no individual quark is observed in the experiment, color confinement states that color charged particles cannot be isolated and therefore cannot be directly observed in normal condition [12]. Asymptotic freedom states that interaction between two color charged particles becomes weaker as the energy scale increases and the distance scale decreases [14].

The Lagrangian of QCD [15] reads

1 L = ψ¯ (iγµ∂ δ − g γµtC AC − m δ )ψ − FA FAµν, q,a µ ab s ab µ q ab q,b 4 µν q X

µ where γ are the Dirac γ-matrices and ψq,a are quark-field spinors for a quark of flavor q and mass mq, with a color-index a that runs from a = 1 to 3 as quarks come in three

¯ µ 1 A Aµν colors. The kinematic energy of quarks is q ψq,a(iγ ∂µδab − mqδab)ψq,b − 4 FµνF and P the potential terms represents the energy that quarks interact with strong interaction field

¯ µ C C q ψq,a(gsγ tabAµ )ψq,b. P 5

Due to the heavy mass of charm quark, the velocities of quarks in the charmonium states are low [2]. The observed spectra of charmonium resonances correspond to different eigenstates of QQ¯ system provides a good probe of the QCD potential in the non-relativistic limit, which is non-relativistic QCD (NRQCD).

The NRQCD potential [2] can be decomposed into two components. Analog to QED po- tential between an electron and a positron, one component of the NRQCD potential between a quark and an antiquark is called short-range NRQCD potential [2]

4 α V (r) = − S s 3 r

where αs is the coupling constant for the strong interaction and r is the distance between the quark and antiquark.

At relatively large distances, the energy density between the quarks containing the gluon

field is constant. The energy stored in the field is thus proportional to the distance between the quarks [2]

Vl(r) κr ≈ where κ 1GeV/fm. Thus, the complete NRQCD potential in the QQ¯ state is ≈

4 α V (r) = − S + κr qq¯ 3 r 6

The production of heavy quark and anti-quark is perturbative while hadronization is non- perturbative, which cannot be calculated from first principle. In the process of hadronization, quarks and anti-quarks undergo a ”cooling down” process such that the hadrons are formed

[16]. The observed hadrons are usually mesons or baryons, which contain two or three valence quarks, respectively. Exotic states have also been discovered where hadrons consist of more than three quarks, e.g. pentaquarks [17].

1.2 Heavy flavor quarkonium

A meson made of a quark and its anti-quark is called quarkonium [18]. The quarkonium formed by heavy quarks, including charmonium and bottomonium, is called heavy flavor quarko- nium.

The existence of charm quark was proposed by Sheldon Lee Glashow and James Bjorken.

The spectra of charmonium states and the decay processes are present in Figure 2. The ground state of charmonium J/ψ meson has a rest mass of 3.0969GeV/c2 and mean lifetime 7.2 10−21s × [18]. It was discovered independently by two research groups from Stanford Linear Accelerator

Center (SLAC) and Brookhaven National Laboratory (BNL). Because this discovery reveals the existence of the fourth known quark, the charm quark, Burton Richter and Chao Chung

Ting were awarded the 1976 Nobel Price.

Hadronic decay modes of J/ψ are strongly suppressed because of the OZI Rule [19]. The branching ratios (BR) of J/ψ main decay modes are [20] :

J/ψ hadrons BR = (87.7 0.5) %; ± J/ψ e+e− BR = (5.94 0.06) %; → ± → 7

J/ψ µ+µ− BR = (5.93 0.06) %; ±

→ 91. Charmonium system 749

91. Charmonium System

Updated in 2012.

Mass (MeV)

4700 X(4660)

4500 ψ (4415) Thresholds: X(4360) 4300 X(4260) π π D s*D s* ψ (4160)

4100 D s*D s ψ (4040) D*D* χ (2P) c2 D s D s π π X(3872) 3900 D D* ψ (3770) (2-+?) DD ψ (2S) 3700 (2S) ηc π0 χ (1P) χ (1P) c2 π π c1 3500 η h (1P) χ (1P) c c0

3300 π π π π π0

η π π 3100 (1S) ηc J /ψ (1S)

2900

J PC = 0− + 1− − 1+ − 0+ + 1+ + 2+ +

The level scheme of the cc states showing experimentally established states with solid lines. Singlet states are called ηc and hc,tripletstates ψ and χcJ ,andunassignedcharmonium-likestatesFigure 2: Spectrum andX.Inparenthesesitissu transitions offfi thecient charmonium to give the radial quantum family numb [18].er and the orbital angular momentum to specify the states with all their quantum numbers. Only observed hadronic transitions are shown; the single photon transitions ψ(nS) γηc(mP ), ψ(nS) γχ (mP ), and χ (1P ) γJ/ψ are omitted for clarity. → → cJ cJ →

1.2.1 J/ψ production mechanism

The heavy mass of charm quark plays a cut-off effect that the perturbative phase of quarko- nium production can be calculated precisely. However, the non-perturbative phase of quarko- 8 nium production can only be accessed by experimental measurements and phenomenological models.

1.2.1.1 color evaporation model

The color evaporation model (CEM) calculates the cross section for quarkonium production in hadron collisions in the following way:

2 4mM 2 dσ ¯ σ[A + B H + X] = FH dmQQ¯ 2 [A + B QQ + X] 2 dm Z4m QQ¯ → → where mM is the mass of meson M containing the heavy quark Q, dσ is the differential cross

¯ section for a QQ pair to be produced in a collision of A and B, mQQ¯ is the invariant mass of

QQ¯ . FH is the fraction factor represent the ratio that invariant mass of QQ¯ less than 2mM, which is assumed to be universal that can be determined by data [21]. As one of the simplest quarkonium production model that was first proposed in 1970’s, the calculation doesn’t take quantum numbers such as angular momentum, spin or color of quarks into account [22, 23].

1.2.1.2 color-singlet model and color-octet model

The quantum states of color-singlet and color-octet read √1 (r¯r+gg¯+bb¯) and rg¯, rb¯, g¯r, gb¯, 3 b¯r, bg¯, √1 (r¯r − gg¯), √1 (r¯r + gg¯ − 2bb¯) respectively [2]. In the color-singlet model, it assumes 2 6 that the quantum state of the pair does not evolve between its production and hadronization, neither in spin, nor in color. The differential production cross section is calculated in the following way: 9

¯ 2 dσ[A + B QQ + X] = dxidxjfi(xi, µF)fj(xj, µF)dσ^i+j QQ¯ +X|ψ(0)| i,j X Z → → where parton distribution functions (PDFs) fi(xi, µF) (fj(xj, µF)) is the number density of the parton of flavour i (j) inside the hadron A (B); xi (xj) is the parton momentum fraction denoted the fraction parton carried from proton; and µF is factorisation scale [24].

Compared with color-singlet model, color-octet model take quantum numbers that angu- lar momentum, spin and color into the calculation of production mechanism and express the hadronization probability of a heavy-quark pair into a quarkonium via long-distance matrix

n elements (LDMEs) < OQ > [25].

¯ n dσ[A + B QQ + X] = dxidyjfi(xi, µF)fj(xj, µF)dσ^i+j QQ¯ +X(µR, µF, µΛ) < OQ > i,j,n X Z → → where n denotes sets of quantum numbers including color, angular momentum and spin; fi(xi, µF) (fi(xi, µF)) is the parton distribution function of parton of i (j) flavour. The cross section is calculated by summing up all component contribution. For the color-singlet and color-octet model, they essentially differs from how many quantum states allowed during the process of quarkonium production. 10 2 10 10 10 10 p+p s = 200 GeV p+p s = 200 GeV p+p s = 200 GeV p+p s = 200 GeV STAR 2012 STAR 2012 STAR 2012 STAR 2012 STAR 2009 MB CEM CGC + NRQCD NLO NRQCD B STAR 2009 HT NLO NRQCD A dy) nb/(GeV/c) T −2 PHENIX −2 −2 2

10dp T p π /(2 σ 2 Bd

10−5 −5 −5 5

3 3 3 3

2 2 2 2 Ratio to 2012 data

1 1 1 1

0 0 0 0 0 5 10 0 5 10 0 5 10 0 5 10 p (GeV/c) p (GeV/c) p (GeV/c) p (GeV/c) T T T T

Figure 3: Top: J/ψ cross section times branching ratio as a function of pT in p+p collisions at

√s = 200GeV. Solid circles, open circles and blue squares are the published results from STAR

[26]; triangles are the published results for |η| < 0.35 from PHENIX [27]. Bars and boxes are statistical and systematic uncertainties, respectively. The curves are CEM (green) [28], NLO

NRQCD A (orange) [29], CGC + NRQCD (blue) [30], and NLO NRQCD B (magenta) [31] theoretical calculations, respectively. Bottom: ratios of these results with respect to the central value [32]. A. ADARE et al. PHYSICAL REVIEW D 85, 092004 (2012) C. J=c p distribution TABLE XIII. Mean transverse momentum squared in T 2 GeV=c of J=c and c 0 for different rapidity and pT ranges. Figure 23 shows the (a) pT and (b) xT 2pT=ps de- ðUncertaintiesÞ are type A and type B, respectively. pendence of the dielectron differential cross¼ section at ffiffiffi 2 2 midrapidity, compared to other p p and p p experi- System pT pT p <5 GeV=c þ þ h ihij T ments [9,21,64–66]. The shapes of the transverse momen- J=c 1:2 < y < 2:23:63 0:03 0:09 3:43 0:02 0:08 tum distributions in Fig. 23(a) follow the well known J=c y < 0j:35j 4:41 Æ 0:14 Æ 0:18 3:89 Æ 0:11 Æ 0:15 ‘‘thermal’’ exponential behavior for p < 2 GeV=c and j j Æ 1:5 Æ Æ 1:5 Æ T c 0 y < 0:35 4:7þ1:05 0:44:7þ1:05 0:4 j j À Æ À Æ a hard-scattering power law behavior at high pT . Fig- ure 23(b) shows that the hard process scales with xT 3 3 PHENIX [17,18]. Attempts are being made to extend the (psEd =d p G xT n)[67] for xT > 0:1 in all collision energies, where¼n ð 5Þ:6 0:2 [65](n is related to the CSM to NNLO. Preliminary NNLO CSM calculations ffiffiffi ¼ Æ number of partons involved in the interaction). A pure performed for pT > 5 GeV=c [17] show a large increase LO process leads to n 4, hence, NLO terms may be in the yield, but still underpredict the experimental results. important in J=c production¼ [68–71]. None of these theoretical models consider the B-meson The pT dependence of the J=c differential cross sec- decay contribution to the J=c yield. The FONLL calcu- tions measured at forward and midrapidity are shown in lation [3] for these decays is also plotted in Fig. 24 and has Fig. 24 along with theoretical calculations where the ab- a reasonable agreement with STAR measurements using solute normalization is determined in the calculations. The J=c -hadron correlations [65]. According to this calcula- CEM and the NRQCD (for pT > 2 GeV=c) provide rea- tion, the B-meson contribution to the measured J=c sonable descriptions of the pT distribution, whereas the inclusive yield is between 2% (1%) at 1 GeV=c and CSM disagrees in both the normalization and the slope of 20% (15%) at 7:5 GeV=c in the mid(forward)-rapidity the pT distribution, indicating that NLO color singlet region with large theoretical uncertainties. intermediate states cannot account for the direct J=c The pT spectrum of the J=c is harder at midrapidity, as production. However, the NLO CSM calculation gives a seen11 from the ratio between the forward and midrapidity good description of the J=c polarization measured by differential cross sections versus pT shown in Fig. 24, bottom panel. Also shown are the forward/midrapidity yield ratios from the theoretical models, using their mean

] 5 2 10 p+p→ J/ψ+X s=200GeV values and assuming that theoretical uncertainties in these 4 NLO CSM (direct J/ψ) 10 NLO NRQCD ratios cancel out. All of the models predict a downward 3 CEM 10 B-meson→ J/ψ+X trend, but the CEM and NRQCD calculations do not follow 102 |y|<0.35 × 1000 a slope as large as the data. 1.2<|y|<2.2 2 [nb/(GeV/c) The mean transverse momentum squared, p , was T 10 h Ti 1 calculated numerically from the pT distribution. The cor- 2 /dydp -1 related uncertainty was propagated to pT by moving σ 10 2 low-p and high-p data points coherentlyh i in opposite d -2 T T T 10

p directions according to their type B uncertainty. -3 π 10 2

/2 Table XIII shows pT with the propagated type A and ll -4 h i 2 B 10 type B uncertainties and pT for pT < 5 GeV=c (to allow 10-5 a direct comparison withh thei previous PHENIX results 1 [72]). As expected, the mean transverse momentum 0.8 squared is larger at midrapidity than at forward rapidity. 0.6 0.4 D. Charmonia ratios and J=c feed-down fractions 0.2 0 The transverse momentum dependence of the c 0= J=c 0246810 yield ratio [Fig. 8(b)] is consistent with that observedð inÞ fwrd./cent. rapidity p [GeV/c] T other experiments. Figure 25 shows the collision-energy dependence of the c 0= J=c yield ratio in low-atomic- FIG. 24 (color online). Transverse momentum dependence of ð Þ Figure 4: PHENIX J/ψ pT spectrum measurement at energy √SNN = 200 GeV comparedmass with fixed-target p A, p p, and p p [9,64,66,73– (top panel) J=c yield in y < 0:35 and 1:2 < y < 2:2 along þ þ þ j j j j 75] collisions. The ratios from p p experiments were different modelwith predictions predictions [33]. based on CSM [17], NRQCD [20], CEM þ calculated using the reported J=c and c 0 cross sections for [52,53], and B-meson decay based on FONLL calculation [3]. p > 5 GeV=c together with their point-to-point uncorre- All models use CTEQ6M [44]. Boxes are systematic uncertain- T lated uncertainties.3 The B meson decay contribution was ties. Theoretical uncertainties are represented as bands. Note that the midrapidity points are scaled up by a factor of 1000. The bottom panel shows the ratio of the forward and central rapidity 3This may be an overestimate of the systematic errors, given The theoreticalpT spectra models and on correspondingJ/ψ production theoretical mechanism providepredictions. very distinct predictionsthat on a good fraction of the J=c and c 0 yields may be correlated.

J/ψ polarization, while have good agreement with J/ψ cross section experiment measurements as shown in Figure 3 and Figure 4. Therefore, the experimental measurement of J/ψ092004-22polarization 12 become a useful tool to distinguish different models and constraints the parameters of theoretical models.

1.2.2 J/ψ polarization

To extract the J/ψ polarization parameters from data, we need to define the polarization reference frame first. As shown in Figure 5, p1 and p2 correspond to the colliding proton’s momentum in the J/ψ rest frame, which defines the production plane. Y axis is perpendicular to the production plane. The difference between Helicity frame and Collins-Soper frame is that the Z axis in Helicity frame is defined along the sum of p1 and p2 vector, while the Z axis in

Collins-Soper frame is defined as the bisector of p1 and p2 vector. Consequently, after Y axis and Z axis is aligned, the X axis is determined. In the present study, we usually measure the polarization parameters in Helicity frame and Collins-Soper frame.

To remove the effect of the arbitrary choice of reference frame, λinv is the specific quantity remaining the same in both the Helicity frame and the Collins-Soper frame.

λθ + 3λφ λinv = (1.1) 1 − λφ 13

Figure 5: Definition of the polarization frames: helicity(HX) and Collins-Soper(CS) frames [34].

The y-axis is perpendicular to the production plane, which is defined by the momenta of the two colliding protons, represented by p1 and p2 respectively.

A particle may be observed preferentially in a state belonging to a definite subset of the possible eigenstates of the angular momentum component Jz along a characteristic quantization axis. When this happens, the particle is said polarized [35]. Polarization of J/ψ indicates the anisotropic spatial distribution of lepton decay from J/ψ with respect to J/ψ momentum. And the differential cross section of J/ψ decay products can be written as the Fourier series expansion as following:

2 ∂ N 2 2 1 + λθcos θ + λφsin θcos(2φ) + λθφsin(2θ)cosφ, (1.2) ∂cosθ∂φ ∝ 14

where the coefficients λθ, λφ and λθφ are parameters determining the probability density dis- tribution of the decay products with respect to the polarization (z^) axis in the J/ψ rest frame.

As illustrated by Figure 6, when λ = +1 the positron prefers moving parallelly to z axis and 1.3.θ J/ψ POLARIZATION when λθ = −1 the positron prefers moving orthogonally to z axis. would mean that the J/ψ is on average unpolarized, an anisotropic distribution would mean that the J/ψ is polarized. Figure 1.3 shows dilepton decay distribution of transversely (a) and longitudinally (b) polarized J/ψ in the natural frame (polarization axis z coincides with the −→J ).

λ FigureFigure 6: The 1.3: extreme Representation cases that of the when dilepton polarization decay distribut parameterion of transverselyθ is equal (a) to and +1 longitu- and -1 respec- tively.dinally The (b) probability polarized J/ thatψ in positron the natural decayframe. from The the probability direction of the is equal lepton to emission the distance in one of the direction is represented by the distance of the corresponding surface point from the origin. [19] corresponding surface from the origin [35].

In an experiment, the coordinate system used for the J/ψ polarization studies, has to be defined. Figure 1.4 shows possible definition of the coordinate system, as seen from the J/ψ rest frame. The z axis is the polar axis, which depends on the chosen reference frame. The x axis lies on the production plane xz (x = y z), which contains momenta of colliding beams, × and the y is the production plane normal. The polar angle, θ,isdefinedastheanglebetween momentum of a lepton from the J/ψ decay (usually it is l+)intheJ/ψ rest frame and the chosen polar axis z.Theazimuthalangle,φ,isdeterminedbytheproductionplane.

1.3.1 Reference frames

There are three reference frames that are usually used for theJ/ψ polarization measurements: helicity [58], Collins-Soper [59] and Gottfried-Jackson [60] frames. In all frames the polar- ization axis z belongs to the production plane (xz). Figure 1.5 shows the production plane definition on the left-hand side, and definitions of the polarization axis z in different rest frames are shown on the right-hand side of the Fig. 1.5

25 15

The integration over cosθ or φ yields the one-dimensional distribution of φ and cosθ re-

spectively,

1 2 W(cosθ) (1 + λθcos θ), (1.3) ∝ 3 + λθ

2λφ W(φ) 1 + cos2φ (1.4) ∝ 3 + λθ

Due to the nature of differential cross section of positron decay from J/ψ is probability which

should be greater than or equal to 0 for every direction. This requirement exerts constraints

on the J/ψ polarization parameters so that the admissible set of J/ψ polarization parameters

is shown in Figure 7.

Eur. Phys. J. C (2010) 69: 657–673 665

Fig. 6 Allowed regions for the decay angular parameters

Figure 7: Allowed regions for the J/ψ polarization parameters [35]. with respect to an axis rotated by 90◦ (2): 5Effectofproductionkinematics on the observed decay kinematics 90 1 1 1 1, 1 ◦ 1, 1 1, 1 1, 0 . (23) | ± ⟩ −−→ 2| + ⟩ + 2| − ⟩∓√2| ⟩ The amplitude of the transition of this mixed state to the “ro- Ideally, the dependence of the polarization on the momen- tated” dilepton state in (5)containsthreetermswithrelative tum components of the produced quarkonium should reflect phases (due to the ϕ dependence of the rotation matrix) giv- the relative contribution of individual production processes ing rise to the observable azimuthal dependence. The same in different kinematic regimes, thereby providing informa- polar anisotropy λ 1/3wouldbemeasuredinthepres- tion of fundamental physical interest. However, the obser- ϑ′ = − ence of a mixture of at least two different processes resulting vations are, in general, affected by some experimental lim- in 50% “transverse” and 50% “longitudinal” natural polar- itations, which must be carefully taken in consideration. ization along the chosen axis. In this case, however, no az- First, the frame-dependent polarization parameters λϑ , λϕ imuthal anisotropy would be observed. As a second exam- and λϑϕ can be affected by a strong explicit kinematic de- ple, we note that a fully “longitudinal” natural polarization pendence (encoded in the parameter δ in (21)), reflecting the (λϑ 1) translates, in a frame rotated by 90 with respect = − ◦ change in direction of the chosen experimental axis (with re- to the natural one, Fig. 5(d), into a fully “transverse” po- spect to the “natural axis”) as a function of the quarkonium larization (λ 1), accompanied by a maximal azimuthal ϑ′ =+ anisotropy (λ 1). In terms of angular momentum, the momentum. Second, detector acceptances and event sam- ϕ′ = − measurement in the rotated frame is performed on a coher- ples with limited statistics induce a dependence of the mea- ent admixture of states, surement on the distribution of events effectively accepted by the experimental apparatus. 90 1 1 1, 0 ◦ 1, 1 1, 1 , (24) To better explain the first problem, let us consider the HX | ⟩ −−→ √2| + ⟩−√2| − ⟩ and CS frames as the experimental and natural frames, re- while a natural “transverse” polarization would originate spectively. We start by calculating the angle between the po- from the statistical superposition of uncorrelated 1, 1 larization axes of the CS and HX frames as a function of the | + ⟩ and 1, 1 states. The two physically very different cases quarkonium momentum. The beam momenta in the “labora- | − ⟩ of a natural transverse polarization observed in the nat- tory” frame (centre of mass of the colliding particles), writ- ural frame, shown in Fig. 5(a), and a natural longitudi- ten in longitudinal and transverse components with respect nal polarization observed in a rotated frame, shown in to the quarkonium direction, are P 1 P 2 P cos Θı = − = ˆ∥ + Fig. 5(d), are experimentally indistinguishable when the az- P sin Θı ,whereP is their modulus and Θ is the angle imuthal anisotropy parameter is integrated out. These ex- ˆ⊥ formed by the quarkonium momentum with respect to the amples show that a measurement (or theoretical calculation) beam axis, defined in terms of the quarkonium momentum consisting only in the determination of the polar parameter p as cos Θ pL/p.Whenboostedtothequarkoniumrest λϑ in one frame contains an ambiguity which prevents fun- = damental (model-independent) interpretations of the results. frame, the two vectors become (neglecting the masses of the colliding particles) P 1′ (γ P cos Θ βγP)ı P sin Θı The polarization is only fully determined when both the po- = − ˆ∥ + ˆ⊥ lar and the azimuthal components of the decay distribution and P 2′ ( γ P cos Θ βγP)ı P sin Θ ı ,whereγ = − − ˆ∥ − ˆ⊥ = are known, or when the distribution is analyzed in at least E/m is the Lorentz factor of the quarkonium state, and two geometrically complementary frames. β p/E 1 1/γ 2.Theunitvectorsindicatingthez = = − ! 16

The rest of the thesis is organized as following:

Chapter 2 introduces Relativistic Heavy Ion Collider (RHIC), the physics tasks of Solenoid

Tracker at RHIC (STAR) experiments, and main detectors used for triggering, track recon- struction, and particle identification.

Chapter 3 explains how the J/ψ signals are reconstructed, including: how to identify elec- trons, how to calculate the invariant mass spectra, remove the combinatorial background and get the signal of J/ψ particle.

Chapter 4 describes how to calculate likelihood based on dataset and efficiency of detectors, locate the optimum point of likelihood and exam the property of estimator via toy Monte Carlo simulation.

Chapter 5 presents the systematic uncertainties coming from different sources, and how to combined them together as overall systematic uncertainty of our measurements.

Chapter 6 draws the conclusion based on empirical measurements results.

Chapter 7 summarizes the J/ψ polarization using STAR 2012 dataset and provides a vista on the J/ψ polarization measurements from p+Au and Au+Au collisions system from STAR

2011 and 2015 datasets. CHAPTER 2

STAR EXPERIMENT

2.1 Relativistic Heavy Ion Collider (RHIC)

The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) is the first machine in the world capable of colliding heavy ions [36]. As shown in Figure 8, the detectors locate in 6 o’clock and 8 o’clock are STAR and PHENIX respectively. The colliding atoms are ionized and striped of outside electrons in the Tandem Van de Graaff, accelerated at Alternating Gradient Synchrotron (AGS), then injected into the RHIC rings and accelerated to desire energy [37]. The superconducting magnets along the ring deflect and focus the beam. The beams of protons and heavier nuclei are accelerated by oscillating electrical field to nearly the speed of light. By convention, one beam is called the ”blue” beam

(clockwise), while the other is called ”yellow” beam (anticlockwise) [37]. When two beams of ions colliding with each other, the constituent quarks and gluons will undergo a break-up and regroup-up. This process provides an important window for us to gain insight into many important physics mechanisms. The measurements conducted at RHIC help to develop cutting edge physics theory and put influential ones in test [38]. With the capability of high precision track reconstruction, momentum measurement, and particle identification at the mid-rapidity region, the STAR experiment is designed for multiple important tasks including the property of quark-gluon plasma (QGP) and proton spin physics [39].

17 18

Figure 8: The complex structure of Relativistic Heavy Ion Collider at Brookhaven National

Laboratory [36].

QGP, also known as quark-gluon soup, is a state of nuclear matter which exists at ex- tremely condition [38]. QGP consists of asymptotically free strong-interacting quarks and gluons. Through collision of Au nuclei, the QGP is generated in the overlapping region with an almond shape. The dynamical evolution of the QGP can be described by hydrodynamics, which can successfully describe experimental results on azimuthal anisotropy of final state par- ticles. The QGP will cool down to form hadrons shortly after its generation [40]. Such a phase 19 transition is illustrated in the phase diagram of nuclear matter in Figure 10, in analogy to the phase diagram of water. Understanding the properties and phase transition of QGP is essential for us to understand the evolution of the early universe [38].

Figure 9: Through collision of Au nuclei, the QGP is generated in the overlapping region with an almond shape. phenomena, 3) providing new opportunities to probe the spectrum of fluctuations in high gluon density matter, and 4) mapping the transition to a classical description of gluonic matter at high density. It is suspected that the rapid formation of almost perfectly liquid hot QCD matter in heavy-ion collisions may be related to the emergence of universal characteristics in high-density gluon matter at zero temperature that is predicted to dominate the low-x component of the nuclear wave function when probed at high energy. To explore this connection, precision measurements of the nuclear wave function at an EIC will be required to complement nuclear collision experiments with small and large nuclei.

3.2 Mapping the QCD phase diagram

When the first protons and neutrons and pions formed in the microseconds-old universe, and when they form in heavy-ion collisions at the highest RHIC energies and at the LHC, they condense out of liquid quark-gluon plasma consisting of almost as much antimatter as matter. Lattice calculations [71–73] show that QCD predicts that, in such an environment, this condensation occurs smoothly as a function of decreasing temperature, with many thermodynamic properties changing dramatically but continuously within a narrow temperature range around the transition temperature T [145 MeV, 163 MeV] [33, 73], 20 c 2 referred to as the crossover region of the phase diagram of QCD, see Fig. 4. In contrast, quark-gluon

Figure 4: A sketch illustrating the experi- mental and theoretical exploration of the QCD phase diagram. Although experi- ments at highest energies and smallest baryon chemical potential are known to cross from a QGP phase to a hadron gas phase through a smooth crossover, lower energy collisions can access higher baryon chemical potentials where a first order phase transition line is thought to exist.

plasma dopedFigure 10: with Phase a diagram sucient and critical excess point of of QGP quarks [41]. over anti-quarks may instead experience a sharp first order phase transition as it cools, with bubbles of quark-gluon plasma and bubbles of hadrons coexisting at a well-defined critical temperature, much as bubbles of steam and liquid water coexist in a boiling pot. The point where the doping of matter over antimatter (parametrized by the net baryon number chemical potential µB) becomes large enough to instigate a first order phase transition is referred to as the QCD critical point. It is not yet known whether QCD has a critical point [74–78], nor where in its phase diagram it might lie. Lattice calculations become more dicult or more indirect or both with increasing µB and, although new methods introduced within the past decade have provided some hints [75,77,79], at present only experimental measurements can answer these questions definitively. The theoretical calculations are advancing, however, with new methods and advances in computational power both anticipated. The phase diagram of QCD, with our current knowledge schematically shown in Fig. 4, is the only

16 21

The recent understanding of proton’s spin suggests it is made of three components, quarks’ spin, gluons’ spin and quark and gluon orbital angular momenta [42]. To investigate the con- tribution from each component, it is required to know the prior knowledge of momentum distribution of the quarks and gluons of proton, which is studied by parton model [43]. To understand the nature of proton’s spin thereby the nature of fundamental particles’ spin, as one of important goals of RHIC, the experiments at RHIC were also designed to conduct the experiments of colliding spin-aligned proton beams.

The installation of Siberian Snake [44] makes it possible for spin physics experiments. The

Siberian Snake is an arranged group of dipole magnets. The anomalous magnetic moment of the proton is 2.792µ , where the nuclear magnetic moment µ = e¯h [45]. When protons enter N N 2mp the Siberian Snake, the magnetic field will exert a torque on the protons due to the interaction between the magnetic field and protons’ intrinsic magnetic moment [46]. It will rotate the orientation of protons’ spin to the desired direction.

2.2 Solenoidal Tracker At RHIC (STAR)

For high energy physics experiments at colliders, detectors usually have an onion-like struc- ture. This kind of hierarchical structure gives detector the capability to distinguish different types of particles and measure their physics quantities such as momentum, energy and so forth.

The center-of-mass energy √s is calculated based on the total energy and momentum of the two colliding particles [2], 2 2 2 2 s = ( Ei) − ( p~i) i=1 i=1 X X 22

The scattered partons are observed as jets or leading hadrons. The momenta of jets or hadrons can be precisely measured and applied to calculate differential cross section [2].

2.2.1 Time Projection Chamber

Time Projection Chamber (TPC) shown in Figure 11 is the main detector at STAR to reconstruct the trajectories of charged particles and help identify particles. The length of the

TPC is 420cm, the outer diameter of the drift volume is 400cm and the inner diameter of the drift volume is 100cm. The TPC is placed inside a solenoidal magnet that provides a uniform homogeneous and non-divergent magnetic field of 0.5 Tesla magnitude. The gas inside TPC is called P10 gas (90% argon, 10% methane by volume) [47].

When energetic charged particles go through the P10 gas, they ionize nearby gas atoms and ionized particles drift in an exerted electrical field of the TPC to the side. Multi-wire proportional chambers (MWPC) are mounted on both sides of the TPC. High voltage is exerted on each wire of the MWPC. When an ionized particle passes by the wire, it causes avalanche and generates a current signal along the wire. The positive ions created in the avalanche induce a temporary image charge on the pads which disappears as the ions move away from the anode wire. The image charge is measured by a preamplifier/shaper/waveform digitizer system. The induced charge from an avalanche is shared over several adjacent pads, so the original track position can be reconstructed to a small fraction of a pad width. There are a total of 136,608 pads in the readout system [48]. Knowing the measured drift velocity of ionized particles, one can calculate the position of the charged particle along the z-axis. The trajectory of the charged particle is then reconstructed by lining up the hits from the TPC. With the magnitude and 23 direction of the magnetic field is known, the momentum and charge sign of energetic charged particles can be determined by calculating the Lorentz force exerted on the particle.

η=-1 η=0

η=1 BEMC

Blue BBC Yellow

TPC

West

East

Figure 11: The complex of Time Projection Chamber [49]. 24

Meanwhile, the charged particle loses its kinematic energy as it ionizes the gas atoms nearby.

The Bethe-Bloch equation calculates the energy loss per unit length traversed [50].

2 2 2 dE Z 1 1 2mec β γ Wmax δ(βγ) − < >= Kz2 [ ln − β2 − ] (2.1) dx A β2 2 I2 2

2 2 where K = 4πNAremec ; A atomic mass of absorber; ze charge of incident particle; Z atomic number of absorber; I mean excitation energy (eV); δ(βγ) density effect correction to ionization energy loss; Tmax is the maximum kinetic energy which can be imparted to a free electron in a single collision [2, 18]. STAR experiments use Bichsel function instead [51].

The energy loss is normalized by calculating nσe with Equation 2.2

log(dE/dx) − log(dE/dx) nσ = measured theoretical (2.2) e σ(log(dE/dx))

where log(dE/dx)theoretical is the theoretical log(dE/dx) calculated by Bethe-Bloch equation; log(dE/dx)measured is calculated by how much extent the trajectory is bend via ionization process and σ(log(dE/dx)) is the variation of log(dE/dx) from the measurements. As shown in

Figure 12, measurements of log(dE/dx) distinguish electron tracks from hadron tracks based on the different characteristic of energy loss. 676 M. Anderson et al. / Nuclear Instruments and Methods in Physics Research A 499 (2003) 659–678

À shows the pT resolution for p and anti-protons in The measured dE=dx resolution depends on the STAR. The figure shows two regimes: at low gas gain which itself depends on the pressure in the momentum, where multiple Coulomb scattering TPC. Since the TPC is kept at a constant 2 mbar dominates (i.e., pTo400 MeV=c for pions, and above atmospheric pressure, the TPC pressure pTo800 MeV=c for anti-protons), and at higher varies with time. We monitor the gas gain with a momentum where the momentum resolution is wire chamber that operates in the TPC gas return limited by the strength of the magnet field and the line. It measures the gain from an 55Fe source. It TPC spatial resolution. The best relative momen- will be used to calibrate the 2001 data, but for the tum resolution falls between these two extremes 2000 run, this chamber was not installed and so we and it is 2% for pions. monitored the gain by averaging the signal for tracks over the entire volume of the detector and 5.8. Particle identification using dE=dx we have done a relative calibration on each sector based on the global average. Local gas gain Energy lost in the TPC gas is a valuable tool variations are calibrated by calculating the average for identifying particle species. It works especially signal measured on one row of pads on the pad well for low momentum particles but as the plane and assuming that all pad-rows measure the particle energy rises, the energy loss becomes less same signal. The correction is done on the pad-row mass dependent and it is hard to separate particles level because the anode wires lie on top of, and run with velocities v > 0:7c: STAR was designed to be the full length of, the pad rows. able to separate pions and protons up to The readout electronics also introduce uncer- 1:2 GeV=c: This requires a relative dE=dx re- tainties in the dE=dx signals. There are small solution of 7%. The challenge, then, is to calibrate variations between pads, and groups of pads, due the TPC and understand the signal and gain to the different response of each readout board. variations well enough to be able to achieve These variations are monitored by pulsing the this goal. ground plane of the anode and pad plane read-out

25

Fig. 11. The energy loss distributionFigure 12: for The primary characteristics and secondary of energy particles loss distribution in the STAR for different TPC as particles a function [47]. of the pT of the primary particle. The magnetic field was 0:25 T: 26 M. Anderson et al. / Nuclear Instruments and Methods in Physics Research A 499 (2003) 659–678 661

The readout system is based on Multi-Wire Table 1 Proportional Chambers (MWPC) with readout Basic parameters for the STAR TPC and its associated hardware pads. The drifting electrons avalanche in the high fields at the 20 mm anode wires providing an Item Dimension Comment amplification of 1000–3000. The positive ions Length of the TPC 420 cm Two halves, created in the avalanche induce a temporary image 210 cm long charge on the pads which disappears as the ions Outer diameter of 400 cm 200 cm radius move away from the anode wire. The image charge the drift volume is measured by a preamplifier/shaper/waveform Inner diameter of 100 cm 50 cm radius the drift volume digitizer system. The induced charge from an Distance: cathode 209:3 cm Each side avalanche is shared over several adjacent pads, to ground plane so the original track position can be reconstructed Cathode 400 cm diameter At the center of the to a small fraction of a pad width. There are a total TPC of 136,608 pads in the readout system. Cathode potential 28 kV Typical Drift gas P10 10% methane, The TPC is filled with P10 gas (10% methane, 90% argon 90% argon) regulated at 2 mbar above atmo- Pressure Atmospheric Regulated at spheric pressure [7]. This gas has long been used in þ2 mbar 2 mbar above atm. TPCs. Its primary attribute is a fast drift velocity Drift velocity 5:45 cm=ms Typical which peaks at a low electric field. Operating on Transverse 230 mm= cm 140 V=cm & 0:5T diffusion (s) pffiffiffiffiffiffiffi the peak of the velocity curve makes the drift Longitudinal 360 mm= cm 140 V=cm velocity stable and insensitive to small variations diffusion (s) pffiffiffiffiffiffiffi in temperature and pressure. Low voltage greatly Number of anode 24 12 per end simplifies the field cage design. sectors The design and specification strategy for the Number of pads 136 608 Signal to noise 20:1 TPC have been guided by the limits of the gas and ratio the financial limits on size. Diffusion of the Electronics 180 ns FWHM drifting electrons and their limited number defines shaping time the position resolution. Ionization fluctuations and Signal dynamic 10 bits range finite track length limit the dE=dx particle Sampling rate 9:4 MHz identification. The design specifications were ad- Sampling depth 512 time buckets 380 time buckets justed accordingly to limit cost and complexity typical without seriously compromising the potential for Magnetic field 0; 70:25 T; Solenoidal tracking precision and particle identification. 70:5T Table 1 lists some basic parameters for the STAR TPC. The measured TPC performance has generally agreed with standard codes such as membraneFigure is 13: operated Main parameters at 28 kV for: theThe STAR end TPC caps [47]. are 40 MAGBOLTZ [8] and GARFIELD [9]. Only for at ground. The field cage cylinders provide a series the most detailed studies has it been necessary to of equi-potential rings that divide the space make custom measurements of the electrostatic or between the central membrane and the anode gas parameters (e.g., the drift velocity in the gas). planes into 182 equally spaced segments. One ring at the center is common to both ends. The central membrane is attached to this ring. The rings are 2. Cathode and field cage biased by resistor chains of 183 precision 2 MO resistors which provide a uniform gradient be- The uniform electric field in the TPC is defined tween the central membrane and the grounded end by establishing the correct boundary conditions caps. with the parallel disks of the CM, the end caps, The CM cathode, a disk with a central hole to and the concentric field cage cylinders. The central pass the Inner Field Cage (IFC), is made of 70 mm 27

2.2.2 Vertex Position Detector and Time Of Flight minum outer cylinder outside by several layers of Kapton tape. The output coaxial connector shield is isolated from the detector housing and the high As shown in Figure 14, the vertex position detector (VPD) exists as two identical detector voltage ground but is indirectly connected via a 1 kΩ resistor. This prevents assemblies,the (inductive) one on the shield east and of the one on coaxial the west signal of STAR. cable Each from of the forming nineteen an channels undesir- used able resonant circuit with the (capacitive) electrostatic shield and detector inhousing each assembly while is maintaining composed ofthe a Pb high converter voltage followed ground by a return fast, plastic path. scintillator which Each VPD assembly consists of two rings of readout detectors and is is read out by a photomultiplier tube (PMT). The signals from the nineteen channels in each mounted to the I-beam that supports the STAR beam pipe. A front view assemblyof one of are the digitized VPD independently assemblies by is showntwo different in Figure sets of 2. electronics The outer [52]. diameterThe start time of the beam pipe at this distance is five inches. An assembly exists as two ofsemi-annular the collision and “clam-shells” the position of that vertex enclose can be calculated the beam based pipe. on the These average are time bo oflted two VPDstogether assemblies. and are held in place by Delrin support blocks which attach toa horizontal mount plate which is clamped to the beam pipe support I-beam. The beam pipe and I-beam are at a diffVPD and TOF erent (dirty) electrical groundthan the experiment, so the Delrin support blocks both hold the assembly in place and electrically isolate it.

front and back plates The VPD exists as two iden9cal beam pipe detector assemblies, one on the east and one on the west of detector STAR. Each of the nineteen

support block detectors used in each beam pipe support assembly is composed of a Pb clamp converter followed by a fast, mount plate plas9c scin9llator which is read out by a photomul9plier Figure 2: On the left is a schematic front view of a VPD assembly, and on the right is a photographOn the of the lei twoFigure VPD is 14: assemblies. a The schema9c complex A one of VPDfoot front long detectors ruler view [52]. is shown of for a scale VPD on the tube(PMT). The signals from right.assembly; and on the right is a photograph of the two the nineteen detectors in each VPD assemblies. A one foot long ruler is shown for assembly are digi9zed Thescale on the right. two assemblies are mounted symmetrically with respect to the center independently by two different of STAR at a distance of 5.7 m. The nineteen detectors in each assembly sets of electronics. subtend approximately half of the solid angle in the pseudo-rapidity range Zvtx = c(Teast −Twest ) / 2 of 4.24 η 5.1. When viewed from the rear and looking towards the center ≤ ≤ of STAR, the detectors are numbered 1-10(11-19) counter-clockwise starting Tstart = (Teast +Twest ) / 2 − L / c 5 11 28

The Time of Flight (TOF) detector covers 2π in azimuth angle and |η| < 1 in pseudo-rapidity

[53]. It extends the limit of particle identification capability to a higher momentum range than

that of the TPC. As the main component of TOF, multi-gap resistive plate chamber (MRPC) is

designed to achieve good time and reasonable position resolutions, where signals are generated

when charged particles travel through the gas and cause the avalanche [54]. Figure 15 depicts

the schematic structure of MRPC [54]. TRTFPooa – Proposal TOF STAR

Honey comb length = 20.8 cm electrode length = 20.2 cm pad width = 3.15cm pad interval = 0.3cm honey comb thickness = 4mm (not shown: mylar 0.35mm)

outer glass thickness = 1.1mm inner glass thickness = 0.54mm gas gap = 220micron Ma PC Board thickness = 1.5 mm y 4 2004 24,

inner glass length = 20.0 cm outer glass length = 20.6 cm PC board length = 21.0 cm 56

honey comb PC board pad electrode (graphite) glass

0 0.5 1.3 7.4 8.4 8.9 9.4 position (cm) 0.8 1.1 8.6 1.0 Figure 23: Two side views of the structure of an MRPC module. The upper(lower) view shows the long(short) edge. The

Figuretwo views 15: are Two not shown side at views the same of thescale. structure of a MRPC module. The upper (lower) view shows 56

the long (short) edge. The two views are not shown at the same scale [53]. 29

Together with the start time of collision recorded by the VPD and the time of arrival recorded by the TOF, the time interval from the start time and the time the particle reaches the TOF can be calculated accordingly. The vertex position of collision of beams is calculated by:

z = c(Teast − Twest)/2 (2.3)

where, Teast and Twest are the time recorded by the VPD, c is the speed of light. The start time is calculated by:

Tstart = (Teast + Twest)/2 − L/c where, the distance L = 5.7m. 1 c∆T = (2.4) β L STAR TOF Proposal – May 24, 2004 101

Distinguishing flow effects from other dynamical effects by comparison of the • identified particle transverse momentum spectra for p+p, d+Au, and Au+Au collisions; distinguishing flow effects arising from the partonic versus the hadronic30 stage of thecollision.

3 β 2 10 1/

1.8 STAR Preliminary MRPC TOF (d+Au) 1.6 2 p 10 1.4 K

1.2 π

10 1 e

0.8

0.6 1 0 0.5 1 1.5 2 2.5 3 3.5 p (GeV/c) Figure 46: Inverse velocity vs momentum from 2.6 million Figure 16: Inverse velocity vs momentum from 2.6 million TOFr+pVPD-triggered events in TOFr+pVPD-triggered events in d+Au collisions. d+Au collisions [53].

x10 -5 0.7 STAR Preliminary 0.6 STAR Preliminary dE/dx vs p from TPC 2 10 ± 3 MRPC TOF PID (d+Au) 0.5 π TPC dE/dx (GeV/cm) 10 1.0

0.3 2 10 0.2 1 0.7 x10-5 ± dE/dx after MRPC TOF PID Cut 2 e 10 0.6 10

0.5 10

0.4 e± 1 0.3 π± 1

0.2 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 p (GeV/c) TPC dE/dx (KeV/cm) Figure 47: Left top: TPCdE/dx vs the momentum in d+Au collisions. Left bottom: TPC dE/dx vs the momentumafterTOFrPIDselection of 1 β < 0.03. Clean electron identification is achieved. Right: dE/dx from| − TPC| after TOFr PID selection (left bottom panel) for 1.0

The analyses of the Run-3 data are ongoing. Fig. 46 shows the particle identifi- cation capabilities as 1/β from time of flight measured by TOFr versus the particle

101 STAR TOF Proposal – May 24, 2004 101

Distinguishing flow effects from other dynamical effects by comparison of the • identified particle transverse momentum spectra for p+p, d+Au, and Au+Au collisions; distinguishing flow effects arising from the partonic versus the hadronic stage of thecollision.

3 β 2 10 1/

1.8 STAR Preliminary MRPC TOF (d+Au) 1.6 2 p 10 1.4 K

1.2 π

10 1 e

0.8

0.6 1 0 0.5 1 1.5 2 2.5 3 3.5 p (GeV/c) Figure 46: Inverse velocity vs momentum from 2.6 million TOFr+pVPD-triggered events in d+Au collisions. 31

x10 -5 0.7 STAR Preliminary 0.6 STAR Preliminary dE/dx vs p from TPC 2 10 ± 3 MRPC TOF PID (d+Au) 0.5 π TPC dE/dx (GeV/cm) 10 1.0

0.3 2 10 0.2 1 0.7 x10-5 ± dE/dx after MRPC TOF PID Cut 2 e 10 0.6 10

0.5 10

0.4 e± 1 0.3 π± 1

0.2 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 p (GeV/c) TPC dE/dx (KeV/cm) Figure 47: Left top: TPCdE/dx vs the momentum in d+Au collisions. Figure 17: Left top: TPC dE/dx vs the momentum in d+Au collisions. Left bottom: TPC Left bottom: TPC dE/dx vs the momentumafterTOFrPIDselection of dE/dx1 vsβ the< momentum0.03. Clean after TOFr electron PID identificationselection of |1β| < is 0.03 achieved.. Clean electron Right: identification dE/dx from| − TPC| after TOFr PID selection (left bottom panel) for 1.0

p < 1.5GeV/c [53]. The analyses of the Run-3 data are ongoing. Fig. 46 shows the particle identifi- cation capabilities as 1/β from time of flight measured by TOFr versus the particle

101 2.2.3 Barrel ElectroMagnetic Calorimeter

In STAR, a sampling calorimeter uses lead and plastic scintillator for the detection of

electromagnetic energy [55]. The calorimeter is constructed from a number of relatively small

modules, which allows the calorimeter to cover the necessary area and be flexible to complex

geometry as well [55]. 32

The geometry and coverage of the Barrel ElectroMagnetic Calorimeter (BEMC) is shown in Figure 18. It covers the pseudo-rapidity range |η| < 1 and full azimuthal angle. The STAR physics program requires that the calorimeter permit the reconstruction of the π0’s and isolated photons at relatively high pT 25 − 30GeV/c and be capable of identifying single electrons and ≈ pairs in intense hadron backgrounds from heavy vector mesons and W and Z decays [55].

In STAR, a sampling calorimeter uses lead and plastic scintillator for the detection of electromagnetic energy [55]. The calorimeter is constructed from a number of relatively small modules, which allows the calorimeter to cover the necessary area and be flexible to complex geometry as well [55].

When energetic charged particles go through a scintillator, scintillator molecules absorb energy and go to excited states. The molecules will release photons and come back to the ground state. The photon signal is led to photomultiplier tubes via optical fibers. After calibration, one can measure the energy that the particle deposits in the BEMC from the amplitude of the signal from the photomultiplier readout [55]. 33

!=-1 !=0 !=1

263.0 cm 223.5 cm

Figure 18: The geometry and coverage of BEMC [49].

In contrast to that hadrons only deposit a fraction of its energy, as shown in Figure 19, electrons deposit all the kinematic energy. By measuring the ratio of momentum over the energy, it help us distinguish electrons from hadrons. Main detectors used in the analysis

Barrel Electro-Magne/c Calorimeter

34

Time-Of-Flight Detector

Time Projec/on Chamber

E/pc Figure 19: The E/pc distribution 14 CHAPTER 3

J/ψ SIGNALS RECONSTRUCTION

3.1 J/ψ reconstruction

To measure the polarization parameters λθ, λφ and λθφ, one need have both the precise measurement of raw yield distribution of polar and azimuthal angular distribution of positron momentum with regard to J/ψ momentum and the knowledge of detectors’ efficiency. There- after, the J/ψ polarization parameters measurement can be decomposed into three main steps

: 1) event reconstruction, 2) spatial distribution of probability density function measurement including central value and statistical uncertainty of polarization parameters, 3)systematic un- certainty measurements with regard to different sources.

The event reconstruction refers to means that reconstruct physics event of interest happened during the collision. In my study, the event of J/ψ decay to dielectron are of significance.

First of all, the observable quantity like electron’s momentum, positron’s momentum, J/ψ’s momentum and so on should be measured precisely in order to reconstruct the event of J/ψ’s decay. Meanwhile, the combinatorial background that the event mistakenly taken as J/ψ decay should also be calculated as well. The technique called ”unlike sign minus like sign” is applied to remove combinatorial background in this study.

The spatial distribution of probability density function measurement involves the likelihood estimation and minimization. Based on maximum likelihood estimation, the likelihood is cal-

35 36 culated according to the definition in combination with efficiency of detectors. The problem of finding the probability density function match the empirical observation best is transformed to the problem of minimization. The ROOT TMinuit package is applied to search for the minimum point of the likelihood.

3.1.1 Datasets

Three STAR datasets in p+p collisions at √s = 200 GeV collected in 2012 are used in this study. The present measurement of J/ψ polarization focuses on the process that J/ψ decays into an electron and a positron. Hence, the capability to identify electron track is the most important part to reconstruct J/ψ mesons.

As listed in Table I, the track quality cuts indicates whether a track is a ”good” track. First of all, the momentum of a track should be great than 0.2 GeV/c because the magnetic field will bend the trajectory of charged particle, if the momentum of the charged particle is too low, it will form a closed circular track inside TPC, which would be rejected by STAR track reconstruction algorithm. The cut on the distance of the closed approach to the primary vertex

(DCA) is set to be less than 1cm due to the short lifetime of J/ψ, which requests the electron tracks that decay from J/ψ particle should be within 1cm away from the vertex position. The number of TPC hits used in track reconstruction (nHitsFit) is required to be larger than 20.

NHitsdEdx indicates how many hits point are used to measure energy loss (dE/dx) of particle during traversal per length [56]. The ratio of nHitsFit/nHitsMax (nHitsRatio), where nHitsMax is the maximum number of points available in track fit, is required to be larger than 0.52 and less than 1.02. 37

As listed in Table II, for electron identification, the cut for the normalized energy loss is

−1.9 < nσe < 3. The BEMC and TOF are also used for the electron identification. The BEMC has a better capability to identify electrons with relatively high momentum (pT > 1GeV/c) . It requests the momentum over energy of the track satisfy that 0.3 < p/E < 1.5. Complementarily, the TOF has a better capability to identify electrons with relatively low momentum. It requests

TOF TOF the TOF matching Ylocal satisfy that |Ylocal| < 2cm and |1/β − 1| < 0.03.

As listed in Table III, triggers define under what condition that the detectors start to record information of interests. The datasets used in this analysis are triggered by minimum- bias (MB), BHT0 and BHT2 trigger respectively. For VPDMB-nobsmd triggered events, it is triggered by a coincidence signal from the VPDs on both sides. The integrated luminosity of VPDMB-nobsmd is 0.029 pb−1 with 734.853 million events. For BHT0*BBCMB*TOF0 triggered events, in combination with the coincidence signal from BBC and TOF, it requires additionally an electron fire high tower (HT) trigger with DSMADC readout greater than 11 and ADC0 readout greater than 180, which corresponds to an energy deposition in the BEMC that is larger than 2.6GeV. The integrated luminosity of BHT0*BBCMB*TOF0 is 1.371 pb−1 with 39.164 million events. Similarly, for BHT2*BBCMB triggered events, in combination with coincidence signal from BBC, it requires an electron fire HT trigger with DSMADC readout greater than 18 and ADC0 readout greater than 300, which corresponds to an energy deposition in the BEMC larger than 4.3GeV. The integrated luminosity of BHT2*BBCMB is 23.550 pb−1 with 36.062 million events. The electron track is called triggered electron. 38

TABLE I: Track quality cuts

0.2 < pT < 30 GeV/c

-1 η 1 ≤ ≤ DCA < 1 cm

nHitsFit 20 ≥ nHitsdEdx 11 ≥ nHitsRatio > 0.52

TABLE II: Electron identification cuts

BEMC electron -1.9 < nσe < 3, pT > 1 GeV/c , 0.3 < p/E < 1.5

TOF TOF electron -1.9 < nσe < 3, |Ylocal| < 2 cm, |1/β − 1| < 0.03

TABLE III: Triggered electron cuts

BHT0 pT >2.6 GeV/c, DSMADC>11, ADC0>180

BHT2 pT >4.3 GeV/c, DSMADC>18, ADC0>300 39

3.1.2 J/ψ reconstruction from data

The process that J/ψ decay to dielectron is a two-body decay. To reconstruct J/ψ meson via its decay products electron and positron, the invariant mass is calculated from the four momenta of the electron and positron, namely [2]

2 µ 2 2 minv = p pµ = E − p (3.1)

Two electrons with opposite charge sign in the same event are paired and the invariant mass is calculated to reconstruct J/ψ candidates. These dielectron pairs are called unlike-sign pairs

, which receives contributions from not only electron-positron pairs from J/psi decay, but also those randomly paired ones, which is referred to as combinatorial background. To remove the combinatorial background, two electrons or positrons with the same sign in the same event are paired, which are called like-sign pairs. By subtracting like-sign distribution from unlike-sign distribution, I can remove the contribution from the combinatorial background and extract the signal distribution. This technique to remove combinatorial background is called ”unlike sign minus like sign”. The invariant mass spectra of dielectron pairs in different pT bins are present in Figure 20. 40

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 2 2 2 2 2

350 0 < p < 2 GeV/c 350 2 < p < 4 GeV/c 350 4 < p < 6 GeV/c 350 6 < p < 8 GeV/c 350 8 < p < 14 GeV/c Unlike T 2 T 2 T 2 T 2 T 2 3.0 < Me+e- < 3.15 GeV/c 3.0 < Me+e- < 3.15 GeV/c 3.0 < Me+e- < 3.15 GeV/c 3.0 < Me+e- < 3.15 GeV/c 3.0 < Me+e- < 3.15 GeV/c Like 300 S(Unlike-Like) = 131.0 300 S(Unlike-Like) = 476.0 300 S(Unlike-Like) = 442.0 300 S(Unlike-Like) = 330.0 300 S(Unlike-Like) = 91.0 Unlike-Like B(Like) = 11.0 B(Like) = 20.0 B(Like) = 39.0 B(Like) = 19.0 B(Like) = 5.0 Fit 250counts/25 MeV/c S/B = 12.2 250counts/25 MeV/c S/B = 24.4 250counts/25 MeV/c S/B = 12.1 250counts/25 MeV/c S/B = 18.4 250counts/25 MeV/c S/B = 21.2

200 200 200 200 200

150 150 150 150 150

100 100 100 100 100

50 50 50 50 50

0 0 0 0 0 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.52.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 2 2 2 2 2 Mon Mar 5 21:37:07 2018 me+e- GeV/c me+e- GeV/c me+e- GeV/c me+e- GeV/c me+e- GeV/c

Figure 20: Invariant mass spectra of dielectron pairs in different pT bins (from left to right: pT

= 0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The black markers (blue filled histograms) are the spectra from unlike-sign (like-sign) charge pairs, while the red marker represent those obtained by subtracting the like-sign spectra from the unlike-sign ones. The latter are fit to Double Crystal functions represented by the red solid curves.

3.1.3 J/ψ decay products cosθ and φ distribution

Using the same method ”unlike sign minus like sign”, I calculate the 2-dimensional for positron from J/ψ candidate decay distribution as a function of (cosθ, φ). The results are shown in Figure 21. 41

−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 1 1 1 1 1 θ θ θ θ θ

0.8 0.8 0.8 0.8 0.8

cos 0.6 0.6cos 0.6cos 0.6cos 0.6cos

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0 0 0 0

−0.2 −0.2 −0.2 −0.2 −0.2

−0.4 −0.4 −0.4 −0.4 −0.4

−0.6 −0.6 −0.6 −0.6 −0.6

−0.8 −0.8 −0.8 −0.8 −0.8 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −11 −11 −11 −11 −11 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 θ φθ φθ φθ φθ φ 0.8 0.8 0.8 0.8 0.8

0.6cos 0.6cos 0.6cos 0.6cos 0.6cos

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0 0 0 0

−0.2 −0.2 −0.2 −0.2 −0.2

−0.4 −0.4 −0.4 −0.4 −0.4

−0.6 −0.6 −0.6 −0.6 −0.6

−0.8 −0.8 −0.8 −0.8 −0.8

− − − − − 1 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 3 Fri Mar 2 16:11:23 2018 φ φ φ φ φ

Figure 21: 2D distribution of J/ψ candidates from data as a function of (cosθ, φ) plane in different pT bins (from left to right: pT =0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The top (bottom) row shows the distributions in the HX (CS) frame.

3.1.4 J/ψ Reconstruction Efficiency from Detector Simulation

The detectors response to track of electrons differently with regard to orientation, momen- tum, energy of the trajectory of electrons and acceptance and trigger of detectors. Hence,the knowledge of detectors’ response is essential for us to interpret datasets collected from the experiment, which is called efficiency of detectors. The GEANT detector simulation and em- bedding technique are used to calculate the efficiency of detectors [57]. The simulated tracks are embedded into the real data tracks information. To correctly represent the detectors’ response, the simulated tracks are weighted according to the real data distribution by a weight function. 42

The Levy weight function is used to weight the pT distribution in the embedding. The Levy function has the following form with parameters A, B and C and the parameters of Levy weight function is set by fitting to pT distribution in dataset.

p 2 2 B Levy(pT ) = √2ApT /(1 + ( pT + 3.0969 − 1.865)/BC) (3.2)

where, A = 339.805, B = 8.96953 and C = 0.210824.

The rapidity distribution is also weighted [58]. The rapidity weight function is:

0.5y2 w = exp(− ) (3.3) y 1.4162

The value of the fitting parameter is: A = 1.42 0.04. ± In pairing procedure, I select EMC electron first, then seek for another candidate from TOF matching. In this way, the TOF efficiency has a dependence on EMC matching efficiency. Then,

I need to calculate the TOF efficiency condition with following formula:

P(TOF) = P(TOF|EMC)P(EMC) + P(TOF|!EMC)P(!EMC) (3.4)

P(TOF)(1 − P(EMC)/r) p(TOF|!EMC) = (3.5) P(!EMC)

−(p ∗p )p2 The TOF matching efficiency is calculated by a function p0e 1 T in 20 bins of |η| < 1. 43

As shown in Figure 22, the overall efficiency of detectors as a function of (cosθ, φ) is calculated by the ratio of Monte Carlo tracks passed all cuts over all Monte Carlo tracks in the embedding.

−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 1 1 1 1 1 θ θ θ θ θ

0.8 0.8 0.8 0.8 0.8

cos 0.6 cos 0.6 cos 0.6 cos 0.6 cos 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0 0 0 0

−0.2 −0.2 −0.2 −0.2 −0.2

−0.4 −0.4 −0.4 −0.4 −0.4

−0.6 −0.6 −0.6 −0.6 −0.6

−0.8 −0.8 −0.8 −0.8 −0.8 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 − − − − − 11 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 31 −3 −2 −1 0 1 2 3 θ θ φ θ φ θ φ θ φ φ 0.8 0.8 0.8 0.8 0.8

cos 0.6 cos 0.6 cos 0.6 cos 0.6 0.6cos

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0 0 0 0

−0.2 −0.2 −0.2 −0.2 −0.2

−0.4 −0.4 −0.4 −0.4 −0.4

−0.6 −0.6 −0.6 −0.6 −0.6

−0.8 −0.8 −0.8 −0.8 −0.8

−1 −1 −1 −1 −1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Mon Mar 5 13:43:20 2018 φ φ φ φ φ

Figure 22: J/ψ reconstruction efficiency from embedding as a function of (cosθ, φ) in different pT bins (from left to right: pT =0-2, 2-4, 4-6, 6-8, 8-14 GeV/c). The top(bottom) row shows the efficiencies in the HX (CS) frame.

In the end, the statistical and systematic uncertainty of polarization parameters are mea- sured, which indicates how precise the measurements are. The statistical uncertainty of polar- ization parameters mainly depends on the how many samples are collected in the dataset. And the systematic uncertainty of polarization parameters are measured mostly with regard to the choice of cuts configuration. CHAPTER 4

EXTRACTION OF J/ψ POLARIZATION

4.1 Maximum Likelihood Estimation of J/ψ Polarization Parameters

The probability density function indicates the spatial distribution of positron decay from

J/ψ, can be written as a Fourier series as following.

2 2 f(θ, φ) 1 + λθcos θ + λφsin θcos(2φ) + λθφsin(2θ)cosφ (4.1) ∝ where θ is polar angle and φ is azimuthal angle as shown in Figure 23. Coefficients of the Fourier series represent the contribution of each component term. Our measurements try to determine the values of these coefficients that match the experimental observation best. The results represent the anisotropic characteristics of probability density function of J/ψ polarization.

44 Measurement of J/! polarization in p+p collisions at √s = 200 GeV through the di-muon channel at STAR

Zhen Liu, for the STAR Collaboration University of Science and Technology of China State Key Laboratory of Particle Detection and Electronics

Abstract Quarkonium production mechanism in elementary collisions has not been fully understood. Experimental data on the J/" cross section in p+p collisions can be described relatively well by several models that are currently available on the market. However, these models differ in their predictions for the J/" polarization. Therefore precise measurements of J/" polarization can provide further constraints on the production models. During the RHIC 2015 run, the STAR experiment recorded a large sample of p+p collisions at # = 200 GeV triggered by the Muon Telescope Detector for charmonium studies via the di-muon decay channel. In this poster, we will present the J/" polarization measurement in the helicity and Collins-Soper reference frames utilizing this data set. The polarization parameters %& and %' are extracted from simultaneous fit to 1-dimensional polar and azimuthal angular distributions of decayed () in the J/" transverse momentum range of 0-5 GeV/c in both frames. The results will be compared with similar measurements in higher transverse momentum region as well as with model calculations.

Motivation and Introduction STAR Experiment

QCD factorization: Muon Telescope Detector • Long distance process: no full-QCD description of quarkonium formation, Time Of Flight o Model dependent; o Input from experiments needed.

• J/" polarization can be analyzed via the angular distribution of the decayed positively charged leptons[1], which can be expressed as:

1 9 9 -(/0#1, *) ∝ 8 (1 + %&/0# 1 + %'#:; 1/0#2* + %&'#:;21/0#*) 3 + %& 1 |% | < 1, % ≤ 1 + % & ' 2 & Time Projection Chamber • %& and %' can be extracted from simultaneous fit to 1-dimensional angular distributions, o %&' vanishes in both integrations. 45 J/ψ polarization reference frame Barrel ElectroMagnetic Calorimeter z quarkonium rest frame production plane y + ℓ z ϑ HX z production GJ • Top right: A schematic view of the entire Muon Telescope Detector (MTD) system. MTD covers 45% plane zCS in * and |,| < 0.5. It is used to trigger on and identify muons which emit less Bremsstrahlung b b 1 2 radiation compared to electrons. x y b b φ collision 1 2 Q • Bottom right: A schematic side-view of the Multi-gap Resistive Plate Chambers with long readout centre quarkonium [2] of mass rest strips (LMRPC) used in the MTD design: time resolution ~100 ps and spatial resolution ~1-2 cm . frame frame

Figure• @ 23:- Thepolar definition angle of production, between polar angle momentumθ and azimuthal of angle positiveφ. lepton in J/" rest frame and polarization axis z. • A - corresponding azimuthal angle. Results and Conclusions • Polarization axis z,

ϕ 1

ϕ 1 inv λ J/ψ λ Statistically, the measurement to extract the polarization parameters from the dataset STAR p+p@200 GeV λ 3 STAR p+p@200 GeV o Helicity (HX) frame: along J/" momentum in center-of-mass frame of colliding beams; STAR Preliminary STAR p+p@200 GeV STAR Preliminary STAR Preliminary - - J/ψ→µ+µ |y| < 0.5 J/ψ→µ+µ- |y| < 0.5 J/ψ→µ+µ |y| < 0.5 is point estimation.o Collins-Soper (CS) frame: bisector of the angle formed by one beam direction and the 0.5 0.5 2

In this study, maximumopposite likelihood direction estimation isof applied the to other extract parametersbeam in of J/ψthepo- J/" rest frame. 1 0 0 larization. The negative log-likelihood is calculated based on the probability density function, • Frame invariant quantity: 0 spatial distribution of positron decay from J/ψ and reconstruction efficiency (cos θ, φ):%& + 3%' −0.5 −0.5 −1 Helicity frame Helicity frame %BCD = systematic uncertainty Collins-Soper frame Collins-Soper frame 1 − % systematic uncertainty (Points shifted for clarify) 2 ' −1 −1 −2 ∂ N 0 2 4 6 0 2 4 6 0 2 4 6 P(cos θ, φ|λθ, λφ, λθφ) = (cos θ, φ|λθ, λφ, λθφ) (cos θ, φ). (4.2) p (GeV/c) ∂cosθ∂φ p (GeV/c) p (GeV/c) T ∗ T T

θ 1 Any arbitrary choice of the experimental observation frame should give the same value of % ; θ 1.5 o BCD θ λ λ

1 λ + STAR p+p@200 GeV p+p→J/ψ+X CSM - direct NLO COM

STAR Preliminary STAR Preliminary + - o Good cross-check on measurements performed in different frames. 7 J/ψ→µ µ |y| < 0.5 1 s = 200 GeV CSM - direct NLO+ + ICEM(prompt) approx. Feed-down 0.5 |y|<0.5 1.2

0 0 0

−0.5 −0.5 STAR p+p@200 GeV Efficiency and Acceptance Corrections −0.5 - J/ψ→µ+µ |y| < 0.5 STAR preliminary Helicity frame published: Helicity frame Collins-Soper frame −1 systematic uncertainty PHENIX |y|<0.35 (2009 data) − systematic uncertainty 1 STAR |y|<1 (2009 data) J/ψ→µ+µ- |y|<0.5 (2015 data) • Raw J/" distributions have to be corrected for the STAR detector acceptance and efficiency. −1 −1.5 0 2 4 6 0 2 4 6 0 1 2 3 4 5 6 p (GeV/c) p (GeV/c) p (GeV/c) • The complication is that the acceptance correction is sensitive to the J/" polarization parameters, T T T which are not known a priori. The measured inclusive J/" polarization: • An iterative procedure is used to overcome this complication. The measured inclusive J/" polarization: • %& and %' parameters are consistent with 0 in HX and CS frames. • %& and %' parameters are consistent with 0 in HX and CS frames. efficiency correction % % #$ = 0,#' = 0 #$,#' • %BCD as a function of pT are consistent between HX and CS frames. factor from embedding • %BCD as a function of pT are consistent between HX and CS frames. No [3, 4] • Newly measured %1 parameter is consistent with the previous publication[3, 4] , even though the *+,-. • Newly measured %1 parameter is consistent with the previous results , even though the get #$ trends seem a bit different at high pT. convergence? Yes trends seem a bit different at high pT. !"#$%: (), (+ *+,-. • Color Singlet Model (CSM) calculation[3] (direct J/") and Color Octet Model[4] (COM) #' • Color Singlet Model (CSM) calculation[3] (direct J/") and Color Octet Model[4] (COM) calculation (direct J/") are in agreement with data while an improved Color Evaporation calculation[5] (direct J/") are in agreement with data while an improved Color Evaporation corrected cos! and " fit and get Model[5] (prompt J/") calculation is touching the upper limit of some data points. % % Model (prompt J/") calculation is touching the upper limit of some data points. distributions in data #$, #' θ θ

ϕ 0.6 ϕ 0.6 θ θ

ϕ 0.6 ϕ 0.6 λ λ λ λ

λ STAR p+p@200 GeV STAR Preliminary STAR Preliminary λ STAR Preliminary STAR Preliminary 1 λ 1 λ 1 STAR p+p@200 GeV STAR Preliminary STAR Preliminary 1 STAR Preliminary STAR Preliminary 800 |y|< 0.5; NLO NRQCD LDMEs1 0.4 700 1600 450 [6] raw signal raw signal |y|<1.0; NLO|y|< 0.5; NRQCD LDMEs1 LDMEs1 0.4 700 0.4 [6] 600 MC simulation 1400 MC simulation 400 0.4 |y|< 0.5; |y|<1.0;NLO NRQCD LDMEs1 LDMEs2 [7] 0.2 600 (scaled) (scaled) |y|<1.0; NLO|y|< 0.5; NRQCD LDMEs2 LDMEs2 1200 350 0.5 0.5 [7] 0.2 500 0.5 0.5 |y|<1.0; LDMEs2 300 - - - - 500 1000 0 µ µ µ µ + + + 400 + 250 0.2 0 raw µ raw µ raw µ raw µ 400 800 0.2 N N N 300 N 200 0 0 −0.2 300 600 150 0 0 −0.2 200 200 400 0 100 0 −0.4 100 100 200 −0.4 50 −0.5 −0.5 − − NLO NRQCD LDMEs1: PRL 114, 092006 (2015) − −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 0.4−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 0.5 0.5 0.6 cosθ 0.1 ϕ cosθ ϕ NLO NRQCD LDMEs2: PRL 110, 042002 (2013) 0.14 J/ψ→e+e- |y|<1 (Run12, bin center) −0.2 −0.6 0.1 + - J/ψ→+e e |y|<1 (Run12, bin center) −0.2 0.35 STAR Preliminary J/ψ→µ µ- |y|<0.5 (Run15, mean p ) + - T −0.8 0.12 0.08 Helicity frame J/ψ→µ µ |y|<0.5 (Run15, mean p ) Helicity frame Collins-Soper frame Collins-Soper frame 0.3 0 < p < 1 GeV/c 0.08 −1 Helicity frame T Helicity frame −1 Collins-Soper frame −0.8 Collins-Soper frame [a.u.] [a.u.] 0.1 T,J/ψ −1 −1 θ θ [a.u.] [a.u.] 0.06 0.25 Collins-Soper frame −0.4 −1 ϕ 0.08 ϕ 0.06 0 1 2 3 4 5 6 7 8 −0.40 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 −01 1 2 3 4 5 6 7 8 0.2 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 /d /d -

- p (GeV/c) p (GeV/c) p (GeV/c) p (GeV/c) µ /dcos µ

/dcos T T T T

0.06 - + - + 0.04 0.15 0.04 p (GeV/c) p (GeV/c) p (GeV/c) p (GeV/c)

STAR Preliminary µ corr µ µ corr µ T T T T + + 2 2 corr µ

corr µ χ 0.04 0 < p < 1 GeV/c χ /ndf = 15.2 / 21 0.1 /ndf = 21.9 / 17 dN T,J/ψ dN 0.02 corrected signal corrected signal λθ = 0.01 ± 0.14 0.02 λθ = -0.46 ± 0.68 dN dN 0.02 0.05 [6] Helicity frame simultaneous fitting λ = 0.10 ± 0.07 simultaneous fitting λ = 0.09 ± 0.06 φ φ Comparison with NRQCD approach[6] : 0 0 0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 Comparison with NRQCD approach : cos(θ) ϕ cos(θ) ϕ • Model calculations for different kinematic regions (|y| < 0.5 and |y| < 1) using two sets of Long • Model calculations for different kinematic regions (|y| < 0.5 and |y| < 1) using two sets of Long Distance Matrix Elements (LDMEs). • An example of the last iteration for 0 < pT < 1 GeV/c. Distance Matrix Elements (LDMEs). • Theoretical calculations are in agreement with data within uncertainties. The substantial • Theoretical calculations are in agreement with data within uncertainties. The substantial difference in J/" polarization at low p when different LDMEs are used, points to the potential difference in J/" polarization at low p whenT different LDMEs are used, points to the potential References of constraining the LDMEs with J/"Tpolarization measurements of better precision in the future. [1] P. Faccioli et al. (2010) Eur. Phys. J. C 69, 657 of constraining the LDMEs with J/" polarization measurements of better precision in the future. [2] C. Yang et al. (2014) Nuclear Instruments and Methods in Physics Research A 762: 1–6 [3] L. Adamczyk et al. (STAR) (2014) Physics Letters B 739 : 180-188 [4] A. Adare et al. (PHENIX) (2010) Phys. Rev. D 82, 012001 Acknowledgement [5] B.A. Kniehl et al. (2017) Phys. Rev. D 96, 054014 (use the matrix elements) [6] Hong-Fei Zhang et al. (2015) Phys. Rev. Lett 114, 092006 This work was supported in part by the National Natural Science Foundation of China (Grant [7] Bin Gong et al. (2013) Phys. Rev. Lett 110, 042002 No.11675186, the sub-topic of the 973 project. (HH2030040069)

The STAR Collaboration drupal.star.bnl.gov/STAR/presentations 46

The likelihood function for a data sample with N observed J/ψ’s is thus

N L(λθ, λφ, λθφ) = P(cos θi, φi|λθ, λφ, λθφ), (4.3) i=1 Y which would reach maximum at the true λθ, λφ, λθφ values. In our study the MLE is obtained by finding the minimum of the negative logarithm of the likelihood function

N −lnL(λθ, λφ, λθφ) = − lnP(cosθi, φi|λθ, λφ, λθφ) (4.4) i=1 X by finding the place where −lnL(λθ, λφ, λθφ) = −lnLmin, i.e.,

∂lnL ∂lnL ∂lnL = = = 0. (4.5) ∂λθ ∂λφ ∂λθφ

The statistical uncertainties are obtained with −lnL = −lnLmin + 1/2 [18].

As shown in Figure 24, the likelihood we calculated based on the dataset from pT range

[6,8]GeV/c is a convex function. I extract the J/ψ polarization parameters’ value by locating the position of the function’s minimum point. The process to find the minimum point of the objective function is called minimization. Broadly speaking, numerical methods for the minimization are methods that approach to the solution via a sequence of calculation. In this measurement, Newton’s method implemented by ROOT Minuit package solves the minimum point of the objective function by constructing the sequence of searching point that approaching to the minimum [59]. Nega1ve log-likelihood func1on

he objec1ve func1on:

47 L(λ) = − ∑ NJ/ψ log( f (cosθ,φ)∗ε(cosθ,φ)) cosθ,φ λ* = argmin(L(λ))

∇L(λ) = 0

Subject to constraint:

f (cosθ,φ) ≥ 0,∀θ,φ FigureLikelihood as a func1on of (λ 24: For instance, the objective function likelihood as aθ function,λφ) at p of (λθ, λTφ[6-8] ) at pT [6-8] GeV/cGeV/c in helicity frame. in helicity frame is convex.

4.2 Bias of the estimator 17 When an estimator is implemented, we should also investigate the properties of the estima- tor. With the raw yield distribution of (cosθ, φ) from datasets and efficiency of detectors, one can construct an estimator as above to estimate the J/ψ polarization parameters. For an esti- mator, there are four aspects of properties we should bear in mind. These are bias, efficiency, consistency and robustness. 48

For the estimation, we usually concern about the difference between the average of collection of parameter’s estimations and the true value of the parameter. The estimator is called unbiased if and only if the mean of estimation of the parameter is equal to the true value of the parameter

[60]. Statistically, if the estimator is unbiased, the mean value extracted by the estimator should agree with the truth.

Efficiency is another important property of the estimator, which indicates the variation of the collection of measurements. If the measurements of parameters is regarded as to hit the position of some target’s bulleye, for two unbiased estimator, the one with a smaller mean square error is a more efficient and desirable one [61].

The bias and variance indicates the accuracy and precision of the estimator. Namely, when estimators have unbiased estimation on parameters that the mean of estimation is equal to the truth of parameters, then variance of the measurements is the further property to decide which estimator is better than the others. The efficient estimator is the minimum variance unbiased estimator (MVUE) [61].

An estimator is called consistent if the collection of estimation Tn of parameter converge in probability to the true value of parameter [61].

Pr{ lim Tn = θ^} (4.6) n

→∞ Robustness of an estimator usually is a relatively vague property. Broadly speaking, it indicates the estimator can still achieve successful and accurate point estimation under certain circum- 49 stances. In our situation, the input of the J/ψ polarization parameters to generate pseudo distribution of cosθ and φ can be outside well-defined domain. Robustness is a desirable property that simplify the calculation and data processing [61].

lambdas central value -0.19 -0.04 0.13 lambdas central value -0.19 -0.04 0.13 lambdas central value -0.19 -0.04 0.13 90 60 100 80

50 Entries 1000 70 Entries 1000 80 Entries 1000

60 40 50 60 Mean −0.1653 ± 0.006495 Mean −0.04414 ± 0.004512 Mean 0.1187 ± 0.003465 30 40 40 30 20 RMS 0.2054 ± 0.004593 RMS 0.1427 ± 0.00319 RMS 0.1096 ± 0.00245 20 20 10 10

0 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 λ λ λ φ θφ θ

lambdas uncertainty lambdas uncertainty lambdas uncertainty

90 200 240 220 80 180 200 160 70 Entries 1000 Entries 1000 Entries 1000 180 140 60 160 120 140 50 Mean 0.201 ± 0.00662 Mean 0.1374 ± 0.005178 Mean 0.108 ± 0.003157 100 120 40 80 100 30 80 RMS 0.2091 ± 0.004681 60 RMS 0.1638 ± 0.003662 RMS 0.09972 ± 0.002232 60 20 40 40 10 20 20 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 σλ σλ σλ φ θφ Fri Jun 15 20:01:58 2018 θ

Figure 25: Distributions of the estimated central value (top row) and statistical uncertainty

(bottom row) for λθ (left), λφ (middle) and λθφ (right) from 1000 pseudo experiments in 6 < pT < 8 GeV/c bin in the HX frame. See text for details. 50

With the probability density function of positron distribution for fixed truth values of J/ψ polarization parameters, we generate a pair of random numbers corresponding to (cosθ, φ).

To take the efficiency of detectors into consideration, a random number is rolled based on 2- dimensional efficiency histogram of (cosθ,φ) to decide whether to accept the entry of (cosθ,φ) generated. The total number of entries is set as the same as that in the datasets. In this way, I produce pseudo data of (cosθ,φ) distribution. Same as data, we apply the maximum likelihood estimator to extract the polarization parameters from these pseudo datasets. For an unbiased estimator, based on the law of large numbers, the extracted polarization parameters’ values should converge to the true value I feed to probability density function asymptotically. This experiment has been repeated for N times (N=1000) and the result from each experiment is

filled in a histogram such that I can compare the mean of extracted polarization parameters and the true polarization parameters I feed in the probability density function beforehand.

With the toy Monte Carlo technique, I can simulate and sample J/ψ decay. Importantly, during this process, the truth of parameters’ values of probability density function is known ahead of each experiment. Thus, it allows us to exam the bias of the estimator by comparing the measured parameters’ values and the truth of parameters’ values.

As shown in Figure 26-Figure 28, the mean of extracted polarization parameters locate within one-sigma area of the true polarization parameters I feed in the probability density function. 51

MB 0 < p < 2 GeV/c in HX frame mcpx_0_0_0 HT0 2 < p < 4 GeV/c in HX frame mcpx_1_1_0 HT2 4 < p < 6 GeV/c in HX frame mcpx_3_2_0 HT2 6 < p < 8 GeV/c in HX frame mcpx_3_3_0 HT2 8 < p < 14 GeV/c in HX frame mcpx_3_4_0 T T T T T Entries 1000 Entries 1000 Entries 1000 Entries 1000 Entries 1000 40 50 20 Mean 0.3951 70 Mean 0.1816 Mean 0.229 80 Mean 0.1725 Mean 0.6857 RMS 0.164 RMS 0.09683 RMS 0.1105 RMS 0.0839 RMS 0.3196 35 18 60 70 40 16 30 60 50 14 25 30 50 12 40 20 40 10 30 20 8 15 30 20 6 10 20 10 4 5 10 10 2 0 0 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4

σλθ σλθ σλθ σλθ σλθ

MB 0 < p < 2 GeV/c in CS frame mcpx_0_0_1 HT0 2 < p < 4 GeV/c in CS frame mcpx_1_1_1 HT2 4 < p < 6 GeV/c in CS frame mcpx_3_2_1 HT2 6 < p < 8 GeV/c in CS frame mcpx_3_3_1 HT2 8 < p < 14 GeV/c in CS frame mcpx_3_4_1 T T T T T Entries 1000 Entries 1000 Entries 1000 Entries 1000 Entries 1000 90 60 Mean 0.8057 Mean 0.2714 Mean 0.2983 Mean 0.1909 Mean 0.2023 25 50 RMS 0.2542 RMS 0.1067 RMS 0.1126 80 RMS 0.06212 RMS 0.09091 50 50 70 20 40 40 60 40

15 30 50 30 30 40 10 20 20 30 20

20 5 10 10 10 10

0 0 0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 σλ σλ σλ σλ σλ Fri Jun 15 23:50:27 2018 θ θ θ θ θ

Figure 26: Statistical uncertainties for λθ from data (vertical red lines) and 1000 pseudo exper- iments (black histogram). Top (bottom): HX (Collins-Soper) frame. From left to right: pT =

0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c.

MB 0 < p < 2 GeV/c in HX frame mcpx_0_0_0 HT0 2 < p < 4 GeV/c in HX frame mcpx_1_1_0 HT2 4 < p < 6 GeV/c in HX frame mcpx_3_2_0 HT2 6 < p < 8 GeV/c in HX frame mcpx_3_3_0 HT2 8 < p < 14 GeV/c in HX frame mcpx_3_4_0 T T T T T Entries 1000 Entries 1000 Entries 1000 Entries 1000 Entries 1000 220 90 90 Mean 0.1915 90 Mean 0.1692 Mean 0.2577 Mean 0.1168 Mean 0.2434 50 RMS 0.06333 RMS 0.0949 RMS 0.1071 200 RMS 0.05194 80 RMS 0.08466 80 80 180 70 70 70 40 160 60 60 60 140 30 50 50 50 120 100 40 40 40 20 80 30 30 30 60 20 20 20 10 40 10 10 20 10 0 0 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 σλφ σλφ σλφ σλφ σλφ

MB 0 < p < 2 GeV/c in CS frame mcpx_0_0_1 HT0 2 < p < 4 GeV/c in CS frame mcpx_1_1_1 HT2 4 < p < 6 GeV/c in CS frame mcpx_3_2_1 HT2 6 < p < 8 GeV/c in CS frame mcpx_3_3_1 HT2 8 < p < 14 GeV/c in CS frame mcpx_3_4_1 T T T T T 220 Entries 1000 Entries 1000 Entries 1000 Entries 1000 120 Entries 1000 300 Mean 0.1489 400 Mean 0.07504 Mean 0.08323 300 Mean 0.1069 Mean 0.1853 200 RMS 0.0336 RMS 0.02999 RMS 0.03262 RMS 0.03323 RMS 0.06356 180 350 100 250 250 160 300 80 140 200 200 250 120 150 60 100 200 150

80 150 100 100 40 60 100 40 50 50 20 50 20 0 0 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 σλ σλ σλ σλ σλ Fri Jun 15 23:51:06 2018 φ φ φ φ φ

Figure 27: Same as Figure 26 but for λφ.

MB 0 < p < 2 GeV/c in HX frame mcpx_0_0_0 HT0 2 < p < 4 GeV/c in HX frame mcpx_1_1_0 HT2 4 < p < 6 GeV/c in HX frame mcpx_3_2_0 HT2 6 < p < 8 GeV/c in HX frame mcpx_3_3_0 HT2 8 < p < 14 GeV/c in HX frame mcpx_3_4_0 T T T T T Entries 1000 200 Entries 1000 Entries 1000 Entries 1000 Entries 1000 100 250 Mean 0.2055 Mean 0.09732 140 Mean 0.1027 Mean 0.09653 60 Mean 0.2886 180 RMS 0.07079 RMS 0.05571 RMS 0.03771 RMS 0.04958 RMS 0.1178 120 80 160 200 50 140 100 40 60 120 150 80 100 30 40 80 60 100 60 20 40 20 40 50 10 20 20

0 0 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 σλθφ σλθφ σλθφ σλθφ σλθφ

MB 0 < p < 2 GeV/c in CS frame mcpx_0_0_1 HT0 2 < p < 4 GeV/c in CS frame mcpx_1_1_1 HT2 4 < p < 6 GeV/c in CS frame mcpx_3_2_1 HT2 6 < p < 8 GeV/c in CS frame mcpx_3_3_1 HT2 8 < p < 14 GeV/c in CS frame mcpx_3_4_1 T T T T T Entries 1000 Entries 1000 Entries 1000 Entries 1000 Entries 1000 100 300 Mean 0.2636 300 Mean 0.09968 250 Mean 0.1002 Mean 0.1019 100 Mean 0.1554 RMS 0.06635 RMS 0.03621 RMS 0.04792 RMS 0.02842 RMS 0.05971 250 80 250 200 80

200 200 60 150 60 150 150 40 100 40 100 100

20 50 20 50 50

0 0 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.020.040.060.08 0.1 0.120.140.160.18 0.2 0.220.24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 σλ σλ σλ σλ σλ Fri Jun 15 23:51:45 2018 θφ θφ θφ θφ θφ

Figure 28: Same as Figure 26 but for λθφ. 52

Sometimes, an estimator has a precise measurement in parameters estimation at a single point doesn’t guarantee that it can achieve good performance over a wide range of domain.

Therefore, I not only compared the truth of polarization parameters and measured value of polarization parameters at a single point that extracted from real dataset, but also scan the another 4 points around with the interval equals to 0.2. That is with the measured polarization parameters’ value centered I also feed the each polarization parameter that +0.2, -0.2, +0.4 and -0.4 to the probability density function of positron decay from J/ψ and conduct the toy

Monte Carlo simulation described above. Thereafter, I compared the measured polarization parameters’ value and their statistical errors with the true polarization parameters’ value and their statistical errors.

As shown in Figure 29 and Figure 30, points of measured polarization parameters and the truth of polarization parameters are present. The measured polarization parameters and their errors are fit with linear function, it is observed that the measured polarization parameters’ value is just around the truth polarization parameters’ value, hence points locate near the line of y = x. The measured polarization parameters’ statistical error is only a little bit offset to the line y = x. This observation indicates our estimator achieve high accuracy in the parameters estimation over a wide range of domain and good accuracy in the parameters’ error estimation over a wide range of domain as well. More importantly, the relationship between the measured parameters and truth of parameters provide us the routine to calibrate the estimator by mapping measured value to the truth. This calibration is applied in our calculation to get the final measurement results. 53

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3 3 3 3 3 3 MB @ 0 < p < 2 GeV/c in HX frame χ2 / ndf 2.547 / 3 HT0 @ 2 < p < 4 GeV/c in HX frameχ2 / ndf 5.429 / 3 HT2 @ 4 < p < 6 GeV/c in HX frameχ2 / ndf 4.842 / 3 HT2 @ 6 < p < 8 GeV/c in HX frameχ2 / ndf 0.2328 / 3 HT2 @ 8 < p < 12 GeV/c in HX frameχ2 / ndf 2.548 / 3 T T T T T p0 1.057 ± 0.02469 p0 1.058 ± 0.01063 p0 1.103 ± 0.02059 p0 1.023 ± 0.01036 p0 1.153 ± 0.04447 p1 0.06308 ± 0.007635 p1 0.0493 ± 0.00458 p1 0.09657 ± 0.007039 p1 0.03449 ± 0.004561 p1 0.00759 ± 0.03911 2 2 2 2 2

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3

φ −3 φ −3 φ −3 φ −3 φ −3

λ −3 −2 λ −1 0 χ2 1 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ2 1 2 3 measurements of φ / ndf 4.569 / 3 measurements of λφ / ndf 2.399 / 3 measurements of λφ / ndf 1.039 / 3 measurements of λφ / ndf 1.178 / 3 measurements of λφ / ndf 6.415 / 3 p0 0.9395 ± 0.01209 p0 0.9511 ± 0.009819 p0 1.008 ± 0.02232 p0 0.9348 ± 0.006718 p0 0.949 ± 0.01512

p1 −0.03732 ± 0.003699 p1 −0.03284 ± 0.002893 p1 −0.02024 ± 0.0138 p1 −0.01024 ± 0.001928 p1 −0.01475 ± 0.005191 2 2 2 2 2 measured measured measured measured measured

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3

φ −3 φ −3 φ −3 φ −3 φ −3 θ −3 −2 −1 0 χ2 1 2 θ −33 −2 −1 0 χ21 2 θ −33 −2 −1 0 χ2 1/ ndf 2 θ −33 −2 −1 0 χ21 2 θ −33 −2 −1 0 χ2 1/ ndf 2 3 measurementsλ of λ / ndf 1.601 / λ 3 measurements of λ / ndf 2.476 /λ 3 measurements of λ 0.1426 /λ 3 measurements of λ / ndf 2.801 /λ 3 measurements of λ 1.031 / 3 θφ λ θφ λ θφ λ θφ λ θφ λ p0 0.9932 ± 0.01233true φ p0 0.9477 ± 0.005194true φ p0 0.977 ± 0.007378true φ p0 0.9616 ± 0.005625true φ p0 1.003 ± 0.01752true φ p1 −0.01115 ± 0.004781 p1 0.001972 ± 0.001441 p1 −0.0003458 ± 0.001894 p1 −0.003187 ± 0.001715 p1 −0.0008559 ± 0.006685 2 2 2 2 2 measured measured measured measured measured

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3 −3 −3 −3 −3 −3 − − − 2 − − − 2 − − − 2 − − − 2 − − − 2 MB 3@ 0 < p <2 2 GeV/c 1in CS frame0 χ / ndf1 2 0.7492 / 3 33 HT0 @ 2 < p < 4 1GeV/c in 0CS frameχ 1/ ndf 2 3.056 / 3 33 HT2 @ 24 < p < 6 1GeV/c in 0CS frameχ 1/ ndf 2 0.6382 / 3 33 HT2 @ 26 < p < 8 1GeV/c in 0CS frameχ 1/ ndf 2 0.6313 / 3 33 HT2 @ 28 < p < 121 GeV/c in0 CS frameχ 1/ ndf 2 1.805 / 3 3 T true λ T true λ T true λ T true λ T true λ p0 0.9851 ± 0.0469 θφ p0 1.023 ± 0.01655 θφ p0 1.007 ± 0.01826 θφ p0 1.007 ± 0.0114 θφ p0 1.063 ± 0.01201 θφ p1 0.1008 ± 0.03026 p1 0.03376 ± 0.01056 p1 0.04313 ± 0.007046 p1 0.02692 ± 0.003929 p1 0.07535 ± 0.01039 2 2 2 2 2

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3

φ −3 φ −3 φ −3 φ −3 φ −3

λ −3 −2 λ −1 0 χ2 1 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ21 2 λ −33 −2 −1 0 χ2 1 2 3 measurements of φ / ndf 0.9722 / 3 measurements of λφ / ndf 7.68 / 3 measurements of λφ / ndf 0.2373 / 3 measurements of λφ / ndf 2.818 / 3 measurements of λφ / ndf 1.416 / 3 p0 0.8942 ± 0.008157 p0 0.9293 ± 0.004031 p0 0.9149 ± 0.004555 p0 0.9126 ± 0.006047 p0 0.9402 ± 0.01055

p1 −0.006153 ± 0.002554 p1 −0.001076 ± 0.001358 p1 −0.0009304 ± 0.00175 p1 −0.007085 ± 0.001895 p1 −0.01393 ± 0.003454 2 2 2 2 2 measured measured measured measured measured

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3 φ − φ − φ − φ − φ −3 3 3 3 3

θ − − − θ − − − θ − − − θ − − − θ − − − 3 2 λ 1 0 χ2 / ndf1 2 33 2 1 λ 0 χ2 1/ ndf 2 5.676 / 3 33 2 1 λ 0 χ2 1/ ndf 2 2.522 / 3 33 2 1 λ 0 χ2 1/ ndf 2 1.49 / 3 33 2 1 λ 0 χ2 1/ ndf 2 6.485 / 3 3 measurementsλ of θφ 2.481 / λ 3 measurements of θφ λ measurements of θφ λ measurements of θφ λ measurements of θφ true λφ true λφ true λφ true λφ true λφ p0 0.9756 ± 0.01481 p0 0.9793 ± 0.00573 p0 0.9545 ± 0.005906 p0 0.9501 ± 0.005836 p0 0.9815 ± 0.009742 ± − ± − ± − ± ± 2 p1 0.006156 0.00593 2 p1 0.0005378 0.001618 2 p1 0.001443 0.001696 2 p1 0.0005641 0.001834 2 p1 0.00146 0.002863 measured measured measured measured measured

1 1 1 1 1

0 0 0 0 0

−1 −1 −1 −1 −1

−2 −2 −2 −2 −2

−3 −3 −3 −3 −3 −3 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 −33 −2 −1 0 1 2 3 λ λ λ λ λ true θφ true θφ true θφ true θφ true θφ

Figure 29: Biases in the central value estimation: the x-axis is the input λ value that are used to generate pseudo data, while the y-axis is the mean of the extracted λ values from 1000 pseudo experiments. From top to bottom, λθ, λφ and λθφ in the HX frame, and λθ,λφ and λθφ in the

CS frame, respectively. From left to right, pT = 0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c, respectively.

The red line is y = x, while the black line is a linear fit y = ax + b to the points. 54

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ θ θ θ θ λ χ2 λ χ2 λ χ2 λ χ2 λ χ2 measurementsσ of σλ / ndf 20.44 / σ 4 measurements of σλ / ndf 27.87 /σ 4 measurements of σλ / ndf 120.2 /σ 4 measurements of σλ / ndf 2.36 /σ 4 measurements of σλ / ndf 7.52 / 4 1.4 θ 1.4 θ 1.4 θ 1.4 θ 1.4 θ p0 0.854 ± 0.007735 p0 1.117 ± 0.02062 p0 0.7052 ± 0.01341 p0 0.9226 ± 0.01196 p0 0.8209 ± 0.006785

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured

1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

φ 0 φ 0 φ 0 φ 0 φ 0 λ 0 0.2 0.4 0.6 0.8 χ2 1 1.2 1.4λ 0 0.2 0.4 0.6 0.8 21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ2 1 1.2 1.4 χ σ σ σ σ σ / ndf 126.4 / σ 4 σ / ndf 28.99 / 4 σ / ndf 28.63 / 4 σ / ndf 14.87 / 4 σ / ndf 21.41 / 4 measurements of λ σ measurements of λ σ measurements of λ σ measurements of λ σ measurements of λ σ 1.4 φ true 1.4λ φ true 1.4λ φ true 1.4λ φ true 1.4λ φ true λ p0 0.8073 ± 0.00708 θ p0 1.1 ± 0.02088 θ p0 0.7979 ± 0.01487 θ p0 0.972 ± 0.01307 θ p0 0.8141 ± 0.007193 θ

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured

1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ φ φ φ φ

θ 0 θ 0 θ 0 θ 0 θ 0 λ 0 0.2 0.4σ 0.6 0.8 χ2 / ndf1 1.2 1.4λ 0 0.2 0.4 σ0.6 0.8 χ21 / ndf 1.2 1.4λ 0 0.2 0.4 σ0.6 0.8 χ21 / ndf 1.2 1.4λ 0 0.2 0.4 σ0.6 0.8 χ21 / ndf 1.2 1.4λ 0 0.2 0.4 σ0.6 0.8 χ2 1/ ndf 1.2 10.221.4 / 4 measurementsσ of λ 4.373 /σ 4 measurements of λ 15.32 /σ 4 measurements of λ 59.1 /σ 4 measurements of λ 3.929 /σ 4 measurements of λ 1.4 θφ true 1.4σλ θφ true 1.4σλ θφ true 1.4σλ θφ true 1.4σλ θφ true σλ p0 0.9032 ± 0.009424 φ p0 1.328 ± 0.02511 φ p0 1.059 ± 0.02054 φ p0 0.9903 ± 0.01363 φ p0 0.8564 ± 0.006987 φ

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured

1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

θ 0 θ 0 θ 0 θ 0 θ 0 λ 0 0.2 0.4 0.6 0.8 χ2 1 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4 σ σ σ σ σ σ σ σ σ measurementsσ of λ / ndf 11.34 / 4 measurements of λ / ndf 14.02 / 4 measurements of λ / ndf 35.03 / 4 measurements of λ / ndf 1.204 / 4 measurements of λ / ndf 11.69 / 4 1.4 θ true 1.4σλ θ true 1.4σλ θ true 1.4σλ θ true 1.4σλ θ true σλ p0 0.8718 ± 0.004309 θφ p0 0.8924 ± 0.01045 θφ p0 0.8972 ± 0.01079 θφ p0 0.9453 ± 0.01411 θφ p0 0.859 ± 0.01027 θφ

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured

1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

φ 0 φ 0 φ 0 φ 0 φ 0 λ 0 0.2 0.4 0.6 0.8 χ2 1 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 χ2 1 1.2 1.4 σ σ / ndf 41.31 /σ 4 σ / ndf 5.335 /σ 4 σ / ndf 3.058 /σ 4 σ / ndf 2.537 /σ 4 σ / ndf 35.68 / 4 measurements of λ σ measurements of λ σ measurements of λ σ measurements of λ σ measurements of λ σ 1.4 φ true 1.4λ φ true 1.4λ φ true 1.4λ φ true 1.4λ φ true λ p0 0.8955 ± 0.005185 θ p0 0.9868 ± 0.01165 θ p0 1.009 ± 0.0132 θ p0 0.9892 ± 0.01375 θ p0 0.8913 ± 0.009229 θ

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured

1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ φ φ φ φ

θ 0 θ 0 θ 0 θ 0 θ 0

λ 0 0.2 0.4 0.6 0.8 2 1 1.2 1.4λ 0 0.2 0.4 0.6 0.8 21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 21 1.2 1.4λ 0 0.2 0.4 0.6 0.8 21 1.2 1.4 measurements of σ χ / ndf 30.44 / 4 measurements of σ χ / ndf 4.483 / 4 measurements of σ χ / ndf 2.917 / 4 measurements of σ χ / ndf 13.73 / 4 measurements of σ χ / ndf 15.47 / 4 σ λ σ σ λ σ σ λ σ σ λ σ σ λ σ 1.4 θφ true 1.4λ θφ true 1.4λ θφ true1.4 λ θφ true 1.4λ θφ true λ p0 0.896 ± 0.004651 φ p0 0.9641 ± 0.01145 φ p0 0.9632 ± 0.01492 φ p0 0.9372 ± 0.01264 φ p0 0.8713 ± 0.01265 φ

1.2 1.2 1.2 1.2 1.2 measured measured measured measured measured 1 1 1 1 1

0.8 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2 0.2

0 0 0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 σ σ σ σ σ true λ true λ true λ true λ true λ θφ θφ θφ θφ θφ Figure 30: Biases in the uncertainty estimation: the x-axis is the RMS of the extracted λ values from 1000 sets of pseudo data, while the y-axis is the mean of the extracted λ uncertainty from

1000 pseudo experiments. From top to bottom, λθ,λφ and λθφ in the HX frame, and λθ,λφ and

λθφ in the CS frame, respectively. From left to right, pT = 0-2, 2-4, 4-6, 6-8 and 8-14 GeV/c, respectively. The red line is y = x, while the black line is a linear fit y = ax to the points. 55

4.3 minimum point position and 1σ contours

The likelihood function is projected to the plane of two parameters as shown above and the 1σ contour and the measured minimum point position are present. The area of 1σ contour indicates 68% confidence interval [18]. And the maximum distance of this contour along each axis suggests the magnitude of statistical uncertainty of each parameter. 56

φ φ φ 1 φ φ 0.8λ 0.4λ λ λ 0.6 λ 2

0.6 0.5 0.4 0.2 1 0.4 0.2 0 0.2 0 0 0 0 −0.5 − −0.2 0.2 −0.2 −1 −1 −0.4 −0.4 −0.4 −2 − −0.6 0.6 −1.5 −0.8 −0.6 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 λ −0.4 −0.2 0 0.2 0.4 λ −1 −0.5 0 0.5 1 λ1.5 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 λ0.8 −1 0 1 2 3 λ θ θ θ θ θ

φ 0.6φ φ φ φ 2 0.6 0.4 λ λ λ λ λ 0.5 1.5 0.4 0.4 0.3 0.4 1 0.2 0.2 0.2 0.3 0.5 0.1 0 0 0.2 0 0 −0.2 0.1 −0.2 − −0.1 0.5 0 −0.4 −0.4 −0.2 −1 −0.1 − 0.6 −0.3 −1.5 −0.6 −0.2 −1 −0.5 0 0.5 1 1.5 2 2.5λ −1 −0.8 −0.6 −0.4 −0.2 0 0.2λ −0.8 −0.6 −0.4 −0.2 0 0.2 0.4λ −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 λ −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3λ θ θ θ θ θ Sat Jun 16 00:05:17 2018

Figure 31: −lnLmin + 0.5 contour (black area) from data as a function of λθ and λφ. Also shown are the λθ and λφ values estimated from data using the MLE.

1.5φ φ φ φ φ

θ θ θ 0.8θ θ 1 λ λ λ λ λ 0.4 1 0.6 2

0.5 0.4 0.2 1 0.5 0.2 0 0 0 0 0

−0.2 −0.2 − −0.5 −0.5 1 −0.4 −0.4 −1 −2 −1 −0.6 −1 −0.5 0 0.5 1 1.5λ −0.4 −0.2 0 0.2 0.4 λ −1 −0.5 0 0.5 1 λ −0.6 −0.4 −0.2 0 0.2 0.4 0.6 λ0.8 −1 0 1 2 3 λ θ θ θ θ θ

φ φ 1 φ φ φ

θ θ θ 1 θ θ 0.4 λ 2 λ λ λ λ 0.4

0.5 0.2 1 0.5 0.2

0 0 0 0 0 −0.2 −0.2 −1 −0.5 −0.4 −0.5 −0.4 −2 − 0.6 −0.6 −3 −1 −1 −2 −1 0 1 2 3 λ −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 λ0.6 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 λ0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 λ −1.2 −1 −0.8 −0.6 −0.4 −0.2 λ θ θ θ θ θ Sat Jun 16 00:05:17 2018

Figure 32: Same as Figure 31 but for λθ and λθφ

φ 0.5φ 1.5φ φ φ

0.8θ θ θ θ 1.2θ

λ λ λ λ 0.4 λ 0.4 1 1 0.6 0.3 0.3 0.8 0.2 0.4 0.6 0.5 0.2 0.1 0.4 0.2 0 0 0.1 0.2 −0.1 0 0 0 −0.5 −0.2 −0.2 −0.2 −0.1 −0.3 −0.4 −1 −0.4 −0.4 −0.2 −0.6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 λ −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 λ0.4 −1.5 −1 −0.5 0 0.5 λ1 −0.3 −0.2 −0.1 0 0.1 0.2 λ0.3 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6λ φ φ φ φ φ

0.6φ φ φ φ φ θ θ θ 0.2θ θ

λ λ λ λ λ 0.4 0.4 0.2 0.3 0.1 0.2 0.1 0.2 0.2 0 0 0.1 0 0 −0.2 −0.1 0 −0.4 −0.1 −0.2 −0.2 −0.6 −0.1 − −0.3 0.2 −0.4 −0.8 −0.2 −0.4 −0.3 −1 −0.6 −0.3 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 λ1 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2λ −0.3 −0.2 −0.1 0 0.1 0.2 λ0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 λ −0.4 −0.2 0 0.2 0.4 0.6 λ φ φ φ φ φ Sat Jun 16 00:05:17 2018

Figure 33: Same as Figure 31 but for λφ and λθφ 57

4.4 Goodness of fit

After the polarization parameters’ value from dataset are extracted, the expected cosθ and φ distribution are calculated with Monte Carlo method. Together with the efficiency of detectors, the extracted polarization parameters’ value are feed in the probability density function to generate the expected cosθ and φ distribution. It is compared with the real cosθ and φ distribution from datasets. As shown in Figure 34 and Figure 35, the expected distribution is consistent with the real distribution in dataset and χ2/ndf indicates the consistency of the results of measurement with regard to dataset and the level of goodness of fit. Meanwhile, the extreme cases that when polarization parameters are equal to 1 or -1 are also shown in Figure 34 and Figure 35. The differences between them suggest how much different can be expected from the experimental observation. 58

0.4 0.4 0.4 0.4 0.4 default fit, {χ^2}/ndf = 12.78/10 default fit, {χ^2}/ndf = 5.96/10 default fit, {χ^2}/ndf = 13.97/10 default fit, {χ^2}/ndf = 7.97/10 default fit, {χ^2}/ndf = 10.24/10 0.35 0.35 0.35 0.35 0.35 λ χ λ χ λ χ λ χ λ χ φ = +1, { ^2}/ndf = 59.32/10 φ = +1, { ^2}/ndf = 55.54/10 φ = +1, { ^2}/ndf = 58.85/10 φ = +1, { ^2}/ndf = 50.77/10 φ = +1, { ^2}/ndf = 51.07/10 0.3 λ χ 0.3 λ χ 0.3 λ χ 0.3 λ χ 0.3 λ χ φ = -1, { ^2}/ndf = 15.42/10 φ = -1, { ^2}/ndf = 25.85/10 φ = -1, { ^2}/ndf = 12.44/10 φ = -1, { ^2}/ndf = 42.85/10 φ = -1, { ^2}/ndf = 10.32/10

0.25 0.25 0.25 0.25 0.25 raw data

0.2 0.2 0.2 0.2 0.2

0.15 0.15 0.15 0.15 0.15

0.1 0.1 0.1 0.1 0.1

0.05 0.05 0.05 0.05 0.05

0 0 0 0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.4 0.4φ 0.4φ 0.4φ 0.4φ φ default fit, {χ^2}/ndf = 10.01/10 default fit, {χ^2}/ndf = 9.64/10 default fit, {χ^2}/ndf = 1.53/10 default fit, {χ^2}/ndf = 27.40/10 default fit, {χ^2}/ndf = 14.95/10 0.35 0.35 0.35 0.35 0.35 λ χ λ χ λ χ λ χ λ χ φ = +1, { ^2}/ndf = 49.36/10 φ = +1, { ^2}/ndf = 129.28/10 φ = +1, { ^2}/ndf = 56.76/10 φ = +1, { ^2}/ndf = 101.22/10 φ = +1, { ^2}/ndf = 53.24/10 0.3 0.3 0.3 0.3 0.3 λ χ λ χ λ χ λ χ λ χ φ = -1, { ^2}/ndf = 42.49/10 φ = -1, { ^2}/ndf = 164.81/10 φ = -1, { ^2}/ndf = 235.86/10 φ = -1, { ^2}/ndf = 84.31/10 φ = -1, { ^2}/ndf = 35.72/10 0.25 0.25 0.25 0.25 0.25

0.2 0.2 0.2 0.2 0.2

0.15 0.15 0.15 0.15 0.15

0.1 0.1 0.1 0.1 0.1

0.05 0.05 0.05 0.05 0.05

0 −3 −2 −1 0 1 2 30 −3 −2 −1 0 1 2 30 −3 −2 −1 0 1 2 30 −3 −2 −1 0 1 2 30 −3 −2 −1 0 1 2 3 Fri Aug 16 14:20:27 2019 φ φ φ φ φ

Figure 34: The comparison of theoretical distribution and raw data distribution in φ. The top

(bottom) row shows the HX (CS) frame.

0.5 default fit, {χ^2}/ndf = 8.29/10 0.5 default fit, {χ^2}/ndf = 1.66/10 0.5 default fit, {χ^2}/ndf = 3.33/10 0.5 default fit, {χ^2}/ndf = 38.39/10 0.5 default fit, {χ^2}/ndf = 14.53/10 λ χ λ χ λ χ λ χ λ χ θ = +1, { ^2}/ndf = 12.31/10 θ = +1, { ^2}/ndf = 17.85/10 θ = +1, { ^2}/ndf = 9.38/10 θ = +1, { ^2}/ndf = 42.89/10 θ = +1, { ^2}/ndf = 14.23/10 0.4 λ χ 0.4 λ χ 0.4 λ χ 0.4 λ χ 0.4 λ χ θ = -1, { ^2}/ndf = 23.56/10 θ = -1, { ^2}/ndf = 88.46/10 θ = -1, { ^2}/ndf = 103.86/10 θ = -1, { ^2}/ndf = 111.05/10 θ = -1, { ^2}/ndf = 57.19/10 raw data 0.3 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1

0− − − − − 0− − − − − 0− − − − − 0− − − − − 0− − − − − 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 cos0.8 θ11 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 cos0.8 θ11 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 cos0.8 θ11 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 cos0.8 θ11 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 cos0.8 θ1 0.5 default fit, {χ^2}/ndf = 1.96/10 0.5 default fit, {χ^2}/ndf = 8.37/10 0.5 default fit, {χ^2}/ndf = 3.07/10 0.5 default fit, {χ^2}/ndf = 7.97/10 0.5 default fit, {χ^2}/ndf = 8.41/10 λ χ λ χ λ χ λ χ λ χ θ = +1, { ^2}/ndf = 2.07/10 θ = +1, { ^2}/ndf = 24.07/10 θ = +1, { ^2}/ndf = 15.40/10 θ = +1, { ^2}/ndf = 25.43/10 θ = +1, { ^2}/ndf = 16.67/10 0.4 0.4 0.4 0.4 0.4 λ χ λ χ λ χ λ χ λ χ θ = -1, { ^2}/ndf = 7.82/10 θ = -1, { ^2}/ndf = 10.31/10 θ = -1, { ^2}/ndf = 6.61/10 θ = -1, { ^2}/ndf = 28.18/10 θ = -1, { ^2}/ndf = 12.45/10

0.3 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1

0−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0−11 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0−11 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0−11 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0−11 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Fri Aug 16 14:20:27 2019 cosθ cosθ cosθ cosθ cosθ Figure 35: The comparison of theoretical distribution and raw data distribution in cosθ. The top (bottom) row shows the HX (CS) frame.

In summary, the probability density function of positron decaying from J/ψ is a Fourier series as a function of cosθ and φ. The characteristics of the probability density function describe the spatial probability density distribution of positrons’ (decay from J/ψ) momentum with respect to J/ψ momentum, which provides us the good probe to distinguish available 59 theoretical quarkonium formation models. I calculate the likelihood function based on dataset collected from the experiment and efficiency of detectors, pin down the minimum point position of the objective function to extract J/ψ polarization parameters. Moreover, toy Monte Carlo simulation is conducted to check the properties of the estimator, which contribute to achieve the reliable measurement result with high accuracy. CHAPTER 5

SYSTEMATIC UNCERTAINTY

In experimental measurements, results of the measurements can be affected by many factors introduced by choice, configuration, threshold and the others. It is important to know their effects along with the measurement results, which also suggests the level of accuracy of mea- surement results. This chapter will introduce main factors that may affect the measurement results and how they are calculated.

5.1 pT Weight function

The choice of pT weight function (see Eq. 3.2), which was obtained by fitting the measured

J/ψ pT spectrum, could to some extent affect the J/ψ polarization parameter estimation. To calculate the systematic uncertainty from the pT weight function, another function PowLaw(pT )

(see Figure 36) is utilized to recalculate the J/ψ reconstruction efficiency with the following parameters. x PowLaw(p ) = A(1 + ( )2)C (5.1) T B where, A = 4.90151, B = 3.30694 and C = -4.86445.The variation of polarization parameters is taken as a systematic uncertainty from the pT weight function.

60 61

2 10 p+p s = 200 GeV STAR 2012 |y|<1 Levy Power-law 10−2 dy) nb/(GeV/c) T dp T p π

/(2 −5 σ

2 10 Bd 0 5 10 15 ψ pJ/ (GeV/c) T Figure 36: The comparison of Levy function and Powerlaw function fit to the measured J/ψ production cross section [62].

5.2 pT smearing

Compared with the invariant mass distribution of J/ψ from the experimental data, the distribution obtained from embedding is narrower. In order to reproduce the invariant mass distribution in embedding to better account for the momentum resolution, the transverse mo- 62 mentum is smeared by adding a component from Double Crystal Ball (DCB) function, which reads

A (B − R)−n, R < −α  ×   2 P(prec, pMC)  − R T T e 2 , −α < R < β ∝   −m C (D + R) , R > β  ×   where  n α2 A = ( )n exp(− ), |α| ∗ 2

n B = − |α|, |α|

2 m m − β C = ( ) e 2 |β| ×

m D = − |β| |β|

rec MC pT − pT σpT R = ( MC − µ)/ pT pT

The parameters are fit by χ2 minimization.

Considering the resolution of pT measurement, pT is smeared using the function fReso(pT mc).

p 2 2 2 fReso(pT mc) = a x + b (5.2) where, a = 0.0035324 and b = 0.00829197 63

The resolution is parameterized by a Double Crystal Ball function and a pT dependent resolution function. In this systematic uncertainty calculation, the parameter a is adjusted to a = 0.0035324 + 0.0003 and a = 0.0035324 - 0.0003.

5.3 DCA cut efficiency

Due to the short lifetime of J/ψ meson [63], to reconstruct the event of J/ψ decay to dielectron, it is required the distance of closest approach DCA < 1cm by default. The distance between the electron track is within 1cm away from vertex.

To calculate the systematic uncertainty from DCA configuration, DCA cut is adjusted from

DCA < 1cm to DCA < 0.8cm and DCA < 1.2cm and the maximum variation of polarization parameters’ value are assigned as systematic uncertainty from DCA cut configuration.

5.4 nHitsFit cut efficiency

It has been introduced that the charged particle’s trajectory is reconstructed from the readout of TPC by lining up the hits position left by the charged particle. The quantity nHitsFit refers to how many hits are used to reconstructed a track. So the nHitsFit cut affects whether a track is defined as a good track and then make a difference in J/ψ reconstruction. To calculate the systematic uncertainty from this aspect, the nHitsFit cut is adjusted to nHitsFit 18, ≥ nHitsFit 19, nHitsFit 22 and nHitsFit 25 and the maximum variation introduced by ≥ ≥ ≥ these changes is taken as systematic uncertainty from nHitsFit cut.

5.5 nHitsDedx cut efficiency

The energy loss per unit length is also calculated based on the readout of TPC. The quantity nHitsDedx indicates how many hits are used in the calculation of dE/dx, which affects the 64

accuracy of nσe measurement. The nHitsDedx cut is adjusted in both data and embedding to nHitsDedx 15 from nHitsDedx 11 and the variation of polarization parameters is ≥ ≥ assigned as a systematic uncertainty.

5.6 p/E cut efficiency

The ratio of momentum over energy p/E is used in the electron identification process. The cut on p/E together with other electron identification cuts will affect the number of electron candidates and their distribution. By adjusting the cut on p/E, we estimate the systematic uncertainty originated from the p/E cut. The p/E cut is adjusted in both data and embedding from 0.3 < p/E < 1.5 to 0.2 < p/E < 1.4 and 0.4 < p/E < 1.6. The maximum variation is taken as a systematic uncertainty.

5.7 ADC0 cut efficiency

The ADC0 cut is changed by multiplying the factor 0.95 or 1.05. For BHT0 trigger, it is changed in both data and embedding to ADC0 > 180 0.95(or1.05). For BHT2 trigger, it ∗ is changed to ADC0 > 300 0.95(or1.05). The maximum variation is taken as a systematic ∗ uncertainty.

5.8 TOF matching efficiency

The systematic uncertainty from TOF matching efficiency is calculated in 20 η bins and

−(p ∗p )p2 fitted by a function p0e 1 T . Parameters are adjusted by their uncertainties and the variation is assigned as the systematic uncertainty from TOF matching efficiency. 65 4 J/ YIELD EXTRACTION 17

Eminus eff Eminus eff Eminus eff hPtEff_0 hPtEff_1 hPtEff_2 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.496 Mean 2.539 Mean 2.489 -1.0<η<-0.9 RMS 1.277 -0.9<η<-0.8 RMS 1.257 -0.8<η<-0.7 RMS 1.249 χ2 / ndf 37.29 / 19 χ2 / ndf 70.98 / 19 χ2 / ndf 21.56 / 19 1.2 p0 1.795 ± 0.443 1.2 p0 4.21 ± 0.72 1.2 p0 3.869 ± 1.675 p1 976.4 ± 3626.0 p1 8.753e+04 ± 1.024e+05 p1 1.454e+05 ± 8.609e+05 p2 0.0495 ± 0.0162 p2 0.06972 ± 0.00595 p2 0.05798 ± 0.01239 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff Eminus eff hPtEff_3 hPtEff_4 hPtEff_5 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.529 Mean 2.459 Mean 2.524 -0.7<η<-0.6 RMS 1.247 -0.6<η<-0.5 RMS 1.273 -0.5<η<-0.4 RMS 1.225 χ2 / ndf 20.25 / 19 χ2 / ndf 37.53 / 19 χ2 / ndf 22.02 / 19 1.2 p0 5.353 ± 0.998 1.2 p0 7.325 ± 1.611 1.2 p0 6.713 ± 1.145 p1 6.894e+06 ± 1.077e+07 p1 3.079e+08 ± 5.717e+08 p1 2.957e+08 ± 2.229e+07 p2 0.05328 ± 0.00388 p2 0.04957 ± 0.00423 p2 0.04738 ± 0.00376 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff Eminus eff hPtEff_6 hPtEff_7 hPtEff_8 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.52 Mean 2.427 Mean 2.405 -0.4<η<-0.3 RMS 1.226 -0.3<η<-0.2 RMS 1.258 -0.2<η<-0.1 RMS 1.261 χ2 / ndf 21.8 / 19 χ2 / ndf 24.55 / 19 χ2 / ndf 21.99 / 19 1.2 p0 5.819 ± 1.757 1.2 p0 4.437 ± 2.620 1.2 p0 3.237 ± 8.191 p1 1.05e+07 ± 3.49e+07 p1 4.858e+05 ± 4.254e+06 p1 6.098e+05 ± 2.971e+07 p2 0.05497 ± 0.00654 p2 0.05823 ± 0.01861 p2 0.04567 ± 0.06425 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff Eminus eff hPtEff_9 hPtEff_10 hPtEff_11 1.4 Entries 24 1.4 Entries 24 1.4 Entries 23 Mean 2.407 Mean 2.447 Mean 2.564 -0.1<η<0.0 RMS 1.281 0.0<η<0.1 RMS 1.26 0.1<η<0.2 RMS 1.24 χ2 / ndf 26.5 / 19 χ2 / ndf 8.799 / 19 χ2 / ndf 8.757 / 18 1.2 p0 4.301 ± 1.301 1.2 p0 4.438 ± 1.017 1.2 p0 6.259 ± 0.778 p1 4.211e+07 ± 1.301e+08 p1 3.577e+07 ± 7.688e+07 p1 9.395e+06 ± 1.185e+07 p2 0.04279 ± 0.00715 p2 0.04347 ± 0.00626 p2 0.05744 ± 0.00334 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff Eminus eff hPtEff_12 hPtEff_13 hPtEff_14 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.438 Mean 2.538 Mean 2.505 0.2<η<0.3 RMS 1.262 0.3<η<0.4 RMS 1.24 0.4<η<0.5 RMS 1.244 χ2 / ndf 24.83 / 19 χ2 / ndf 17.67 / 19 χ2 / ndf 9.636 / 19 1.2 p0 4.166 ± 0.775 1.2 p0 5.045 ± 3.563 1.2 p0 4.183 ± 0.574 p1 8.583e+06 ± 8.979e+05 p1 2.532e+06 ± 2.570e+07 p1 2.652e+06 ± 2.679e+05 p2 0.04416 ± 0.00613 p2 0.05462 ± 0.01738 p2 0.0471 ± 0.0052 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff Eminus eff hPtEff_15 hPtEff_16 hPtEff_17 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.482 Mean 2.526 Mean 2.512 0.5<η<0.6 RMS 1.262 0.6<η<0.7 RMS 1.238 0.7<η<0.8 RMS 1.263 χ2 / ndf 18.57 / 19 χ2 / ndf 14.31 / 19 χ2 / ndf 28.89 / 19 1.2 p0 5.635 ± 1.180 1.2 p0 7.053 ± 1.674 1.2 p0 8.506 ± 2.451 p1 4.979e+07 ± 1.309e+08 p1 1.045e+09 ± 2.794e+09 p1 1.672e+10 ± 5.104e+10 p2 0.04787 ± 0.00511 p2 0.04522 ± 0.00426 p2 0.04356 ± 0.00352 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eminus eff Eminus eff hPtEff_18 hPtEff_19 1.4 Entries 24 1.4 Entries 24 Mean 2.512 Mean 2.546 0.8<η<0.9 RMS 1.274 0.9<η<1.0 RMS 1.269 χ2 / ndf 71.66 / 19 χ2 / ndf 37.76 / 19 1.2 p0 12.48 ± 3.86 1.2 p0 24.08 ± 9.59 p1 3.589e+12 ± 7.595e+12 p1 1.971e+14 ± 5.784e+14 p2 0.04149 ± 0.00363 p2 0.04258 ± 0.00391 1 1

0.8 0.8

0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) T T

Figure 15: TOF matching eciency of electron in ⌘ range (-1,1). Figure 37: Electrons TOF matching efficiency in different η range as a function of pT . 66 4 J/ YIELD EXTRACTION 16

Eplus eff Eplus eff Eplus eff hPtEff_0 hPtEff_1 hPtEff_2 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.528 Mean 2.528 Mean 2.489 -1.0<η<-0.9 RMS 1.264 -0.9<η<-0.8 RMS 1.253 -0.8<η<-0.7 RMS 1.248 χ2 / ndf 39.47 / 19 χ2 / ndf 48.24 / 19 χ2 / ndf 20.49 / 19 1.2 p0 1.602 ± 0.296 1.2 p0 3.953 ± 0.554 1.2 p0 2.336 ± 0.219 p1 108.8 ± 254.4 p1 2.691e+04 ± 2.069e+04 p1 140.5 ± 167.8 p2 0.05918 ± 0.01675 p2 0.07634 ± 0.00517 p2 0.08302 ± 0.01085 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff Eplus eff hPtEff_3 hPtEff_4 hPtEff_5 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.48 Mean 2.492 Mean 2.459 -0.7<η<-0.6 RMS 1.244 -0.6<η<-0.5 RMS 1.237 -0.5<η<-0.4 RMS 1.256 χ2 / ndf 26.92 / 19 χ2 / ndf 18.84 / 19 χ2 / ndf 13.01 / 19 1.2 p0 3.065 ± 0.674 1.2 p0 2.493 ± 0.684 1.2 p0 1.632 ± 0.483 p1 5436 ± 14852.0 p1 200.1 ± 631.2 p1 3.567 ± 11.399 p2 0.06526 ± 0.01014 p2 0.08432 ± 0.01860 p2 0.09642 ± 0.02680 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff Eplus eff hPtEff_6 hPtEff_7 hPtEff_8 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.528 Mean 2.545 Mean 2.387 -0.4<η<-0.3 RMS 1.263 -0.3<η<-0.2 RMS 1.247 -0.2<η<-0.1 RMS 1.264 χ2 / ndf 15.62 / 19 χ2 / ndf 30.24 / 19 χ2 / ndf 41.17 / 19 1.2 p0 3.158 ± 0.640 1.2 p0 2.969 ± 0.765 1.2 p0 3.282 ± 0.964 p1 1563 ± 2340.9 p1 3.711e+04 ± 1.414e+05 p1 1.166e+06 ± 4.850e+06 p2 0.08399 ± 0.00701 p2 0.05349 ± 0.01158 p2 0.04452 ± 0.00906 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff Eplus eff hPtEff_9 hPtEff_10 hPtEff_11 1.4 Entries 23 1.4 Entries 23 1.4 Entries 23 Mean 2.514 Mean 2.543 Mean 2.531 -0.1<η<0.0 RMS 1.26 0.0<η<0.1 RMS 1.248 0.1<η<0.2 RMS 1.239 χ2 / ndf 22.98 / 18 χ2 / ndf 10.98 / 18 χ2 / ndf 23.76 / 18 1.2 p0 3.751 ± 12.506 1.2 p0 4.501 ± 2.002 1.2 p0 3.731 ± 1.181 p1 5e+05 ± 2.8e+07 p1 3.579e+07 ± 2.164e+08 p1 1.469e+06 ± 6.765e+06 p2 0.05182 ± 0.08934 p2 0.04404 ± 0.00750 p2 0.04722 ± 0.00908 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff Eplus eff hPtEff_12 hPtEff_13 hPtEff_14 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.469 Mean 2.47 Mean 2.45 0.2<η<0.3 RMS 1.263 0.3<η<0.4 RMS 1.262 0.4<η<0.5 RMS 1.261 χ2 / ndf 18.39 / 19 χ2 / ndf 18.59 / 19 χ2 / ndf 16.65 / 19 1.2 p0 2.999 ± 0.457 1.2 p0 4.486 ± 1.493 1.2 p0 3.41 ± 0.81 p1 2.114e+06 ± 6.151e+06 p1 1.493e+06 ± 7.312e+06 p1 3.615e+04 ± 1.065e+05 p2 0.03619 ± 0.00614 p2 0.05271 ± 0.00989 p2 0.05759 ± 0.00698 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff Eplus eff hPtEff_15 hPtEff_16 hPtEff_17 1.4 Entries 24 1.4 Entries 24 1.4 Entries 24 Mean 2.518 Mean 2.49 Mean 2.492 0.5<η<0.6 RMS 1.247 0.6<η<0.7 RMS 1.25 0.7<η<0.8 RMS 1.254 χ2 / ndf 19.63 / 19 χ2 / ndf 29.04 / 19 χ2 / ndf 23.97 / 19 1.2 p0 6.047 ± 1.427 1.2 p0 4.539 ± 0.555 1.2 p0 6.256 ± 1.759 p1 1.116e+06 ± 2.341e+06 p1 1.121e+06 ± 6.456e+04 p1 4.214e+07 ± 1.245e+08 p2 0.06438 ± 0.00544 p2 0.05379 ± 0.00448 p2 0.05206 ± 0.00521 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) p (GeV/c) T T T Eplus eff Eplus eff hPtEff_18 hPtEff_19 1.4 Entries 24 1.4 Entries 24 Mean 2.527 Mean 2.549 0.8<η<0.9 RMS 1.28 0.9<η<1.0 RMS 1.3 χ2 / ndf 90.35 / 19 χ2 / ndf 57.15 / 19 1.2 p0 8.34 ± 2.47 1.2 p0 9.366 ± 3.207 p1 1.724e+10 ± 3.117e+10 p1 2.145e+10 ± 2.967e+09 p2 0.04508 ± 0.00407 p2 0.04788 ± 0.00460 1 1

0.8 0.8

0.6 0.6

TOF matching efficiency 0.4 TOF matching efficiency 0.4

0.2 0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 p (GeV/c) p (GeV/c) T T

Figure 14: TOF matching eciency of positron in ⌘ range (-1,1). Figure 38: Positron TOF matching efficiency in different η range as a function of pT . 67

In Equation 3.5, r = P(TOF)/P(TOF|EMC). Figure 39 shows this ratio of electron from this study and the ratio of pion from low luminosity runs used in 2011 D∗ analysis [64]. The correlation between TOF and EMC efficiencies depends on their geometrical acceptances. At high pT , where the tracks are closer to straight lines, we expect the correlation independent of pT , as Figure 39 shows. Therefore, a constant extrapolation for the ratio r is applied to pT > 1GeV/c. We use the ratio r of electrons in Figure 39 to calculate the J/ψ efficiency and assign the difference between this efficiency and the one calculated by the r from pions as a 5 J/ YIELD EXTRACTION 28 uncertainty.

1.2

1

(TOF|EMC) 0.8 ∈

0.6 (TOF)/

∈ 0.4 electron 0.2 pion 0 0 1 2 3 4 p GeV/c T

Figure 39: Ratio ofFigure TOF 26: efficiency Ratio of over TOF TOF eciency efficiency over TOF condition eciency oncondition no EMC on no matching for EMC matching for electron and pion in p+p MB at 200 GeV. Pion is taken electron and pion in p+p MB at 200 GeV.from Pion David’s is taken D* 2011AN. from David’s D∗ 2011 analysis [64]. 68

5.9 TOF 1/β cut efficiency

1/β distribution Tof_mean 1.01 Entries 18 Mean 1.449 RMS 1.013 1.005

1

0.995

0.99

0.985

0.98 0.5 1 1.5 2 2.5 3 3.5 4

Mon Oct 16 13:29:42 2017 p GeV/c Figure 40: The mean of the Gaussian function sampling the 1/β distribution. 69

1/β σ distribution

0.02

0.019

0.018

0.017

0.016

0.015

0.014

0.013

0.012

0.011 0.5 1 1.5 2 2.5 3 3.5 4

Mon Oct 16 16:13:37 2017 p GeV/c Figure 41: The sigma of the Gaussian function sampling the 1/β distribution.

The Gaussian function is used to sample the 1/β distribution and calculate TOF 1/β cut efficiency. The mean of Gaussian function is the bin content of Figure 40 at corresponding momentum and the variation of Gaussian function is the bin content of Figure 41 at corre- sponding momentum. Using the Gaussian function to roll a random number, if the random number is within the 1/β cut, then this track passes the 1/β cut. In the systematic uncertainty 70 calculation, the mean or the variation of Gaussian function is adjusted in embedding by its uncertainty separately. The maximum variation from these changes is assigned as a systematic uncertainty from 1/β cut efficiency.

5.10 nσe cut efficiency

In the embedding, a Gaussian function is used to sample the nσe distribution by setting the nσe value to each track, where the mean and variance of the Gaussian function is determined by photonic electron study. The default choice of mean and variance of the Gaussian function is from the fit by a constant (see Figure 42). In the systematic uncertainty calculation, the mean and variance of the Gaussian function is fit by the function y = kx + b. In this way, the pT dependence of the mean and variance of the Gaussian function are taken into the consideration.

And the maximum of absolute difference get from these new efficiency is assigned as systematic uncertainty from the nσe cut configuration. 71

mean of Gaussian function and fit by POL0 mean of Gaussian function and fit by POL1

µ 1 µ 1 nsigma_mean_0 nsigma_mean_linear Entries 8 Entries 8 Mean 7.417 Mean 7.417 0.5 RMS 3.481 0.5 RMS 3.481 χ2 / ndf 4.311 / 7 χ2 / ndf 4.05 / 6 p0 −0.3947 ± 0.0066 p0 −0.3839 ± 0.0222 p1 −0.002503 ± 0.004906 0 0

−0.5 −0.5

−1 −1

−1.5 −1.5 4 6 8 10 12 14 16 4 6 8 10 12 14 16 p GeV/c p GeV/c T T

sigma of Gaussian function and fit by POL0 sigma of Gaussian function and fit by POL1

σ 4 σ 4 nsigma_sigma_0 nsigma_sigma_linear Entries 8 Entries 8 3.5 Mean 7.698 3.5 Mean 7.698 RMS 3.61 RMS 3.61 3 χ2 / ndf 6.484 / 7 3 χ2 / ndf 6.184 / 6 p0 0.8711 ± 0.0051 p0 0.8619 ± 0.0176 2.5 2.5 p1 0.002173 ± 0.003971

2 2

1.5 1.5

1 1

0.5 0.5

4 6 8 10 12 14 16 4 6 8 10 12 14 16 p GeV/c p GeV/c Tue Jan 30 12:20:29 2018 T T

Figure 42: Top left is the mean of Gaussian function with the fit by POL0. Top right is the mean of Gaussian function with the fit by POL1. Bottom left is the sigma of Gaussian function with fit by POL0. Bottom right is the sigma of Gaussian function with the fit by POL1 72

5.11 Rapidity weight function

The systematic uncertainty from rapidity weight is calculated by adjusting the parameters

of rapidity weight function based Barbara’s study [58]. The parameter is adjusted in embedding

by its uncertainty and the variation of polarization parameter measurement is assigned as a

systematic uncertainty.

5.12 Systematic uncertainty combination

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

θ 2 φ 2 φ 2 2 θ λ λ inv λ λ

1.5 λ 1.5 λ 1.5 λ 1.5 λ θ in Helicity frame φ in Helicity frame θφ in Helicity frame inv in Helicity frame

1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

default p smearing −0.5 −0.5 −0.5 −0.5 T Dedx DCA NHitsFit −1 −1 −1 −1 Dsmadc 1/β p/E σ n e −1.5 −1.5 −1.5 −1.5 weight TOF|!EMC TOF matching 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 rapidity8 10

φ − φ − − θ −2 2 2 2 θ λ λ 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 inv 0 2 4 6 8 10 λ p GeV/c p GeV/c p GeV/cλ p GeV/c T T T T 1.5 λ 1.5 λ 1.5 λ 1.5 λ θ in Collins-Soper frame φ in Collins-Soper frame θφ in Collins-Soper frame inv in Collins-Soper frame 1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5

−1 −1 −1 −1

−1.5 −1.5 −1.5 −1.5

−2 −2 −2 −2 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 p GeV/c p GeV/c p GeV/c p GeV/c T T T T

Figure 43: Systematic uncertainty of λθ, λφ,λθφ and λinv in Helicity and Collins-Soper frame

from different sources.

Wed Aug 14 18:38:41 2019 73

The systematic uncertainties from different sources in different pT bins are shown in Fig-

ure 43.

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

θ 2 φ 2 φ 2 2 θ λ λ inv λ

λ

1.5 λ 1.5 λ 1.5 λ 1.5 λ θ in Helicity frame φ in Helicity frame θφ in Helicity frame inv in Helicity frame

1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5

−1 −1 −1 −1

−1.5 −1.5 −1.5 −1.5

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

φ − φ − − θ −2 2 2 2 θ λ λ 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 inv 0 2 4 6 8 10 λ

λ

p bin 0 p bin 0 p bin 0 p bin 0 T T T T 1.5 λ 1.5 λ 1.5 λ 1.5 λ θ in Collins-Soper frame φ in Collins-Soper frame θφ in Collins-Soper frame inv in Collins-Soper frame 1 1 1 1

0.5 0.5 0.5 0.5

0 0 0 0

−0.5 −0.5 −0.5 −0.5

−1 −1 −1 −1

−1.5 −1.5 −1.5 −1.5

−2 −2 −2 −2 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 p bin 0 p bin 0 p bin 0 p bin 0 T T T T

Figure 44: Systematic uncertainty combination of λθ, λφ,λθφ and λinv in Helicity and Collins-

Soper frame.

Figure 44 shows the combined systematic uncertainty, which is square root of summation

of all the systematic uncertainty square.

Wed Aug 14 18:38:42 2019 CHAPTER 6

RESULTS ON J/ψ POLARIZATION

As shown in Figure 45 and Figure 46, the J/ψ polarization parameters have been extracted in the dielectron channel (open circles) in both the Helicity and Collins-Soper frames. These are the first measurement of λφ, λθφ and λinv in the Helicity frame, and the first J/ψ polarization measurements of λθ, λφ, λθφ and λinv in the Collins-Soper frame for p+p collisions at √s =

200GeV at STAR experiment. The dielectron and dimuon results are shown as open and

filled circles, and are consistent with each other in the overlapping pT range [65]. The J/ψ’s polarization results do not exhibit significant transverse or longitudinal polarization with little dependence on pT .

As shown in Figure 45, there are four theoretical models compared with experimental mea- surements. These are the theoretical results calculated by the improved color evaporation model (ICEM) [66], NRQCD with two sets of LDMEs denoted as ”NLO NRQCD1” [67] and

”NLO NRQCD2” [68] respectively and color glass condensate (CGC) [69] in conjunction with

NRQCD. Because the theoretical predictions and experimental measurements are all close to the the line that λ = 0, both theoretical models and experimental results don’t suggest promi- nent J/ψ polarization. Among these model calculations compared to the experimental result, the CGC+NRQCD achieves the best agreement overall. As shown in Figure 46, the λinv values measured in the two frames are consistent with each other within experimental uncertainties and consistent with the CGC+NRQCD calculations within uncertainties. 74 75

TABLE IV: χ2/NDF and the corresponding p-values between data and different model calcu- lations. Model χ2/NDF p-value ICEM [66] 13.28/9 0.150 NRQCD1 [67] 48.81/32 0.029 NRQCD2 [68] 42.99/32 0.093 CGC+NRQCD [69] 32.11/46 0.940

2 2 STAR (a) p+p s = 200 GeV (b) - J/ψ→e+e-, HX, |y|<1.0 J/ψ→e+e , CS, |y|<1.0 + - 1 J/ψ→µ+µ-, HX, |y|<0.5 1 J/ψ→µ µ , CS, |y|<0.5 θ λ 0 0

ICEM(prompt) 1.2

0 0 ϕ λ

−0.5 −0.5

0 5 10 (e) 0 5 10 (f) 0.5 0.5

ϕ 0 0 θ λ (prompt) −0.5 NLO NRQCD1: |y|<1.0 |y|<0.5−0.5 (direct) NLO NRQCD2: |y|<1.0 |y|<0.5 CGC+NRQCD: |y|<1.0 |y|<0.5 0 5 10 0 5 10 p (GeV/c) p (GeV/c) T T

Figure 45: The J/ψ polarization parameters (from top to bottom: λθ, λφ, λθφ) as a function of pT in the Helicity (left) and Collins-Soper (right) frames. 76

3 J/ψ→e+e-, HX, |y|<1.0 J/ψ→e+e-, CS, |y|<1.0 J/ψ→µ+µ-, HX, |y|<0.5 2 J/ψ→µ+µ-, CS, |y|<0.5 CGC+NRQCD, |y|<1.0 CGC+NRQCD, |y|<0.5 1 inv λ

0

−1 STAR p+p s = 200 GeV 0 5 10 p (GeV/c) T

Figure 46: The J/ψ polarization parameters (from left to right: λθ, λφ, λθφ) as a function of pT in the Helicity (top) and Collins-Soper (bottom) frames. CHAPTER 7

SUMMARY AND OUTLOOK

Precise measurement of anisotropic property of J/ψ polarization can enhance the under- standing of physics on quarkonium production mechanism. The presented measurement in this thesis measurement of J/ψ polarization utilizes the advanced detectors at STAR for track recon- struction and particle identification. The estimation on J/ψ polarization parameters is based on maximum likelihood estimation. Properties of the estimator is analyzed by Monte Carlo technique. Compared with theoretical predictions, the theory CGC together with NRQCD best describes the result of measurements. Measurements of J/ψ polarization in p+Au and

Au+Au collisions may provide insights into cold and hot nuclear matter effects on quarkonium production, which has been used extensively to study the properties of Quark-Gluon Plasma.

The further deeper and more comprehensive result can be expected from the J/ψ polarization measurements in p+Au and Au+Au collisions at √sNN = 200GeV using the data taken in 2011 and 2015.

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NAME Siwei Luo

EDUCATION B.S., Physics, Northeast Forestry University, Harbin, Heilongjiang Province, China 2011 M.S., Physics, Southern Illinois University, Carbondale, Illinois, U.S. 2013

CONFERENCES Siwei Luo “J/ψ polarization measurement in p+p collisions at √s = 200 GeV at STAR experiment” American Physics Society (2015).

Siwei Luo “J/ψ polarization measurement in p+p collisions at √s = 200 GeV at STAR experiment” Quark Matter (2017).

Siwei Luo “Measurement of J/ψ polarization in p+p at √s = 200 GeV by the STAR experiment” American Physics Society Prairie Section Fall Meeting 2017 (2017).

AWARDS American Physics Society-DNP Travel Award April 2015 Meeting

Student Support from 11th Hadron Collider Physics 2016 Summer School organizers

Young Scientist Support for Accommodation from Quark Matter 2017 organizers

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