Quarkonium Suppression Using 3+1D Anisotropic Hydrodynamics

QUARKONIUM SUPPRESSION USING 3+1D ANISOTROPIC HYDRODYNAMICS

A dissertation submitted to Kent State University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Brandon Krouppa

August, 2018

c Copyright

Except for previously published materials Dissertation written by

Brandon Krouppa

BS, Jacksonville University, 2013

MA, Kent State University, 2015

PhD, Kent State University, 2018

Approved by

, Chair, Doctoral Dissertation Committee Dr. Michael Strickland

, Members, Doctoral Dissertation Committee Dr. Spyridon Margetis

Dr. Declan Keane

Dr. Khandker Quader

Dr. Edgar Kooijman

Dr. Diane Stroup

Accepted by

, Chair, Department of Physics Dr. James T. Gleeson

, Dean, College of Arts and Sciences Dr. James L. Blank Table of Contents

Table of Contents ...... iii

List of Figures ...... vi

List of Tables ...... xiii

List of Publications ...... xv

Acknowledgments ...... xvi

1 Introduction ...... 1

1.1 Units and notation ...... 1

1.2 The standard model ...... 3

1.3 Quantum Electrodynamics (QED) ...... 4

1.4 Quantum Chromodynamics (QCD) ...... 6

1.5 The running coupling in QED and QCD ...... 7

1.6 Hadrons ...... 8

1.6.1 Heavy quarkonia ...... 10

1.7 The QCD phase diagram ...... 12

1.8 Heavy-ion collisions ...... 13

1.8.1 Stages of heavy-ion collisions ...... 14

1.8.2 Centrality ...... 15

1.8.3 A hydrodynamical description of the quark-gluon plasma ...... 18

1.9 Quarkonium suppression ...... 20

1.9.1 Other suppression eﬀects ...... 23

iii 2 The heavy quark potential ...... 24

2.1 Heavy quarkonia ...... 24

2.2 A perturbative inspired heavy-quark potential (Strickland-Bazow) ...... 25

2.3 A lattice QCD vetted heavy-quark potential (Rothkopf) ...... 26

2.4 Non-equilibrium corrections to the lattice-vetted heavy quark potential . . . 34

2.5 Solving for binding energies ...... 35

2.5.1 Strickland-Bazow potential discretizations ...... 37

2.5.2 Rothkopf potential discretizations ...... 38

2.6 Discussion on diﬀerence between potentials ...... 38

3 3+1d anisotropic hydrodynamics ...... 40

3.1 Anisotropic hydrodynamics ...... 40

3.1.1 The anisotropic equation of state ...... 46

3.2 aHydro initial conditions ...... 47

3.2.1 Determining the initial temperature-like scales ...... 49

4 Computing the suppression factor RAA ...... 52

4.1 Computing RAA ...... 52

4.1.1 Regenerative eﬀects ...... 56

5 Results for quarkonium suppression ...... 69

5.1 Charmonium ...... 69

5.2 Bottomonium ...... 70

6 Discussion & outlook ...... 79

6.1 Charmonium ...... 79

6.2 Bottomonium ...... 80

6.3 Outlook ...... 81

iv 7 Appendix ...... 82

7.1 Finite initial anisotropy parameter ...... 82

7.2 Adjustment of the formation time of heavy quarkonia ...... 83

7.3 Model sensitivity of the ﬁnal cutoﬀ temperature for survival probability inte-

gration ...... 84

Bibliography ...... 85

v List of Figures

1.1 The Standard Model as of June 2018...... 4

1.2 The three fundamental vertices found in QCD. On the left we see the three-

gluon vertex, in the middle we see the four-gluon vertex, and on the right we

see the quark-gluon vertex. Figure taken from Reference [1]...... 6

2 1.3 A summary of the QCD running coupling constant αs(Q ) from [2]...... 9

1.4 Various types of hadron examples and their classiﬁcation...... 9

1.5 A plot of the pNRQCD-based heavy quark potential for three temperatures.

The real part seen in this plot starts with the vacuum Cornell potential at

T = 0 GeV, and as the temperature increases, the quarkonia are subject to

melting...... 11

1.6 The phase diagram, as currently understood, with various high-energy exper-

iments outlined to probe various parts of the diagram. Figure taken from

[3]...... 12

1.7 Stages of heavy-ion collisions. Figure taken from [4]...... 14

1.8 The transverse proﬁle of two colliding nuclei. The overlap is measured by the

impact parameter, b, the distance between the center of the two nuclei. When

calculated with the Glauber model, the overlapping region can be quantiﬁed

by the number of partons participating in the collision, Npart...... 16

1.9 The number of participants, Npart, as a function of the impact parameter, b,

for various collision systems studied in this work...... 17

vi 1.10 Charge-hardon multiplicity as a function of pseudorapidity, compared to AL-

ICE 2.76 TeV/nucleon collisions. The percentage ranges given in the ﬁgure

are various centrality classes, with 0 5% relating to the 5% most central − collisions, and so on. Figure taken from [5], with ALICE data provided from

[6, 7]...... 19

1.11 As a function of the dimuon invariant mass, we see three resonances appear in

proton-proton reference data, and then disappear in Pb-Pb collisions at 5.02

TeV. This suppression of states is thought to be due to the formation of the

QGP. Figure taken from [8]...... 21

1.12 An estimate of shadowing (a cold nuclear matter eﬀect) on Υ(1S) suppression

at 2.76 TeV/nucleon LHC collisions of Pb-208 nuclei...... 23

2.1 The in-medium heavy quark potential [real part (left) and imaginary part

(right)] in full QCD with Nf = 2 + 1 light quark ﬂavors based on ensembles

by the HotQCD collaboration (colored points, shifted for better readability).

By adjusting the Debye mass mD, the lattice QCD values of the real part are

adequately reproduced with the Gauss-law parameterization as shown with

the solid lines, at all temperature and length scales. Theoretical error bars (the

shaded region around the solid lines) are due to the uncertainty of the Debye

mass ﬁt. The imaginary part shows good agreement at high temperatures

and small distances, while at temperatures close to the crossover transition,

deviations from the lattice data are seen. Note that the crossover temperature

on these lattices due to the relatively large pion mass of mπ 300MeV lies ≈

at TC = 172.5 MeV...... 31

vii 2.2 The Re[V] (left) and Im[V] (right) part of the in-medium heavy quark poten-

tial which has been vetted against lattice QCD. The diﬀerent lines correspond

to diﬀerent values of the Debye mass of the medium. The vacuum parameters

αs, σ, and c at mD = 0 are tuned to reproduce the PDG charmonium and bot-

tomonium spectra. String breaking is enforced at rsb = 1.25 fm. Ultimately,

only one parameter governs the temperature-dependence which modiﬁes the

vacuum (Cornell) potential. Such thermal eﬀects lead to Debye screening

of the real part of the potential and induce a ﬁnite imaginary part which

asymptotes at large distances...... 34

2.3 Comparison between Strickland-Bazow Υ(1S) and Rothkopf Υ(1S) for the

cases of an isotropic, ξ = 0, plasma (left), and a slightly oblate, ξ = 1 plasma.

In the Rothkopf model we observe weaker binding and a slightly stronger

screening eﬀect from the imaginary part of the binding energy. Here Tc = 192

MeV which then sets the scale of the temperature-like scales of the calculation.

The discontinuities seen in Im[V] are due to the real part going negative. . . 39

3.1 Plotted with the dimensionless time variablew ¯ = τ = τT , we see varying τR 5η/s

L/ T initial conditions converge rather quickly to an attractor. Figure taken P P from Reference [9]...... 41

3.2 Shear tensor corrections calculated using second-order viscous hydrodynamics

Reference [10]...... 42

3.3 From the Glauber model, we calculate the transverse energy density proﬁle

for a given impact parameter, b = 4.5...... 48

viii 3.4 Using a boost-invariant plateau with Gaussian tails, we can phenomenolog-

ically model the longitudinal proﬁle for the initial energy density produced

in an ultra-relativistic heavy-ion collision for RHIC 200 GeV/nucleon Au-Au

collisions and LHC 2.76 TeV/nucleon and 5.02 TeV/nucleon Pb-Pb collisions. 50

4.1 Heavy-quarkonium production at the CMS and ATLAS experiments [11]. . . 53

4.2 cc¯ production across multiple collision energies and experiments [12]. . . . . 60

4.3 Using this plot, we can derive the factors going from 1.6 < y < 2.4 to y < | | | | 1.6 (x1.5), and 2.5 < y < 4.0 to y < 1.6 (x2). Linear interpolation was used | | | | to ﬁll in the gaps from mid-rapidity to ultra-forward rapidity 2.5 < y < 4.0 | | [13]...... 61

4.4 J/ψ production in 5.02 TeV p-p collisions at CMS [14]...... 62

dσ 4.5 cc¯ dy production across two energies and a range of forward rapidities. With interpolation to mid-rapidity, this ﬁgure is used to derive the multiplicative

factors in going from 1.6 < y < 2.4 to y < 1.6 (x1.5), and 2.5 < y < 4.0 | | | | | | to y < 1.6 (x2) [15]...... 63 | |

4.6 χcJ production cross sections as a function of transverse momentum. Note

+ − that B = (χcJ J/ψγ) (J/ψ µ µ ) [16]...... 64 B → ·B → 4.7 Diﬀerential cross sections for the Υ(nS) states [17]...... 65

4.8 Diﬀerential cross sections for the Υ(nS) states [18]...... 66

4.9 Production cross sections for χb(mP ) states given as a ratio to Υ(1S) produc-

tion cross sections [19]...... 67

5.1 Raw suppression for the J/ψ, ψ(2S), and χc(1P ) charmonium states for LHC

2.76 TeV/nucleon Pb-Pb collisions. For this ﬁgure the states are modeled

with the Strickland-Bazow potential. The lack of suppression for the ψ(2S)

state is due to the large formation time...... 70

ix 5.2 Inclusive RAA calculation for the J/ψ state in 2.76 TeV/nucleon Pb-Pb colli-

sions at the LHC, as calculated by the Strickland-Bazow potential...... 71

5.3 Raw suppression for Υ(nS) and χb(mP ) states from LHC 2.76 TeV/nucleon

Pb-Pb collisions. With both potentials we see a sequential ordering of suppres-

sion and smooth transitions in RAA from peripheral to central collisions. The

order of sequential suppression is the same in both potentials, except for the

pileup of largely-suppressed states, which is more evident with the Rothkopf

potential. This pileup is due to a “halo” survival eﬀect for bottomonia formed

near the edge of the QGP...... 72

5.4 Bottomonium suppression for RHIC 200 GeV/nucleon collisions as seen by

the STAR collaboration. The Strickland-Bazow potential features an error

band which corresponds to varying values of η/s while the Rothkopf potential

features a band corresponding to uncertainty in the Debye mass calculation

as a function of temperature. STAR data taken from [20]...... 73

5.5 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). Comparisons between our model calculations and

bottomonium suppression seen at LHC 2.76 TeV/nucleon Pb-Pb collisions as

viewed by the CMS collaboration. CMS data taken from [21]...... 74

5.6 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). In LHC 2.76 TeV/nucleon Pb-Pb collisions, we

make a comparison between model calculations and bottomonium suppres-

sion data gathered from the ALICE collaboration. ALICE data taken from

[22]...... 75

x 5.7 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). In this ﬁgure, we can state within reason, there

is no signiﬁcant rapidity dependence for Υ(1S) suppression for LHC 2.76

TeV/nucleon collisions. With some tension from forward-rapidity ALICE

data, the more strongly suppressed Rothkopf Υ(1S) provides a reasonable

description of suppression. CMS and ALICE data taken from [21, 22]. . . . . 76

5.8 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). For transverse momentum comparisons, we see no

signiﬁcant pT dependence in the data while the model calculations show a

decreasing suppression for larger pT due to larger formation times eﬀectively

shielding formed bottomonia from thermal suppression. CMS data taken from

[21]...... 76

5.9 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). Predictions made with the Strickland-Bazow po-

tential agree quite well with CMS data at 5.02 TeV/nucleon Npart data for

all observed Υ(nS) states. Not seen is the Υ(3S) conﬁdence intervals for a

completely suppressed Υ(3S). CMS data taken from [8]...... 77

5.10 Strickland-Bazow potential calculation (left), compared to the Rothkopf po-

tential calculation (right). For 5.02 TeV/nucleon collisions, we see the Strickland-

Bazow potential calculation successfully predicts ALICE data. The Rothkopf

potential calculation slightly overestimates suppression. ALICE data taken

from [23]...... 77

xi 5.11 The Strickland-Bazow potential calculation (left) predicted CMS and ALICE

rapidity data successfully. The Rothkopf potential calculation (right) slightly

overestimates Υ(1S) suppression, but explains Υ(2S) and Υ(3S) suppression

successfully. Note that conﬁdence levels for a fully suppressed Υ(3S) state

are not shown here. CMS and ALICE data taken from [8, 23]...... 78

5.12 There is a slight increase in Υ(1S) experimental data as well as the Strickland-

Bazow potential model calculation (left). The Rothkopf potential model calcu-

lation describes Υ(1S) for low-pT , but overall seems to overestimate suppres-

sion for the Υ(1S). Both potential calculations describe Υ(2S) suppression

for this given collision system. CMS data taken from [8]...... 78

7.1 We make an adjustment to the initial geometry of the collision by deforming

the initial momentum-space distribution to an oblate spheroidal distribution. 82

7.2 As we adjust the formation time of the bottomonium states, the inclusive RAA

is aﬀected by 10%. Adjusting this quantity measures the “shielding” eﬀect ∼ the formation time has for heavy quarkonia...... 83

7.3 The survival probability is cut oﬀ at a given temperature, Tf . As we adjust

Tf , we see a small eﬀect on ﬁnal inclusive Υ(1S) RAA which becomes less

sensitive as the cutoﬀ temperature decreases...... 84

xii List of Tables

1.1 The relationship between impact parameter, b, and centrality class, c, includ-

ing the average impact parameter for each centrality class in a √sNN = 2.76

TeV/nucleon Pb-Pb collision modeled using optical Glauber. Table taken

from [24]...... 17

3.1 Λ0 values (in GeV) for various collision energies, ξ0, and η/s...... 51

4.1 Feed down fractions which have been averaged over pT , for the J/ψ and ψ(2S)

states used in the determination of the ﬁnal measured yields [11]. Note that

the feed-down fractions for the ψ(2S) are estimates due to lack of measured

feed down data...... 55

4.2 Feed down fractions which have been averaged over pT , for the Υ(nS) states

used in the determination of the ﬁnal measured yields [11]. Note that the

feed-down fractions for the Υ(3S) are estimates due to lack of measured feed

down data...... 55

4.3 cc¯ production cross sections calculated from Figure 4.2. The mid-rapidity

value for 200 GeV collisions was taken directly from [25]. The mid-rapidity

value for 2.76 TeV collisions was taken directly from [26]. The mid-rapidity

value for 5.02 TeV collisions was derived via interpolating data given in Figure

4.2. The interpolation yields a 40% increase from 2.76 TeV to 5.02 TeV. . . . 60

J/ψ 4.4 Fixed via a ratio of cc¯ = 0.6%, we compute the diﬀerential cross sections of J/ψ states...... 62

σ(ψ(2S)) 4.5 Fixed via a ratio of σ(J/ψ) = 0.17, we compute the cross sections of ψ(2S) states...... 63

4.6 Cross sections for the χc(1P ) state computed via χc/ψ(2S) = 1.7%...... 65

xiii dσ 4.7 Diﬀerential cross sections dy from [27] for the Υ(1S) and Υ(2S) states. Υ(3S) states are computed from [17] for 2.76 TeV and [18] for 5.02 TeV collisions.

For RHIC collision energy, we take an extrapolated value for the diﬀerential

cross section...... 66

4.8 Calculated cross sections from [27] for the Υ(1S) and Υ(2S) states. Υ(3S)

states are computed from [17] for 2.76 TeV and [18] for 5.02 TeV collisions.

For ALICE rapidity cuts, the stated rapidity coverage for dimuons (from [28])

is 2.5 < y < 4.0...... 67

4.9 Calculated cross sections for the χb(mP ) states...... 68

xiv List of Publications

1. B. Krouppa, R. Ryblewski and M. Strickland, “Bottomonia suppression in 2.76 TeV

Pb-Pb collisions,” Phys. Rev. C 92, no. 6, 061901 (2015) doi:10.1103/PhysRevC.92.061901

[arXiv:1507.03951 [hep-ph]].

2. B. Krouppa and M. Strickland, “Predictions for bottomonia suppression in 5.023

TeV Pb-Pb collisions,” Universe 2, no. 3, 16 (2016) doi:10.3390/universe2030016

[arXiv:1605.03561 [hep-ph]].

3. B. Krouppa, R. Ryblewski and M. Strickland, “Bottomonium suppression in heavy-

ion collisions,” Nucl. Phys. A 967, 604 (2017) doi:10.1016/j.nuclphysa.2017.05.073

[arXiv:1704.02361 [nucl-th]].

4. B. Krouppa, A. Rothkopf and M. Strickland, “Bottomonium suppression using a lattice

QCD vetted potential,” Phys. Rev. D 97, no. 1, 016017 (2018) doi:10.1103/PhysRevD.97.016017

[arXiv:1710.02319 [hep-ph]].

xv Acknowledgments

I am eternally grateful for the love and support I have received from those who made this achievement possible. There are too many stories of encouragement, prayer, listening, and sharing of wisdom, to include in this section, but I sincerely hope each of you know the impact you had on my life, and the fact that I could not have done this without you.

I would like to thank my advisor, Dr. Michael Strickland, for his guidance, support, and supervision during my doctoral research. I also extend a thank you to the collaborators who have made this work possible, including each person who pushed our understanding of physical laws governing the universe a little further. This work was made possible by the

Department of Energy Oﬃce of Science.

I would also like to thank my parents for their love and support in every endeavor I take on, whether professionally or personally.

Deo ac Veritati (For God and for Truth),

Brandon Krouppa

xvi Chapter 1

Introduction

In this chapter I will give a brief introduction to high-energy physics. The standard model is discussed along with an overview of elementary particles and the quantum physics which governs such particles. I will then introduce the quark-gluon plasma (QGP), and the hydrodynamics which eﬀectively models ultrarelativistic heavy-ion collisions. I will end with the description of heavy quarks and mesons which contain a heavy quark-antiquark pair, i.e. heavy quarkonia, and general phenomenology of such mesons in the QGP.

1.1 Units and notation

In the following work we utilize a natural system of units which simplifies notation significantly. The main feature of these units provide simplification as c = ~ = kb = 1. The occasional translation of units uses a factor of ~c and is worth listing as such conversions are frequently used this work

~c = 0.1973269788(12) GeV fm. (1.1)

Timescales in this work are typically denoted by fm/c. We make a note of the following unit in SI units to provide an ordinary perspective of timescales seen in heavy-ion collisions

1 fm/c 3.33 10−24 s. (1.2) ≈ ×

1 Other conversions to SI units can be done in the following manner

1 MeV = 1.602177 10−13 J, (1.3) ×

1 MeV/c2 = 1.78266 10−30 kg, (1.4) ×

1 MeV = 1.16045 1010 K. (1.5) ×

Providing further insight, one can make conversions to ordinary SI units

The region where we can deﬁne a critical temperature for the creation of a quark-gluon • 12 plasma is T0 150 MeV, which is 1.74 10 K. ∼ ∼ ×

The mass of the proton is 938 MeV = 1.67 10−27 kg. • ∼ ×

The proper time the quark-gluon plasma evolves before hadronization is 10 fm/c • ∼ 3 10−23 s. ∼ ×

It takes 8.263 104 J 5.157 1017 MeV to heat a cup of water from room • ∼ × ∼ × temperature ( 294 K) to its standard pressure boiling point at 373.15 K. ∼

In this work, we use the particle-physics Minkowski-space metric, i.e. “mostly negative,”

µν µ g = diag(+1, 1, 1, 1). As such, the four-velocity is normalized with uµu = 1. Milne − − − coordinates in Minkowski space are deﬁned by xµ = (τ, x, y, ς), which is sometimes seen as

µ 2 2 x = (τ, x⊥, ς), with x⊥ = (x, y). Longitudinal proper time in this space is τ = √t z , − spacetime rapidity ς = arctanh(z/t), with the full metric being ds2 = dτ 2 dx2 dy2 τ 2dς2. − − − In addition to the spacetime rapidity ς, we will make use of the pseudorapidity, which is deﬁned as θ η ln tan , (1.6) ≡ − 2

2 where θ is the angle between the particle three-momentum p and the beam-line axis.1 As a function of three-momentum, one can write the pseudorapidity as

where pL is the longitudinal momentum, i.e. the momentum component along the beam-line

axis. One can also simplify things by using the ultrarelativistic limit where m p, such that E p. Finally, we can deﬁne rapidity as ≈

1 E + p p y = ln L = arctanh L (1.8) 2 E pL E −

where E is the energy of the particle, E = pm2 + p2. Given such a deﬁnition, one can

complete the set of equations in relation to other quantities such as E = mT cosh(y), pT = pT ,

p 2 2 and pL = mT sinh(y), where mT = pT + m is the transverse mass.

1.2 The standard model

The four fundamental forces which govern the standard model are gravity, the strong

force, the electromagnetic force, and ﬁnally the weak force. Below I will give a brief descrip-

tion of each fundamental force.

The theory of gravity is largely described by the curvature of the spacetime “mesh” that

makes up the whole universe. While the macroscopic principles of the theory of gravitation

are successfully described by this, gravity remains unquantized and is not properly under-

stood in conjunction with the other three fundamental forces. The weak force is responsible

for interactions in nuclear matter, including the description of radiative decay of unstable

nuclei. The carriers of this force are the W and Z bosons. The electromagnetic force is

carried by the massless photon and pertains to other radiative eﬀects largely known as the

1Note that in ultrarelativistic heavy-ion collisions, backscattering is completely negligible.

3 Figure 1.1: The Standard Model as of June 2018. electromagnetic spectrum. Quantization of this force has led to the theory of Quantum Elec- trodynamics (QED) and is regarded as one of the more successful quantizations, mainly due

1 to the nature of the running coupling being small at low momentum transfer, i.e. αQED . ≈ 137 The strong force is carried by the gluon and the length scale of this force probes quarks,

fundamental fermionic particles in the standard model. The quantization of the strong force

is described in the theory of Quantum Chromodynamics (QCD).

1.3 Quantum Electrodynamics (QED)

As explained in Section 1.2, Quantum Electrodynamics is the fundamental quantization

of the electromagnetic ﬁeld and the leptons that couple to it, e.g. electrons. The Lagrangian

4 for QED is made from two parts, the Dirac part which is given by

¯ Dirac = ψ(i∂/ m)ψ, (1.9) L −

where ψ is the Dirac spinor, ψ¯ is the conjugate of the Dirac spinor and is deﬁned as ψ¯ ψ†γ0, ≡ µ and we have employed the Feynman slash notation, where in general, A/ γ Aµ. The ≡ electromagnetic portion of the QED Lagrangian is

1 µν µ Maxwell = FµνF AµJ , (1.10) L −4 −

where Fµν = ∂µAν ∂νAµ is the ﬁeld strength tensor for the electromagnetic ﬁeld, Aµ is the − vector potential, and J µ is the 4-current source term. To get the full QED Lagrangian, we

sum Equations (1.9) and (1.10), with J µ = eψγ¯ µψ −

¯ 1 µν ¯ QED = ψ(i∂/ m)ψ FµνF + eψAψ./ (1.11) L − − 4

We can further simplify by introducing the covariant derivative

Dµ ∂µ ieAµ. (1.12) ≡ −

With the covariant derivative notation, the QED Lagrangian can be written as

¯ 1 µν QED = ψ(iD/ m)ψ FµνF . (1.13) L − − 4

5 1.4 Quantum Chromodynamics (QCD)

The QCD Lagrangian is

Nf X 1 = ψ¯ (iD/ m )ψ F a F µν, (1.14) QCD i i i 4 µν a L i=1 − −

where the subscript i denotes ﬂavors of quarks and thus Nf is the number of quark ﬂavors

a with masses mi. We can deﬁne the ﬁeld strength Fµν as

a a a abc b c F = ∂µA ∂νA + gf A A , (1.15) µν µ − µ µ ν where A are the QCD ﬁelds and f abc are the structure constants of the SU(3) group.

Figure 1.2: The three fundamental vertices found in QCD. On the left we see the three-gluon vertex, in the middle we see the four-gluon vertex, and on the right we see the quark-gluon vertex. Figure taken from Reference [1].

Two key aspects of the theory of QCD are conﬁnement and asymptotic freedom. Conﬁne- ment is the principle which keeps quarks bound together. As quarks separate with distance, ever increasing amounts of energy are required to further separate the pair from one another.

The second property of QCD, asymptotic freedom, is the opposite of conﬁnement. Asymp- totic freedom describes the fact that the QCD coupling constant becomes weak at high energies. Quantum chromodynamics also introduces additional degrees of freedom known as

6 color charge, as the preﬁx “chromo” suggests.

The quark model was developed in 1964, independently, by Gell-Mann and Zweig. Ver- ification of the model came rapidly with multiple experiments, most notably with the observation of the Ω− particle at Brookhaven National Laboratory (BNL) [29], and with deep inelastic scattering at the Stanford Linear Accelerator Center (SLAC). With those two experiments, physicists have known that nuclear matter, i.e. protons and neutrons, are not elementary particles. The constituents of protons and neutrons, now known as up and down quarks, make up just two of the six flavors of quarks which have been experimentally observed. Heavier flavors of quarks were theorized along with the quark model in the 1960s, and since then have been observed in a wide range of experiments. The quark model was expanded with the inclusion of a force carrier, the gluon, completing the theory of QCD largely as it stands today. As the present-day Standard Model stands, we have experimentally observed six quarks, six leptons, four gauge bosons, and one scalar boson. For this work we will primarily focus two of the six quarks seen in Figure 1.1, namely the charm and bottom quarks.

1.5 The running coupling in QED and QCD

Derived from ﬁrst-principle calculations [30, 31], we know the one-loop running coupling in QED to be

α(Q0) α(Q) = 2 , (1.16) α(Q0) Q 1 log 2 − 3π Q0 and the one-loop running coupling in QCD is

αs(Q0) αs(Q) = 2 , (1.17) αs(Q0) 2Nf Q 1 + 11 log 2 4π − 3 Q0

7 where Q is the momentum transfer scale which is used in the renormalization to compute the one-loop coupling, and Q0 is a reference scale which acts as an initial condition for the running coupling. This initial condition is gathered from experiment where scatterings can measure the coupling, or theoretical calculations which utilize a first-principles approach such as lattice QCD. The difference between the QED and QCD running coupling is found in the denominator of Equations (1.16) and (1.17). For 11 2Nf /3 > 0, i.e. Nf < 33/2 = 16.5, − the sign is opposite from that of QED. While QED flourished with theoretical methods such as perturbation theory due to the small coupling constant at low momentum transfer, i.e. αQED 1/137, QCD only sees this advantage for large temperatures with ordinary ≈ nuclear matter being described with other methods. The coupling for QCD, αs, has been studied extensively and a summary of measurements and theoretical efforts to calculate the running coupling have led to Figure 1.7. In Figure 1.7, we see the trend we expect from the phenomenon of confinement, notably when the momentum transfer Q becomes large, the coupling constant in QCD becomes small. We also see strong agreement between experiment and the one-loop running coupling theory curve.

1.6 Hadrons

Under normal circumstances, quarks are conﬁned inside bound states called hadrons.

Hadrons are bound by the strong force which is described by QCD. There are various classes of hadrons which correspond to how many quarks make up the bound state, and can be seen in Figure 1.4. Mesons are made from quark-antiquark pairs, qq¯ and are largely the study of this work. Baryons are a combination of three quarks and make up the bulk of “ordinary” nuclear matter we see today, i.e. protons and neutrons. Other exotic hadrons have been observed in experiment such as the tetraquark (made of four quarks) and the pentaquark

(ﬁve quarks). The tetraquark is currently discussed to either by a molecular particle, i.e. a bound set of mesons, or a true bound state where gluons are exchanged between a quark

8 April 2016 2 τ decays (N3LO) αs(Q ) DIS jets (NLO) Heavy Quarkonia (NLO) – 0.3 e+e jets & shapes (res. NNLO) e.w. precision fits (N3LO) (–) pp –> jets (NLO) pp –> tt (NNLO) 0.2

0.1 QCD αs(Mz) = 0.1181 ± 0.0011 1 10 100 1000 Q [GeV]

2 Figure 1.3: A summary of the QCD running coupling constant αs(Q ) from [2]. and the other three quarks in the state.

quark

Antiquark

Figure 1.4: Various types of hadron examples and their classiﬁcation.

Perturbative approaches to describe hadrons are, in large part, unsuccessful due to regime not being valid. However, we’ll see a few special properties which allow us to use perturbative approaches to extract physics from mesons formed with heavy quarks.

9 1.6.1 Heavy quarkonia

Heavy quarkonia are bound states of heavy quarks with their antimatter heavy anti-quark

partner, so they are generally characterized as mesons. Heavy quarks are the three most

massive quarks, q = c, b, t , the charm quark, bottom quark, and top quark, respectively. { } The charm and bottom quark have been seen with their antimatter partners to form charmo-

nia and bottomonia, however the top quark has not been observed to form the hypothetical

particle toponia, due to the ultra-short lifetime of the top quark. Therefore modern physics

understanding suggests toponia may never be formed.

Heavy quarkonia were ﬁrst detected in 1974 with an excess of dimuons corresponding

to a dimuon invariant mass of 3.1 GeV, now known as the J/ψ meson which is a bound ∼ state between a charm quark and an anti-charm quark. Lately, heavy quarkonia have been

produced and studied using many collision systems at RHIC and the LHC.

Heavy quarkonia are rather small states with state radii less than 1 fm. Moreover,

their relative velocity is small and can be described with an inherent separation of scales,

2 mq mqv prel mqv Ebind. This hierarchy of scales along with small bound state ∼ | | ∼ radii allow us to exploit the heavy quarkonia with potential-based non-relativistic QCD

(pNRQCD), especially since pNRQCD approaches are well-deﬁned for r < 1/mD

It is now known that the heavy-quark pair potential is complex-valued, with the imaginary part of the potential stemming from the state’s proclivity to dissociate when the surrounding temperature dampens gluonic interactions, eﬀectively causing a quark to no longer “see” its antimatter partner. However, we can get a basic understanding of thermal eﬀects on heavy quarkonia by examining the real part of the potential.

In Figure 1.5, we see the real part of a potential extracted from potential-based non- relativistic QCD (pNRQCD) heavy-quark potential. At T = 0 GeV, we obtain the Cornell

10 ��

��

) �� ]( � [ ��

-�� = �� = �� = �� -�� (��)

Figure 1.5: A plot of the pNRQCD-based heavy quark potential for three temperatures. The real part seen in this plot starts with the vacuum Cornell potential at T = 0 GeV, and as the temperature increases, the quarkonia are subject to melting.

potential which is generally given by

a V (r) = + br, (1.18) −r

where a and b are parameters related to the strong coupling constant and QCD string

tension, respectively. Its general features are a linear part for large r which corresponds to

QCD conﬁnement, followed by the 1/r part, known as the Coulombic part, which is induced

by the one-gluon exchange between the quark-antiquark pair.

Theoretical and phenomenological models to predict heavy-quark production utilize a

spectrum of techniques from eﬀective theories to pNRQCD approaches which lean more on

ﬁrst principle approaches. Recent studies from lattice QCD suggest pNRQCD models give

accurate descriptions of heavy quarkonium.

11 In more recent studies, a lattice-vetted potential has been used to compute the eﬀects of the QGP on heavy quarkonia. This was an eﬀort to more properly treat the heavy-quark potential in a hot QGP medium. Our work resulted in comparisons between theoretical models and experimental data seen in RHIC and LHC collisions at √sNN = 200 GeV/nucleon, 2.76

TeV/nucleon, and 5.02 TeV/nucleon [32].

1.7 The QCD phase diagram

Figure 1.6: The phase diagram, as currently understood, with various high-energy experiments outlined to probe various parts of the diagram. Figure taken from [3].

As mentioned above, normally quarks are confined inside hadrons, however, if one heats or compresses a system of hadrons, one finds that there can be a transition to a deconfined state. This can be summarized in the phase diagram of QCD, as seen in Figure 1.6. In the QCD phase diagram, two dependencies are typically seen: temperature and the baryon

12 chemical potential, which is related to the net baryon density. The main discussion of the

QCD phase diagram lies within the transition between hadronic matter and the QGP. For larger net baryon density and relatively low temperatures, there’s believed to be a ﬁrst order phase transition. However, as the net baryon density is lowered toward zero, there’s a smooth crossover between hadronic matter and the QGP. Between the ﬁrst order phase transition and the crossover, there’s a possibility for a critical point (as common among other phase diagrams in physics), so the search for such a point is currently underway at the Relativistic Heavy-Ion Collider (RHIC) with their Beam Energy Scan program [33].

Lattice studies performed suggest this region might be marked with a critical temperature of Tc = 150 170 MeV [34, 35]. The focus of this work is to discuss a theory model − which describes an observable seen at two colliders which probe ultra-low net baryon density and high temperatures with heavy-ion collisions. This region of the QCD phase diagram is relevant to understanding the early universe in accordance with the big bang theory where, initially, the universe was hot and neutral with nearly equal numbers of baryons and anti- baryons. As the early universe cooled, matter started dominating over antimatter and is now in a region where T 0 and µB 1 GeV. ∼ ∼

1.8 Heavy-ion collisions

Heavy-ion collisions are used to study various physical principles and theories such as

QED, QCD, the early universe, and other fundamental particle interactions which are used to provide a more comprehensive picture of the physics governing the universe. Observables such as elliptic ﬂow, charged-hadron multiplicity, heavy quarkonium suppression, etc. are used to describe the hot QGP generated in the short timescales, and each probe a diﬀerent aspect of the QGP. Below I will give an overview of heavy-ion collisions and how we model them using a realistic hydrodynamics framework.

13 1.8.1 Stages of heavy-ion collisions

Freezeout time τ > 10 fm/c

Hot Hadron Gas 6 < < 10 fm/c time

Equilibrium QGP beam direction 2 < < 6 fm/c

Non-equilibrium QGP 0.3 < < 2 fm/c

Semi-hard particle production 0 < < 0.3 fm/c beam direction

Figure 1.7: Stages of heavy-ion collisions. Figure taken from [4].

Various stages of heavy-ion collisions can be deﬁned by the dominant features during the

timescale.

Before impact. At this time, the heavy ions are traveling close to the speed of light in

opposite directions. In the lab frame the two nuclei are highly Lorentz contracted with

γ = 1/p1 v2/c2. Collisions at RHIC and at the Large Hadron Collider (LHC) at CERN − see γ-values which are approximated by γ √sNN /2, which lead to γ 100 for √sNN = 200 ' '

GeV Au-Au collisions. For larger collision energies at the LHC, we see for √sNN = 5020

GeV γ 2500. While the nuclei are Lorentz contracted in the lab frame, it is important ' to consider the time it takes for the nuclei to pass through one another. As an example,

for Pb-208 in LHC 5.02 TeV/nucleon collisions, we can use R = (1.12A1/3 0.86A−1/3) − to compute the radius which is R 6.49 fm, which will be the “longest” case for central ' collisions. The duration of the collision is R/γ = 13/2500 0.05 fm/c, and increases for ∼ ' lower collision energies.

14 Semi-hard particle production. In this stage, 0 < τ < 0.3 fm/c, semi-hard particles are

produced, including many of the heavy-quark pairs studied in this work. At this stage it is

required to accurately describe the non-equilibrium modes of the QGP.

Pre-equilibrium. Immediately following the nuclear impact, the system is characterized

by a strong momentum-space anisotropy given by a negative ratio L/ T , where L is the P P P

local rest frame pressure in the longitudinal (beam) direction, and T is the local rest frame P pressure in the transverse direction. This timescale is also marked by strong color ﬁelds and studied theoretically using color-glass condensate and glasma models [36, 37, 38, 39, 40, 41].

Equilibrium. This stage is described as being close to equilibrium and locally thermalized.

In this timescale, pressure gradients are small enough to provide a consistent study of the

QGP using anisotropic hydrodynamics, a hydrodynamical model which incorporates local pressure anisotropies in its 3+1d evolution.

Freeze-out. Simply put, this stage is marked by the transition from QGP to free streaming nuclear matter. Collective ﬂow of particles and inelastic collisions are ceased, and the timescale between collisions becomes larger than the expansion timescale, τcoll τexp. Par- ≥ ticles during this time become free streaming and subsequently travel to particle detectors roughly unimpeded.

1.8.2 Centrality

The determination of impact parameter is not measured directly, but rather modeled.

In this work we will use the optical Glauber model, which is employed to map the impact parameter, b, to various centrality classes. The optical Glauber model describes nuclei with smooth nucleon densities in which individual partons are not located at speciﬁc spatial coordinates. There are models which track individual partons such as Monte Carlo Glauber, however, averaging over (108+) events smooths out the randomness associated with lumpy O

15 partons in nuclear distributions. In experiments, 0% centrality corresponds to the most- central (overlapping) collisions, i.e. b = 0 (see Figure 1.9, while 100% centrality maps to peripheral collisions where the nuclei pass by each other. In the later case, peripheral collisions occur when the impact parameter reaches a threshold which depends on the colliding system. Table 1.1 contains the relationship between the impact parameter, b, and centrality class, c, for a given collision system with √sNN = 2.76 TeV/nucleon Pb-Pb collisions using optical Glauber. One can also make a connection to the number of participants in a given collision, Npart, via the Glauber model.

Figure 1.8: The transverse proﬁle of two colliding nuclei. The overlap is measured by the impact parameter, b, the distance between the center of the two nuclei. When calculated with the Glauber model, the overlapping region can be quantiﬁed by the number of partons participating in the collision, Npart.

16 �� -�� -�� -��

��

� � � �� (��)

Figure 1.9: The number of participants, Npart, as a function of the impact parameter, b, for various collision systems studied in this work.

cmin cmax bmin bmax b cmin cmax bmin bmax b h i h i 0. 0.05 0 3.473 2.315 0.5 0.6 10.983 12.031 11.515 0.05 0.1 3.473 4.912 4.234 0.6 0.7 12.031 12.995 12.519 0.1 0.2 4.912 6.946 5.987 0.7 0.8 12.995 13.893 13.449 0.2 0.3 6.946 8.507 7.753 0.8 0.9 13.893 14.795 14.334 0.3 0.4 8.507 9.823 9.181 0.9 1. 14.795 20 15.608 0.4 0.5 9.823 10.983 10.414

Table 1.1: The relationship between impact parameter, b, and centrality class, c, including the average impact parameter for each centrality class in a √sNN = 2.76 TeV/nucleon Pb-Pb collision modeled using optical Glauber. Table taken from [24].

To calculate Npart, we start with the Woods-Saxon nuclear proﬁle function.

n n (r) = 0 , (1.19) A 1 + e(r−R)/d

−3 where n0 = 0.17 fm is the central nucleon density and is determined via the normalization

R 3 1/3 −1/3 limA→∞ d r nA(r) = A, R = (1.12A 0.86A ) fm is the nuclear radius, and d = 0.54 −

17 fm is the skin depth of the nucleus [42]. Using Eq. (1.19), we construct the thickness function

Z ∞ p 2 2 2 TA(x, y) = dz nA x + y + z . (1.20) −∞

We then utilize the impact parameter, b, to oﬀset the two nuclei.

nAB(x, y, b) = TA(x + b/2, y)TB(x b/2, y). (1.21) −

We can then integrate Equation (1.21) over the transverse plane to obtain Npart as a function of b. Z ∞ Npart(b) = dx dy nAB(x, y, b). (1.22) −∞

We now have a framework for computing centrality classes and the alternatively-seen Npart and will use such calculations extensively to translate b to Npart, as well as average over

Npart to obtain centrality classes and centrality-averaged heavy quarkonium suppression. In

Figure 1.9, we see the relation between the number of participants, Npart, and the impact parameter, b, for three collision systems explored by this work.

1.8.3 A hydrodynamical description of the quark-gluon plasma

Dissipative hydrodynamical codes, which describe the full 3+1d evolution of the temperature- like scale and the degree of momentum-space anisotropy, can be used to simulate the background evolution of the hot media generated in ultrarelativistic heavy-ion collisions from just before the formation times of heavy quarkonia until the plasma becomes free streaming and the QGP “freezes out” into cold matter. The momentum-space anisotropy, ξ(x), characterizes the breaking of symmetry between the longitudinal (beam-line) direction and the transverse plane. Momentum-space anisotropies in the local rest frame of the QGP directly aﬀect

p 2 2 the partonic one-particle distribution function, faniso(x, p) = fiso( pT + [1 + ξ(x)]pz/Λ(x)),

18 where Λ(x) is the local temperature-like scale of the QGP. The resulting local rest frame distribution is spheroidal in momentum space. This particular anisotropy parameter was found in studies to be the dominant dissipative eﬀect in hydrodynamic simulations [10] and results in important modiﬁcations of the in-medium heavy-quark potential.

Analysis of heavy quarkonium suppression and regeneration provide insight into certain hydrodynamic parameters like the initial temperature and the initial momentum-space anisotropy of the plasma, as well as the transport coeﬃcient η/s which is the shear viscosity to entropy density ratio. Initial studies have shown η/s to have a lower bound of 1/4π and

η/s is generally believed to be between 1/(4π) and 3/(4π). Recent phenomenological work to describe the experimentally observed hadron spectra with quasiparticle aHydro have pinned down this value to be approximately 2/(4π) [43], which is consistent with theory calculations which compute η/s using second-order hydrodynamics using NLO equations [44].

300 0-5% 40-50% (a) (b) 250 1500 5-10% 200

η 10-20% 50-60% d / 1000 150 20-30% dN 100 60-70% 500 30-40% 50 70-80% aHydro ALICE 0 0 80-90% - 6 - 4 - 2 0 2 4 6 - 6 - 4 - 2 0 2 4 6 η η

Figure 1.10: Charge-hardon multiplicity as a function of pseudorapidity, compared to ALICE 2.76 TeV/nucleon collisions. The percentage ranges given in the ﬁgure are various centrality classes, with 0 5% relating to the 5% most central collisions, and so on. Figure taken from [5], with ALICE− data provided from [6, 7].

As we will see in our model, bottomonium suppression within this framework is consistent with other work which suggests 4πη/s = 2. As seen in Figure 1.10, aHydro has been

19 shown to reproduce charged-hadron multiplicity as a function pseudorapidity for ALICE 2.76

TeV/nucleon collisions at the LHC. Dissipative hydrodynamical codes provide critical insight

into the interaction of a heavy quarkonium state with a hot medium which is anisotropic in

momentum space.

1.9 Quarkonium suppression

The idea of heavy quarkonia deconﬁning in a hot media can trace its origin to Matsui

and Satz [45]. In their pioneering work, Matsui and Satz suggested that the hot medium

generated in URHICs resulted in color screening for closed cc¯ states. Ultimately, their work

would go on to suggest that suppression of the J/ψ charmonium state in URHICs would

indicate an unambiguous signature for the formation of a new state of matter known as

the quark-gluon plasma. This suppression is due to the formation of the QGP, a state of

matter where hadronic particles “melt,” resulting in a sharp increase in the relative number

of degrees of freedom. High-energy heavy ion collisions would form the heavy-quark pairs,

but the hot QGP around heavy quarkonium states would dampen the exchange gluon and

cause the cc¯ pair to dissociate. This eﬀect is known as thermal suppression. Due to the short

formation times and relatively long decay times in the dimuon channel, heavy quarkonium

production is considered to be a reliable probe of the QGP temperature. The QGP is

characterized by “melting” of hadrons into their constituent parts known as quarks and

gluons. While ordinary hadronic matter contains light quarks (q = u, d ), the energies found { } in URHICs are high enough to generate signiﬁcant amounts of heavy quarks (q = s, c, b ) { } formed with their antimatter quark pairs (¯q = s,¯ c,¯ ¯b ). Recent URHIC experiments at { } Brookhaven National Laboratory’s Relativistic Heavy Ion Collider (RHIC) and CERN’s

Large Hadron Collider (LHC) study heavy ions and the resulting hot and deconﬁned matter that is generated in the process.

In Figure 1.11, we see the number of dimuon events detected by the CMS collaboration as a

20 -1 ×103 PbPb 368 µb (5.02 TeV)

9 µµ p < 30 GeV T µµ CMS 8 |y | < 2.4 µ p > 4 GeV T 7 µ |η | < 2.4 6 Centrality 0-100% PbPb Data Total fit 5 Background 4 RAA scaled 3 Events / (0.1 GeV) 2 1 0 8 9 10 11 12 13 14

mµ+µ- (GeV)

Figure 1.11: As a function of the dimuon invariant mass, we see three resonances appear in proton-proton reference data, and then disappear in Pb-Pb collisions at 5.02 TeV. This suppression of states is thought to be due to the formation of the QGP. Figure taken from [8]. function of the dimuon invariant mass. In scaled proton-proton reference data (dashed line), we see three peaks appear which correspond to the Υ(1S), Υ(2S), and Υ(3S) bottomonium states. This is compared to Pb-Pb collisions at the same energy (solid line with data points) where we see peaks appear for the Υ(1S) and Υ(2S), but they are strongly suppressed. The

Υ(3S) has not be observed in signiﬁcant amounts in Pb-Pb 5.02 TeV/nucleon collisions as the collection of events around the Υ(3S) mass is well within the background for the CMS experiment. The suppression of states is known as heavy quarkonium suppression and is believed to be a strong signature of the formation of a QGP.

21 Based on ﬁts to other observables such as pion spectra, hydrodynamic simulations of

√sNN = 2.76 TeV/nucleon Pb-Pb collisions show the initial temperatures of the generated

QGP are T0 500 600 MeV, which are consistent with the work presented in this model. ∼ − With recent calculations of bottomonium state widths, it is suggested that the ground state

Υ(1S) may survive temperatures through 600 MeV. The indication that some Υ(1S) states deconfine in the hot QGP show the presence of an additional physical effect leading toward further decay of heavy quarkonia in the QGP. Furthermore, the benefits of working with heavy quarkonia are derived from their bound states being dominated by short range physics, as well as their relatively large mass which suppresses statistical regeneration of states based on large numbers of open heavy qq¯ pairs (for Au-Au 200 GeV/nucleon collisions at RHIC).

The presence of regeneration can be clearly seen in experimental data for the J/ψ state

when moving from 200 GeV/nucleon Au-Au collisions at RHIC, to 2.76 TeV/nucleon Pb-

Pb collisions at the LHC. With the recent upgrades at the LHC to expand nucleus-nucleus

collisions to 5.02 TeV/nucleon, bottomonia show hints of regeneration in some experiments,

while others show more suppression of states. Even with stronger thermal suppression from

the hotter QGP that has been formed, small, but signiﬁcant, amounts of regeneration may

be possible for bottomonia at 5.02 TeV/nucleon Pb-Pb collisions.

The model was then extended to make predictions for 5.02 TeV/nucleon Pb-Pb collisions

at LHC [46, 47]. This was done by using extrapolations which determined the increase in

the initial temperature of the generated QGP and bottomonium distribution throughout the

QGP compared to lower energy collisions. The accuracy of such predictions compared to

subsequent experimental measurements were critical in understanding the lack of a regen-

eration eﬀect for bottomonia, as the increase in energy did not result in less suppression in

most experimental data [47].

22 EPS09 NLO shadowing, Pb-Pb 2.76 TeV, R. Vogt, Priv. Comm. 1.4

1.2

1.0 AA

R 0.8 L s 1 H

U 0.6

0.4

0.2

0.0 -4 -2 0 2 4 y

Figure 1.12: An estimate of shadowing (a cold nuclear matter eﬀect) on Υ(1S) suppression at 2.76 TeV/nucleon LHC collisions of Pb-208 nuclei.

1.9.1 Other suppression eﬀects

Heavy quarkonium production can be modified by other effects such as cold nuclear matter effects and co-movers. For the purposes of this work, we will be neglecting such effects, as they are less important for bottomonium suppression.

One of these effects, cold nuclear matter shadowing, has been calculated using EPS09 nuclear parton distribution functions with NLO shadowing and is seen in Figure 1.12. It’s noted that the effect for the rapidity window using this model predicts a maximal 10% ∼ effect for raw Υ(1S) RAA. Therefore, to good approximation, we can ignore cold nuclear matter effects for the purposes of this work for the bottomonium states. Small collision systems such as p-Pb at the LHC suggest cold-nuclear matter effects such as shadowing play an important role for charmonium [48]. However, since this work focuses on nuclear collisions and bottomonium, we can neglect such effects due to limited impact on final suppression.

23 Chapter 2

The heavy quark potential

2.1 Heavy quarkonia

Heavy quarkonia are a hard probe of the quark-gluon plasma due to their formation early in the plasma’s proper time evolution, and exit into the hadronization stage of the QGP.

Heavy quarkonia then decay into dilepton pairs and then are detected by particle detectors surrounding the collision region of the particle collider. The mass scales of the relatively large charm and bottom quarks ensure that charmonium and bottomonium evolve in the plasma without losing vital information regarding properties of the QGP such as the temperature proﬁle during the whole 3+1d evolution.

Heavy quarkonia have long been discussed as having a complex potential which is the result of gluonic dampening from the hot quark-gluon plasma. A “proper” heavy quark potential should feature the two main properties of QCD, namely confinement and asymptotic freedom. If such a potential is not derived via first principles, e.g. lattice QCD, free parameters should be fit in an attempt to phenomenologically describe certain features in experiment.

The presence of a hydrodynamical attractor (see Fig. 3.1) in momentum-space is critical for not only the QGP evolution description using hydrodynamics, but also the modeling of heavy quarkonia as they are formed in the non-equilibrium phase of the plasma. As we will see in each respective section, the presence of this anisotropy has been computed in the

Strickland-Bazow potential analytically to all orders in the momentum-space anisotropy, ξ, for the real part and to ﬁrst order in the imaginary part of the potential. Recent calculations have extended this result to include an analytic calculation of the imaginary part to all orders

24 in ξ [49].

2.2 A perturbative inspired heavy-quark potential (Strickland-Bazow)

In previous work seen in References [50, 51], and factoring in the signiﬁcant presence

of momentum-space anisotropies in the QGP, the free-energy-based heavy quark potential

models, as originally described by Karsch, Matsui, and Satz, do not reproduce quarkonium

suppression data from experiments at both RHIC or the LHC. For this reason the pNRQCD

model used herein, also known as the perturbative-inspired (Strickland-Bazow) heavy quark

potential, utilizes the internal energy of the quarkonium pair U = F + TS to provide the

model potential.

The masses of heavy quarks allow charmonium and bottomonium pairs to be treated

using pNRQCD methods which allow for relativistic corrections. The model presented herein

represent the short- and medium- range gluonic screening of the heavy-quark potential in a

plasma which is anisotropic in momentum-space [51, 52, 53, 54, 55]. The long-range eﬀects of

the potential are modeled by the original Karsh–Mehr-Satz (KMS) form for the free energy.

The real part internal-energy-based model, in it’s ﬁnal form, is given by,

a −µr 2σ −µr −µr 0.8σ Re[V] = (1 + µr)e + [1 e ] σre 2 , (2.1) − r µ − − − mqr

where µ = (ξ, θ)mD [51, 53, 54], with θ quantifying the angle between the beam-line G direction and the line connecting the quark–antiquark pair, mD is the isotropic leading-

2 order Debye mass, a = 0.385, σ = 0.223 GeV [56], for q = c we have mc = 1.29 GeV, the mass of the charm quark, and for q = b, we have mb = 4.7 GeV, the mass of the bottom quark. The ﬁnal term factors in the correction for the ﬁnite quark mass. Though this term is not important for bottomonium states, it is included for historical continuity with previous works. Recent progress with the imaginary part include an expansion which is accurate to

25 all orders in ξ. However, for the model presented in this work we utilize a small-ξ expansion of the heavy-quark potential [55, 57, 58].

n o Im[V] = αsCFT φ(ˆr) ξ[ψ1(ˆr, θ) + ψ2(ˆr, θ)] , (2.2) − −

wherer ˆ = rmD, φ(ˆr) is deﬁned as

Z ∞ z sin (zrˆ) φ(ˆr) = 2 dz 2 2 1 , (2.3) 0 (z + 1) − rˆ

and ψ1 and ψ2 are deﬁned as follows

Z ∞ z 3 2 sin(z rˆ) 2 ψ1(ˆr, θ) = dz 2 2 1 sin θ + (1 3 cos θ)G(ˆr, z) , (2.4) 0 (z + 1) − 2 z rˆ −

Z ∞ 4 3 z 2 2 sin(z rˆ) 2 ψ2(ˆr, θ) = dz 2 3 1 3 cos θ + (1 3 cos θ)G(ˆr, z) . − 0 (z + 1) − 3 − z rˆ − (2.5)

2.3 A lattice QCD vetted heavy-quark potential (Rothkopf)

In this section I will describe a lattice QCD vetted, non-relativistic potential-based description of heavy quarkonium in a QGP which can trace its roots from a systematic treatment of heavy quarkonium in QCD from the eﬀective ﬁeld theories non-relativistic QCD

(NRQCD) and potential NRQCD [59, 60]. The inherent separation of scales between the rest mass of the heavy quark, mc,b, the medium temperature, and the characteristic scale of

QCD, ΛQCD, are exploited to adequately simplify the physical picture of heavy quarkonium.

One does not have to consider a full quantum ﬁeld theoretical boundary value problem for

Dirac ﬁelds, but rather use an initial-value problem for two-component Pauli spinors which follows from NRQCD. Following, this non-relativistic theory is matched to terms of coupled

26 color singlet ψS(r, t) and the color octet ψO(r, t) wave functions which results in pNRQCD.

In the latter pNRQCD treatment, the interaction among heavy quarks, and the interaction with the hot medium, is obtained in both potential and non-potential contributions. The speciﬁc scale hierarchy of this problem means the potential contributions dominate, therefore the evolution of heavy quarkonium reduces to a Schr¨odingerequation.

In realistic high-temperature environments, such as heavy-ion collisions at RHIC and the

LHC, perturbation theory cannot be used to fully determine the matching coeﬃcients to obtain the potential. Withal, vital insight has been gained by pNRQCD with the hard-loop approximation [57, 60]. Moreover, the in-medium potential must be complex at large temperatures, as discussed with an isotropic medium in [57] and a medium which is anisotropic in momentum space in [53, 58, 55, 49]. Therefore, purely real potential models such as the color singlet free energies, or the internal energies, are not valid descriptions of heavy quarkonium physics in a medium.

The heavy quark potential in this section is related to the rectangular Wilson loop which is a real-time QCD quantity. The relation is matched with a correlation function in the eﬀective theory pNRQCD and the underlying microscopic QCD theory which follow identical physical content at the appropriate scale. In the case of a lattice QCD vetted potential, the unequal time correlation function of a heavy quarkonium singlet state can be identiﬁed with the following Wilson loop in the static limit

Z D E m→∞ D h µ a aiE ψS(r, t)ψS(r, 0) W(r, t) = Tr exp ig dx AµT . pNRQCD QCD ≡ −

Since the Wilson loop obeys a simple equation of motion [57]

i∂tW(r, t) = Φ(r, t)W(r, t), (2.6)

27 with a space- and time-dependent complex function Φ(r, t), the potential picture is valid, considering Φ asymptotes toward a constant value at late times, i.e. time independent. In general this value is complex, and the analogous potential is formally deﬁned as

i∂tW (t, r) VQCD(r) = lim . (2.7) t→∞ W (t, r)

In its current form, Equation (2.7) is not amenable to a non-perturbative lattice QCD evaluation which is evolved in Euclidean time which is an unphysical quantity. To counteract this, one must take a detour via spectral decomposition of the Wilson loop which relates the

Euclidean and Minkowski domains [61, 62],

Z Z W (τ, r) = dωe−ωτ ρ (ω, r) dωe−iωtρ (ω, r) = W (t, r). (2.8) ↔

Inserting Equation (2.8) into Equation (2.7) speciﬁes that both the real and imaginary part of the potential are related to the position and width of the lowest lying peak within the

Wilson loop spectrum. If the potential picture is applicable, the Wilson loop spectrum contains a well-deﬁned lowest-lying peak skewed Lorentzian form, from which the potential values can be extracted via a χ2-ﬁt [63].

With the recent success of a Bayesian approach [64], the extraction of spectral functions from Euclidean lattice data has only been possible as of late due to an inverse problem.

Reconstruction robustness was improved significantly compared to attempts made in the past with the Maximum Entropy Method [62]. One considers Wilson line correlators fixed in the Coulomb gauge, instead of the Wilson loop on the lattice. This practice produces results which are free from a class of divergences impeding the numerical determination of the Wilson loop. With this method, potential values have been extracted in quenched QCD based on the Wilson action [65], as well as for QCD with Nf = 2 + 1 light quark flavors based on

28 recent work by the HotQCD collaboration. Both cases conﬁrmed the appropriateness of the potential picture at all temperatures considered due to the well-deﬁned peak of Lorentzian shape from the Wilson spectral functions.

Two more steps are needed before utilizing discrete potential values obtained from lattice

QCD, which have been explicitly laid out in [66, 67]. First, we parameterize values of the potential with an analytic formula which allows for the evaluation of Re[V] and Im[V] at intermediate separation lengths not resolved by lattice methods. Second, we correct the parameters in the analytic parameterization for artifacts remaining from ﬁnite volume and

ﬁnite lattice spacing, as the continuum extrapolated lattice QCD determination of the in- medium heavy quark potential has yet to be achieved.

The parameterization to an analytic expression we used is one based on a generalized

Gauss law for the vacuum heavy quark potential. Lattice QCD studies have shown, for phenomenologically relevant distances, that the quarkonium vacuum (T = 0) potential is reproduced by the Cornell ansatz which consists of a Coulombic region dominating the small-r physics, and a linearly rising term which is consistent with confinement in QCD. The effects of a running coupling constant in QCD are mimicked by allowing the linear terms to contribute down to the smallest distances between the quark pair. Following the Gauss law example, we consider the divergence of the auxiliary (color) electric field E = qra−1rˆ emerging from either the Coulombic, a = 1, q = αs, [αs] = 1 part, or the string-like part − with a = 1, q = σ, [σ] = GeV2.

E = 4π q δ(r). (2.9) ∇ ra+1

The parameters which enter this expression characterize the non-perturbative vacuum physics of the heavy quarkonium bound state are the strong coupling αs, the string tension σ, and a constant shift c. It is noted that the CF factor is absorbed into the deﬁnition of the strong

29 2 g CF coupling, i.e. αs = 4π . One can perform a Fourier transform on Gauss’ law and modify the right-hand side by dividing it with an in-medium permittivity, . Such a treatment, which is a well-known prescription in classical electrodynamics, can be used to introduce the eﬀects of a thermal medium. This permittivity of a QCD medium which has been introduced is computed in hard-thermal loop perturbation theory,

2 2 −1 p p mD ε (p, mD) = 2 2 iπT 2 2 2 . (2.10) p + mD − (p + mD)

Non-perturbative physics of the bound state are encoded in the Cornell potential which is driven by a weakly-coupled gas of quarks and gluons. The combination of Equations (2.10) and (2.9) leads to integro-diﬀerential equations for the in-medium modiﬁed Coulombic and string-part of the vacuum potential as discussed in [68]. With a complex in-medium permittivity, the in-medium potential is also complex. Unlike the purely perturbative computations, the in-medium potential considered here receives a contribution to its real and imaginary part from the string-like portion of the Cornell potential. The expressions for the Coulombic part are

e−mDr Vc(r) = αs mD + + iT φ(mDr) , (2.11) − r with

Z ∞ z sin(xz) φ(x) = 2 dz 2 2 1 , (2.12) 0 (z + 1) − xz which coincide with the results of Reference [57].

30 �� shifted lQCD Nf =2+1 asqtad data & T=0.86TC T=0.95TC T=1.06TC � Gauss-law fit T=1.19TC T=1.34TC T=1.41TC

� �� T=1.66TC

] �

�� ][

� ][ �� ] �� [ � �� [ � �

T≈0(β=6.9) T≈0(β=7.48) T=0.86TC ��

� T=0.95TC T=1.06TC T=1.19TC

T=1.34TC T=1.41TC T=1.66TC � �� [��] �[��]

Figure 2.1: The in-medium heavy quark potential [real part (left) and imaginary part (right)] in full QCD with Nf = 2 + 1 light quark ﬂavors based on ensembles by the HotQCD collaboration (colored points, shifted for better readability). By adjusting the Debye mass mD, the lattice QCD values of the real part are adequately reproduced with the Gauss- law parameterization as shown with the solid lines, at all temperature and length scales. Theoretical error bars (the shaded region around the solid lines) are due to the uncertainty of the Debye mass ﬁt. The imaginary part shows good agreement at high temperatures and small distances, while at temperatures close to the crossover transition, deviations from the lattice data are seen. Note that the crossover temperature on these lattices due to the relatively large pion mass of mπ 300MeV lies at TC = 172.5 MeV. ≈

31 The additional and string-like contribution, on the other hand, reads

1 1 Γ[ 4 ] σ Γ[ 4 ] σ ReVs(r) = 3 D 1 √2µr + , (2.13) − 2 3 −2 4 √π µ 2Γ[ 4 ] µ

4 2 for the real part, where the parameter µ = mDσ/αs characterizes the strength of the in- medium modiﬁcation. For the imaginary part we obtain

2 σmDT ImVs(r) = i ψ(µr) = iαsT ψ(µr), (2.14) − µ4 −

where ψ corresponds to the following Wronskian

Z x 2 ψ(x) = D−1/2(√2x) dy ReD−1/2(i√2y)y φ(ymD/µ) 0 Z ∞ 2 +ReD−1/2(i√2x) dy D−1/2(√2y)y φ(ymD/µ) x Z ∞ 2 D−1/2(0) dy D−1/2(√2y)y φ(ymD/µ). − 0

A key characteristic of the above expressions is once the vacuum parameters of the Cornell

potential are ﬁxed, the remaining temperature dependent parameter is the Debye mass mD.

The parameterization to analytic form was done in a straightforward fashion, however, it relies on some assumptions and needs to be vetted with lattice QCD data before proceeding.

It has been shown that lattice values of the resulting potential, with the Gauss law parameterization, are signiﬁcantly reproduced in both quenched [65] and full QCD simulations with

Nf = 2 + 1 light flavors [66]. The agreement can be seen in Fig.2.1. After fixing the three parameters αs, σ, and c with low-temperature ensembles, the Re[V] is fitted by tuning the

Debye mass mD. After mD is ﬁxed, the given parameterization predicts the imaginary part of the potential Im[V]. The prediction results in agreement with quenched QCD simulations at high temperatures and becomes less accurate for temperatures around the phase transition

32 at zero net baryon chemical potential. To date, there has not been a vigorous determination of the imaginary part of the potential using full QCD. However, the values extracted in the procedure above show good agreement with the Gauss-law parameterization at high and intermediate temperatures [66]. The Debye mass related to the full QCD in-medium potential show coherent deviations from perturbative predictions in the temperature regime relevant to this study, TC < T < 3TC .

Due to lattice artifacts mentioned above, the potential as it stands may not be directly applied to phenomenological computations, despite the already successful parameterization which reproduces the lattice QCD in-medium potential. The parameters αs, σ and c could be determined from ﬁrst principles if one were to perform a continuum extrapolation of the

T = 0 potential in the thermodynamic limit. Instead, here we narrow these parameters phenomenologically such that the vacuum bottomonium spectrum is reproduced below the

D-meson and B-meson threshold for charmonium and bottomonium, respectively. The mass of the correct quark masses is the renormalon subtracted mass which then reﬂect the repro- duction of the spectra. They take on the value mRS0 = 4.882 0.041GeV for the bottom b ± quark and, for the charm quark, we compute the value to be mRS0 = 1.472 0.008GeV. c ± With this parameterization. With the vacuum potential in full QCD only robustly reproduced up to distances of r 1 fm, we enforce, by hand, asymptotics which are ﬂat due to ≈ string breaking at rSB = 1.25 fm. With all the above, the superlative set of parameters are given by

c = 0.1767 0.0210 GeV, αs = 0.5043 0.0298, √σ = 0.415 0.015 GeV. (2.15) − ± ± ±

Finally, the Debye mass is computed in a manner which is self-consistent with the dynamical evolution of the QGP medium and use its value to apply the in-medium modiﬁcation of the vacuum potential with the parameters given above. Figure 2.2 shows the real part of

33 Real-part of the heavy-quark potential from the Gauss-Law Imaginary-part of the heavy-quark potential from the Gauss-Law ��

��

] �� ][ �� ] ][

�� [ ��

mD=0 mD=400MeV - �� [ � �� mD=0 mD=400MeV α_s=0.504(29) -�� mD=100MeV mD=500MeV mD=100MeV mD=500MeV σ =0.415(15)GeV mD=200MeV mD=600MeV �� mD=200MeV mD=600MeV c=-0.177(21)GeV mD=300MeV mD=300MeV �� -�� [��] �[��]

Figure 2.2: The Re[V] (left) and Im[V] (right) part of the in-medium heavy quark potential which has been vetted against lattice QCD. The different lines correspond to different values of the Debye mass of the medium. The vacuum parameters αs, σ, and c at mD = 0 are tuned to reproduce the PDG charmonium and bottomonium spectra. String breaking is enforced at rsb = 1.25 fm. Ultimately, only one parameter governs the temperature-dependence which modifies the vacuum (Cornell) potential. Such thermal effects lead to Debye screening of the real part of the potential and induce a finite imaginary part which asymptotes at large distances.

the potential (left) and the imaginary part of the potential (right) which has been used for

various values of the Debye mass. To determined which values of the Debye mass are signif-

icant in the evolution of heavy quarkonium, it is noted that lattice QCD studies show, in a

thermal QCD medium close to the crossover transition temperature, the ratio is mD/T 1 ≈

and grows to mD/T 2 as temperature approaches T = 2TC . ≈

2.4 Non-equilibrium corrections to the lattice-vetted heavy quark potential

The Debye mass in an isotropic medium is deﬁned in the following manner

2 Z ∞ 2 g 2 dfiso mD = 2 dp p , (2.16) −2π 0 dp

where p is deﬁned in the following manner which follows the breaking of symmetry along

the longitudinal axis and transverse plane, p p2 = p2 + p2. Lattice QCD studies to date ≡ ⊥ z

34 are restricted to isotropic systems where an explicit temperature can describe the medium.

In the case of the anisotropy used in this study, we instead have a temperature-like scale.

The eﬀect of an anisotropic parameter, ξ, on the Strickland-Bazow potential derived from

pNRQCD has been analytically calculated which obtained the following result

µ(θ)−4 2b(a 1) + (1 + ξ)1/8 c(θ)(1 + ξ)d = 1 + ξ a − b 1 + 2 , (2.17) mD − (3 + ξ) (1 + eξ )

with a = 16/π2, b = 1/2, d = 3/2, e = 1/3, and the angular-dependence is factored into a

function which is deﬁned as

3π2 cos(2θ) + (9 + 4√3 4√6)π2 + 64(√6 3) c(θ) = − − . (2.18) 4√3(√2 1)π2 16(√6 3)) − − −

The implementation is handled by tabulating the real and imaginary parts of the potential

as a function of mD and r, and then simply replacing mD with µ in the isotropic potential to extend the potential for plasma anisotropies.

2.5 Solving for binding energies

The algorithm from References [69, 70] is used to solve the resulting 3d Schrödinger equation on a regular lattice by transforming to imaginary time and using the finite difference time domain (FDTD) method. Using this method, we compute the real and imaginary parts of the binding energy over a range of temperature-like scales, Λ, from 144 MeV to 1037 MeV.

For each Λ, we compute the real and imaginary binding energies for a range of anisotropies,

ξ, from 0.3 to 200, with an irregular spacing which accounts for the fact that the majority − of the time the system probes small values of ξ.

Determination of wave functions of the heavy quarkonia are performed computationally

35 by solving the eigenvalue problem found in the time-independent Schr¨odingerequation

ˆ Hφν(x) = Eνφν(x), (2.19) where Hˆ is the Hamiltonian and is given in the form used to solve two-body problems

2 ˆ H = ∇ + V (x) + m1 + m2, (2.20) − 2µ

where m1 and m2 are the masses of the quarks, µ = m1m2/(m1 + m2) is the reduced mass of the quarkonium system which reduces to mq/2 since m1 = m2 = mq, and the potential

V (x) is the complex-valued potential mentioned above, V = Re[V] + iIm[V]. The index ν found on eigenfunctions φν and energies Eν represent a combination of all quantum numbers, however, we will utilize the standard labeling for heavy quarkonia, i.e. nS and mP for s- and p-wave heavy quarkonia with n and m being integers. To obtain the eingenfunctions, we solve the 3-dimensional Schr¨odingerequation

∂ i ψ(x, t) = Hψˆ (x, t), (2.21) ∂t

which can be expanded in terms of the eigenfunctions φν,

X −iEν t ψ(x, t) = cνφν(x)e . (2.22) ν

To extract the lowest-lying states, we perform a Wick rotation, τ it. Using this rotation ≡ to imaginary time, Equations (2.21) and (2.22) are written as

∂ ψ(x, τ) = Hψˆ (x, τ), (2.23) ∂τ −

36 X −Eν τ ψ(x, t) = cνφν(x)e . (2.24) ν

This process of using a Wick rotation eﬀectively projects out the lowest-lying states since as

imaginary time is evolved, the excited-state wave functions become exponentially dampened.

The real and imaginary binding energies are extracted using [51]

φν V∞(θ) φν Eν,bind Eν m1 m2 h | | i , (2.25) ≡ − − − − φν φν h | i

where

V∞(θ) lim Re[V(θ, r)], (2.26) ≡ |r|→∞

and for the case of the Rothkopf potential we use the last point in r for V which is a ∞ good approximation as the potential is tabulated for distances up to 8 fm. Negative values

of Im[Ebind] only occur for large values of ξ in the Strickland-Bazow potential, which is a

consequence of the small-ξ expansion. Large values of ξ correspond to a nearly free streaming

quark-gluon plasma, so it is expected that the widths of quarkonium states return to vacuum

values, which are on the order of keV, which eﬀectively allows us to set Im[Ebind] = 0 for ∼ this speciﬁc case.

2.5.1 Strickland-Bazow potential discretizations

For charmonia, we use a N 3 = 2563 lattice with lattice spacing a = 0.15 GeV−1 0.03 ≈ fm and a = 0.20 GeV−1 0.04 fm for a total box length of L = Na 7.68 fm and ≈ ≈ L = Na 10.08 fm, for s-wave and p-wave states, respectively. For bottomonia, we use a ≈ N 3 = 2563 lattice with lattice spacing a = 0.1 GeV−1 0.02 fm and a = 0.15 GeV−1 0.03 ≈ ≈ fm for a total box length of L = Na 5.04 fm and L = Na 7.56 fm, for Υ(nS) and ≈ ≈

χb(mP ) states, respectively. The potential is put onto this lattice and evolved with the

3-dimensional Schr¨odinger equation in imaginary time. The grid is ﬁlled with wave function

37 which is randomly initialized and then evolved in imaginary time until the ground state

converges within a given tolerance for changing binding energies in time. The imaginary-

time step size for the solving algorithm is taken to be ∆τ = a2/8 which has been shown to

keep the temporal evolution stable while retaining a reliable physical picture of the state.

Using snapshots of the wave function we can project out the ﬁrst- and second-excited states

which are low-lying in the wave function due to the imaginary time evolution. We also

employ symmetry constraints which allow the code to solve for the s-wave quarkonia with

symmetric initial conditions and likewise we use antisymmetric initial conditions for the p-

wave quarkonia. This property is exploited using the fact symmetry cannot be broken by

the Hamiltonian evolution [69].

2.5.2 Rothkopf potential discretizations

To solve the Schr¨odingerequation with this potential, we employ the same 3-dimensional

code in which we solve the Schr¨odingerequation in imaginary time. We use a N 3 = 2563

lattice with grid spacing a = 0.15 GeV−1 for Υ(nS) states and a = 0.175 GeV−1 for the

χb(mP ) states. Since non-equilibrium corrections to this potential are handled on the potential level, the process for extracting the binding energies is the same for both potentials studied in this work.

2.6 Discussion on diﬀerence between potentials

The Strickland-Bazow potential is formulated using the internal energy of the heavy quarkonium pair which provides a much stronger binding when the potential is used to calculate suppression of states in heavy ion collisions [51]. While the Rothkopf potential is more weakly bound compared to the internal energy formulation, the Rothkopf potential remains bound stronger than the free energy description of heavy quarkonia.

From the results of Fig. 2.3, we see more-weakly bound Υ(1S) in the Rothkopf binding

38 �� ◆ ◆

● Strickland-Bazow, Real Part ● Strickland-Bazow, Real Part �� ■ Strickland-Bazow, Imaginary Part Υ(��) ■ Strickland-Bazow, Imaginary Part Υ(��) ◆ Rothkopf, Real Part ξ=� ◆ Rothkopf, Real Part ξ=� ▲ Rothkopf, Imaginary Part ▲ Rothkopf, Imaginary Part ● �� ▲ ▲ �� ▲ ▲ ◆ ▲ ▲ ◆ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ �� ● ▲ �� ◆ ▲ ▲ ▲ ▲ ● ▲ ▲ ◆ ▲ ● ◆ ● ◆● ● ◆ ● ● �� ◆ �� ◆ ● ● ◆ ▲ ● ● ◆ �� [ �� ] �� [ �� ] ● ◆ ● ◆ ● ◆ ● ■ ■ ◆ ● ■ ■ ■ ● ▲ ◆ ● ▲ ◆ ● ▲ ■ ■ �� ● ▲ ▲ ■ ■ �� ◆ ● ▲ ◆ ● ▲ ■ ■ ■ ◆ ● ▲ ▲ ■ ◆ ● ▲ ■ ■ ■ ◆ ●▲ ■ ■ ■ ◆ ▲ ●▲ ● ■ ■ ■ ◆ ▲ ▲ ● ● ■ ■ ■ ▲ ◆ ● ■ ■ ■ ▲ ◆ ● ■ ■ ■ ■ ▲ ◆ ■ ■ ●■ ▲ ▲ ◆ ●■ ●■ ■ ▲ ▲ ■ ■ ■ ◆ ● ● ▲ ◆■ ■ ■ ■ ● ● ▲ ■▲ ■ ■ ■ ◆ ● ● ▲ ▲ ■ ■ ■ ■ ■ ◆ ● ● ● ■ ▲■ ▲■ ■ ◆ ● ● ● ▲ ■▲ ■▲ ■ ■ ■ ◆ ● ● ● ■ ▲■ ▲■ ▲ ◆ ● ● ● ■ ▲■ ▲■ ▲■ ■▲ ■ ◆ ● ● ● ● ��▲ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆● ◆● ◆● ◆● ◆● ◆● ◆● ◆● ◆● ��▲ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆● ◆●

� � � � � � � � � � � � Λ/�� Λ/��

Figure 2.3: Comparison between Strickland-Bazow Υ(1S) and Rothkopf Υ(1S) for the cases of an isotropic, ξ = 0, plasma (left), and a slightly oblate, ξ = 1 plasma. In the Rothkopf model we observe weaker binding and a slightly stronger screening eﬀect from the imaginary part of the binding energy. Here Tc = 192 MeV which then sets the scale of the temperature- like scales of the calculation. The discontinuities seen in Im[V] are due to the real part going negative.

energies for both of isotropic and anisotropic plasma. We also see the Rothkopf potential

results in a stronger imaginary part which corresponds to stronger screening eﬀects in the

hot QGP. Therefore, we expect to see more suppression for the Rothkopf heavy quarkonia

compared to Strickland-Bazow heavy quarkonia. The jump for each Im[V] is due to con-

vergence issues when Re[V] < 0. Such regions of discontinuity are not factored into the

RAA calculation as we will come to see later in this paper. Finally, the sensitivity for the

momentum-space anisotropy parameter, ξ, is diﬀerent for the Strickland-Bazow Υ(1S) and

the Rothkopf Υ(1S). This is due to the non-equilibrium correction being substituted for the

Debye mass mD for the Rothkopf potential.

39 Chapter 3

3+1d anisotropic hydrodynamics

3.1 Anisotropic hydrodynamics

Throughout this study, a hydrodynamic background is used to simulate ultrarelativistic heavy-ion collisions. This hydrodynamic background consists of massless conformal quarks and gluons which obey an ideal equation of state which reduces to εiso = 3Piso in the isotropic state. Due to rapid expansion along the longitudinal beam-line axis of the colliding nuclei, compared to the relatively slow initial transverse expansion, one expects high-energy heavy- ion collisions to result in a plasma which is anisotropic in momentum space. In fact, recent studies have shown, with this degree of freedom established in the equations which govern the background evolution, regardless of the initial condition of L/ T , i.e. the local rest P P frame (LRF) pressure in the longitudinal (beam-line) direction and the LRF pressure in the transverse plane direction, respectively, the QGP evolution converges to an anisotropic

“attractor” which drives the approach toward isotropic equilibrium (see Figure 3.1) at late times [9].

The presence of such an attractor is critical as heavy quarkonia are formed rather early in the QGP evolution (τ < 1 fm/c). Across a wide range of initial L/ T , we see the presence P P of a momentum-space anisotropy is important for the hydrodynamical background. Further- more, an observable like quarkonium suppression can probe the early time dynamics of the

QGP, for example, the initial energy density and the initial momentum-space anisotropy of the QGP.

40 1.0 aHydro attractor(Tinti Matching) NS 0.8 Numerical solution

0.6 T /  L

 0.4

0.2

0.0 0.5 1 2 5 w

Figure 3.1: Plotted with the dimensionless time variablew ¯ = τ = τT , we see varying τR 5η/s L/ T initial conditions converge rather quickly to an attractor. Figure taken from Reference [9].P P

The following one-particle distribution function is used in the typical, anisotropic, “Romatschke-

Strickland” form 2 p 2 2 faniso(p , ξ, Λ) = fiso p + ξ(p ˆn) , Λ , (3.1) ·

where Λ is transverse momentum scale, which characterizes the temperature-like scale of

the plasma, and ξ is the momentum-space anisotropy parameter. For the case of breaking symmetry between the longitudinal axis and the transverse plane, ξ is given by the following

relation where ˆn = z. 1 p2 ξ = h ⊥i 1 (3.2) 2 p2 − h zi

where pz p ˆz and p⊥ p pz. ≡ · ≡ −

41 A spheroidal treatment of the momentum-space anisotropy of the plasma is an approximation, as there are two other diagonal components of the anisotropic tensor as well as three oﬀ-diagonal components. The eﬀect of each component of this tensor has been studied in the context of viscous hydrodynamics and is shown in detail in Figure 3.2.

Σ 0.1 Au+Au, b=7 fm SM-EOS Q 〉

/(e+p) 0 ∆

mn 0.004 τx 〉 π 〈π τy π ττ

/(e+p) π 2 ηη 0

mn xy τ π π -0.1 〈π -0.004 ∆ mn mn initialized by π =2ησ mn 0 5 10 initialized by π =0 τ−τ0(fm/c) 0 5 10 τ−τ0(fm/c)

Figure 3.2: Shear tensor corrections calculated using second-order viscous hydrodynamics Reference [10].

Figure 3.2 shows the evolution of the shear tensor components from a realistic second- order viscous hydrodynamics simulation as a function of proper time. Relative to the other curves, two curves stand out much more than the rest, which are Σ and τ 2πηη.1 Here Σ is the sum of the spacelike components xx and yy, i.e. Σ πxx + πyy, and the third spacelike ≡ component, zz, being found in the τ 2πηη curve, i.e. τ 2πηη = πzz. The correction to the transverse pressure is simply Σ/2 and πzz provides the correction for the longitudinal pressure.

This alone provides the motivation for breaking the symmetry between the longitudinal axis

1Here η is the spatial rapidity.

42 and the transverse plane. The transverse plane also contains an anisotropy which is the

curve ∆ πxx πyy. The quantity ∆ is smaller than the two curves mentioned prior up to ≡ − around 7 fm/c after the initial proper time, which also roughly corresponds to the time at which freezeout will occur in the aHydro simulation of minimum-bias collisions. Therefore, for the purposes of this study, we can ignore shear tensor corrections beyond an anisotropy which characterizes the breaking of symmetry between the longitudinal beam-line axis and the transverse plane.

This spheroidal description of momentum-space anisotropy creates three cases for the plasma: ξ < 0 describing a prolate plasma, ξ = 0 describing an isotropic plasma, and ξ > 0 describing a plasma that is oblate in momentum-space. The isotropic distribution function

−E/T is given by Boltzmann statistics, i.e. fiso = e .

The equations of motion for the anisotropic system are derived by starting from kinetic theory and the assumption that the distribution function of the plasma is known. This is done by taking moments of the Boltzmann kinetic equation in the relaxation-time approximation

(RTA) µ µ p uµ p ∂µf = , (3.3) τeq

where τeq is the local relaxation time of the plasma which can depend on proper time, and

uµ is the local four-velocity of the plasma. The zeroth moment of the Boltzmann kinetic

equation produces the particle production equation

µ µ µ Neq N ∂µN = uµ − , (3.4) τeq

which shows a system where particles are not conserved. The equilibrium particle ﬂux is

deﬁned as follows

µ µ Neq = nequ , (3.5)

43 and likewise,

N µ = nuµ (3.6)

The ﬁrst moment of the Boltzmann kinetic equation yields a set of partial diﬀerential equa-

tions which ensures energy-momentum conservation

µν ∂µT = 0, (3.7)

where at leading-order, the energy-momentum tensor has the following form for a plasma

which has a spheroidal anisotropy

µν µ ν µν µ ν T = (ε + P⊥)u u P⊥g (P⊥ Pk)z z . (3.8) − − −

When used together, Equations (3.4) and (3.7) result in a set of ﬁve partial diﬀerential

equations

ν Dµε = (ε P⊥)θu + (P⊥ Pk)uνDzz , (3.9) − − −

ν DzPk = (P⊥ Pk)θz + (ε + P⊥)zνDuu , (3.10) − " # u⊥ p⊥ ⊥P⊥ ν Duu⊥ = · ∇2 + DuP⊥ + (P⊥ Pk)uνDzz , (3.11) −ε + P⊥ u⊥ − ux 1 Du = 2 (ux∂y uy∂x)P⊥, (3.12) uy uy(ε + P⊥) −

Duξ 3DuΛ 1 h 3/4 p i = θu + 1 (ξ) 1 + ξ . (3.13) 2(1 + ξ) − Λ τeq − R

In these equations, note that the subscript denotes two-dimensional vectors in the trans- ⊥ verse plane, e.g. ⊥ (∂x, ∂y). Also note that the convective derivative is given by ∇ ≡ µ µ µ Dµ u ∂µ, the longitudinal derivative is Dz z ∂µ, and the expansion scalars are θu ∂µu ≡ ≡ ≡ µ and θz ∂µz . Without any symmetry constraints, we have the following parameterization ≡

44 of the four-velocity uµ and four-vector zµ in the lab frame,

µ u = (u0 cosh ϑ, u⊥, u0 sinh ϑ) (3.14)

zµ = (sinh ϑ, 0, cosh ϑ) (3.15) where the local longitudinal rapidity of the ﬂuid is given by ϑ. A normalization condition is

µ applied to the four-velocity u uµ = 1 resulting in

q 2 u0 = 1 + u⊥, with (3.16)

q 2 2 u⊥ u + u . (3.17) ≡ x y

With the parameterizations given in Equations (3.14) and (3.15), quantities appearing in

Equations (3.9)-(3.13), ˆ Du = u⊥ ⊥ + u0L1, (3.18) · ∇

ˆ ˆ θu = ⊥ u⊥ + L1u0 + u0L2ς, (3.19) ∇ ·

ˆ Dz = L2, (3.20)

ˆ θz = L1ϑ, (3.21)

ν ˆ uνDzz = u0L2ϑ, (3.22)

ν ˆ zνDuu = u0 u⊥ ⊥ + u0L1 ϑ, (3.23) − · ∇ where the two linear diﬀerential operators are given by

ˆ ∂ς L1 = cosh(ς ϑ)∂τ sinh(ς ϑ) (3.24) − − − τ

45 ˆ ∂ς L2 = sinh(ς ϑ)∂τ cosh(ς ϑ) . (3.25) − − − τ

Lastly, the following relation is used between the relaxation time τeq and transport coeﬃcient, shear viscosity to entropy density ratioη ¯ η/s, ≡

5¯η τ = . (3.26) eq 2T

3.1.1 The anisotropic equation of state

In order to properly close the dynamical equations above, an equation of state is needed to relate the energy density and pressure of the QGP. For this work, we use an ideal equation of state which relates the two as = 3P . The standard kinetic theory deﬁnitions are used E for the particle four-current, N µ, and the energy-momentum tensor, T µν,

Z d3p N µ pµf, (3.27) ≡ (2π)3p0

Z d3p T µν pµpνf. (3.28) ≡ (2π)3p0

With these deﬁnitions in Equations (3.27) and (3.28) and the tensor decompotions given in

Equations (3.5) and (3.8), the thermodynamics quantities of the system are given by

n n(Λ, ξ) = iso(Λ) , (3.29) √1 + ξ

ε(Λ, ξ) = (ξ)εiso(Λ), (3.30) R

P⊥(Λ, ξ) = ⊥(ξ)Piso(Λ), (3.31) R

Pk(Λ, ξ) = (ξ)Piso(Λ), (3.32) k

46 where niso, εiso, and Piso are the isotropic particle density, energy density, and pressure, respectively, and the anisotropic special functions required are

1 1 tan−1 √ξ (ξ) + , (3.33) R ≡ 2 1 + ξ √ξ

3 1 + (ξ2 1) (ξ) ⊥ − R , (3.34) R ≡ 2ξ ξ + 1 3 (ξ + 1) (ξ) 1 k R − . (3.35) R ≡ ξ ξ + 1

The factoring of the anisotropic parts from the thermodynamic quantities of the system

is a consequence of the conformal ﬂuid assumed in this work. Extensions to a non-ideal

equation of state within the framework of quasiparticle anisotropic hydrodynamics have

been developed and shown to reproduce particle-spectra, elliptic ﬂow coeﬃcients, etc. from

ultra-relativistic heavy-ion collisions [43].

3.2 aHydro initial conditions

We consider two symmetric lead nuclei (A = 197 for RHIC Au-Au collisions and A = 208

for Pb-Pb LHC collisions). Each nuclei is modeled using a Woods-Saxon distribution for the

transverse nuclear proﬁles. For one nuclei, we have

n n (r) = 0 , (3.36) A 1 + e(r−R)/d

−3 where n0 = 0.17 fm is the central nucleon density and is determined via the normalization

R 3 1/3 −1/3 limA→∞ d r nA(r) = A, R = (1.12A 0.86A ) fm is the nuclear radius, and d = 0.54 − fm is the skin depth of the nucleus [42]. Using Eq. (3.36), we construct the thickness function

Z ∞ p 2 2 2 TA(x, y) = dz nA x + y + z . (3.37) −∞

47 We can then build the overlap density function for the two colliding nuclei with centers

separated by an impact parameter b chosen to be along thex ˆ direction, b = bxˆ,

nAB = TA(x + b/2, y)TB(x b/2, y). (3.38) −

The overlap density function is used as the probability weight for bottomonium production

in the transverse plane and is used to calculated the number of participants for a given

impact parameter b.

��

�

� y ( fm )

-�

-�� -�� -� � � �� x(fm)

Figure 3.3: From the Glauber model, we calculate the transverse energy density proﬁle for a given impact parameter, b = 4.5.

For aHydro initial conditions we use a smooth linear combination, κbinary = 0.145 for

RHIC Au-Au collisions and κbinary = 0.15 for LHC Pb-Pb collisions, of Glauber wounded-

nucleon and binary collision scaling to set the initial energy density proﬁle of the transverse

48 inel plane in the QGP [51]. The inelastic cross section is taken to be σNN = 42 mb for RHIC

inel 200 GeV Au-Au collisions, σNN = 62 mb for LHC Run One 2.76 TeV Pb-Pb collisions, and

inel ﬁnally σNN = 67 mb for LHC Run Two 5.02 TeV Pb-Pb collisions. In the spatial rapidity direction, a boost-invariant plateau is used at central rapidities along with Gaussian tails at large rapidities, which is consistent with limited fragmentation [71]. This is explicitly given by the following function,

( ς ∆ς)2 f(ς) exp − − 2 Θ( ς ∆ς) , (3.39) ≡ − 2σς | | −

with ∆ς = 1.0 and σς = 1.3 used for RHIC 200 GeV collisions, and ∆ς = 2.5 and σς = 1.4 used for LHC Run One 2.76 TeV collisions, both of which are tuned to reproduce the experimental pseudorapidity distributions of charged particles. For LHC Run Two 5.02

TeV collisions, we made predictions based on [72]. Using [72], one ﬁnds that going from 2.76

TeV collisions to 5.02 TeV collisions, there is a 12% increase in the plateau halfwidth which gives ∆ς5.02 TeV = 2.8, and no observable diﬀerence in the Gaussian halfwidth which results

5.02 TeV in σς = 1.4. Diﬀerences between longitudinal distributions among experiments can be seen in Figure 3.4.

The combination of a Glauber profile in the transverse plane along with Equation (3.39) for the longitudinal profile, we now have a full description of the initial 3+1d energy density profile for the QGP modeled using aHydro.

3.2.1 Determining the initial temperature-like scales

The initial temperatures for √sNN = 2.76 TeV collisions were fixed against soft hadron production using aHydro [73, 43]. Since these initial temperatures were fixed, we needed a self-consistent way to determine whether scans for the temperature-like scale Λ0 would result in reliable numbers. To do this, we run the aHydro data files through a code which finds the

49 ��

��

�� -� � � ��

Figure 3.4: Using a boost-invariant plateau with Gaussian tails, we can phenomenologically model the longitudinal proﬁle for the initial energy density produced in an ultra-relativistic heavy-ion collision for RHIC 200 GeV/nucleon Au-Au collisions and LHC 2.76 TeV/nucleon and 5.02 TeV/nucleon Pb-Pb collisions.

freezeout hypersurface at a given temperature, and then integrate the number of conformal

partons on the surface given by

Z µ N = dΣµu ndens(ξ, Λ) (3.40)

n (Λ) √iso where ndens is the anisotropic number density and is given by ndens = 1+ξ . This number would become the target for adjusting η/s and the initial momentum-space anisotropy

parameter, ξ0, i.e. we hold the ﬁnal gluon number ﬁxed as we vary the initial conditions.

However, during this work the LHC expanded their heavy-ion runs to 5.02 TeV/nucleon,

and little information was available about the initial temperatures in LHC Run Two Colli-

sions, so we made predictions based on a few simple assumptions about the QGP.

50 √sNN 0.2 TeV 2.76 TeV 5.02 TeV ξ 0 0 10 50 0 10 50 0 10 50 4πη/s 1 0.442 0.613 0.744 0.552 0.765 0.925 0.641 0.888 1.076 2 0.440 0.608 0.739 0.546 0.752 0.909 0.632 0.869 1.053 3 0.439 0.609 0.742 0.544 0.748 0.906 0.629 0.863 1.046

Table 3.1: Λ0 values (in GeV) for various collision energies, ξ0, and η/s.

We assume that the QGP created forms a rough ellipsoidal shape with end cap area A

and total initial volume V0 = τ0A. The energy of a particular collision is given by the beam

energy √sNN. However, only a certain fraction of beam energy is turned into energy in the

QGP system. This is given by E = γ√sNN, where γ is the fraction associated with QGP

formation from a heavy-ion collision. We assume that γ is constant as a function of center

of mass beam energy √sNN from 2.76 TeV/nucleon to 5.02 TeV/nucleon. We start with the

fact,

0 ENN QGP = √sNN, (3.41) E V0 ∝

0 4 and as we understand T √sNN, we can then draw conclusions that lead to the EQGP ∝ 0 ∝ following equation in which

1/8 T0 = sNN. (3.42)

We then took the initial temperature from 2.76 TeV Pb-Pb collisions for 4πη/s = 1 and

5.02 TeV 1/4 ξ0 = 0 and scaled it by the fraction 1.16. As a result, the isotropic temperature 2.76 TeV ≈ was increased by 16%, then the other temperature-like scales for aHydro runs for 4πη/s =

1, 2, 3 and finite initial anisotropy ξ0 were completed using the procedure outlined above, { } by fixing final particle multiplicity.

Using the above procedure, we were able to determine the initial temperature-like scales

(seen in Table 3.1) which eﬀectively determine the initial energy density proﬁles in the

aHydro simulation of the QGP.

51 Chapter 4

Computing the suppression factor RAA

4.1 Computing RAA

The nuclear modiﬁcation factor for nucleus-nucleus collisions, RAA, is generically deﬁned

Nparticles,AA RAA , (4.1) ≡ nbinary-collisions Nparticles,pp

where Nparticles,AA is the number of particles detected in AA collisions, Nparticles,pp is the

number of particles detected in pp collisions, and the scaling term, nbinary-collisions, is how

many parton pairs collide in the AA collision.

Computing RAA in our framework requires knowledge of the energy density and momentum- space anisotropy, ξ, from the realistic 3+1d background simulation code, aHydro, and com- putation of the heavy-quark potential to obtain the complex-valued binding energies as a function of energy density and ξ. The two are folded together in a code which utilizes the

following procedure to calculate RAA. We assume the following initial transverse momentum

distribution of heavy-quarkonia,

tanh (0.35pT ) f0(m, pT ) = N 2 2 5.7/2 , (4.2) (m + pT )

where N is some normalization which is determined using experimental cross sections, m is

the mass of the heavy-quark state, and pT is the transverse momentum of the state. The

speciﬁc form of this distribution, namely the ﬁts which detail the factor of 0.35 and the power

of 5.7 to the heavy quarkonium energy, come from ﬁts to the heavy quarkonium spectra in

proton-proton collisions seen at the LHC, as seen in Figure 4.1.

52 ��

� ��

�� [ �� / �� ] � �� / ��

� σ �� -�

� �� (��)

Figure 4.1: Heavy-quarkonium production at the CMS and ATLAS experiments [11].

The data seen in Figure 4.1 is for Υ(1S) production, however it is outlined in Reference

[11] that each Υ(nS) state and charmonium state follows a similar shape. Therefore this pT

ﬁt seen in Equation (4.2) is used for all heavy-quarkonium states and across all heavy-ion

collision systems.

The following rate equation, ∂τ ni = Γi [ni neq,i], is used to compute the survival − − probability of a state as we sample the whole QGP via integration, where i is the state being

evolved. The function neq,i is the result of a regeneration component which is explicitly

outlined in Section 4.1.1. We use the binding energies from the Schr¨odingercode to extract

a heavy quark state breakup rate which is given by

  2Im[Ebind(τ, x⊥, ς)] Re[Ebind(τ, x⊥, ς)] > 0 Γ(τ, x⊥, ς) = (4.3)  γdis Re[Ebind(τ, x⊥, ς)] 0, ≤

53 where the instantaneous breakup rate is a function of space and time. It is given by 2Im[Ebind] for states which are “bound” as determined by the real part of the binding energy and, when the state is unbound, we choose a particular breakup rate of 10 GeV which rapidly dissociates the heavy quarkonium state.

The formation time of each state is computed using its transverse momentum as τform(pT ) =

0 0 γτform = ET τform/M, where M is the mass of the state. Therefore, the lower integration limit is max(τform, τ0), where τ0 is the initial proper time for plasma evolution which is taken to

0 be 0.3 fm/c, and τform is taken to be inversely proportional to the vacuum binding energy

0 [74] and thus, τform = 0.2, 0.4, 0.6, 0.4, 0.6, and 0.6 fm/c, for Υ(1S), Υ(2S), Υ(3S), χb(1P ),

χb(2P ), and χb(3P ), respectively, for bottomonia. For charmonia we make the same as-

0 sumption for the formation time of each state so that τform = 0.3, 3.9, and 0.8 fm/c, for J/ψ,

ψ(2S), and χc(1P ), respectively [75].

Transverse momentum cuts were implemented via the assumption that the transverse momentum distribution of charmonia and bottomonia states is proportional to a distribution

ﬁtted to heavy quarkonium state production in proton-proton collisions,

R pT,max 2 2 2 5.7/2 dpT RAA(pT , x⊥, ς) tanh (0.35pT )/(pT + M ) R (x , ς) pT,min (4.4) AA ⊥ R pT,max 2 2 2 5.7/2 ≡ dp tanh (0.35pT )/(p + M ) pT,min T T

Followed by averaging RAA over the transverse plane

R nAA(x⊥)RAA(x⊥, ς) x⊥ RAA(ς) R (4.5) h i ≡ nAA(x⊥) x⊥

Centrality averaging is performed by converting the impact parameter, b, to centrality, C, using the Glauber model, and then integrating over the centrality cut with a probability distribution proportional to e−C/20 with 0 < C < 100 [76].

Primordial heavy quarkonium states are not measured in detectors, but rather the decay

54 dileptons, so the process of feed down must be factored in to compute the inclusive RAA for each state observed in experiment; currently limited to the charmonium states J/ψ and ψ(2S), and the bottomonium states Υ(1S), Υ(2S), and Υ(3S). Feed down has been measured at the ATLAS, CMS, and LHCb experiments at the LHC and have obtained the following values listed in Tables 4.1 and 4.2, which are the values we report when averaged over the transverse momentum, pT . While the process of feed up due to in-medium excitation, e.g. Υ(1S) Υ(2S) exists, such a process in perturbation theory between color singlet states → must occur via exchanges between two gluons. However, such processes are suppressed. In perturbation theory, transitions between color singlets Υ states must happen via the exchange of two gluons due to selection rules and therefore are unlikely.

Charmonium feed down fractions J/ψ J/ψ 0.76 - - ψ(2S)→ J/ψ 0.08 ψ(2S) ψ(2S) 0.5 → → χc(1P ) J/ψ 0.16 χc(1P ) ψ(2S) 0.5 → →

Table 4.1: Feed down fractions which have been averaged over pT , for the J/ψ and ψ(2S) states used in the determination of the ﬁnal measured yields [11]. Note that the feed-down fractions for the ψ(2S) are estimates due to lack of measured feed down data.

Bottomonium feed down fractions Υ(1S) Υ(1S) 0.668 - - - - Υ(2S) → Υ(1S) 0.086 Υ(2S) Υ(2S) 0.604 - - Υ(3S) → Υ(1S) 0.010 Υ(3S) → Υ(2S) 0.043 Υ(3S) Υ(3S) 0.5 → → → χb(1P ) Υ(1S) 0.170 - - - - → χb(2P ) Υ(1S) 0.051 χb(2P ) Υ(2S) 0.309 - - → → χb(3P ) Υ(1S) 0.015 χb(3P ) Υ(2S) 0.044 χb(3P ) Υ(3S) 0.5 → → →

Table 4.2: Feed down fractions which have been averaged over pT , for the Υ(nS) states used in the determination of the ﬁnal measured yields [11]. Note that the feed-down fractions for the Υ(3S) are estimates due to lack of measured feed down data.

Feed down fractions are implemented by ﬁrst computing the “raw” suppression for each

55 state, followed by using a linear combination formula

inclusive X raw,i RAA = fiRAA (4.6) i

raw,i th where RAA is the raw (pre-feed-down) suppression of the i state, fi is the feed down fraction which are listed in Tables 4.1 and 4.2. Equation (4.6) computes the inclusive feed down suppression for each state we wish to calculate and compare to experimental data.

4.1.1 Regenerative eﬀects

Throughout the evolution of the quark-gluon plasma, there’s a chance that a closed quark pair (i.e. a bound quarkonium state) will dissociate into the plasma due to thermal effects from the QGP. This is one of the main mechanisms behind quarkonium suppression and is known as thermal suppression. There is also a chance the now open pair of quarks will form into a bound state via a process called recombination, or form a new bound state via a process called regeneration. The details of which effect comes into play are not important, so we will call both effects regeneration.

We utilize the following rate equation which folds together the non-equilibrium spa- tiotemporal evolution of the quark gluon plasma along with binding energy data calculated by solving the Sch¨odingerequation for the heavy-quark potential

dn(τ, x) h i = Γ(T (τ, x)) n(τ, x) neq(T (τ, x)) , (4.7) dτ − −

explains the balance between the local number density of closed heavy quarkonia and the

equilibrium number density of states given by Boltzmann statistics, which for large tem-

peratures is a very good approximation for Fermi-Dirac statistics. The primary feature of

this regeneration model is the local description of heavy quarkonia within the QGP. Such

56 a description can provided critical insight into experimental observables seen across quarko-

nium production in the QGP. The equilibrium number of heavy quarkonia is given by the

following Z d3q n (T, γ ) = 3γ2(T ) f(m; T ), (4.8) eq q q (2π)3

where γq is the fugacity factor driving statistical regeneration of heavy-quark closed states,

i.e. heavy quarkonia. Its calculation is detailed in the following manner. The speciﬁc form

of neq is given by

˜ 3 2 neq(T, m) = 4πNT mˆ eqK2(m ˆ eq), (4.9)

˜ 3 wherem ˆ eq = m/T , and N Ndof/(2π) , with Ndof being the number of degrees of freedom. ≡ Since the suppression code integrates over space and transverse momentum, we must use

the local unintegrated distribution function to compute the equilibrium distribution at each

point in space and proper time. This is given by

3 pp2 + m2 f (m, p , τ, x) = γ2e−y T (4.10) eq T (2π)3 q T (ξ, Λ)

where y is the spatial rapidity direction in the QGP, and the temperature, T , is given by

T (ξ, Λ) = Λ (ξ)1/4.1 (4.11) R

We assume all heavy quark pairs are generated in the initial hard scatterings of partons in

high energy collisions. The number of heavy quark pairs is determined phenomenologically

in order to accurately pin down the regeneration component for each colliding system and is

given by pp σqq¯ Nqq¯ = Ncoll(b), (4.12) σinel

1A factor of ~c should be used for unit conversion.

57 pp where σqq¯ is the heavy quarkonium production cross section from proton-proton collisions,

σinel is the inelastic cross section for nucleus-nucleus collisions, and Ncoll(b) is the number

is binary collisions and is a function of the impact parameter, b, for a given AA collision.

In this work, Ncoll is calculated using the smooth Glauber model. We start with Equation

(3.38), and proceed to calculate Ncoll by integrating over the spatial transverse plane with

the multiplicative factor σinel,

Z Ncoll(b) = σinel TA(x + b/2, y)TB(x b/2, y), (4.13) x⊥ −

where TA(x + b/2, y) and TB(x + b/2, y), depend on nuclear properties via the mass numbers

A and B, respectively, and are given, in general, by Equation (3.37).

We assume the number of heavy quark pairs remains ﬁxed throughout the evolution of

the QGP and the only occurrences involving these pairs are transitions between open states

and closed states. The model which details this evolution is described by

1 I1(Nop) Nqq¯ = Nop + Nhid, (4.14) 2 I0(Nop)

where I0 and I1 are modiﬁed Bessel functions of the ﬁrst kind and the functions Nop and

Nhid are the number of open and hidden heavy quark states. Equation (4.14) is derived from

the assumption [77] that

1 2 Nqq¯ = γqNop + γ Nhid + , (4.15) 2 q ···

where the factors of the heavy quark fugacity factor, γq, have been factored out of Nop and

Nhid to show the original expansion used in [77], and other terms which include second-

2 and third-order factors of γq are negligible. This is due to quark contributions to various

2At the time of Reference [77], tetraquark and pentaquark states were hypothetical particles and have yet to be observed in experiments.

58 states which have particle densities orders of magnitude below those of open heavy ﬂavor and quarkonium states. It was shown in [78] that, when using the canonical ensemble, multiplicities of open heavy ﬂavor are multiplied by an additional ‘canonical suppression’ factor, which is explicitly given by

c.e. I1(Nop) 3 Nop = Nop . (4.16) I1(Nop)

The functions Nop and Nhid are given by the following expressions

Z d3p N = γ V d f q(p; T ), and (4.17) op q coll b (2π)3

X Z d3p N = γ2V d f states(p; T ), (4.18) hid q coll states (2π)3 states where Vcoll is the volume of the colliding system and is understood to be a function of the proper time τ, db is the degeneracy describing the number of colors and matter/anti-matter,

q i.e. db = 6 2, f (p; T ) is the free quark distribution at a given momentum p and temperature · T , and f states(p; T ) is similar for a given closed state, i.e. heavy quarkonium state.

Charmonium cross sections

Production of cc¯ pairs has been measured extensively across many collision energies and experiments. In Figure 4.2, we see both experimental data and theory curves. From this plot, we interpolate to achieve known cc¯ cross sections for all relevant collision energies. The cross section for 2.76 TeV collisions at ALICE are reported as the total cross section σcc¯, so the conversion to mid-rapidity density means dividing by a factor of 7-8 [79]. For the purposes of this study we divide by 7.5 which then establishes a fair error-bar to include in further calculations. Dividing by the σcc¯ value reported for 2.76 TeV collisions at ALICE, one

3Here we can use the canonical ensemble due to the lack of baryon density, i.e. B = 0.

59 b) ALICE (total unc.) µ ALICE extr. unc.

( 4

10c LHCb (total unc.) c PHENIX σ STAR HERA•B (pA) E653 (pA) E743 (pA) 3 10 NA27 (pA) NA16 (pA) E769 (pA) NLO (MNR)

102

10 102 103 104 s (GeV)

Figure 4.2: cc¯ production across multiple collision energies and experiments [12].

dσ finds the mid-rapidity density of cc¯ production to be dy = 0.64 µb. Squaring the error bars in quadrature produces an error bar of 0.06 µb and thus the final mid-rapidity differential

dσ cross section used for 2.76 TeV collisions is dy = 0.64(6) µb. For further continuation of errors, we will assume a 10% variation in reported cross sections. ± dσ Diﬀerential Cross Section dy 0.2 TeV 2.76 TeV 5.02 TeV pp cc¯ + X [mb] ( y < 1.6) 0.17(2) 0.64(6) 0.90(8) pp → cc¯ + X [mb] (1| .6| < y < 2.5) 0.11(1) 0.43(4) 0.60(6) pp → cc¯ + X [mb] (2.5 < |y| < 4.0) 0.09(1) 0.32(3) 0.45(5) → | | Table 4.3: cc¯ production cross sections calculated from Figure 4.2. The mid-rapidity value for 200 GeV collisions was taken directly from [25]. The mid-rapidity value for 2.76 TeV collisions was taken directly from [26]. The mid-rapidity value for 5.02 TeV collisions was derived via interpolating data given in Figure 4.2. The interpolation yields a 40% increase from 2.76 TeV to 5.02 TeV.

We calculate forward rapidities for RHIC & LHC collisions by assuming a 150% increase

from 1.6 < y < 2.5 to y < 1.6 and a 200% increase from 2.5 < y < 4.0 to y < 1.6, | | | | | | | |

60 based on data seen from LHCb in Figure 4.5.

b) e+e•, s=7 TeV (±4% luminosity) µ 8 µ+µ•, s=7 TeV (±5.5% luminosity) e+e•, s=2.76 TeV (±3% luminosity) ALICE pp + •

/dy ( µ µ , s=2.76 TeV (±3% luminosity)

ψ 7 J/ σ

d 6

1 open: reflected 0 •5 •4 •3 •2 •1 0 1 2 3 4 5 y

J/ψ We compute the ratio cc¯ by first reading off the central value from Figure 4.3. This value is dσJ/ψ = 3.74 0.68 µb for mid-rapidity. Dividing the cc¯ yield from 2.76 TeV collisions dy ± J/ψ 3.74µb at mid-rapidity, we obtain cc¯ = 0.64mb = 0.58%. This ratio is verified with 5.02 TeV data

dσ from the CMS experiment. We take data in Figure 4.4 and integrate over pT to derive dy for the mid-rapidity window (summing the curves up to y = 1.5). When we calculate the

J/ψ 3.81 µb ratio, we ﬁnd cc¯ = 0.90 mb = 0.42%. This is with integration cutting oﬀ at 3 GeV/c. Lower

pT contributions would increase this fraction slightly toward the calculated value 0.6%. ∼ dσJ/ψ dσJ/ψ We also know dy = 3.4 µb at CDF 1.96 TeV collisions and dy = 7.5 µb at 7 TeV

dσ ALICE collisions [80, 81]. Interpolating these values gives dy = 5.90 µb for 5.02 TeV, which J/ψ when used to determine the cc¯ ratio, gives a ratio of 0.66%. With all the above evidence,

61 28.0 pb-1 (pp 5.02 TeV) 103 102 CMS 10 Prompt J/ψ 1

−1

b/ GeV/c) 10 µ 10−2 dy ( 10T −3

/dp −4

σ 10 2 Forward rapidity − 5 0 < y < 0.9 10 CM B d 0.9 < y < 1.5 (x10) CM −6 1.5 < y < 1.93 (x100) 10 CM 1.93 < y < 2.4 (x1000) CM 10−7 0 5 10 15 20 25 30 p (GeV/c) T

Figure 4.4: J/ψ production in 5.02 TeV p-p collisions at CMS [14].

J/ψ it is reasonable to apply cc¯ = 0.6% to cross sections at all energies.

dσ Diﬀerential Cross Section dy 200 GeV 2.76 TeV 5.02 TeV pp J/ψ + X [µb] ( y < 1.6) 1.02(10) 3.84(38) 5.40(54) pp → J/ψ + X [µb] (1| .6| < y < 2.5) 0.66(7) 2.58(26) 3.60(36) pp → J/ψ + X [µb] (2.5 < |y| < 4.0) 0.54(5) 1.92(19) 2.70(27) → | | J/ψ Table 4.4: Fixed via a ratio of cc¯ = 0.6%, we compute the diﬀerential cross sections of J/ψ states.

In 2014, the LHCb experiment at CERN measured the ratio of ψ(2S) states to J/ψ states at √s = 7 TeV. They found the ratio to be σ(ψ(2S)) = 0.17 0.02 [82]. Based on this ratio, σ(J/ψ) ± and with the assumption the ratio does not change signiﬁcantly as a function of √s, we can calculate the cross section of ψ(2S) states from derived J/ψ cross sections.

Finally, we turn our attention to the χc(1P ) state. ATLAS has measured χcJ production cross sections using non-prompt J/ψ states observed in √s = 7 TeV collisions.

From Figure 4.6, the cross sections for the χcJ states are calculated and then summed. The

62 ] 1.75 pPb, Pbp mb

[ LHCb

∗ . pp rescaled

σ 1 50 y √s = 8.16TeV

d NN d prompt J/ψ 1.25

1.00

0.75

0.50

0.25 0 < pT < 14GeV/c 0.00 5 0 5 − y∗

dσ Figure 4.5: cc¯ dy production across two energies and a range of forward rapidities. With interpolation to mid-rapidity, this ﬁgure is used to derive the multiplicative factors in going from 1.6 < y < 2.4 to y < 1.6 (x1.5), and 2.5 < y < 4.0 to y < 1.6 (x2) [15]. | | | | | | | | dσ Diﬀerential Cross Section dy 200 GeV 2.76 TeV 5.02 TeV pp ψ(2S) + X [µb] ( y < 1.6) 0.17(2) 0.65(7) 0.92(9) pp → ψ(2S) + X [µb] (1| .6| < y < 2.5) 0.11(1) 0.44(4) 0.61(6) pp → ψ(2S) + X [µb] (2.5 < |y| < 4.0) 0.09(1) 0.33(3) 0.46(5) → | | σ(ψ(2S)) Table 4.5: Fixed via a ratio of σ(J/ψ) = 0.17, we compute the cross sections of ψ(2S) states.

ﬁnal summed cross section is divided by 1.5 to convert it into a diﬀerential cross section

dσ dσ dy = 22 nb. Using a J/ψ diﬀerential cross section of dy = 7.5 µb [81] and a ratio of σ(ψ(2S)) σ(J/ψ) = 0.17, we compute the “generic” diﬀerential cross section for the ψ(2S) state of

dσ dy = 1275 nb. Using these two ﬁnal numbers, the ratio χc/ψ(2S) is calculated to be 1.7%.

Using this ratio, we can compute the cross sections for the χc(1P ) state.

63 J/ψ ATLAS Non•prompt |y | < 0.75 1 •1 s = 7 TeV L dt = 4.5 fb Data χ ∫ c1

[nb/GeV] Data T Isotropic Decay χc2 10•1 )/dp cJ χ (

σ •2 d 10 × B

10•3

FONLL b X → χc1 •4 FONLL b X 10 → χc2

10 20 30 χ p c [GeV] T

Figure 4.6: χcJ production cross sections as a function of transverse momentum. Note that + − B = (χcJ J/ψγ) (J/ψ µ µ ) [16]. B → ·B →

Bottomonium cross sections

Values for the diﬀerential cross section for the Υ(1S) and Υ(2S) states are taken from

[27] and the total cross sections are veriﬁed from the sources within. The Υ(3S) production

cross sections are calculated from two other sources. For 2.76 TeV collisions, the approximate

dσ differential cross section dy is read off from Figure 4.7. We assume a 50% drop in differential cross section going from CMS to ALICE, due to different rapidity cuts.

As for 5.02 TeV collisions, dσ is read oﬀ from Figure 4.8 for the Υ(3S) state. With B dy a Particle Data Group value of (Υ(3S) µ+µ−) = 2.18% [2], we compute the following B → Υ(3S) diﬀerential cross sections (see Table 4.7).

According to Figure 4.9, production of P-wave bottomonia states can be measured with respect to the Υ(1S) state which is known very well. Using the pT -averaged value of 9

64 dσ Diﬀerential Cross Section dy 200 GeV 2.76 TeV 5.02 TeV pp χc(1P ) + X [nb] ( y < 1.6) 0.29(3) 1.11(11) 1.56(16) → | | pp χc(1P ) + X [nb] (1.6 < y < 2.5) 0.19(2) 0.75(8) 1.04(10) → | | pp χc(1P ) + X [nb] (2.5 < y < 4.0) 0.15(2) 0.56(6) 0.78(8) → | |

Table 4.6: Cross sections for the χc(1P ) state computed via χc/ψ(2S) = 1.7%.

pp 5.4 pb-1 s = 2.76 TeV

(nb) 1 CMS σ dy d

10−1

ϒ(1S) ϒ(2S) ϒ(3S)

10−2 0 0.5 1 1.5 2 |y|

Figure 4.7: Diﬀerential cross sections for the Υ(nS) states [17].

GeV/c (except for the ﬁrst available data point for the χb(3P ) state), we can read oﬀ the following values

χ (1P ) b = 17% R Υ(1S) χ (2P ) b = 5% R Υ(1S) χ (3P ) b = 0.75% R Υ(1S)

Using these values, we can now construct our table of production cross sections for the

χb(mP ) states.

65 pp 28.0 pb-1 (5.02 TeV)

µµ p < 30 GeV/c 10 T CMS Preliminary

1 (nb) σ dy d B 10−1 Υ(1S) Υ(2S) Υ(3S) 10−2

0 0.5 1 1.5 2 µµ |y |

Figure 4.8: Diﬀerential cross sections for the Υ(nS) states [18].

dσ Differential Cross Section dy 0.2 TeV 2.76 TeV 5.02 TeV pp Υ(1S)( y < 0.5) [nb] 2.35 - - pp → Υ(1S)(|y| < 2.4) [nb] - 30.3 57.6 pp → Υ(1S) (2| .5| < y < 4.0) [nb] - 15.1 28.8 pp → Υ(2S)( y < 0.5) [nb] 0.77 - - pp → Υ(2S)(|y| < 2.4) [nb] - 10.0 19.0 pp → Υ(2S) (2| .5| < y < 4.0) [nb] - 5.0 9.5 pp → Υ(3S)( y < 0.5) [nb] 0.13 - - pp → Υ(3S)(|y| < 2.4) [nb] - 0.002 0.004 pp → Υ(3S) (2| .5| < y < 4.0) [nb] - 0.001 0.002 → dσ Table 4.7: Differential cross sections dy from [27] for the Υ(1S) and Υ(2S) states. Υ(3S) states are computed from [17] for 2.76 TeV and [18] for 5.02 TeV collisions. For RHIC collision energy, we take an extrapolated value for the differential cross section.

66 Full Cross Section 0.2 TeV 2.76 TeV 5.02 TeV pp Υ(1S)( y < 0.5) [nb] 2.35 - - pp → Υ(1S)(|y| < 2.4) [nb] - 145.44 276.48 pp → Υ(1S) (2| .5| < y < 4.0) [nb] - 22.65 43.20 pp → Υ(2S)( y < 0.5) [nb] 0.77 - - pp → Υ(2S)(|y| < 2.4) [nb] - 48.00 91.20 pp → Υ(2S) (2| .5| < y < 4.0) [nb] - 7.50 14.25 pp → Υ(3S)( y < 0.5) [nb] 0.13 - - pp → Υ(3S)(|y| < 2.4) [nb] - 0.01 0.02 pp → Υ(3S) (2| .5| < y < 4.0) [nb] - 0.002 0.003 → Table 4.8: Calculated cross sections from [27] for the Υ(1S) and Υ(2S) states. Υ(3S) states are computed from [17] for 2.76 TeV and [18] for 5.02 TeV collisions. For ALICE rapidity cuts, the stated rapidity coverage for dimuons (from [28]) is 2.5 < y < 4.0.

Figure 4.9: Production cross sections for χb(mP ) states given as a ratio to Υ(1S) production cross sections [19].

67 Full Cross Section 0.2 TeV 2.76 TeV 5.02 TeV pp χb(1P )( y < 0.5) [nb] 0.40 - - → | | pp χb(1P )( y < 2.4) [nb] - 24.72 47.00 → | | pp χb(1P ) (2.5 < y < 4.0) [nb] - 3.85 7.34 → pp χb(2P )( y < 0.5) [nb] 0.12 - - → | | pp χb(2P )( y < 2.4) [nb] - 7.27 13.82 → | | pp χb(2P ) (2.5 < y < 4.0) [nb] - 1.13 2.16 → pp χb(3P )( y < 0.5) [nb] 0.02 - - → | | pp χb(3P )( y < 2.4) [nb] - 1.09 2.07 → | | pp χb(3P ) (2.5 < y < 4.0) [nb] - 0.16 0.32 →

Table 4.9: Calculated cross sections for the χb(mP ) states.

68 Chapter 5

Results for quarkonium suppression

In this work, the main goal is to try to explain experimental charmonium and bottomo-

nium suppression data seen in ultra-relativistic heavy-ion collisions by eﬀectively modeling

the QGP generated and the heavy quarkonium found in such collisions, namely as a function

of the local energy density and the degree of momentum-space anisotropy. Comparisons to

experiments including STAR at RHIC, and CMS and ALICE at the LHC, show that the

thermal suppression model with regeneration describe data collected by these experiments

in multiple windows including Npart, which is mapped from centrality classes in experiment

using the Glauber model, the spatial rapidity y, and the transverse momenta of heavy quark

states. The span of collision energies investigated in this work go from 200 GeV/nucleon

Au-Au collisions at RHIC, to 2.76 TeV/nucleon Pb-Pb collisions at the LHC, and ﬁnally to

5.02 TeV/nucleon Pb-Pb collisions at the LHC.

5.1 Charmonium

In Figure 5.1, we see the raw suppression for the charmonium states J/ψ, ψ(2S), and

χc(1P ). The main features are the strong suppression seen from J/ψ, and the lack of

suppression seen in the ψ(2S) state. This lack of suppression is due to the large formation

time of the ψ(2S). In this model, the state can be formed early in the hot medium, so any proper time formation after the aHydro time τ0, is an approximation which shields the state

from pre-formation suppression eﬀects. The χc(1P ) state also has a relatively large formation

time compared to the initial time for the aHydro simulation, so peripheral collisions do not

see any suppression since, when the state is formed, the ﬁnal integration temperature for the

69 survival probability has reached its threshold.

�� ψ(��) |�| < �� πη/�=�

�� = �� <� � < �� -��

��

� �/ψ ��

�� χ�(��)

��

Figure 5.1: Raw suppression for the J/ψ, ψ(2S), and χc(1P ) charmonium states for LHC 2.76 TeV/nucleon Pb-Pb collisions. For this ﬁgure the states are modeled with the Strickland- Bazow potential. The lack of suppression for the ψ(2S) state is due to the large formation time.

In Figure 5.2, we see the inclusive feed down calculation for J/ψ RAA which include direct

suppression from J/ψ and then feed down eﬀects from the ψ(2S) and χc(1P ) states.

5.2 Bottomonium

The suppression for each of the six bottomonium states is shown in Figure 5.3 for 2.76

TeV/nucleon collisions at the LHC.

The ﬁrst inclusive (feed down) RAA result to present (Figure 5.4) is the diﬀerence between

the Strickland-Bazow internal energy potential mentioned in Section 2.2 and the Rothkopf

potential mentioned in Section 2.3.

Each potential has three model calculation curves, but the diﬀerences in the bands vary

between potentials. The Strickland-Bazow potential’s band comes from the aHydro shear

70 �� /ψ �πη/�=� |�| < �� πη/�=� �� = �� πη/�=� ��<� < ��

��

Figure 5.2: Inclusive RAA calculation for the J/ψ state in 2.76 TeV/nucleon Pb-Pb collisions at the LHC, as calculated by the Strickland-Bazow potential.

viscosity to entropy density ratio, η/s. The bottom curve corresponds to 4πη/s = 1, the

middle with 4πη/s = 2, and the top curve is for 4πη/s = 3. There is no signiﬁcant η/s

dependence for the Rothkopf potential RAA. For the Rothkopf calculation we vary the Debye

mass mD by 15% to provide a modest band when producing ﬁnal suppression calculations. ± The lowest Rothkopf curve corresponds to an increase of 15%, the middle is the normal

Debye mass as a function of temperature as provided in the model, and the top curve is

a 15% reduction in the Debye mass. The ﬁnal eﬀect on inclusive feed down RAA is that

of a weaker-binded Υ(1S) for the Rothkopf potential compared to the Strickland-Bazow

potential. This weaker binding makes the bottomonia more susceptible to thermal eﬀects

seen in the QGP. While the binding in the Rothkopf potential is weaker, it is still stronger

than the free energy approach taken in previous studies [51]. For 200 GeV/nucleon Au-

Au collisions at RHIC, we see model calculations are consistent with STAR data for the

Rothkopf potential.

71 �� |�| < �� |�| < �� πη/�=� �πη/�=�

�� = �� = �� <� � < �� <� � < �� -��

�� Υ(��) �� Υ(��) ��

�� χ�(��) ��

Υ(��) χ�(��) ��

χ�(��) χ�(��) Υ(��) ��

Figure 5.3: Raw suppression for Υ(nS) and χb(mP ) states from LHC 2.76 TeV/nucleon Pb- Pb collisions. With both potentials we see a sequential ordering of suppression and smooth transitions in RAA from peripheral to central collisions. The order of sequential suppression is the same in both potentials, except for the pileup of largely-suppressed states, which is more evident with the Rothkopf potential. This pileup is due to a “halo” survival eﬀect for bottomonia formed near the edge of the QGP.

Next we compare to 2.76 TeV/nucleon Pb-Pb collisions at CMS where both potential

models describe the experimental data quite well in Figure 5.5. In the Strickland-Bazow

potential there’s a slight underestimation of suppression for the excited Υ(2S) state for cen-

tral collisions. The Rothkopf model does reasonably well with both states for this particular

collision system.

The ALICE experiment also calculated bottomonium suppression as a function of cen-

trality and comparisons between data and our model can be seen in Figure 5.6. ALICE data

at forward rapidity suggests bottomonia are strongly suppressed when compared to CMS

suppression data at the same collision energy. The main diﬀerence between the two exper-

iments are the rapidity windows, with CMS collecting data in the mid-rapidity ( y < 2.4) | | region and ALICE collecting data in the forward rapidity (2.5 < y < 4.0) region. The dra-

matic diﬀerence between the two experiments might be due to many eﬀects which include

statistics, pp reference data, or other eﬀects which cause suppression in varying kinematic

pictures such as cold nuclear matter eﬀects.

72 ��

�� -��

��

�� Υ(��) �πη/�=�(��) �� |�| < �� = �� <� � < ��

Figure 5.4: Bottomonium suppression for RHIC 200 GeV/nucleon collisions as seen by the STAR collaboration. The Strickland-Bazow potential features an error band which corresponds to varying values of η/s while the Rothkopf potential features a band corresponding to uncertainty in the Debye mass calculation as a function of temperature. STAR data taken from [20].

Bottomonium suppression as a function of spatial rapidity, as seen in Figure 5.7, is

measured at the CMS experiment for mid-rapidity and the ALICE experiment for forward

rapidity at the LHC. Both potential models describe CMS data, however some lingering

tension between our model calculations and ALICE forward rapidity data exists as we un-

derestimate the suppression with the Strickland-Bazow potential, and just barely touches

the data with the Rothkopf potential at forward rapidity, while mid-rapidity calculations

with the Rothkopf potential explain CMS data within the mD error band.

As a function of the transverse momentum, pT , in Figure 5.8 both potentials describe

Υ(1S) suppression. For the Υ(2S) state, the Strickland-Bazow potential underestimates

suppression while the Rothkopf potential explains the data well with its systematic weaker

binding.

73 �� πη/ = �πη/�=� � � � | | < �πη/�=� � �� |�| < �� πη/�=� �� = �� = �� Υ(��) �� <� � < �� <� � < �� Υ(��) ��

�� Υ(��) Υ(��) ��

�� Υ(��) Υ( ) �� Υ(��) Υ(��)[��] ��

Figure 5.5: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). Comparisons between our model calculations and bottomonium suppression seen at LHC 2.76 TeV/nucleon Pb-Pb collisions as viewed by the CMS collaboration. CMS data taken from [21].

For LHC Run Two comparisons we ﬁrst look at Figure 5.9 for 5.02 TeV/nucleon collisions

at CMS where the Strickland-Bazow potential successfully predicted new data that was

preliminarily released in February 2017. Not shown in this ﬁgure is the Υ(3S) which CMS

projects to be absent from the spectra in varying conﬁdence levels. Our model calculation

for the Υ(3S) is consistent with a “halo” survival eﬀect in the QGP where a certain amount

of heavy quarkonium states are formed on the colder edge of the plasma and in large part

are free from thermal eﬀects from the QGP. With the Rothkopf potential we see a slight

overestimation of RAA for the Υ(1S) and Υ(2S) states.

In Figure 5.10 we look at 5.02 TeV/nucleon ALICE data [23]. The Strickland-Bazow

potential describes data from ALICE in the Npart window while the Rothkopf potential

slightly overestimates suppression for central collisions. This is unlike 2.76 TeV/nucleon

data where the model calculation and experimental data disagree.

As seen in Figure 5.11, the Strickland-Bazow potential describes rapidity data from the

CMS and ALICE experiments well for the Υ(1S) and Υ(2S) states. The Rothkopf potential

slightly predicts more suppression than the experimental data suggests. Once again the

74 �� Υ(��) Υ(��) πη/ = � � � �� πη/�=� ��<�< �� πη/�=� ��<�< �� = �� πη/�=� �� = �� <� � < �� <� � < ��

��

Figure 5.6: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). In LHC 2.76 TeV/nucleon Pb-Pb collisions, we make a comparison between model calculations and bottomonium suppression data gathered from the ALICE collaboration. ALICE data taken from [22].

Strickland-Bazow potential reasonably describes data as seen in Figure 5.12 as a function of

pT . For larger pT the Rothkopf potential slightly overestimates suppression of bottomonium

states.

75 �πη/�=� �-��% �πη/�=� �� πη/�=� �� -��% �� = �� πη/�=� �� = �� <� � < �� (��) Υ(��) �� <� � < �� (��) �<� < �� (��) Υ(��) �� <� � < �� (��) Υ(��) ��

�� Υ(��) �� [Υ(��) ��] � �

��

Υ(��) Υ(��) �� Υ(��) Υ( ) �� Υ(��)[��] �� |�| |�|

Figure 5.7: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). In this ﬁgure, we can state within reason, there is no signiﬁcant rapidity dependence for Υ(1S) suppression for LHC 2.76 TeV/nucleon collisions. With some tension from forward-rapidity ALICE data, the more strongly suppressed Rothkopf Υ(1S) provides a reasonable description of suppression. CMS and ALICE data taken from [21, 22].

�� πη/�=� �� = �� = �� πη/�=� �πη/�=� �- ��% - % �πη/�=� |�| < �� |�| < ��

�� Υ(��) �� Υ(��) ��

Υ(��)

�� Υ(��)

�� (��) �� (��)

Figure 5.8: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). For transverse momentum comparisons, we see no signiﬁcant pT dependence in the data while the model calculations show a decreasing suppression for larger pT due to larger formation times eﬀectively shielding formed bottomonia from thermal suppression. CMS data taken from [21].

76 �� πη/�=� �πη/�=� �� |�| < �� πη/�=� �� |�|<�� = �� πη/�=� �� = �� <� < �� Υ(��) �� <� � < �� Υ(��) �� =Υ(��) ��=Υ(��) �� Υ(��) �� Υ(��)

�� Υ(��) Υ(��) Υ(��) Υ(��)[��] ��

Figure 5.9: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). Predictions made with the Strickland-Bazow potential agree quite well with CMS data at 5.02 TeV/nucleon Npart data for all observed Υ(nS) states. Not seen is the Υ(3S) conﬁdence intervals for a completely suppressed Υ(3S). CMS data taken from [8].

�� Υ(��) Υ(��) �πη/�=� �� πη/�=� ��<�< �� πη/�=� ��<�< �� = �� πη/�=� �� = �� <� � < �� <� � < ��

��

Figure 5.10: Strickland-Bazow potential calculation (left), compared to the Rothkopf potential calculation (right). For 5.02 TeV/nucleon collisions, we see the Strickland-Bazow potential calculation successfully predicts ALICE data. The Rothkopf potential calculation slightly overestimates suppression. ALICE data taken from [23].

77 �πη/�=� �πη/�=� �� -��% �πη/�=� �� -��%

�� = �� πη/�=� �� = �� Υ(��) �� <� � < �� (��) �<� � < �� (��) Υ(��) �� <� < �� (��) �� <� � < �� (��) � Υ(��) ��

�� Υ(��) Υ(��) ��

�� Υ(��) �� Υ(��) Υ(��) Υ( )[ ] �� |�| |�|

Figure 5.11: The Strickland-Bazow potential calculation (left) predicted CMS and ALICE rapidity data successfully. The Rothkopf potential calculation (right) slightly overestimates Υ(1S) suppression, but explains Υ(2S) and Υ(3S) suppression successfully. Note that con- ﬁdence levels for a fully suppressed Υ(3S) state are not shown here. CMS and ALICE data taken from [8, 23].

�� πη/�=� �� = �� = �� πη/�=� �πη/�=� �- ��% - % �πη/�=� |�| < �� |�| < ��

�� Υ(��) �� Υ(��)

Υ(��) �� Υ(��)

�� (��) �� (��)

Figure 5.12: There is a slight increase in Υ(1S) experimental data as well as the Strickland- Bazow potential model calculation (left). The Rothkopf potential model calculation describes Υ(1S) for low-pT , but overall seems to overestimate suppression for the Υ(1S). Both potential calculations describe Υ(2S) suppression for this given collision system. CMS data taken from [8].

78 Chapter 6

Discussion & outlook

6.1 Charmonium

While the inclusion of a local regeneration model makes suppression calculations possible for all experimental physics ranges for centrality, rapidity, and pT , we have run into a limitation in our current model. We assume the heavy quarkonia are formed early in the

QGP proper time evolution around the initial time τ0. This makes state formation times which are roughly inversely proportional a good approximation as many of the states, primarily for bottomonia, are formed near the initial proper time τ0. Some states, however, are weakly bound in the QGP, so the calculation results in large formation times which can be much larger than the initial proper time. In peripheral collisions, the QGP only has a short evolution time before the plasma becomes nearly free streaming which turns off thermal suppression effects for heavy quarkonia. Therefore, any state formed after the initial hydrodynamics simulation time will be shielded by thermal suppression effects as times before the formation time are not factored into the survival probability calculation. The final effect on inclusive J/ψ suppression is limited to an 8% contribution from the ψ(2S), so inclusive J/ψ results are possible to estimate at this time. However, raw suppression calculations results in curves which are not sequentially suppressed and furthermore, the inclusive suppression for ψ(2S) is not possible to show at this time due to the lack of suppression. Experimental results show inclusive ψ(2S) states are more suppressed than inclusive J/ψ states for all centrality.

79 6.2 Bottomonium

Model calculations for RAA as a function of Npart show a smooth variation with increasing centrality of AA collisions. This is consistent with the picture of no thresholds where the

QGP suddenly turns on with increasing energy density. Rather, we see a smooth transition where the QGP rises in temperature. Likewise, the thermometer description of bottomonium is not so simplistic as there are no discrete thresholds for melting where states drop off from the spectrum once a specific energy density is reached. Smooth transitions from no nuclear modification in peripheral collisions to the strong suppression seen in central collisions can also be described by the realistic description of the QGP using aHydro. Energy density is not the same everywhere in the QGP, even in the region where the nuclei overlap due to the

“thickness” function description of the modeled nuclei in the Glauber model.

The dependence of RAA as a function of spatial rapidity is flat for the range of rapidity viewed in experiments and thus this model. This feature can be traced back to energy density profile which features a plateau region for mid-rapidities along with Gaussian tails for forward rapidities. For higher-energy collisions at the LHC as seen in Run One and more-so in Run Two, we see a slight decrease in suppression in for mid-rapidity which is consistent with a hotter medium, where a b¯b pair is more likely to be produced and later recombine via regenerative effects if suppressed in the first place.

Likewise in the transverse momentum picture we do not see a signiﬁcant dependence for heavy-quarkonium suppression. This contrasts with the indiction that bottomonium are regenerated for currently-studied AA collisions. However, with the current error estimations seen in experiment, it’s possible regeneration is stronger than data suggests. Further analysis of bottomonium draws our attention to the move from 2.76 TeV to 5.02 TeV Pb-Pb collisions at the LHC. Centrality-averaged RAA for the Υ(1S) state decreases by 20% as measured ∼ by CMS [8], while the ALICE experiment sees 20% increase in RAA [22, 23] indicating ∼

80 the significance of regeneration which is not seen in the model presented in this work. The differences between experiments will hopefully be worked out with upgrades at the LHC which will improve statistics of heavy quarkonium measurements. Until then, it’s clear that both experimental data seen from both experiments and the model calculations presented in this work suggest bottomonium production may be subjected to an increasingly significant regeneration component at around 5.02 TeV/nucleon Pb-Pb collisions at the LHC. With an increase in collision energy, cold nuclear matter effects will become less significant and thermal effects will present a cleaner probe of bottomonium suppression.

6.3 Outlook

The charmonium suppression results shed light on the need for further work which factors quantum-statistical evolution to the states formation. Furthermore, the inclusion of cold nuclear matter effects for charmonium suppression have a stronger effect than that on bottomonium. While the charmonium results I show are a good estimate, further effects can modify final calculations.

Overall, the model presented in this work provides a realistic picture of the QGP with anisotropic hydrodynamics and likewise an eﬀective framework which models heavy quarkonium using a potential-based treatment. Together, we can, in large part, adequately describe the heavy quarkonium suppression seen in ultra-relativistic heavy-ion collisions at RHIC and the LHC. The hard probe qualities of heavy-quarkonium narrow the theory which describes the QGP throughout the entire proper time evolution.

81 Chapter 7

Appendix

7.1 Finite initial anisotropy parameter

In Figure 7.1, we adjust the initial momentum-space distribution to an oblate spheroidal

distribution. There’s reason to believe the QGP starts oﬀ in this anisotropic state due to

the initial geometry of the collision where there is rapid expansion along the longitudinal

(beam) axis, and, initially, a slow transverse expansion.

�� Υ(��)

|�| < �� ξ� =� �πη/�=� �� ξ� = �� <� < �� ξ� = �� = ��

��

Figure 7.1: We make an adjustment to the initial geometry of the collision by deforming the initial momentum-space distribution to an oblate spheroidal distribution.

As seen in Figure 3.1, the initial L/ T quickly converges to an attractor solution. A P P hard probe like heavy quarkonia, due to their early formation times, is sensitive to solutions

82 before the attractor solution is reached. However, since the solutions converge rapidly, the

ﬁnal eﬀect on RAA is small.

7.2 Adjustment of the formation time of heavy quarkonia

As discussed in Section 6.1, the formation time in this model is inversely proportional

to the binding energy of each state. This assumption works well for a state which is formed

early in the QGP evolution and hydrodynamics simulation ( 0.2 fm/c). However, a state ∼

formed later than the initial proper time of the QGP, τ0, is eﬀectively shielded from the

thermal suppression eﬀects seen in the hot QGP.

��

Υ(��) �πη/�=� �� = ��

�<� � < ��

τ = �� /� �� τ�� = �� /�

τ�� = �� /�

��

Figure 7.2: As we adjust the formation time of the bottomonium states, the inclusive RAA is aﬀected by 10%. Adjusting this quantity measures the “shielding” eﬀect the formation time has for heavy∼ quarkonia.

In Figure 7.2, we vary the formation time by a generous 50% for the Υ(1S) state. ± Due to the aforementioned early formation time, the “shielding” eﬀect is reduced and the

inclusive RAA is limited to a 10% band. ± 83 7.3 Model sensitivity of the ﬁnal cutoﬀ temperature for survival probability

integration

Finally, we look at the cutoﬀ temperature for the survival probability calculation. As we

integrate numerically, the code ignores points which are below a temperature value which

corresponds to heavy quarkonium suppression turning oﬀ.

��

Υ(��) �πη/�=� �� = ��

�<� � < ��

� = �� = ��

�� = ��

��

Figure 7.3: The survival probability is cut off at a given temperature, Tf . As we adjust Tf , we see a small effect on final inclusive Υ(1S) RAA which becomes less sensitive as the cutoff temperature decreases.

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