The beginning and end of heavy ion collisions using uranium beams and bose einstein correlations as probes of the collision fireball

See text for details.KLN curves are normalized such that, for central Au+Au, integration of the entropy density over the transverse plane produces the sameresult as the Glauber model.. 5

THE BEGINNING AND END OF RELATIVISTIC HEAVY ION COLLISIONS: USING URANIUM BEAMS AND BOSE-EINSTEIN CORRELATIONS AS PROBES OF THE COLLISION FIREBALL

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Anthony Joseph Kuhlman, Jr., M.S.

* * * * * The Ohio State University

Physics

In this work, we begin by examining the possibility of using collisions betweenlarge deformed nuclei, such as uranium, at the Relativistic Heavy Ion Collider (RHIC)facility at Brookhaven National Laboratory We present calculations that highlightthe advantages of such an endeavor over the current gold-gold (Au+Au) program.These calculations are examined both within a Glauber model framework and using

a color glass condensate (CGC) type picture

We first compute the initial entropy densities that would be produced in uranium (U+U) collisions and compare with those currently available in central gold-gold (Au+Au) collisions We find that the maximum entropy density available intip-on-tip U+U collisions is approximately 40% greater than that available in centralAu+Au, providing a long lever arm to test the observed ideal fluid behavior of theelliptic flow

uranium-Additionally, we show that U+U collisions are capable of producing interactionregions with deformations comparable to those of peripheral Au+Au (x = 0.25),but with sources that are larger by a factor of two The longer path lengths andlarger densities of these sources are shown to increase the energy loss of fast partonstraversing the medium by up to a factor of two, with a similar accentuation of thedifference in energy loss between in-plane and out-of-plane paths This enhancesthe resolution available to analyze the path length dependence of this energy loss,

allowing for a fundamental test of our understanding of Quantum Chromodynamics(QCD).

We illustrate a possible event selection process for these collisions We find that

by placing strict cuts on the number of spectators produced in the collision and thecharged particle multiplicity, we are able to reliably select for a given orientation ofthe nuclei

This procedure is analyzed via a Monte Carlo simulation designed to examine theeffects of detector inefficiencies on event selection After implementing event-by-eventfluctuations of the charged particle multiplicity, we show that by placing tight cuts onspectator number and multiplicity, i.e., 0.5% total events, we are able to select eventswith either large or small average deformation We also show that by relaxing thecut on spectator number we improve the statistics of the experiment with a minimaleffect on the control of average initial deformation

We close our examination of U+U collisions by estimating the effects of binningand inefficiencies of the zero degree calorimeters on the maximum transverse particledensity available in these events We find that for the smearing used in our MonteCarlo simulation the effects are somewhat dramatic, reducing a potential 40% increase

in particle density to approximately 23% For larger smearings, we find even greaterreductions We illustrate the source of these reductions with calculations of theaverage transverse particle density for the most restrictive bin

The second portion of this dissertation focuses on the use of Bose-Einstein relations to obtain information about the spatial and temporal extent of the fireballproduced in these collisions We begin by detailing the development of a computerprogram designed to compute the azimuthally sensitive Hanbury Brown-Twiss (HBT)

cor-radii for a given source function We present an overview of the algorithm involved

in our program and discuss its actual implementation

We then present a number of analytic calculations that are then compared tothe output of this program We show that in this analytic limit, the code is quiteaccurate, computing the components of the spatial correlation tensor to within 1%

of the analytic solution Finally, we show how symmetries of the emission functioncan be employed to reduce the amount of time necessary to compute the azimuthaldependence of the HBT radii

The final chapter focuses on the development of a non-relativistic formalism todescribe the effects of final state interactions on the measured radii We present thisformula and examine its behavior in various limits We then pursue approximations

to this expression to produce a more practical result In the limit of a weak interactionwith the medium, we produce a surprisingly straightforward result

For Cindy, Celie, and Joey

There are many people who contributed in one way or another to my graduateeducation, all of whom deserve my thanks Over the six years that I’ve been here,various people have made their impact on me, some in subtle ways, others moreobvious Because of the passage of time and my lackluster memory, I’m sure there arethose that I have forgotten, and I immediately apologize for not explicitly thankingthem in this space There are some, though, whose impact has been so deep andlong-lasting that it will never be forgotten

First among these is my wife, Cindy Without her, I would have nothing She hasbeen a constant source of encouragement and support during my studies She’s alsobeen a standard of patience and tolerance during my late nights and fits of misplacedfrustration She has made me a better man, and for that I am grateful I could nothave done this without her

I must also thank my children, Cecilia and Joseph, for their patience I know thatthey don’t always understand why work sometimes has to come before playtime, butthey tolerate it and love me anyway They have taught me responsibility and shown

me the importance of being a good example Their smiles always pick me up after along day

I also thank Jeremy Bergeson and Ivan Tornes for their friendship and support.Without them, I wouldn’t have lasted through my first year in graduate school They

helped me through the early years of Electrodynamics and Quantum Mechanics, andwere there to turn to later as well Ivan’s odd fascination with all things computer-related helped me immensely when I was faced with a technical hurdle in my pro-gramming I knew he could be counted on to be so intrigued by some minor oddity

in the code that he would eventually help me solve it With Jeremy I always hadsomeone to turn to with seemingly mundane physics questions that needed a deeplevel of insight More often than not, his level thinking would quickly lead him tothe correct answer But perhaps more importantly, Ivan and Jeremy were alwaysthere for the extracurricular projects, whether it be launching paper airplanes overthe laboratory next door or drywalling a basement

Of course, I’m also indebted to my adviser, Ulrich Heinz His wisdom and guidancehas made me a better scientist and a more critical thinker He has shown me the value

of a calm, measured examination of a problem, and his expertise has been invaluable.There are other faculty at Ohio State whose contributions, though relatively mi-nor, have been keenly felt These include Richard Furnstahl, Michael Lisa, CiriyamJayaprakash, Ralf Bundschuh and Bernard Mulligan The depth of their knowledgeand their enthusiasm for physics has made working here all the more exciting Ofcourse, I also owe a debt of gratitude to Richard Furnstahl, Michael Lisa and JuliaMeyer for agreeing to serve on my committee

Finally, I thank my parents and my brother, Tom My mother and father instilled

a love of science and mathematics in me at an early age, and have always encouraged

me in my education Tom has always provided a reference for me to measure myselfagainst and, as a fellow physicist, a sympathetic ear

To everyone mentioned here I extend my sincerest gratitude Without you, mygraduate education would not have been as rewarding as it has been for the past sixyears.

August 2, 1975 Born - Evansville, IN

1997 B.A Music Engineering Technology,

Ball State University

2001 B.S Physics,

Otterbein College

2001-2004 Fowler Fellow,

The Ohio State University

2004-2006 Graduate Research Assistant,

The Ohio State University

2005 M.S Physics, The Ohio State

Univer-sity

2006-Present Distinguished University Fellow,

The Ohio State University

deforma-Anthony Kuhlman, Ulrich Heinz and Yuri V Kovchegov, “Gluon saturation effects

in ultra-relativistic U+U collisions,” Physics Letters B 638 171 (2006)

Anthony Kuhlman, “The scientific promise of a future uranium-uranium collisionprogram at the Relativistic Heavy Ion Collider Facility,” Proceedings of the 2005Edward F Hayes Graduate Research Forum

TABLE OF CONTENTS

Page

Abstract ii

Dedication v

Acknowledgments vi

Vita ix

List of Tables xiv

List of Figures xvi

Chapters: 1 INTRODUCTION 1

1.1 Signatures of the quark gluon plasma 3

1.1.1 Parton energy loss and jet quenching 4

1.1.2 “RHIC serves the perfect liquid”–Elliptic flow and ideal hy-drodynamics at RHIC 7

1.2 Hanbury Brown-Twiss interferometry of the fireball 10

1.2.1 Basic formalism 11

1.2.2 Final state interactions in HBT interferometry 16

1.2.3 Experimental measurement 17

1.2.4 The “HBT puzzle” 23

1.3 Overview 24

2 ULTRA-RELATIVISTIC URANIUM-URANIUM COLLISIONS 27

2.1 Introduction 27

2.2 Entropy density 34

2.2.1 Glauber model 34

2.2.2 Color glass condensate 40

2.3 Parton energy loss 50

2.4 Event selection 56

2.5 Monte Carlo simulation 61

2.5.1 Increases in transverse particle density 68

2.6 Conclusions and future directions 77

3 A PROGRAM TO COMPUTE THE SPATIAL CORRELATION TEN-SOR COMPONENTS OF THE PARTICLE-EMITTING SOURCE IN RHIC COLLISIONS 83

3.1 Importance Sampling 86

3.2 Stratified Sampling 89

3.3 An Overview of the VEGAS Algorithm 93

3.4 Implementation of VEGAS 98

4 ANALYTIC CALCULATIONS OF THE SPATIAL CORRELATION TEN-SOR FOR ULTRARELATIVISTIC HEAVY ION COLLISIONS 102

4.1 Model with Constant Size 109

4.2 Model with τ -Dependent Transverse Size 111

4.3 Comparison with Numerical Results 118

5 FOURIER EXPANSION OF THE SPATIAL CORRELATION TENSOR 124 5.1 Symmetries of the Emission Function 125

5.2 Fourier Expansion of the Tensor Components 126

5.3 Computation of the Fourier Coefficients 128

5.4 Optimizing the calculation 131

5.5 Conclusion 137

6 FINAL STATE INTERACTIONS IN TWO-PION INTERFEROMETRY 140 6.1 Development of the formalism 142

6.1.1 Two-particle probability with no interaction 152

6.1.2 Time-independent two-particle interaction, no medium inter-action 152

6.1.3 Time-independent medium interaction, no two-particle po-tential 160

6.1.4 Time-independent medium and two-particle interactions 162

6.1.5 Summary of key results 169

6.1.6 Time dependence of the two-particle probability 171

6.2 Calculation of the single-particle probability 173

6.3 Approximations 174

6.3.1 The smoothness approximation 174

6.3.2 Neglecting the two-body interaction 179

6.4 Conclusions 182

7 Conclusion 185

Appendices: A Evaluation of Quadratic Averages 190

B Evaluation of τ Integrals Containing Even Powers of R 194

Bibliography 197

LIST OF TABLES

text for a complete explanation of the approximations used to acquirethese results 108

Here, δ = ∆ττ0 111

transverse size 112

decreasing R For elements such as h˜z2i, the listing simply indicatesthat the corresponding expressions from Table 4.4 should be subtractedaccordingly The remaining expressions are the resulting expansions to

O(δ2) 120

Table 4.6 These results were obtained with 106 calls per function 122

from Table 4.6 These results were obtained using 106 calls per function.122

the text versus those calculated using the exact expressions for cosh(η−

Y ) and ul, for the case of constant R 123

4.10 Comparison of results obtained using the approximations described inthe text versus those calculated using the exact expressions for cosh(η−

Y ) and ul, for the case of decreasing R 123

5.2 “Realistic” parameters used in the calculation of Fourier coefficients 131

5.3 Computed Fourier coefficients for the spatial correlation tensor through

n = 10, using parameters in Table 5.2 Calculation used 105 functioncalls per iteration 132

5.4 Computed Fourier coefficients for the spatial correlation tensor through

n = 10, using parameters in Table 5.2 Calculation used 105 functioncalls per iteration 133

listed here are the lowest order expressions that provide an able” value for χ2

“accept-ν 135

described in equations (6.110) and (6.112), respectively These ages are computed in the xyz-coordinate system of the source, where

struc-LIST OF FIGURES

along the beam direction, with the colliding portions of the two nucleirepresented by the red region The impact parameter, b, describes theseparation of the centers of the two nuclei 6

1.2 pT-integrated v2 scaled by the eccentricity of the overlap region versusthe transverse particle density 9

1.3 In this schematic illustration, sources a and b emit particles along fourpossible paths (black) These particles are subsequently detected indetectors 1 and 2 13

1.7 Final vs initial eccentricities of the particle emission region at RHIC.Larger initial eccentricities correspond to more peripheral collisions 22

and hydrodynamic/cascade (Soff) model calculations of HBT radiiwith measured values from RHIC Data for π+π+(π−π−) are indicated

by closed (open) symbols 24

2.1 Illustration of full-overlap collisions between uranium nuclei Such lisions require b = 0 with the long axes of the nuclei coplanar andaligned so that their angles with the beam direction satisfy Φ1 = ±Φ2 33

col-2.2 Charged particle multiplicity per participant pair versus number ofparticipants 37

Au+Au collisions of varying centrality and full-overlap U+U at √sN N =

200 GeV 38

U+U and (c) Au+Au with b = 7 fm Contours are given as fractions

of participants for Au+Au and (inset) full-overlap U+U collisions at

√s

GeV (dashed), pT,cutof f = 2Qmax

(solid gray) Results from Glauber model for full-overlap U+U areindicated by dotted line 42

2.6 Transverse entropy density profiles for central Au+Au (top panels), on-tip U+U (center panels) and edge-on-edge U+U (bottom panels) at

tip-√s

from the KLN model with pT,cutof f = 3 GeV (dashed gray), pT,cutof f =2Qmax

s (solid black) and pT,cutof f → ∞ (solid gray) See text for details.KLN curves are normalized such that, for central Au+Au, integration

of the entropy density over the transverse plane produces the sameresult as the Glauber model 44

at√s

pT,cutof f = 2Qmax

and for full-overlap U+U collisions Results are from KLN model with

pT,cutof f = 2Qmax

Con-tours are given as fractions of the maximum entropy density produced

in tip-on-tip uranium 48

2.9 Entropy density contours for (a) tip-on-tip U+U, (b) edge-on-edgeU+U and (c) Au+Au with b = 5.5 fm Shown are results from the

(bottom panels) Contours are plotted as fractions of the maximum

2.10 Glauber model calculation of energy loss of a fast parton versus tricity, assuming quadratic dependence on path length, i.e., equation(2.18) Shown are results for Au+Au (gray) and full-overlap U+U(black) Energy loss for in-plane (out-of-plane) direction is given bysolid (dashed) curves Top panel: time independent density (ρ(t) =

(ρ(t) = ρ0τ 0

τ See text for discussion 52

2.11 Same as Figure 2.10 but with linear path length dependence for energyloss (see equation (2.19)) 54

2.12 KLN model calculation of energy loss for a fast parton along in-plane(solid) and out-of-plane (dashed) paths assuming quadratic (left pan-els) and linear (right panels) path length dependence Our calculation

black Top panels: time independent density Bottom panels: Timedependent density See text for discussion 55

2.13 Multiplicity distribution for full-overlap uranium collisions resultingfrom Glauber model calculation Gaussian curves illustrate individualdistributions for fixed Φ from Φ = π/2 (leftmost distribution) to Φ = 0(rightmost distribution) in 10 degree increments Individual gaussiansare not drawn to scale 58

2.14 Left Panel: Glauber model calculation of multiplicity distribution forfull-overlap U+U collisions Vertical bands indicate cuts according

to the indicated percentages Right panel: Eccentricity distributionsresulting from the multiplicity cuts shown at left 59

2.15 Multiplicity distribution for full-overlap uranium collisions resulting

2.16 Eccentricity distributions resulting from cuts on the multiplicity butions shown in Figure 2.15 Black (gray) curves correspond to cuts

distri-on the 10% of events with the lowest (highest) multiplicity Shown aredistributions from Glauber model (dotted), KLN model with scaling

200 GeV Calculation relies on Glauber model prescription of equation(2.23) 63

2.18 Multiplicity distributions resulting from selecting the 5% (dotted) and0.5% (solid histogram) of the events of Figure 2.17 with the lowestnumber of spectators Vertical lines represent 5% lowest (left) andhighest (right) multiplicity cuts 64

2.19 Eccentricity distributions of 5% lowest (black) and highes (gray) plicity events from distributions of Figure 2.18 Shown are results from

2.20 Spectator distribution for U+U collisions at √s

shows entropy density contours (s = 20, 40, 60, 80, 100 fm−3 from theoutside inward) for a large eccentricity (x = 0.325) event included in a

“5% spectator cut” data sample This event has an impact parameter

2.21 Minimum bias multiplicity distribution obtained from KLN model with

pT,cutof f → ∞ (black) Also shown is the result from the Glauber modelcalculation (gray) 67

2.22 Multiplicity distribution obtained by taking 0.5% of the events in thedistribution of Figure 2.21 with the lowest number of spectators (redhistogram) This result is compared to the distribution for full-overlap

2.23 Eccentricity distributions obtained by taking the 5% lowest (black) and5% highest (gray) multiplicity events from the distribution shown inFigure 2.22 68

2.24 Left panel: Calculation of D1

S

dN dy

E

Ge-Vusing equation (2.26) for listed cuts on multiplicity and spectator

S

dN dy

Efor Au+Au resulting from Monte Carlosimulation These calculations include event-by-event fluctuations of

2.25 Left panel: Average transverse particle density for U+U collisions at

√s

specta-tor number, using the probability distribution for multiplicity given inequation (2.20) and with no smearing of the spectator number Rightpanel: Ratio of the average transverse particle density for U+U to that

of Au+Au (cf right panel of Figure 2.24) 72

2.26 Ratio DS1dNdyE

U U/DS1dNdyE

AA as a function of cuts on multiplicity andspectator number, assuming smearing technique of reference [96] Seetext for details 73

2.27 Ratio DS1dNdyE

U U/DS1dNdyE

AA as a function of cuts on multiplicity andspectator number, with gaussian smearing of multiplicity using σmult=

2.28 Left panel: Average transverse particle density in the (0.5%,0.5%) bin

√s

S

dN dy

E

U U/D1 S

dN dy

E

(0.5%,0.5%) bin versus σmult 75

2.29 Left panel: Average transverse area in the (0.5%,0.5%) bin versus σmult

GeV Right panel: Average multiplicity in the (0.5%,0.5%) bin for

2.30 Charged particle multiplicity distribution resulting from a 0.5% tator cut with σspec= 0, σ2

spec-mult = 0.6¯n (blue) and σspec = 10, σmult = 70(orange) 76

2.31 Eccentricity distributions resulting from taking the 5% of events in thegreen distribution of Figure 2.30 with the highest (gray) and lowest(black) multiplicity 76

2.32 Schematic representation of a misaligned collision between two nium nuclei In such a collision, the center-of-mass momentum isnonzero, leading to a shift in the average rapidity of the particles pro-duced by the collision 80

near the center of the integration region a.) Initial grid b.) Adjustedgrid 96

5.1 Plots of calculated points together with fits for selected tensor nents (a) h˜x2i, (b) h˜x˜yi, (c) h˜y˜ti, (d) h˜t2i, (e) h˜z˜ti, (f) h˜x˜ti 134

points Here, the optimization routine requested an accuracy of 1%.Note that, in the case where the points are spread over the full range,

an incorrect value is returned for N < 8, whereas a correct result isreturned for only four points when the points are constrained to lie inthe first quadrant 136

CHAPTER 1

INTRODUCTION

Immediately after the Big Bang, the universe existed in an unimaginably hot anddense state Under those conditions, matter as we know it did not exist Instead, theuniverse was permeated by quark-gluon plasma (QGP), a state of matter in whichthe quarks and gluons that are ordinarily confined within the boundaries of particlessuch as protons and neutrons are able to move about a much larger region of space.This deconfinement of quarks and gluons requires sufficiently high energy densityand temperature As the universe cooled below a temperature of approximately 170MeV, the quarks and gluons coalesced to form hadrons (a generic name for particlesother than quarks and gluons that experience the strong force), such as protonsand neutrons After approximately three minutes, temperatures had cooled enough

to permit the formation of simple atomic nuclei [1] The matter was still ionized,however, making the universe opaque to electromagnetic radiation Approximately

400000 years after the Big Bang, conditions became favorable for the capture of freeelectrons by the existing nuclei to form electrically neutral atoms At that time, theuniverse became transparent The photons that had been trapped in the plasma werethen free to roam about the universe, giving rise to the cosmic microwave background

One of the goals of physics is to understand the behavior of the universe at thevery earliest times, all the way back to the Big Bang if possible By training ourtelescopes on the deepest reaches of space, we look further and further back in time,

to times much closer to the very beginning Unfortunately, the fact that the cosmoswas opaque to light for 400000 years means that the very early moments of theuniverse’s formation will never be accessible to direct observation To learn aboutthe primordial universe requires a different tack

We can approach this question by attempting to form a QGP in the laboratory.This is no easy feat, though At temperatures below 170 MeV, calculations indicatethat quarks and gluons are forever trapped within the confines of their parent hadrons,

a radius of roughly 1 fm To free them requires creating a region in space with anenormous energy density

The energy density of a proton is approximately 1 GeV/fm3 If the energy density

in a region of space exceeds this value the situation becomes roughly equivalent toproducing a collection of protons with overlapping radii At that point, the quarksand gluons, referred to collectively as partons, are free to leave their “bubble” andvisit other places This signals the formation of QGP

To produce the necessary energy densities, we rely on enormous particle collidersthat accelerate large atomic nuclei to velocities near the speed of light and then smashthem into one another, depositing an tremendous amount of energy into a very smallvolume At Brookhaven National Laboratory’s (BNL) Relativistic Heavy Ion Collider(RHIC), nuclei are accelerated to 0.99995c and allowed to collide with a center ofmass energy of √sN N = 200 GeV In calculations we present in this dissertation, weestimate that the peak energy density produced in a head-on, or central, collision

between two gold nuclei (Au+Au) at this energy is approximately 25 GeV/fm3, wellabove the threshold of ∼ 1 GeV/fm3 mentioned above.

These collisions presumably form a region of hot, dense matter that is surrounded

by vacuum The produced matter then does what one might expect in such a uation; it explodes The expected lifetime of a QGP produced in the laboratory istherefore very short, on the order of 10−23 seconds That time is roughly equivalent

sit-to the amount of time necessary for a beam of light sit-to traverse the diameter of agold nucleus Direct detection of these plasmas is immediately ruled out Instead,

we must rely on the particles that are emitted from the exploding fireball to conveyinformation about the source from which they emerged

1.1 Signatures of the quark gluon plasma

What, then, is the tell-tale signature of the formation of the quark gluon plasma?

It appears that in current experiments, there is not a single “smoking gun” that, takenalone, proves conclusively QGP formation A variety of possible QGP signatures havebeen observed in experiments at RHIC, including the suppression of the charmonium(bound states of charm quarks and antiquarks) states J/ψ, ψ0 and χc, enhancement

of the production of strange particles relative to proton-proton collisions, sion of particles with large transverse momenta (so-called high-pT particles), and theobservation of transverse flow of the produced matter Taken individually, however,these observations are somewhat ambiguous; though they may point to QGP forma-tion, other possible explanations have not been entirely excluded Only when takentogether does a convincing picture begin to emerge, but this process of consolidation

suppres-is still ongoing

Reviews of the data collected since RHIC began operation in 2000 led all four

of the experiments at RHIC to the same conservative conclusion: it is clear that thestate of matter produced in these collisions is new and behaves in a manner previouslyunobserved, but whether this new state is, in fact, the long sought after quark gluonplasma remains an open question [2 5] Despite the ambiguities, these observations

do allow for rather strong statements about the nature of the produced matter Two

of these, suppression of high-pT jets and the transverse flow of the matter, pointtowards a fireball composed of particles that behave collectively and possessing anunprecedented density Here is how:

1.1.1 Parton energy loss and jet quenching

In collisions between particles at relativistic energies, occasionally a quark or gluon

is produced that has a very large transverse momentum, where transverse describesthe portion of momentum directed perpendicular to the beam direction As thisparton flies away from the interaction point, the distance between it and its neighborsincreases Here, the strong force acting between the parton and its neighbors behaves

in a unique way: as the distance increases, the force acting on the escaping parton alsoincreases, much like pulling on a rubber band At some point, the energy stored inthe color field between the partons becomes large enough to form a quark-antiquarkpair and the “rubber band” breaks, forming two particles where originally there wasonly one This can happen many times over, forming a collection of particles thatemerge from the collision in a jet

From momentum conservation, the formation of a parton with large transversemomentum requires the formation of a second particle with momentum of equal

magnitude but opposite direction Thus, we should expect jets to occur in pairsdirected in opposite directions In fact, in proton-proton (p+p) and deuterium-gold(d+Au) collisions at RHIC, this is exactly what is observed [6 8].

If a QGP is formed, any high-pT particle that is produced in the collision willhave to traverse some portion of an extremely dense medium before it can escapeand will inevitably suffer both elastic and inelastic collisions along its trajectory Wetherefore expect the particle to suffer energy losses as it leaves the medium, leading

us to further conclude that the production of high-pT particles should be suppressedrelative to p+p collisions

This expectation was confirmed by the RHIC experiments The production ofhigh-pT particles is suppressed in Au+Au collisions relative to their p+p counterparts[9; 10] Measurements of jets, however, yield a surprise If the experiment triggers

on a fast particle and then measures the azimuthal distribution of the other highmomentum particles produced in the collision, the jet of particles that travel in thesame direction as the trigger particle (the near-side jet) is readily observed Themomentum conserving jet directed antiparallel to the trigger particle (the away-sidejet), however, is completely suppressed [6 8], implying that the density of the medium

is so large as to completely absorb the energy of the away-side jet Moreover, ifone examines the azimuthal correlation of low-momentum particles with the highmomentum trigger particle, one finds a surplus of these low momentum particlesdirected away from the trigger particle [11], implying that fast moving partons becomenearly thermalized as they travel through the medium It is only at extremely highmomenta that evidence of the back-to-back nature of the jets is recovered [12]

One expects that this energy loss is path length dependent The vast majority

of collisions are similar to that depicted in Figure 1.1 In these peripheral collisions,i.e., those with nonzero impact parameter b, only the edges of the nuclei interact,forming a football shaped overlap area In these collisions, we can examine the effect

of different path lengths by studying jets emitted perpendicular to the plane defined

by the impact parameter and the beam direction (the “out-of-plane” direction) versusthose emitted parallel to the impact parameter (“in-plane”) In this case, it is foundexperimentally that for the out-of-plane jets the away side jet is again completelysuppressed while the in-plane jets show a clear excess of particles at ∆φ ≈ π [13],indicating a strong path length dependence of the parton energy loss just as expected

1.1.2 “RHIC serves the perfect liquid”–Elliptic flow and ideal

hydrodynamics at RHIC

If the matter formed in these collisions thermalizes, then it can be described interms of thermodynamic quantities such as temperature and pressure In particular,the pressure would be expected to be very high in the center of the collision wherethe matter is hottest and densest, leading to large pressure gradients In periph-eral collisions like those indicated schematically in Figure 1.1, these gradients wouldnecessarily be stronger in the in-plane direction than out-of-plane The end result

of this is that as the medium emits particles, it will tend to emit more momentuminto the in-plane directions, leading to an anisotropy in the azimuthal distribution ofmomentum

We can write the momentum distribution in terms of its Fourier expansion [14],

EdN

d3p =

12π

partic-of proton-proton collisions whose particle production is known to be isotropic, thenany anisotropy in the momentum distribution must be due to interactions betweenthe produced particles [15] The momentum anisotropy, then, is driven by the initialspatial anisotropy of the interaction region and provides evidence that the producedmatter behaves collectively

To predict values for observables such as elliptic flow and single particle spectra,

it is necessary to first posit a model that describes the evolution of the producedmatter from the time of its formation until kinetic freezeout, the point at which

the particles cease to interact If the medium is assumed to fully thermalize, then,given a set of initial conditions, the evolution can be described in terms of relativistichydrodynamics As a first approximation, we might assume that the QGP has avanishing viscosity and obeys the laws of ideal hydrodynamics.

The remarkable fact is that this radical approach describes the behavior of theQGP at RHIC with astonishing accuracy In all but the most peripheral collisions,ideal hydrodynamics is able to accurately reproduce the single particle inclusive mo-mentum spectra for the vast majority of particles produced in the collision [16–18].Deviations from the data are observed for large impact parameters and for pT &1.5-

2 GeV [19; 20] This is not entirely unexpected For very peripheral collisions, it

is likely that the energy densities obtained are insufficient to allow the medium toproperly thermalize; in such a situation, ideal hydrodynamics would be expected to

be a poor approximation In the latter case, high-pT partons would be expected toundergo too few scatterings as they traversed the medium, once again leading toincomplete thermalization of these particles [17]

Hydrodynamics is also very successful in reproducing the elliptic flow data tained at RHIC For transverse momenta below approximately 1.5-2.0 GeV, hydro-dynamics predicts the value of v2 with good accuracy Moreover, the observed restmass dependence of v2 [21–23] is nicely predicted by hydrodynamics [17; 20], givingsupport to the conclusion that the produced matter at RHIC behaves as an idealfluid

ob-These empirical successes and strict theoretical limitations on the viscosity of theQGP produced at RHIC [24] that place the ratio of the viscosity to the entropy

0 5 10 15 20 25 30 35 0

0.05 0.1 0.15 0.2 0.25

lab

E /A=158 GeV, NA49

SdNch/dy, where S is thearea of the overlap region Hydrodynamics predicts that the dependence of v2 onthe eccentricity of the fireball should manifest itself in a linear relationship betweenthe two quantities Thus, ignoring the weak dependence on beam energy depicted

by the three horizontal bars of Figure 1.2, v2/ should be constant as a function oftransverse particle density The data points obtained in the lower energy experiments(the leftmost points) obviously fall well below the predictions from hydrodynamics.The points corresponding to the most energetic collisions at RHIC, however, areseen to agree quite nicely with the predictions This behavior is generally considered

evidence that the matter produced at the lower beam energies was not completelythermalized and therefore possessed a larger viscosity than that seen at RHIC, therebylowering the observed value of v2.

Since the elliptic flow is driven by the rescatterings of the produced particles, andsince the vanishing mean free path of the ideal fluid limit provides the maximumamount of scattering, the hydrodynamic prediction for v2 should be viewed as anupper limit Though the actual value of this upper limit depends somewhat onthe equation of state used to describe the quark gluon plasma, an examination ofdifferent equations of state indicate that the size of this effect is small [28] Thefeature of Figure 1.2 that gives pause, then, is that the elliptic flow as a function

of S1dNch/dy shows no sign of saturation as it approaches the hydrodynamic limit.The relationship between v2 and particle density actually appears to be a linear onethat just happens to coincide with the predicted value at the RHIC data points.Thus, it is critical to test the behavior of elliptic flow to higher transverse particledensities to determine whether the data settle into the predicted value In Chapter

2, we present one possible method of accomplishing this test using collisions betweenuranium nuclei at the RHIC facility In this way, we estimate that we could increasethe transverse particle density by as much as 40%, providing a crucial test of the idealfluid behavior of the elliptic flow

1.2 Hanbury Brown-Twiss interferometry of the fireball

Despite the many successes of the ideal hydrodynamic model, it is not withoutits problems One of the most glaring failures of the model is its inability to accountfor femtoscopic measurements of the fireball obtained through momentum correlation

measurements [29] These measurements are based on the technique of intensity terferometry, developed in the 1950s by Hanbury Brown and Twiss (HBT) [30; 31]

in-to improve the angular resolution available in-to measure the diameter of astronomicalradio sources It was later realized by Goldhaber, et al., [32] that intensity inter-ferometry could also be used to probe the spatial structure of the particle emissionregion in proton-proton collisions With the advent of the heavy ion program at theAGS in the mid-1980s, this technique was quickly adapted to explore the fireballsproduced in collisions between atomic nuclei

We note that there are actually a variety of different correlation measurementsthat are made of the nuclear fireball, including multiparticle and non-identical particlecorrelations In this work, however, we will focus on identical particle correlationmeasurements and, in particular, measurements made on pions

1.2.1 Basic formalism

In traditional amplitude interferometry, like that used in a Fabry-Perot or son interferometer, incoming amplitudes are first summed, with the square of theresult giving the intensity at the detector Referring to Figure 1.3, we have twosources, a and b, emitting particles into four possible channels If the amplitude at a

Michel-is given by α and that at b by β, then the total amplitude at detector 1 Michel-is given by[33]

A1 = 1

L(αe

i(p a r a1 +φ a )+ βei(pb r b1 +φ b ) (1.3)

The intensity at detector 1 is then given by

I1 = 1

L2(|α|2+ |β|2+ 2<{αβ∗ei[(pa r a1 −p b rb1)+(φ a −φ b )]) (1.4)

It is only the last term of this equation that can reveal any information about theseparation of the sources, R If particle emission from these sources is chaotic, i.e.,the phases φa and φb vary randomly as a function of time, then any time averaging

of this signal will eliminate this final term and, with it, any information about thesource separation

If, however, we average the product of the intensities I1 and I2, then the story isslightly different In this case, after time averaging we are left with

hI1I2i = hI1ihI2i + L24|α|2|β|2cos(pa(r1a− r2a) + pb(r2b− r1b)), (1.5)making the correlation function

C(pa, pb) = 1 + |α|2|β|2

(α2+ β2)2 cos(pa(r1a− r2a) + pb(r2b− r1b)) (1.6)

In its most basic guise, a two-particle correlation measurement is simply the ratio

of the two-particle yield to the product of the single particle yields, i.e.,

C(pa, pb) = dN/d

3pa d3pb

(dN/d3pa)(dN/d3pb). (1.7)

In practice, the numerator of this equation is obtained by taking a particular set

of collision events and tallying the number of pairs produced in each event withthe desired momenta The denominator, on the other hand, is usually formed by asimilar accumulation of pairs with the desired momenta, but in this case the individualparticles of the pair are taken from different events

Theoretically, the two-particle correlation function can be connected to the time information of the source In the absence of strong and Coulomb final stateinteractions, the correlation function is given by [33]

space-C(K, q) = 1 ± R d

4x d4y eiq·(x−y)S(x, K)S(y, K)

R d4x S(x, K + q/2) R d4y S(y, K − q/2), (1.8)where K = 12(pa + pb) and q = pa− pb The function S(X, K), referred to as theemission function, describes the probability of emitting a particle with momentum Kfrom the point X

In what is known as the “smoothness approximation”, the momentum dependence

of the emission functions is assumed to be weak The correlation function is thenapproximated by

Experimentally, it is common to parameterize the correlation function in terms ofthe HBT radii,

q=0

Referring to equation (1.9), we see that the radii are directly related to the moments

of the emission function

Rij2 = ∓ 12∂

2C(K, q)

∂qi∂qj

q=0

= hxixji − hxiihxji, (1.12)where hf(x)i ≡ R d4x f (x)S(X, K)/R d4x S(X, K) Thus, the HBT radii carryvaluable information about the temporal and spatial extent of the source

Na¨ıvely, one might expect that, through measurement of the correlation function,

we might obtain all possible information about S(X, K) by simply inverting equation(1.9) However, since pa and pb are on-shell momenta, we are limited in our ability

to recover the emission function To see this, consider the inner product

which is the case for identical particles Because of this, we have only three dent components of q and the HBT radii mix the space and time information of thesource

indepen-Frequently, equation (1.13) is used to eliminate q0 in terms of q,

x y

z

K⊥

o s

l

Figure 1.4: Illustration of the osl and xyz coordinate systems

where β = K/K0 Using this result, we obtain for the HBT radii

Rij2 = h(xi− βix0)(xj − βjx0)i − hxi− βix0ihxj − βjx0i (1.15)

We are free to choose any convenient system of coordinates for q In this sertation, we will rely on the out-side-long (osl) system of coordinates, in which onecoordinate axis is oriented parallel to the transverse pair momentum pair momentum

dis-K⊥ (out), another parallel to the beam direction (long) and the third perpendicular

to the other two (cf Figure 1.4) In this system, the pair velocity β is given by(β⊥, 0, βl)

The moments of the emission function, however, are more straightforwardly puted in the coordinate system of the source The HBT radii in the osl coordinatesare then given in terms of the spatial correlation tensor Sµν = hxµxνi−hxµihxνi (with

1.2.2 Final state interactions in HBT interferometry

Equation (1.8) is only valid in the absence of strong and Coulomb interactions

In reality, the emitted pions interact with one another and with the remaining sourceafter their emission The inclusion of these final state interactions makes the con-nection between the HBT radii and the space-time structure of the source far lessclear

In the presence of a two-particle interaction, the correlation function is usuallytaken to be given approximately by [37–39]

C(K, q) = R d4X d4y S X +y2, K
S X − y

2, K
|φq

2 (y) |2

|R d4X S(X, K)|2 , (1.17)i.e., the plane waves of equation (1.8) are replaced by the properly symmetrizedrelative wavefunction φq

2 (y) that describes the behavior of the relative coordinate y

in the presence of the governing potential The wavefunction then acts as a distortinglens, modifying the view of the original emission function We note that the integral

on the right hand side of equation (1.17) is written in the frame where K = 0

The Coulomb interaction is felt not only by same sign pairs but also by pairs

of opposite sign Since the former do not experience symmetrization effects, thestudy of correlations between π+π− pairs can reveal information about the size ofthe correction necessary to unfold the Coulomb effects from those arising from Bose-Einstein statistics [33]

By considering classical trajectories for the emitted particles [40], it is expectedthat the interaction of the particles with the remainder of the positively charged fire-ball will decrease (increase) the apparent source size of a static source in the sidewarddirection for positively (negatively) charged pairs This estimate is in agreement withprevious calculations that employ equation (1.17) [41; 42] The fact that the mea-sured radii for π+π+ and π−π− at RHIC are nearly identical [43–47], however, seem

to indicate that the effects of this interaction are negligible at these collision energies

It is therefore neglected in hydrodynamic simulations

tracks, and multiple tracks incorrectly identified as a single track Additionally, allcuts applied to the signal distribution must also be applied to B(K, q).

The measured correlation functions are then traditionally fit to a gaussian eterization,

param-C(K, q) = 1 ± λ exp−q2

sR2s− q2oR2o− ql2R2l − qoqsRos2 , (1.19)under the assumption that the source is itself Gaussian and that the only source ofcorrelation is quantum statistics [45] Here, λ is known as the coherence parameterand parameterizes the degree to which the emission of particles from the source ischaotic For a fully chaotic source λ = 1 while a coherent source is described by

λ = 0

In these experiments, the reaction plane of the collision is determined from theanisotropic flow [14] As a result, the sign of the impact parameter is ambiguous;after aligning events according to their reaction planes, there are equal numbers ofcontributions from events with b and −b Under these circumstances the othercross terms, Rsl and Rol vanish [48] They are therefore excluded from equation(1.19) Moreover, in any azimuthally integrated analysis, additional symmetries lead

to Ros = 0 as well [45; 48]

In practice, both Coulomb interactions and deviations of the correlation functionfrom a true Gaussian introduce measurable effects in the evaluation of the HBTparameters Non-Gaussian effects are usually ignored, though recent analysis of pionHBT radii by STAR uses a more complicated expansion of the correlation function

in terms of Hermite polynomials [45;49] This produces a somewhat better fit to thedata, but the effect is no more than 10% for each of the radii [45]

The effects of Coulomb interactions, though, are relatively strong In particular,the two-particle interaction reduces the number of pairs at low relative momentum,leading to a reduction in the correlation function for small q The standard method

of correcting for this interaction is to modify equation (1.19) by introducing theweighting function [43–47]

Kcoul(qinv) = R d3r |ψc(r)|2S(r)

R d3r |ψpl(r)|2S(r), (1.20)where ψc(r) is the Coulomb wavefunction, ψpl(r) is the properly symmetrized planewave, S(r) is the distribution of relative separation r at emission, and qinv =p(q0)2− q2.The correlation function is then taken to be

C(K, q) = Kcoul(qinv) 1 + λ exp−q2

sRs2− qo2Ro2− q2lR2l − qoqsR2os
(1.21)This procedure actually overcorrects for the Coulomb interactions since it correctsall pairs, even those that are not primary pairs [50] Because of this, the STARcollaboration has employed two alternative correction procedures, the dilution andBowler-Sinyukov methods [45] In the dilution method, the effect of the correction

Kcoul(qinv) is weakened by limiting its effect to only those pairs that experience theCoulomb interaction f ,

Kcoul0 (qinv) = 1 + f (Kcoul(qinv) − 1) (1.22)C(K, q) = K0

coul(qinv) 1 + λ exp−q2

sRs2− qo2Ro2− q2lR2l − qoqsR2os
(1.23)

It seems reasonable to assume that the particles that participate in the interactionshould also be those that experience quantum interference, thus f should be approx-imately equal to λ [45]

In the presence of. ..

in the presence of the governing potential The wavefunction then acts as a distortinglens, modifying the view of the original emission function We note that the integral

on the right hand...

The moments of the emission function, however, are more straightforwardly puted in the coordinate system of the source The HBT radii in the osl coordinatesare then given in terms of the spatial