Bayesian parameter estimation for relativistic heavy-ion collisions
BBayesian parameter estimation forrelativistic heavy-ion collisions
Jonah E. Bernhard
Ph.D. dissertationAdvisor: Steffen A. BassDepartment of Physics, Duke UniversityApril 19, 2018 a r X i v : . [ nu c l - t h ] A p r bstract I develop and apply a Bayesian method for quantitatively estimating prop-erties of the quark-gluon plasma (QGP), an extremely hot and dense stateof fluid-like matter created in relativistic heavy-ion collisions.The QGP cannot be directly observed—it is extraordinarily tiny andephemeral, about 10 − meters in size and living 10 − seconds before freez-ing into discrete particles—but it can be indirectly characterized by match-ing the output of a computational collision model to experimental obser-vations. The model, which takes the QGP properties of interest as inputparameters, is calibrated to fit the experimental data, thereby extracting aposterior probability distribution for the parameters.In this dissertation, I construct a specific computational model of heavy-ion collisions and formulate the Bayesian parameter estimation method,which is based on general statistical techniques. I then apply these toolsto estimate fundamental QGP properties, including its key transport coeffi-cients and characteristics of the initial state of heavy-ion collisions.Perhaps most notably, I report the most precise estimate to date ofthe temperature-dependent specific shear viscosity η/s , the measurement ofwhich is a primary goal of heavy-ion physics. The estimated minimum valueis η/s = 0 . +0 . − . (posterior median and 90% uncertainty), remarkablyclose to the conjectured lower bound of 1 / π ’ .
08. The analysis alsoshows that η/s likely increases slowly as a function of temperature.Other estimated quantities include the temperature-dependent bulk vis-cosity ζ/s , the scaling of initial state entropy deposition, and the durationof the pre-equilibrium stage that precedes QGP formation.i ontents
Abstract i1 Introduction 12 A pragmatic overview of heavy-ion collisions 6 ii Introduction I magine: One day, strolling through the countryside, you happen uponthe ruins of an old, burned-down building. Little remains of the originalstructure; the charred rubble is haphazardly piled, the wood battered bydecades of wind and rain. Yet, the foundation is mostly intact. A fractionof a wall still stands. You can’t help but wonder: What did the buildinglook like, what was its purpose? And how—and why—did the fire start?This is the allegory I often use when attempting to convey the diffi-culty of a much different problem: How can we measure the properties ofthe quark-gluon plasma, a highly excited and transient state of matter thatcannot be observed directly? Quark-gluon plasma (QGP) can only be cre-ated on Earth in ultra-relativistic collisions of heavy nuclei, and even thenonly in microscopic droplets that almost instantly disintegrate into a showerof particles, which are detected eons (relatively speaking) after the originalcollision. From those particles—the “ashes”—we wish to infer not only thata QGP was, in fact, the source, but also its precise properties.What connects these mysteries is that we can only observe the finalstate of the system, not its original state nor its transformation. How canwe turn back the clock? In the case of the burned-down building, perhapswe could ignite some other structures, observe how they burn, and comparethe results to the discovered ruins. That could get expensive, but the basicstrategy is sound: We need a way to replay history, then we can match theoutcomes to our limited observations.Enter the modern computational model. The generic setup is as fol-lows: We have a physical system with an observed final state (experimentaldata), a set of undetermined parameters which characterize the system, anda computer model of said system. The model takes the parameters as input,1
HAPTER 1. INTRODUCTION η/s —the dimensionless ratio of the shear viscosity to entropy density.Early observations pointed towards a small η/s , meaning that the QGPis nearly a “perfect” fluid (which would have zero viscosity). Meanwhile,a purely theoretical calculation posited that the minimum specific shearviscosity is 1 / π , or approximately 0.08. Subsequent studies, comparingviscous relativistic hydrodynamics models (which take η/s as an input pa-rameter) to experimental measurements of collective behavior, showed thatthe specific shear viscosity is likely nonzero, and within roughly a factor ofthree of the conjectured 1 / π limit.Model-to-data comparison is broadly applicable in a variety of scientificdisciplines. We can acquire stunning images of distant galaxies, but giventhe timescale of most galactic processes, these images are effectively frozensnapshots. Galactic evolution models help unravel the life cycle of galaxies,all the way back to their formation.Recently, the LIGO Scientific Collaboration used observations of grav-itational waves combined with numerical simulations of general relativityto quantify properties of binary black hole mergers, such as the masses ofthe original black holes, the amount of energy radiated, and the distance tothe event. (More recently, a binary neutron star merger was also detected.)These quantitative conclusions are highly nontrivial and, in my opinion,underappreciated. After all, the observed gravitational waveforms—whileastonishingly impressive in their own right—do not directly indicate thatthe source was a black hole merger, much less provide its precise properties. HAPTER 1. INTRODUCTION estimates of the truevalues. This is the terminology I’ll most often use for the remainder of thiswork (“parameter estimation” is even in the title!).Given this inherent inexactness, it is crucial to strive for not only the“best-fit” parameters but also faithful assessments of their associated uncer-tainties.All of this calls for a rigorous, systematic approach. To this end, weframe the problem in terms of Bayes’ theorem.In the Bayesian interpretation, our complete knowledge of the param-eters is contained in the posterior probability distribution on the modelparameters. To obtain the posterior distribution, we first encode our initialknowledge as the prior distribution, for example we probably know a rea-sonable range for each parameter. Then, for any given parameter values,we compute the likelihood by evaluating the model and calculating the fitto data, folding in any sources of uncertainty from the model calculationand experimental measurements. Finally, we invoke Bayes’ theorem, whichstates that the posterior is proportional to the product of the likelihood andthe prior: posterior ∝ likelihood × prior . From the posterior, we can extract quantitative estimates of each parame-ter, their uncertainties, and any other statistical metrics; visualizations canreveal detailed structures and relationships among parameters.Let us step back for a moment and consider what happened here. Thelikelihood and posterior have quite different meanings, even when they aremathematically equivalent (for example if the prior is constant). The like-lihood is the probability of observing the evidence given a proposed set ofparameters; in other words, if we assume certain values of the parameters aretrue, then what is the probability of a universe where the resulting model cal-culations and experimental data exist together? The posterior incorporatesour prior knowledge and reverses the conditionality: it is the probability of
HAPTER 1. INTRODUCTION marginal distributions, obtained for any given parameter by marginalizing over (inte-grating out) all the rest. Importantly, this folds in the remaining uncertaintyof the marginalized parameters, for instance if the estimates of several pa-rameters are correlated, the uncertainty in each parameter contributes tothe uncertainty of the others.Marginalization necessitates calculating multidimensional numerical in-tegrals, for which Monte Carlo techniques usually perform best. Markovchain Monte Carlo methods are the canonical choice for sampling posteriordistributions; this entails roughly a million evaluations of the posterior, plusor minus a few orders of magnitude, depending on the problem at hand andwhich quantities are desired. So unless the model runs rather quickly, therequired computation time is prohibitive. A model that runs in a tenth ofa second would take a little over a day for a million evaluations; heavy-ioncollision models need at least a few thousand hours (consisting of tens ofthousands of individual events, each of which runs in a few minutes on aver-age), which translates to a total time of over a hundred thousand years! (Inpractice, some degree of parallelization would reduce this, but not enough.)One strategy to dramatically reduce the required computation time isto use a surrogate model, or emulator, that predicts the output of the truemodel in much less time than a full calculation. The surrogate is trainedon the input-output behavior of the true model, then used as a stand-induring Monte Carlo sampling. Gaussian process emulators are a commonchoice, since they perform well in high dimensions, do not require any as-sumptions about the parametric form of the model, and naturally providethe uncertainty of their predictions. I n this dissertation, I develop a complete framework for applying Bayesianparameter estimation methods to quantitatively estimate the propertiesof the quark-gluon plasma created in ultra-relativistic heavy-ion collisions.I begin by laying the groundwork in chapter 2, reviewing the history ofheavy-ion physics and surveying its current status, focusing on the aspectsmost relevant to this work. In chapter 3, I go in depth on the computationalmodeling of heavy-ion collisions. I present several original contributions tothe modeling landscape, and assemble a specific set of models to be usedlater for parameter estimation. Then, in chapter 4, I describe the Bayesian HAPTER 1. INTRODUCTION
A pragmatic overview ofheavy-ion collisions A pair of lead nuclei hurtle towards each other, circling in opposite direc-tions around the 27 kilometer ring of the Large Hadron Collider (LHC)near Geneva, Switzerland. Stripped of their electrons, the positively-chargedions have been accelerated to virtually the speed of light by the collider’spowerful electromagnetic field. They are nearly-flat discs due to relativisticlength contraction.The discs collide, depositing their kinetic energy in a nucleus-sized areaand creating temperatures T ∼
300 MeV, or about 3 × K, over 100,000times hotter than the core of the Sun (1 . × K [1]). By around 1 fm /c after the collision, the quarks and gluons that made up the original protonsand neutrons in the lead nuclei have escaped and formed an extremely hotand dense state of fluid-like matter known as quark-gluon plasma (QGP) [2,3]. Quarks and gluons are elementary particles, the constituents of compos-ite particles called hadrons, of which protons and neutrons are examples.In normal matter, quarks and gluons exist only in hadrons, confined bythe strong nuclear force, which also binds protons and neutrons into nuclei.The theory of the strong force, quantum chromodynamics (QCD), stipulatesthat—under normal conditions—particles must have neutral “color” charge,the QCD analog of electric charge (the prefix chromo means color). There A fm /c is the time it takes light to travel a femtometer (10 − meter, usually ab-breviated fermi or fm), which works out to approximately 3 × − seconds. This is aconvenient unit of time in the context of heavy-ion collisions, since most particles moveat a significant fraction of the speed of light and the typical length scale is several fermi. HAPTER 2. A PRAGMATIC OVERVIEW
Time −−−−−−−−−−−−−−−−−→
Figure 2.1
A rendering of the stages of a heavy-ion collision. From left to right,nuclei approach each other and collide, the QGP medium forms and expands whileparticles are emitted, and the QGP dissipates as the hadron gas expands. Visual-ization originally created by Hannah Petersen and modified by the author for thiswork.
Back to the lead-lead collision , where the Lorentz-contracted nuclei arereceding along the z -axis with the created droplet of QGP between them.Bjorken outlined the basic collision spacetime evolution in 1982 [5]. TheQGP is located near the origin, expanding hydrodynamically in both thetransverse ( x - y ) plane and the longitudinal ( z ) direction; at any given z position, the fluid has approximate longitudinal velocity z/t . As the nucleicontinue to recede, the fluid forms at later times further from z = 0, roughlyon a spacetime hyperbola defined by a constant “proper time” τ ≡ p t − z ∼ /c. (2.1) HAPTER 2. A PRAGMATIC OVERVIEW η s ≡
12 log t + zt − z , (2.2)which specifies the position along proper-time hyperbolas. This is a conve-nient kinematic variable since Lorentz boosts simply add as a function ofrapidity, i.e. a boost of η As followed by a second of η Bs is equivalent to a singleboost of η As + η Bs . zt QGPHRG 𝜏 = f m / c 𝜏 = f m / c Figure 2.2
Spacetime diagram of a heavy-ion collision. The nuclei (blue discs)propagate along the z -axis at the speed of light and collide at z = t = 0. The quark-gluon plasma (QGP, orange region) medium forms at proper time τ ∼ /c andconverts to a hadron resonance gas (HRG, blue region) around τ ∼
10 fm /c . The system is approximately invariant under Lorentz boosts near centralrapidity ( η s ∼ γ > HAPTER 2. A PRAGMATIC OVERVIEW T ∼ w hen ordinary substances are subjected to variations in tempera-ture or pressure, they will often undergo a phase transition : a physical change from one state to another. At normal atmospheric pressure, for example, water suddenly changes from liquid to vapor as its temperature is raised past 100° C; in a word, it boils. Water also boils if the temperature is held fixed and the pres-sure is lowered—at high altitude, say. The boundary between liquid and vapor for any given substance can be plotted as a curve in its phase diagram , a graph of tem-perature versus pressure. Another curve traces the boundary between solid and liquid. And depending on the substance, still other curves may trace more exotic phase transitions. (Such a phase diagram may also require more exotic variables, as in the figure).One striking fact made apparent by the phase diagram is that the liquid-vapor curve can come to an end. Beyond this “critical point,” the sharp distinction between liquid and vapor is lost, and the transition becomes continuous. The location of this critical point and the phase boundaries represent two of the most fundamental characteristics of any substance. The critical point of water, for example, lies at 374° C and 218 times nor-mal atmospheric pressure. The schematic phase diagram shown in the figure shows the different phases of nuclear matter predicted for various combinations of temperature and baryon chemical potential. The baryon chemical potential determines the energy required to add or remove a baryon at fixed pres-sure and temperature. It reflects the net baryon density of the matter, in a similar way as the temperature can be thought to determine its energy density from micro-scopic kinetic motion. At small chemical potential (corresponding to small net baryon density) and high temperatures, one obtains the quark-gluon plasma phase; a phase explored by the early universe dur-ing the first few micro-seconds after the Big Bang. At low tempera-tures and high baryon density, such as those encountered in the core of neutron stars, the predictions call for color-superconduct-ing phases. The phase transition between a quark-gluon plasma and a gas of ordinary hadrons seems to be continuous for small chemical potential (the dashed line in the figure). However, model studies sug-gest that a critical point appears at higher values of the potential, beyond which the bound-ary between these phases becomes a sharp line (solid line in the figure). Experimentally verifying the location of these fundamental “landmarks” is central to a quantitative understanding of the nuclear matter phase diagram.Theoretical predictions of the loca-tion of the critical point and the phase boundaries are still uncertain. However, several pioneering lattice QCD calculations have indicated that the critical point is located within the range of temperatures and chemical potentials accessible with the current RHIC facility, with the envi-sioned RHIC II accelerator upgrade, and at existing and future facilities in Europe (i.e., the CERN SPS and the GSI FAIR). Indeed, the recent discovery of the quark-gluon plasma at RHIC gives evidence for the expected continuous transition (dashed line in the figure) from plasma to hadron gas. Physicists are now eagerly anticipat- ing further experiments in which nuclear matter will be prepared with a broad range of chemical potentials and temperatures, so as to explore the critical point and the phase boundary fully. As the experiments close in, for example, the researchers expect the critical point to announce itself through large-scale fluctuations in several observables. These required inputs will be achieved by heavy-ion collisions spanning a broad range of collision energies at RHIC, RHIC II, the CERN SPS and the FAIR at GSI.The large range of temperatures and chemical potentials possible at RHIC and RHIC II, along with important technical advantages provided by a collider coupled with advanced detectors, give RHIC scien-tists excellent opportunity for discovery of the critical point and the associated phase boundaries. Search for the Critical Point: “A Landmark Study”
Quark-Gluon Plasma
The Phases of QCD T e m pe r a t u r e Hadron Gas
Early Universe
Future FAIR Experiments
Future LHC Experiments
NuclearMatterVacuum
ColorSuperconductor
Critical Point
Current RHIC Experiments R H I C E n e r g y S c a n C r o ss o v e r Baryon Chemical Potential ~170 MeV 0 MeV 900 MeV0 MeV
Neutron Stars s t o r d e r p h a s e t r a n s i t i o n Schematic.QCD.phase.diagram.for.nuclear.matter ..The.solid.lines.show.the.phase.boundaries.for.the.indicated.phases ..The.solid.circle.depicts.the.critical.point ..Possible.trajectories.for.systems.created.in.the.QGP.phase.at.different.accelerator.facilities.are.also.shown . The Phases of Nuclear Matter
Figure 2.3
Schematic of the QCD phase diagram [10].
Following convention, the diagram is a function of temperature T and baryonchemical potential µ B , which quantifies the net baryon density, where posi-tive µ B means more baryons than antibaryons. Given the crossover at zero µ B and the first-order transition at zero temperature, it is logical to drawa first-order phase boundary terminating in a critical point at some ( T, µ B )combination [11], however, current experimental evidence for the existenceof a QCD critical point is inconclusive.Our quantitative knowledge of the crossover phase transition at zero µ B derives from lattice QCD calculations of the equation of state, which con-nects the system’s various thermodynamic quantities: temperature, pres- HAPTER 2. A PRAGMATIC OVERVIEW µ B is smallenough that it may be approximated as zero—this is the case at the LHC,for example. See subsection 2.2.2 for more on the equation of state.Heavy-ion collisions trace various trajectories through the phase dia-gram, beginning as a QGP at high temperature and eventually cooling intoa hadron gas, undergoing either a crossover or first-order phase transitiondepending on the value of µ B . Higher energy collisions have a larger initialtemperature and smaller baryon chemical potential, thus, different energycollisions probe different regions of the phase diagram.The conversion back to particles (hadronization) completes by propertime τ ∼
10 fm /c . The system is now a hadron resonance gas (HRG), con-sisting of mostly pions—the lightest hadron—but also protons, neutrons,and a slew of other species, including many unstable resonances. The gascontinues to expand and cool as particles scatter off each other and reso-nances decay into stable species. Soon after hadronization, the decays andother chemical interactions complete, freezing the composition of the system(“chemical freeze-out”). Around temperature T ∼
120 MeV, the system isdilute enough that scatterings cease, freezing all particle momenta (“kineticfreeze-out”). A few nanoseconds later, the particles stream into the exper-imental detector, where they are recorded as tracks to be processed intoobservable quantities.
This is the broad picture of ultra-relativistic heavy-ion collisions. Ofcourse, none of it can be observed directly—the system is far too minisculeand ephemeral, and free quarks and gluons cannot be detected directly dueto QCD color confinement. Much of what we know is inferred by matchingcomputational collision models to experimental observations. The primarygoal of the present work is to perform this model-to-data comparison insystematic fashion, and make quantitative statements on the physical prop-erties of the QGP and precisely what transpires in heavy-ion collisions. Forthe remainder of this chapter, I introduce the experimental observations keyto this comparison and describe the properties we wish to measure.
In this section, I review the current heavy-ion collision experiments andthe primary experimental signatures of the strongly-interacting quark-gluonplasma.
HAPTER 2. A PRAGMATIC OVERVIEW There are two particle accelerators with ongoing heavy-ion programs: theRelativistic Heavy-ion Collider (RHIC) at Brookhaven National Lab in Up-ton, NY and the aforementioned Large Hadron Collider (LHC), operatedby the European Organization for Nuclear Research (CERN) near Geneva,Switzerland (the accelerator ring intersects the French-Swiss border).RHIC has been operational since 2000, colliding assorted combinationsof nuclear species including gold, uranium, copper, aluminum, protons,deuterons, and helium-3 at center-of-mass energies ranging from √ s = 7 . √ s = 2 . √ s = 2 .
76 and 5.02 TeV [14–23] suitable for direct comparison with compu-tational models. The other heavy-ion experiments at the LHC are ATLASand CMS; at RHIC there is STAR, PHENIX, PHOBOS, and BRAHMS(although these all stand for something, most are fairly contrived and theacronyms are used almost exclusively).
Nearly two decades into the RHIC era, there is unequivocal evidence thata strongly-interacting phase of QCD matter is created in heavy-ion collisions The acronym RHIC is colloquially pronounced like the name “Rick”, and as a result,is used in speech like a name, e.g. people say “at RHIC” instead of “at the
RHIC”, eventhough the latter formally makes more sense. Meanwhile, the acronym LHC is pronouncedsimply as its letters spelled out, and so people usually say “at the L-H-C”. I will use theacronyms here as they are colloquially spoken.
HAPTER 2. A PRAGMATIC OVERVIEW Figure 2.4
Event display of a Pb-Pb collision in the ALICE detector [12], whichhas a toroidal shape measuring 16 × ×
26 m [13]. The beam coincides withthe central axis of the toroid and collisions occur in the center. As particles areemitted, they propagate through the various layers of the experimental apparatusand are recorded as tracks, represented here as lines. Left: perspective view, right:beam-axis view. [24–28]. In the following subsections, I review the experimental signaturesof the QGP, emphasizing the observables that I will later use to estimateQGP properties. Among the most straightforward observable quantities from high-energy col-lisions are the number of produced particles (multiplicity) and the amountof produced energy. But they should not be overlooked, for despite (andperhaps because of) their simplicity, these observables connect to the ba-sic thermal properties of the QGP, and serve as important constraints forcomputational models.Particle and energy yields are typically reported per unit rapidity y orpseudorapidity η . Not to be confused with the spacetime rapidity η s , thesequantities have similar form but operate on the energy-momentum vectorrather than the spacetime position. The rapidity is defined as [29] y ≡
12 log E + p z E − p z . (2.3) HAPTER 2. A PRAGMATIC OVERVIEW η ≡ − log (cid:2) tan( θ/ (cid:3) = 12 log | p | + p z | p | − p z (2.4)is sometimes more accessible, since it only depends on the polar angle of themomentum vector relative to the beam axis (cos θ = p z / | p | ). The rapidityand pseudorapidity are equal in the ultra-relativistic limit, p (cid:29) m . Midrapidity yields
Particles emitted transverse to the beam, i.e. at midrapidity (near η = 0),are the purest sample of matter produced in the collision.Figure 2.5 shows the charged particle multiplicity per unit pseudorapid-ity, dN ch /dη , in the central rapidity unit, | η | < .
5, from ALICE measure- æ part N Æ æh / d c h N d Æ æ pa r t N Æ NN s Pb-Pb, = 5.02 TeV NN s p-Pb, = 2.76 TeV (x1.2) NN s Pb-Pb, = 2.76 TeV (x1.13) NN s pp, | < 0.5 h | Figure 2.5
Charged-particle multiplicity at midrapidity perparticipant pair as a function of the number of participants[14, 20, 30, 31]. The circle diagrams show the approximatenuclear overlap of the collision depending on N part . HAPTER 2. A PRAGMATIC OVERVIEW √ s = 2 .
76 and 5.02 TeV and proton-protonand proton-lead collisions for comparison [14, 20, 30, 31]. The multiplici-ties are shown as a function of the number of participating nucleons, N part ,and scaled by participant pair, N part /
2. A “participant” is a nucleon thatengages in inelastic collision processes, as opposed to a spectator, which con-tinues down the beam pipe unaffected. More “central” collisions, i.e. thosewith small impact parameter and more complete nuclear overlap, have moreparticipants; “peripheral” collisions with large impact parameter have fewerparticipants. The maximum number of participants for a collision of
Pbnuclei is 416.Due to the high energy of the collision, many more particles are producedthan the original number of nucleons. In the most central collisions with themost participants, N part ∼ dN ch /dη ∼ 〉 part N 〈 ( G e V ) 〉 η / d c h N d 〈 / 〉 η / d T E d 〈 = 2.76 TeV NN s Pb − ALICE Pb = 200 GeV NN s STAR Au Au = 200 GeV NN s PHENIX Au Au
Figure 2.6
Average transverse energy per charged particle atmidrapidity as a function of the number of participants [21, 32–34].
HAPTER 2. A PRAGMATIC OVERVIEW E T = X i E i sin θ i , (2.5)where E i and θ i are the total energy and angle with respect to the beam,respectively, of particle i . Transverse energy is closely related to charged-particle production and has a similar trend as a function of N part . Figure2.6 shows the average transverse energy per charged particle at midrapidityfor lead-lead collisions at 2.76 TeV and RHIC gold-gold collisions at 200GeV [21, 32–34]; the ratio is constant within uncertainty as a function of N part , but clearly higher-energy collisions produce more transverse energyper particle. Centrality
I have already mentioned the concept of centrality and its relation to N part ,but, as the primary classifier of heavy-ion collision events, it warrants adedicated discussion. Centrality categorizes events based on a final-stateobservable that quantifies the amount of matter produced in the collision,such as N ch or E T . Geometric properties of the initial state, such as N part and the impact parameter b —which are not directly measurable—can thenbe connected to centrality and estimated using a geometric initial conditionmodel. Figure 2.7
Centrality determination of Pb-Pb collisions at2.76 TeV by ALICE [14, 35]. The histogram is the distri-bution of the VZERO amplitude, apportioned into centralitypercentile bins, and the red line is the Glauber model fit.
HAPTER 2. A PRAGMATIC OVERVIEW Table 2.1
Average charged-particle multiplicityat midrapidity and estimated average number ofparticipants for the centrality bins in figure 2.7 [14].Centrality % h dN ch /dη i h N part i ±
60 382 . ± . ±
49 329 . ± . ±
37 260 . ± . ±
23 186 . ± . ±
15 128 . ± . ± . ± . ± . ± . ± . ± . ± . ± . Formally, centrality is the fraction of the nuclear interaction cross section σ above some threshold of particle or energy production, for example c ( N THRch ) ≈ σ Z ∞ N THRch dσdN ch dN ch , (2.6)where N THRch is a threshold number of charged particles. Thus, if N THRch is close to the maximum number of particles that an event can produce,then only a small fraction of the differential cross section dσ/dN ch will beabove the threshold, so the centrality fraction will be small. Inversely, ifthe threshold is low, then most of the cross section will be above it, so thecentrality fraction will be large.To construct centrality bins, experiments run a large number of events,sort them by the chosen observable, and then apportion the events intopercentile bins. Figure 2.7 shows the centrality bins determined by ALICE,which defines centrality by the VZERO amplitude [35]. The ALICE VZEROdetector covers the forward and backward pseudorapidity ranges 2 . < η < . − . < η < − . N part in figures 2.5 and 2.6 are in fact centrality bins in disguise.The following graphic summarizes the relationship between centrality HAPTER 2. A PRAGMATIC OVERVIEW N part and impact parameter b ), where R is thenuclear radius and A the mass number (number of nucleons):100% centrality N part ∼ b ∼ R
0% centrality N part ∼ Ab ∼ Pseudorapidity distributions
In addition to midrapidity yields, experiments have measured particle pro-duction as a function of pseudorapidity, for example: d N c h / d η η -5 -4 -3 -2 -1 0 1 2 3 4 5 ALICE Data (symmetrised)ReflectedUncorr. syst. unc.Corr. syst. unc.Pb–Pb √ s NN = Figure 2.8
Charged-particle pseudorapidity density in several centrality binsmeasured by ALICE for Pb-Pb collisions at 5.02 TeV [36].
Boost invariance asserts that these distributions should be flat near midra-pidity, which is the case for the central rapidity unit | η | < . η = 0 has lessangular coverage. For example the central unit | η | < . ◦ , or about 30% of the total angular space, while 2 < | η | < . ◦ . Identified particle yields
The observables discussed to this point do not differentiate among the var-ious hadronic species created in heavy-ion collisions, such as pions, kaons,
HAPTER 2. A PRAGMATIC OVERVIEW dN/dy .The ratios of various identified particle yields provide insights on chemi-cal freeze-out, expected to occur shortly after the QGP medium hadronizes.A simple description of particle production is the statistical hadronizationmodel [37, 38], which assumes that particles are thermally produced in thegrand canonical ensemble, so each species’s yield is controlled by its Boltz-mann factor e − m/T and spin degeneracy. M u l t i p li c i t y d N / d y Data, ALICE, 0 10%Statistical model = 0 MeV, V=5380 fm b µ Fit: T=156 MeV, = 1 MeV b µ T=164 MeV, =2.76 TeV NN sPb Pb + π π + K K s0 K K* φ p p Λ Ξ + Ξ Ω + Ω d H Λ H Λ Figure 2.9
Statistical hadronization model fit to identified par-ticle yields in central Pb-Pb collisions at 2.76 TeV [39]. Data fromALICE [16–18, 40, 41].
Figure 2.9 shows statistical model fits to hadron yields in central lead-lead collisions at 2.76 TeV [39]. The primary fit has two free parameters:the temperature and effective system volume (to normalize overall particleproduction), with the baryon chemical potential fixed to zero. In this model,the various yields are well-described—with the possible exception of protons,which are somewhat overpredicted—and the best-fit temperature T = 156MeV is within the QCD transition region, consistent with a prompt chemicalfreeze-out. The alternate fit has µ B = 1 MeV and a higher temperature, HAPTER 2. A PRAGMATIC OVERVIEW γ s ∼ .
6, while for nucleus-nucleus collisions γ s ∼
1, implyingfull chemical equilibrium [42, 43]. Interestingly, the best-fit temperature isconsistently 155–170 MeV for all collision systems.The total chemical equilibrium in heavy-ion collisions—as opposed tostrangeness suppression in other systems—is an important signal of QGPformation, since, in the plasma, strange-antistrange pairs can be produceddirectly from pairs of free quarks and gluons, and these processes equilibratewithin the timescale of heavy-ion collisions [44, 45]. These avenues are notavailable in hadronic systems, so small collision systems (that don’t createQGP) cannot produce as much strangeness.
Transverse momentum distributions
A standard measurement in high-energy collisions is the distribution of par-ticle production as a function of the transverse momentum, p T = q p x + p y .These distributions, often called p T spectra, are usually reported as some-thing like d N/ ( N ev πp T dp T dy ), meaning a histogram of particle countsbinned by p T , per unit rapidity, averaged over the events in a centrality bin.The factor 1 / πp T corrects for the phase space density d p T = 2 πp T dp T dφ ,since p T is effectively a polar or cylindrical radius.Figure 2.10 shows typical transverse momentum distributions for severalidentified particle species measured by ALICE [16–18, 40]. The distributionsare approximately thermal in the hydrodynamic region, p T (cid:46) p T and an exponential tail. The height of each curve isproportional to the yield, which simply follows the mass hierarchy, whilethe slope relates to the kinetic freeze-out temperature and rate of transverseexpansion. Notably, the strange baryons (Λ, Ξ, Ω) have shallower slopes andlonger tails than the other species, indicating higher effective kinetic freeze-out temperatures; they cease interacting earlier in the hadron gas expansiondue to their smaller scattering cross sections.The effective kinetic freeze-out temperature and transverse expansion ve- HAPTER 2. A PRAGMATIC OVERVIEW p T [GeV] -2 -1 d N / ( N e v π p T d p T d y ) [ G e V − ] π + pK + ΛΞ − Ω − ALICE Pb-Pb 2.76 TeV10–20% centrality
Figure 2.10
Transverse momentum distributions (histograms) for the labeledidentified particles at midrapidity measured by ALICE [16–18, 40]. locity may be estimated by fitting spectra to the so-called “blast-wave” func-tion [46], which incorporates thermal particle production and hydrodynamicflow. As shown in figure 2.11, the average transverse flow velocity h β T i in-creases significantly with particle production, meaning that central collisions h /d ch N d æ T bÆ = 2.76 TeV NN sALICE, = 200 GeV NN sSTAR, (a) h /d ch N d ( G e V ) k i n T = 2.76 TeV NN sALICE, = 200 GeV NN sSTAR, (b) Figure 2.11
Average transverse flow velocity h β T i (left) and effective kineticfreeze-out temperature T kin (right) from blast-wave fits to transverse momentumspectra. Parameters from ALICE [16] and STAR [25] are shown together as func-tions of midrapidity charged-particle production dN ch /dη . HAPTER 2. A PRAGMATIC OVERVIEW T kin decreases with centrality, presumably because as the system density increases, it mustcool more before particles stop interacting. In central collisions, T kin ∼ ∼
155 MeV, corrob-orating the picture that hadrons continue to scatter for some time after thechemical composition is fixed. Compared to LHC, the RHIC flow velocitiesand temperatures are uniformly smaller given the same number of producedparticles, reflecting the less explosive system created at lower beam energy.
The observation of collective behavior is arguably the most compelling evi-dence that a strongly-interacting quark-gluon plasma is created in heavy-ioncollisions.Collectivity manifests as anisotropies in the azimuthal transverse mo-mentum distribution [47], dN/dφ , where φ = arctan2( p y , p x ). Why wouldsuch anisotropy occur? Consider the diagram of a noncentral collision onthe left: b Initial geometry Final momentum xyz 𝜙 Figure 2.12
Left: Asymmetric overlap region created by a pair of nuclei (circles)colliding with impact parameter b . Right: The resulting anisotropic transverseparticle emission. The nuclei collide with impact parameter b along the x -direction, creatingan asymmetric almond-shaped overlap region where the hot and dense QGPmedium forms. This shape generates a steeper pressure gradient along the x -direction compared to y , since the same total pressure change—from thecentral pressure to surrounding vacuum—occurs over a shorter distance.The pressure gradients then drive fluid dynamical expansion preferentiallyin the x -direction, and as the medium freezes into hadrons, it imparts thatanisotropic momentum to the emitted particles, as shown on the right ofthe figure. Ultimately, the observed transverse momentum distribution will HAPTER 2. A PRAGMATIC OVERVIEW φ = 0 and π . Figure 2.13
Anisotropic expan-sion of a strongly-interacting degen-erate Fermi gas [48].
A similar phenomenon has been directly ob-served in a rather dissimilar strongly-interactingsystem: an ultra-cold, degenerate gas of Fermioniclithium atoms. In the experiment [48], the gas isheld by an asymmetric optical trap, then releasedand allowed to expand; figure 2.13 shows snapshotsof the expanding gas from t = 0 . HAPTER 2. A PRAGMATIC OVERVIEW Anisotropic flow coefficients
To quantify transverse momentum anisotropy, we expand the azimuthal dis-tribution as a Fourier series [50, 51] dNdφ ∝ ∞ X n =1 v n cos (cid:2) n ( φ − Ψ n ) (cid:3) , (2.7)where the flow coefficient v n is the magnitude of n th-order anisotropy andthe event-plane angle Ψ n is the corresponding phase; figure 2.14 shows atypical Fourier decomposition. π/ π π/ π φdN/dφv = 0 . v = 0 . v = 0 . Figure 2.14
Fourier decomposition of an azimuthal particle distribution into flowharmonics v n . The gray histogram is the “observed” distribution (randomly gen-erated, not real experimental data), the colored lines are the Fourier components,and the black line is the total distribution. The flow coefficients, or harmonics, are given by v n = (cid:10) cos (cid:2) n ( φ − Ψ n ) (cid:3)(cid:11) , (2.8)where the average runs over particles (in a p T bin) and events (in a cen-trality bin). In particular, v is called directed flow, v elliptic flow, and v triangular flow. Fluctuations
The simplified collision geometry shown in figure 2.12 explains only the exis-tence of even-order anisotropy; the almond shape would drive strong elliptic
HAPTER 2. A PRAGMATIC OVERVIEW v and contribute to higher-order even harmonics ( v , v , . . . ), but can-not account for triangular flow v or any other odd harmonics. Triangularflow, universally observed at RHIC [52, 53] and LHC [15, 22, 54, 55], is thusattributed to event-by-event fluctuations in the collision geometry [56]. b = 0 fm b = 8 fm Figure 2.15
Initial collision geometry created by fluctuating nucleon positions.Blue and orange circles represent nucleons from each projectile nucleus; dark circlesare participants, light are spectators. Left: ultra-central collision, right: interme-diate centrality.
Above, the impact of nucleon position fluctuations on overlap geometry;the right side is a more realistic version of the perfect almond shape, whilethe left side shows that even perfectly central collisions may have spatialanisotropy. These irregular overlap regions have nonzero ellipticity, trian-gularity, and higher-order deformations, which together drive all orders ofanisotropic flow.
Cumulants
The definition of the flow coefficients (2.8) depends on the event-plane anglesΨ n , characteristics of the initial collision geometry which are therefore notexperimentally observable. To circumvent this, flow coefficients are typicallyestimated via multiparticle azimuthal correlations, or cumulants [57–62].Since collectivity induces particle correlations in momentum space, the flowcan be extracted from measured correlation functions without knowledge ofthe event plane.Figure 2.16 shows typical two-particle correlation functions measuredby the CMS experiment [63]. They are histograms of ∆ φ and ∆ η , thedifferences in azimuthal angle and pseudorapidity between pairs of particles, HAPTER 2. A PRAGMATIC OVERVIEW η ∆ -4 -2 0 2 4 φ ∆ φ ∆ d η ∆ d pa i r N d t r g N -1 b µ = 3.9 int CMS L η ∆ -4 -2 0 2 4 φ ∆ φ ∆ d η ∆ d pa i r N d t r g N η ∆ -4 -2 0 2 4 φ ∆ φ ∆ d η ∆ d pa i r N d t r g N η ∆ -4 -2 0 2 4 φ ∆ = 2.76 TeV NN sPbPb η ∆ -4 -2 0 2 4 φ ∆ η ∆ -4 -2 0 2 4 φ ∆ η ∆ -4 -2 0 2 4 φ ∆ < 3.5 GeV/c trigT η ∆ -4 -2 0 2 4 φ ∆ η ∆ -4 -2 0 2 4 φ ∆ η ∆ -4 -2 0 2 4 φ ∆ < 1.5 GeV/c assocT η ∆ -4 -2 0 2 4 φ ∆ η ∆ -4 -2 0 2 4 φ ∆ Figure 2.16
Two-particle correlation functions in various centrality bins for Pb-Pb collisions at 2.76 TeV measured by CMS [63]. The height of each (∆ φ, ∆ η ) bin isproportional to the number of observed charged-particle pairs in the bin. Each pairconsists of a trigger particle and an associated particle with transverse momenta inthe annotated ranges for p trig T and p assoc T , respectively. where the height of each (∆ φ, ∆ η ) bin is proportional to the number ofobserved charged-particle pairs with those differences. In all but the mostperipheral collisions, there is a pronounced ridge structure at ∆ φ ∼
0. Thefact that this “near-side ridge” extends to long range in ∆ η , and that itdisappears in peripheral collisions, is taken as a signal of collective behavior.A similar long-range “away-side ridge” forms at ∆ φ ∼ π in mid-centralcollisions as a result of elliptic flow. The peak at ∆ φ ∼ ∆ η ∼ HAPTER 2. A PRAGMATIC OVERVIEW h k i denote the single-event k -particle azimuthalcorrelation function, then the two- and four-particle correlations are [60] h i = (cid:10) e in ( φ − φ ) (cid:11) = 1 P M, M X i = j e in ( φ i − φ j ) , h i = (cid:10) e in ( φ + φ − φ − φ ) (cid:11) = 1 P M, M X i = j = k = l e in ( φ i + φ j − φ k − φ l ) , (2.9)where M is the event multiplicity and P M,k = M ! / ( M − k )! is the numberof k -particle permutations, e.g. P M, = M ( M − ,P M, = M ( M − M − M − . (2.10)The two-particle correlation function for a centrality bin is hh ii = (cid:10)(cid:10) e in ( φ − φ ) (cid:11)(cid:11) = P N events i P M i , h i i P N events i P M i , , (2.11)where the outer average is performed over all events in the centrality bin,weighted by each event’s number of permutations. The definition of hh ii isanalogous.To see how the correlation functions relate to the flow coefficients, firstadd and subtract the event plane to the azimuthal angles: hh ii = (cid:10)(cid:10) e in [( φ − ψ n ) − ( φ − ψ n )] (cid:11)(cid:11) . (2.12)Now, as long as φ and φ are only correlated via the event plane, i.e. onlydue to collective flow, the inner average factorizes [61]: hh ii ≈ (cid:10)(cid:10) e in ( φ − ψ n ) (cid:11)(cid:10) e − ( φ − ψ n ) (cid:11)(cid:11) = h v n i . (2.13)(The imaginary parts vanish by symmetry.) Analogously, hh ii ≈ h v n i , etc.In reality, other physical processes besides collective flow, such as jetsand resonance decays, can induce particle correlations, which is why theabove relations are only approximate. Certainly, some fraction of the away-side ridge is attributable to back-to-back jets. When estimating flow viamultiparticle correlations, it is crucial to remove as much of these “nonflow”effects as possible. Using four-particle (or even higher-order) correlations isone way to suppress nonflow. HAPTER 2. A PRAGMATIC OVERVIEW c n { k } be the n th-order cumulant from k -particle correlations, and specifically [58] c n { } = hh ii ,c n { } = hh ii − hh ii . (2.14)Finally, defining v n { k } as the estimate of the flow coefficient v n from thecumulant c n { k } : v n { } = q c n { } ,v n { } = q − c n { } . (2.15)Expressions for the six- and eight-particle cumulants v n { } and v n { } alsoexist but are rather lengthy, so I omit them here. Each flow cumulantprovides a different estimate of the underlying flow. Notice that, in theabsence of nonflow and statistical fluctuations, invoking equation (2.13) gives v n { } ≈ q v n = v n ,v n { } ≈ q − [ v n − v n ) ] = v n . (2.16)However, since these effects generally are present, each flow cumulant willin general be different.Rather than evaluate the k -particle correlation functions via explicitnested loops over particle permutations—which may be feasible for two-or four-particle correlations, but quickly becomes unreasonable for six oreight—one typically uses Q -vectors, defined as [60] Q n = M X i =1 e inφ i . (2.17)Each single-event correlation h k i can be analytically expressed in terms of Q -vectors, for example, the square of Q n is equivalent to a sum over pairs: | Q n | = M X i,j =1 e in ( φ i − φ j ) = M + M X i = j e in ( φ i − φ j ) , (2.18)and comparing to equation (2.9) immediately gives h i = | Q n | − MM ( M − . (2.19) HAPTER 2. A PRAGMATIC OVERVIEW
28A somewhat longer derivation yields [60] h i = | Q n | + | Q n | − < [ Q n Q ∗ n Q ∗ n ] − M − | Q n | + 2 M ( M − M ( M − M − M − . (2.20)Thus, all correlation functions can be evaluated with O ( M ) complexity in-stead of O ( M k ). The Q -vector method obviates the need to store lists ofall particles for each event; only the multiplicity M and the Q n (a fewcomplex numbers) are required. It also provides several other benefits toexperiments, such as dealing with nonuniform detector acceptance [60–62]. n v η∆ {2, | v |>1} η∆ {2, | v |>1} η∆ {2, | v {4} v {6} v {8} v η∆ {2, | v |>1} η∆ {2, | v |>1} η∆ {2, | v {4} v η∆ {2, | v |>1} η∆ {2, | v ALICE Pb Pb Hydrodynamics (a) R a t i o /s(T), param1 η /s = 0.20 η (b) Hydrodynamics, Ref.[25] v v v Centrality percentile R a t i o Figure 2.17
Integrated flow calculated with multi-particle cumulants as a function of centrality for Pb-Pb collisions at 2.76 and 5.02 TeV measured by AL-ICE [15, 22]. Symbols are data as indicated in the leg-end; bands are predictions from a hydrodynamics model.The lower panels show the ratios of the two-particle cu-mulants v n { , | ∆ η | > } between beam energies (sym-bols) and the corresponding model predictions of theratios (bands) [ v in panel (b) and v , v in panel (c)].References [25, 27] annotated in the figure are [64, 65]. Integrated flow
The flow coefficients v n integratedover transverse momentum quantifythe overall azimuthal anisotropy in acentrality class.Figure 2.17 shows the centralitydependence of integrated flow cumu-lants, calculated up to eight parti-cles, for 2.76 and 5.02 collisions mea-sured by ALICE [15, 22]. The no-tation v n { , | ∆ η | > } means two-particle cumulants with a pseudora-pidity gap, i.e. limited to pairs of par-ticles separated by at least one unit ofpseudorapidity. This helps suppressnonflow, since azimuthal correlationscaused by resonance decays, jets, etctend to be short range in η .Elliptic flow v shows strong de-pendence on centrality due to the cor-relation with increasing impact pa-rameter and initial-state anisotropy.It increases until about 50% central-ity, above which it decreases, pre-sumably because the QGP medium,while highly eccentric in these periph-eral collisions, does not survive longenough for the flow to fully develop. HAPTER 2. A PRAGMATIC OVERVIEW v { } > v { } ≈ v { } ≈ v { } ,implying that the two-particle cumulant contains some nonflow despite the η gap, but the four-particle cumulant is sufficient to suppress this nonflow.Meanwhile, triangular and quadrangular flow v , v have much weakercentrality dependence since they are driven mostly by initial-state fluctua-tions. In the most central bin, v is much closer to v , v since in this case,the impact parameter is small and the overlap roughly circular, so v is alsodriven largely by fluctuations.The bottom panels plot the ratios of the two-particle cumulants between5.02 and 2.76 TeV (symbols). In general, flow increases at the higher energydue to the hotter, longer-lived medium. Elliptic flow increases slightly outto intermediate centrality and more significantly in peripheral bins. Theincrease in v is also slight, while v is somewhat more pronounced (althoughthe absolute increase is still small, but since the baseline is small the relativechange is large). | > . } h D { , | n v hD {2, | v |>1} hD {2, | v |>1} hD {2, | v 2.76 TeV |>1} hD {2, | v |>1} hD {2, | v |>1} hD {2, | vALICE Pb-Pb ) c (GeV/ T p | > . } h D { , | n v Figure 2.18
Differential two-particle flow cu-mulants in 0–5% and 30–40% centrality for Pb-Pb collisions at 2.76 and 5.02 TeV measured byALICE [22].
The figure includes hydrodynamicmodel predictions of the two-particleflow cumulants at 5.02 TeV and theratios between beam energies [64, 65].Overall, the model describes the dataexceptionally well.
Differential flow
Flow coefficients may also be measuredas a function of transverse momentum, v n ( p T ), called differential flow.Figure 2.18 shows differential flowcumulants for the two LHC beam ener-gies [22]. In central 0–5% collisions, allmeasured harmonics have similar mag-nitude, with v and v becoming largerthan v at higher p T . However, inte-grated v is still largest, since most par-ticles reside in the low p T region where v is slightly higher; more precisely,the integrated flows are the integralsof these curves, weighted by the trans-verse momentum distribution. In mid-central 30–40% collisions, v is much HAPTER 2. A PRAGMATIC OVERVIEW v and v at all p T .There is little change in differential flow between the two beam energies,but as shown above, integrated flow increases slightly with energy. This isbecause the mean transverse momentum is larger, so particles shift to higherregions of the differential flow curves.The differential flow of identified particles [66], figure 2.19 left side, ex-hibits the characteristic “mass splitting”: Lighter particles (such as pions)have more flow at low p T , while heavier particles (such as protons) havemore flow at high p T . This occurs because all particles originate from thesame expanding source and thus share a common average velocity, so heavierparticles have higher p T (see figure 2.10), and consequently, the underlyinganisotropic flow activates at higher p T for heavier species [67]. The masssplitting also manifests in v and v [68]. = 2.76 TeV NN s ALICE 20 30% Pb Pb | > . } η ∆ { SP , | v ) c (GeV/ T p ± π ± K K pp+ φ Λ + Λ + Ξ + Ξ + Ω + Ω NN s ALICE 20 30% Pb Pb q n / | > . } η ∆ { SP , | v ) c (GeV/ q n )/ m T m ( ± π Kpp+ φΛ + Λ + Ξ + Ξ + Ω + Ω Figure 2.19
Differential elliptic flow of identified hadrons in 20–30% centralityfor Pb-Pb collisions at 2.76 TeV measured by ALICE [66]. Left: standard p T -differential v . Right: v as a function of the transverse kinetic energy ( m T − m ),both scaled by the number of constituent quarks n q . (The underlying data are thesame in both plots.) The right-side plot is a test of quark deconfinement. It shows the samedata as on the left, but scaled by the number of constituent quarks, n q = 2for mesons and 3 for baryons, and as a function of the transverse kinetic en-ergy per constituent, ( m T − m ) /n q , where m T = q m + p T is the transversemass. The curves collapse much closer together at low p T , signaling thatcollective flow develops partially when the medium consists of free quarks,which then coalesce into hadrons. It is particularly compelling that the φ and proton, a meson and baryon with similar mass (1019 and 938 MeV), HAPTER 2. A PRAGMATIC OVERVIEW
Other flow observables
Besides the standard flow observables summarized here, a number of otherflow-related quantities have been measured, including distributions of event-by-event flows [74], correlations between flow harmonics [23, 75], event-planecorrelations [76], the pseudorapidity dependence of flow [77], and more.
Small collision systems
Recent experimental results show unambiguous signatures of collective be-havior in high-multiplicity events of small collision systems, such as proton-nucleus and even proton-proton. Perhaps the clearest sign is the appearanceof a long-range, near-side ridge in high-multiplicity bins. Figure 2.20 com-pares two-particle correlation functions for Pb-Pb and p-Pb collisions in thesame multiplicity bin, which corresponds to about 60% centrality for Pb-Pb hD -4 -2 0 2 4 (r ad i an s ) fD fD d hD d pa i r N d t r i g N < 260 offlinetrk N £ = 2.76 TeV, 220 NN s(a) CMS PbPb < 3 GeV/c trigT assocT hD -4 -2 0 2 4 (r ad i an s ) fD fD d hD d pa i r N d t r i g N < 260 offlinetrk N £ = 5.02 TeV, 220 NN s(b) CMS pPb < 3 GeV/c trigT assocT Figure 2.20
Two-particle correlation functions for Pb-Pb (left) and p-Pb (right)collisions in the same multiplicity bin measured by CMS [78].
HAPTER 2. A PRAGMATIC OVERVIEW v { k } up to eight particles ( k = 8) [80, 81] andsimilar v { } in p-Pb as Pb-Pb [78, 82].It remains an open question whether the observed collective behaviororiginates from hydrodynamic flow, an initial state effect, or something else[83–85]. The quantities discussed to this point are all bulk observables, meaningthey describe the soft particles with p T (cid:46) O (100 GeV) (dependingon the beam energy), by hard scatterings early in the collision evolution.These high- p T particles then propagate through and interact with the hot (GeV) T p AA R and lumi. uncertainty AA T|<1 h | (5.02 TeV PbPb) -1 b m (5.02 TeV pp) + 404 -1 CMS
CMS 5.02 TeVCMS 2.76 TeV
ALICE 2.76 TeVATLAS 2.76 TeV
Figure 2.21
Nuclear modification factor R AA for chargedparticles in central (0–5%) Pb-Pb collisions measured at √ s = 5 .
02 TeV by CMS [86] and at 2.76 TeV by CMS [87],ALICE [88], and ATLAS [89].
HAPTER 2. A PRAGMATIC OVERVIEW p T hadrons—and heavyquarks (charm or bottom).One of the simplest germane observables is the nuclear modification fac-tor R AA , which quantifies the modifications to transverse momentum dis-tributions in nucleus-nucleus ( AA ) collisions relative to proton-proton ( pp )collisions. It is defined as R AA = dN AA /dp T h N coll i dN pp /dp T , (2.21)i.e. the ratio of the AA spectrum to the pp spectrum, scaled by the averagenumber of binary nucleon-nucleon collisions h N coll i . The denominator isa null hypothesis: the hypothetical spectrum if AA collisions were simplya superposition of pp collisions. Thus, if AA collisions did not produce aQGP medium, R AA would equal one. As shown in figure 2.21, R AA forcharged particles is experimentally less than one out to very high p T ; this“suppression” is taken as evidence of medium effects.In this work, I focus on quantifying bulk properties of the QGP, so hardprocesses are not directly relevant. But entire subfields of heavy-ion physicsare devoted to theoretical and experimental study of various hard processes;see, for example, recent reviews of jets [90] and heavy quarks [91]. Let us now turn our attention to the physical properties of hot and denseQCD matter—the quark-gluon plasma and the initial state that leads toits formation—the precise determination of which is a central goal of thiswork. Many of the salient properties are defined in the context of viscousrelativistic hydrodynamics, summarized below.
Viscous relativistic hydrodynamics
The bulk dynamics of the QGP are well-described by viscous relativistichydrodynamics, whose main equations of motion derive from conservationof energy and momentum: ∂ µ T µν = 0 (2.22)where T µν = e u µ u ν − ( P + Π)∆ µν + π µν (2.23) HAPTER 2. A PRAGMATIC OVERVIEW e , P , and u µ are the local energy density,pressure, and flow velocity, respectively, of the fluid, ∆ µν = g µν − u µ u ν is theprojector transverse to the flow velocity, π µν is the shear viscous pressuretensor, and Π is the bulk viscous pressure.An ideal (inviscid) fluid has five independent dynamical quantities: theenergy density, pressure, and three components of flow velocity. Four ofthese are determined by the conservation equations (2.22), and the fifth bythe equation of state P ( e ).The viscous pressures π µν and Π, which account for dissipative correc-tions to ideal hydrodynamics, introduce six additional independent quanti-ties. The shear tensor is traceless ( π µµ = 0) and orthogonal to the flow ve-locity ( π µν u ν = 0), so only five of its ten components are independent. Thebulk pressure, a scalar, effectively adds to the thermal pressure as ( P + Π)in T µν . In a simple relativistic generalization of Navier-Stokes theory, theseterms connect to the fluid flow as [92] π µν = 2 ησ µν , Π = − ζθ, (2.24)where η and ζ are the shear and bulk viscosity, σ µν = ∇ h µ u ν i is the velocityshear tensor, and θ = ∇ · u is the expansion rate. Notation: ∇ h µ u ν i = ( ∇ µ u ν + ∇ ν u µ ) − ( ∇ · u )∆ µν , where ∇ ν = ∆ µν ∂ ν is the gradient inthe local rest frame. However, the instantaneous connection between thefluid flow and viscous pressures leads to acausal signal propagation, so theseequations do not suffice for relativistic hydrodynamics. Israel and Stewartsolved this problem [93, 94] with relaxation-type equations of the form τ π ˙ π µν + π µν = 2 ησ µν + . . . ,τ Π ˙Π + Π = − ζθ + . . . , (2.25)where τ π and τ Π are timescales over which the viscous pressures relax to theNavier-Stokes limit. A recent derivation from the relativistic Boltzmannequation yields [95–97] τ π ˙ π h µν i + π µν = 2 ησ µν + 2 π h µα ω ν i α − δ ππ π µν θ + φ π h µα π ν i α − τ ππ π h µα σ ν i α + λ π Π Π σ µν + φ Π π µν ,τ Π ˙Π + Π = − ζθ − δ ΠΠ Π θ + φ Π + λ Π π π µν σ µν + φ π µν π µν , (2.26)where π h µν i = ∆ µναβ π αβ , using the double projector ∆ µναβ = (∆ µα ∆ νβ +∆ µβ ∆ να − ∆ µν ∆ αβ ), and ω λρ = ( ∇ λ u ρ − ∇ ρ u λ ) is the vorticity tensor. These equa-tions include all terms up to second order in the viscous pressures as well HAPTER 2. A PRAGMATIC OVERVIEW transport coefficients ; the shear and bulk viscosity, η and ζ , are the first-order transport coefficients. Nothing here is unique to the QGP—or any other fluid. Within a hydro-dynamic description, all that distinguish any given fluid are its transportcoefficients and equation of state. In the remainder of this section, I discussthese quantities and how they relate to the QGP.
Broadly, transport coefficients characterize the dynamical properties of afluid, such as its response to external forces. They are in general functionsof temperature and, in the case of QGP, chemical potential.
Viscosity
Shear viscosity η measures a fluid’s resistance to shear strain. A low-viscosity fluid is generally strongly-interacting, efficiently transmits shearstrain through itself, and its constituents have a short mean free path; onthe other hand a nearly-ideal (weakly-interacting) gas has large viscositybecause its constituents do not scatter enough to convey the informationthat a strain is being applied.The “quality” of a fluid is quantified by its specific shear viscosity, thedimensionless ratio to the entropy density, η/s . The entropy density s isa proxy for the number density, so η/s is in a sense the viscosity per unit(an intensive quantity). The QGP specific shear viscosity is of particularinterest since it is believed to be small—nearly zero—meaning that the QGPis nearly a “perfect” fluid. The measurement of the temperature-dependentspecific shear viscosity ( η/s )( T ) is a primary goal of heavy-ion physics [98].It has been famously conjectured, based on a string theory calculationapplicable to a wide range of strongly-interacting quantum field theories,that the minimum possible specific shear viscosity is η/s ≥ / π ’ . η/s of 1 / π –2 . / π ’ . .
20. These results of course do not confirm theconjecture—the uncertainty is quite large, and even if the bound is correct,it would apply only to highly idealized systems, so the actual measured Dimensionless in natural units with (cid:126) = k B = 1. To convert to SI units, multiply by (cid:126) /k B ’ . × − K s.
HAPTER 2. A PRAGMATIC OVERVIEW T/T c η / s Water P c / P c P c Helium P c / P c P c pQCD H R G QGP / π Figure 2.22
Specific shear viscosity η/s of different fluids as a function of tem-perature relative to each fluid’s critical temperature T c . The colored lines representcommon fluids, water and helium, at various pressures relative to their critical pres-sures P c , as annotated. These curves were computed from NIST data [104] with theentropy standardized so that it is zero at zero temperature, S ( T = 0) = 0, usingstandard-state thermochemistry data [105]. The hadron resonance gas (HRG) areais based on a recent study [106], and the perturbative QCD (pQCD) area on aparametrization in the high-temperature limit [107]. The QGP area is motivatedby numerous studies, e.g. [100–103], and results of this work that will be presentedin chapter 5. The locations and shapes of all areas are approximate. The dashedline denotes the conjectured bound 1 / π [99]. QGP value wouldn’t necessary be exactly 1 / π . Nothing special happens tohydrodynamics when η/s drops below 1 / π .What is clear, however, is that the QGP is much closer to perfection thanmost ordinary fluids. Figure 2.22 compares our knowledge of the QGP η/s to common fluids, whose properties have been measured and are tabulatedby NIST [104, 105]. The QGP η/s is about an order of magnitude smallerthan those of water and helium, which are O (1) near their critical temper-atures. From the NIST data, we see that η/s generally reaches a minimumnear T c , either as a continuous curve, a cusp, or a discontinuous jump, de-pending on whether the pressure is above, equal to, or below the criticalpressure, respectively. A similar functional form likely manifests for QCD HAPTER 2. A PRAGMATIC OVERVIEW
10 fm I d e a l η / s = .
100 200 300 350
Temperature [MeV] τ = 1 fm /c /c /c /c Figure 2.23
Comparison of ideal and viscous hydrodynamics applied to a fluctu-ating intermediate centrality event. Shown is the time evolution, starting at propertime τ = 1 fm /c , of the temperature profile at midrapidity with η/s = 0 (top) and0.2 (bottom). matter: Below the QCD transition temperature, various calculations withhadron resonance gas (HRG) models point to η/s decreasing with tempera-ture [106]; in the high-temperature limit, where perturbative QCD (pQCD)is applicable, calculations show an increasing function of temperature [107].Near T c , neither the HRG model or pQCD is reliable, so we must rely oncomparisons of hydrodynamic model calculations to data.How does shear viscosity impact hydrodynamics, and what is the connec-tion to experimental observations of heavy-ion collisions? Most apparent isthe effect on collectivity: increasing η/s reduces collective behavior and flow.As shown in figure 2.23, shear viscosity washes out small-scale structures andinduces a more isotropic system, which would have smaller anisotropic flowcoefficients v n . Thus, flow coefficients are the primary QGP viscometer.The referenced studies [100–103] estimated η/s by running hydrodynamic HAPTER 2. A PRAGMATIC OVERVIEW η/s and comparing the resulting v n tocorresponding experimental data. Bulk viscosity ζ is related to the fluid expansion rate, as can be seen bythe way it enters the evolution equation (2.26). It influences QGP evolu-tion primarily by suppressing the radial expansion rate, translating into areduction in the transverse momentum of emitted particles (however, bulkviscosity makes no qualitative difference in the appearance of the hydrody-namic medium, which is why it isn’t represented in figure 2.23). Recently, itwas shown that a nonzero specific bulk viscosity ζ/s is necessary for hydro-dynamic models to simultaneously describe mean transverse momentum andflow [108], and other phenomenological studies have demonstrated that ζ/s modifies the transverse momentum spectra and, to a lesser extent, collectiveflow [109–112].As for the temperature dependence, ( ζ/s )( T ) is not well-known but isgenerally expected to peak near the QCD transition temperature and falloff on either side, based on calculations below T c [113], near [114, 115], andabove [116]. This picture is consistent with an approximate result from ki-netic theory [111], ζ/η ≈ / − c s ) , where c s is the speed of sound. Inthe high-temperature limit c s ≈ /
3, so ζ (cid:28) η , and near T c it is smaller, c s ∼ . η/ζ ∼ . ζ/s ∼ . ∼ ζ/s (cid:38) . T c can inducenegative-pressure bubbles in the hydrodynamic medium (“cavitation”) [117]. Other hydrodynamic coefficients
Besides the first-order transport coefficients η and ζ , the viscous evolutionequations (2.26) contain several second-order coefficients, coupling coeffi-cients, and relaxation times. The second-order and coupling coefficientshave been computed in the limit of small masses [97], and it is reasonable toexpect that varying them would not have a strong impact on hydrodynamicevolution. The shear and bulk relaxation times τ π and τ Π are important forcausal viscous relativistic hydrodynamics, but empirically, their specific val-ues do not have much impact on physical observables [118, 119] [and section5.1]. HAPTER 2. A PRAGMATIC OVERVIEW An equation of state (EoS) interrelates a system’s various thermodynamicquantities: temperature, energy density, pressure, etc. From a fluid dy-namical perspective, an EoS P = P ( e ) is required to close the system ofconservation equations (2.22).The QCD EoS has been computed numerically using modern latticeQCD techniques. These calculations are complex and extremely computa-tionally expensive (some of the largest NERSC computational allocationsare given to lattice QCD groups), and lattice QCD is an entire field untoitself. For a recent review of lattice techniques emphasizing applications toheavy-ion collisions see reference [120].Lattice calculations begin by evaluating the QCD partition function Z on a hypercubic spacetime lattice of size N σ N τ , where N σ and N τ are thenumber of spatial and temporal steps. Lattice sites are separated by latticespacing a , which relates to the temperature and volume by T = 1 / ( aN τ ) and V = ( aN σ ) . The calculation is repeated with different lattice sizes, usuallywith a fixed ratio of spatial and temporal steps N σ /N τ , and the results areextrapolated to the continuum and thermodynamic limits: a → N τ → ∞ , V → ∞ .Constructing the EoS then hinges on the trace of the energy-momentumtensor, or trace anomaly, Θ µµ = e − P , from which all other thermodynamicquantities can be computed. The trace anomaly is defined on the lattice byΘ µµ = − TV d log Zd log a (2.27)and related to the pressure as [121]Θ µµ T = e − PT = T ddT (cid:18) PT (cid:19) , (2.28)which, after integration, furnishes the pressure explicitly: P ( T ) T = P T + Z TT dT Θ µµ T , (2.29)where P is the pressure at reference temperature T . This reference point isusually computed using a hadron resonance gas (HRG) model, which sums In natural units with (cid:126) = c = 1, pressure and energy density have the same units astemperature to the fourth power, so e.g. P/T is dimensionless. To convert from units offm − to the more intuitive GeV/fm , multiply by (cid:126) c ’ .
197 GeV fm.
HAPTER 2. A PRAGMATIC OVERVIEW s = ( e + P ) /T follow immediately.Below, thermodynamic quantities from the HotQCD Collaboration’s re-cent lattice calculation of the EoS at zero net baryon density in (2+1)-flavorQCD, that is, with two light quarks (of equal mass) and a heavier strangequark [6]. ε /T HRG non-int. limitT c Figure 2.24
The QCD equation of state calculated by theHotQCD Collaboration [6]. The colored bands show the nor-malized pressure, energy density, and entropy density as afunction of temperature, where the width of the bands rep-resents the uncertainty; the solid lines show the correspondingquantities from the HRG model. The vertical band denotesthe crossover region, T c = 154 ± e/T = 95 π / These calculations establish a crossover deconfinement transition region T c =154 ± C V = ∂e/∂T ,near T c but no peak or discontinuity. HAPTER 2. A PRAGMATIC OVERVIEW µ B ) thanks tomodern lattice techniques. Indeed, I will not estimate the EoS in this work,but rather use the lattice results directly in a hydrodynamic model. Butit is reasonable to doubt that the lattice EoS, which is calculated in theinfinite-time and infinite-volume limits, applies to the extremely transientand tiny QGP created in heavy-ion collisions. A recent study [122] testedthis as part of a Bayesian model-to-data comparison using a hydrodynamicmodel, RHIC and LHC data, and similar parameter estimation methods asin chapter 4. They simultaneously varied 14 parameters, including initial ��������� ��� ��� ��� ��� � � � ����������������������� ��� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � ���� � � � � ��� ��� ��� ��� ��� ��� ������������������������������� ��� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � ���� � � � � � ������� Figure 2.25
Left: Randomly sampled, unconstrained equations of staterepresented by the squared speed of sound, c s = ∂P/∂e , as a functionof temperature. Right: Equations of state sampled from the posteriordistribution, constrained by data. The thick red lines represent the rangeof lattice calculations and the green line is the HRG EoS, to which allsamples connect at 165 MeV. Figure from [122]. HAPTER 2. A PRAGMATIC OVERVIEW
Hydrodynamics describes the spacetime evolution of the QGP medium, butnot how it forms. At the very least, some other physical process(es) mustaccount for the initial energy density immediately after the collision, and,assuming the system takes a short time τ ∼ /c to begin behavinghydrodynamically, another dynamical process must describe this early stage.This also makes sense mathematically: The hydrodynamic equations aredifferential equations and therefore require an initial condition.But the initial state, as the earliest stage of heavy-ion collisions, is cre-ated under the most extreme conditions and is the furthest from what weobserve. Besides hindering the fundamental goals of characterizing and mod-eling the initial state itself, this has far reaching consequences for quanti-fying the transport properties of the QGP. For example, one of the mostwell-known studies estimating η/s [101] found ( η/s ) / (4 π ) ≈ .
5, ascribingmost of the uncertainty range to competing models of the initial state.Why would the initial state affect estimates of medium properties like η/s ? One crucial way is through the anisotropy of the initial energy den-sity. Suppose model A tends to produce more anisotropic initial states thanmodel B, then it will generally lead to larger anisotropic flow coefficients v n ,and if the goal is to match an experimental measurement of v n , model Awill require a larger η/s to suppress its anisotropy (in fact, this is roughlywhat happened in the study just mentioned). The precise initialization ofthe other dynamical quantities, namely the fluid flow velocity and viscouspressures, will also, in general, impact the QGP medium evolution and finalstate.To rigorously estimate QGP properties, we must therefore also considerthe variability and uncertainty in the initial state. These degrees of freedombroadly fall into two categories: characteristics of energy and/or entropydeposition immediately after the collision, and the subsequent dynamicalproperties of the system prior to QGP formation. HAPTER 2. A PRAGMATIC OVERVIEW Energy and entropy deposition
Let us break this down even further: How is nuclear density distributed innuclei, and given a pair of colliding nuclei with known density distributions,what energy (or entropy) is deposited?The problem can be factorized like this because ultra-relativistic colli-sions occur on such a short timescale. As a rule of thumb, since the massof a nucleon is about 1 GeV and the total energy of a massive particle is E = γm , the Lorentz factor for a relativistic nucleon is approximately itskinetic energy in GeV. At the LHC, where the beam energy is √ s = 2 . γ > ρ ( r ) ∝
11 + exp (cid:0) r − Ra (cid:1) , (2.30)which is basically a sphere with a blurry edge, where the nuclear radius R and surface thickness (or skin depth) a are measured for many nuclei, e.g.for Pb, R = 6 .
62 fm, a = 0 .
546 fm [123]. In Monte Carlo models, nucleonpositions are randomly sampled, with the radii sampled from the full radialprobability P ( r ) ∝ r ρ ( r ) and the angles sampled isotropically (assuming aspherical nucleus).This already raises a question: How are the nucleon positions correlated?It is reasonable to assume that nucleons cannot occupy the same spatial lo-cation, so one simple way to insert correlations is to impose a minimumdistance between nucleons. When ALICE estimates initial state propertiesfor centrality bins, they vary the minimum distance from 0 to 0.8 fm as partof the systematic uncertainty [35]. More realistic nucleon-nucleon correla-tions have also been implemented [124] and shown to influence anisotropicflow in ultra-central collisions [125].Another unknown degree of freedom is the effective size and shape ofnucleons—“effective” because although some physical properties of nucle-ons are independently measured, they don’t necessarily directly connect toenergy deposition in ultra-relativistic collisions. For example the protonroot-mean-square charge radius is 0.88 fm [126], while the gluon radius ismuch smaller, approximately 0.4 fm [127], and either (or neither) could berelevant in this context. The impact of the nucleon size on the overall initial HAPTER 2. A PRAGMATIC OVERVIEW w = 0 . w = 0 .
88 fm f m Figure 2.26
Fluctuating initial conditions with the Gaussian nucleon width w set to the gluon radius (0.4 fm, left) and the proton charge radius (0.88 fm, right).The nucleon positions are the same in both cases; only the size is different. condition can hardly be overstated, as shown in figure 2.26. Smaller nu-cleons create more compact structures with higher peak temperatures andsteeper gradients, driving increased anisotropic flow and radial expansion,which has ramifications for estimating shear and bulk viscosity. In general,initial state and medium properties are intrinsically entangled, and it is im-portant to estimate them simultaneously while propagating all sources ofuncertainty.As for the nucleon shape, the transverse profile is often chosen to be aGaussian: T nucleon ( x, y ) = Z dz ρ nucleon ( x, y, z ) = 12 πw exp (cid:18) − x + y w (cid:19) , (2.31)where T nucleon is the beam-integrated density, or thickness, and w is theeffective nucleon width. Although unproven, the Gaussian shape is compu-tationally convenient and satisfies reasonable physical limits. More empiricalprofiles are possible, for example one could use the charge distribution fromthe measured electric form factor [128], with the caveat that charge densitymay not directly translate to energy deposition in relativistic collisions.Thickness is the relevant quantity at sufficiently high energy because ofthe aforementioned Lorentz contraction and instantaneity of the collision.The full nuclear thickness of the projectile nuclei shall be denoted T A and HAPTER 2. A PRAGMATIC OVERVIEW T B , where T A is the sum of T nucleon for all nucleons in nucleus A .Now, restating the question posed earlier: Given T A and T B , what isthe resulting transverse energy density e ( x, y ) [or entropy density s ( x, y )]at midrapidity? (In terms of initializing hydrodynamics, either energy orentropy density is acceptable, since they can be interconverted via the equa-tion of state. For the remainder of this section I will only say energy forbrevity.)A simple model of energy deposition is the Glauber model [129, 130],which in its Monte Carlo formulation, deposits energy for each participantnucleon (sometimes called wounded nucleons) and each binary nucleon-nucleon collision, with the fraction of energy apportioned to binary colli-sions controlled by a phenomenological parameter α . Despite its basis onmostly geometrical arguments, the Glauber model has semi-quantitativelyfit a variety of experimental measurements and is the de facto standard forunfolding centrality bins. MC-Glauber MC-KLN IP-Glasma
Figure 2.27
Transverse energy density generated by various initial conditionmodels, as labeled. Figure from [131].
A more theoretically motivated initial state formalism is the color glasscondensate (CGC) [132], an effective field theory based on gluon saturationat high energy. The Monte Carlo Kharzeev-Levin-Nardi (MC-KLN) model[133, 134], a CGC implementation, described the centrality dependence ofyields and elliptic flow, but has fallen out of favor due to it overpredicting thedifference between elliptic and triangular flow [119, 135] [and section 5.1].Notably, the IP-Glasma model [131, 136], which combines the CGC-basedimpact parameter dependent saturation model (IP-Sat) [127, 137] with clas-sical Yang-Mills dynamics of the produced glasma (gluon plasma) fields, hasprecisely described a wide array of observables, including integrated flowharmonics, differential flow, and event-by-event flow distributions [138].These models produce quite different initial energy density—even given
HAPTER 2. A PRAGMATIC OVERVIEW T A and T B —and as with the nucleon width, thisvariability can be difficult or impossible to disentangle from QGP mediumproperties. An alternative approach is to parametrize energy deposition asa function of thickness, retaining as much meaningful flexibility as possible,and constrain the parametrization simultaneously with medium properties.This was part of the strategy in some previous Bayesian parameter estima-tion studies [122, 139, 140], and is the goal of the parametric initial conditionmodel TRENTo [141], developed by myself and fellow Duke graduate studentJ. Scott Moreland. See section 3.1 for a detailed description of TRENTo. Dynamics and thermalization
It is generally assumed that, for hydrodynamics to be valid, the systemmust be in (approximate) local thermodynamic equilibrium (although somerecent work calls this into question [142]), and that the system is not bornin equilibrium. Since hydrodynamic evolution must begin early, by around τ ∼ /c , in order to leave sufficient time for the observed collective flowto build up before freeze-out, another dynamical process must rapidly drivethe system to equilibrium. This suggests a two-stage approach in which anenergy deposition model describes the system immediately after the collision,at time τ = 0 + , then a pre-equilibrium model handles the dynamics priorto QGP formation.It should be noted that pre-equilibrium dynamics are not strictly re-quired for computational models; hydrodynamics can be initialized directlywith the energy density and with zero initial flow and viscous pressures. Inthis interpretation, the energy deposition model provides the energy at thehydrodynamic starting time, not at τ = 0 + , skipping any pre-equilibriumstage. Although not the most realistic, this scheme has been used in nu-merous studies and does not preclude a good description of the data, butit may force the initial conditions to somehow compensate for the lack ofpre-equilibrium evolution.The most rudimentary dynamical pre-equilibrium model is free stream-ing [143, 144], wherein the system is treated as an expanding, noninteractinggas of massless partons (quarks and/or gluons). During free streaming, theenergy density smooths out and radial flow increases, ultimately translatingto larger mean p T . At a variable time τ fs , the system undergoes a suddenequilibration and hydrodynamic evolution begins; this instantaneous transi-tion from zero coupling to strong coupling cannot be the physical reality—itshould be gradual—but the model can nonetheless help bracket the maxi-mum pre-equilibrium time. A multiparameter analysis of transverse momen- HAPTER 2. A PRAGMATIC OVERVIEW τ = 0 .
01 fm /c τ = 1 fm /c τ = 2 fm /c f m Figure 2.28
Free streaming evolution of the transverse energy density for a typ-ical fluctuating initial condition (nucleon width w = 0 . τ to account for longitudinal expansion. tum and flow data found τ fs ≈ . /c and that increased free streamingtime correlates with decreased η/s [145]. See section 3.2 for a mathematicaldescription of free streaming.Other theories of pre-equilibrium dynamics include weakly-coupled effec-tive kinetic theory [146], which ought to be more realistic than free streaming(zero coupling) and may be a viable alternative for computational purposes;the IP-Glasma model [131, 136], in which the system is evolved by solvingthe classical Yang-Mills equations, another weakly-coupled theory; and vari-ous strong coupling approaches, e.g. [147], although so far these have limitedcomputational utility. Computational models ofheavy-ion collisions T he past two decades have seen considerable progress towards a “standardmodel” of the bulk dynamics of relativistic heavy-ion collisions [148–155]. Now well-established, the following multistage approach mirrors thepresumptive true collision spacetime evolution:1. Initial conditions: generates the energy density immediately after thecollision.2. Pre-equilibrium: simulates the dynamics until QGP formation.3. Viscous relativistic hydrodynamics: calculates the evolution of the hotand dense QGP medium, including its collective expansion, cooling,and transition to a hadron gas.4. Particlization: converts the hydrodynamic system into a microscopicensemble of hadrons.5. Boltzmann transport: computes hadronic scatterings and decays untilfreeze-out.Over the next five sections, I assemble a set of these models tailored for use inBayesian parameter estimation. Of the various stages, I have developed new,original code for the initial conditions, pre-equilibrium, and particlizationcomponents. I close the chapter with some details on performing large-scalemodel-to-data comparison.Before proceeding, it’s worth emphasizing that there are other viableapproaches for modeling heavy-ion collision dynamics, such as using a parton48 HAPTER 3. COMPUTATIONAL MODELS
Initial condition models are responsible for generating the energy or entropydensity immediately after the collision. In this work, I use TRENTo, a para-metric initial condition model developed by myself and fellow Duke graduatestudent J. Scott Moreland. The next few subsections introduce the model,demonstrate some of its capabilities, and illustrate why its flexibility makesit ideal for parameter estimation and uncertainty quantification; text andfigures have been adapted from our publication:J. S. Moreland, J. E. Bernhard, and S. A. Bass, “Alternative ansatz towounded nucleon and binary collision scaling in high-energy nuclear colli-sions”, Phys. Rev.
C92 , 011901 (2015), arXiv: .The model is publicly available at https://github.com/Duke-QCD/trento . TRENTo is an effective model, intended to generate realistic Monte Carloinitial entropy profiles without assuming specific physical mechanisms forentropy production, pre-equilibrium dynamics, or thermalization.Suppose a pair of projectiles labeled
A, B collide along beam axis z , andlet ρ part A,B be the density of nuclear matter that participates in inelastic colli-sions. Each projectile may then be represented by its participant thickness˜ T A,B ( x, y ) = Z dz ρ part A,B ( x, y, z ) . (3.1)The construction of these thickness functions will be addressed shortly; first,we postulate the following:1. The eikonal approximation is valid: Entropy is produced if ˜ T A and ˜ T B eikonally overlap.2. There exists a scalar field f ( ˜ T A , ˜ T B ) which converts projectile thick-nesses into entropy deposition. HAPTER 3. COMPUTATIONAL MODELS f is proportional to the entropy created at mid-rapidity andat the hydrodynamic thermalization time: f ∝ dS/dy | τ = τ . (3.2)It should provide an effective description of early collision dynamics: it neednot arise from a first-principles calculation, but it must obey basic physicalconstraints.Perhaps the simplest such function is a sum, f ∼ ˜ T A + ˜ T B , in fact this isequivalent to a wounded nucleon model since the present thickness functions(3.1) only include participant matter. The two-component Glauber ansatzadds a quadratic term to account for binary collisions, i.e. f ∼ ( ˜ T A + ˜ T B ) + α ˜ T A ˜ T B .However, recent results from ultra-central uranium-uranium collisionsat RHIC [158, 159] show that particle production does not scale with thenumber of binary collisions, excluding the two-component Glauber ansatz[160]. Therefore N one-on-one nucleon collisions should produce the sameamount of entropy as a single N -on- N collision, which is mathematicallyequivalent to the function f being scale-invariant: f ( c ˜ T A , c ˜ T B ) = c f ( ˜ T A , ˜ T B ) (3.3)for any nonzero constant c . Note, this is clearly broken by the binary collisionterm ( α ˜ T A ˜ T B ). We will justify this constraint later in the text; for themoment we take it as a postulate.With these constraints in mind, we propose for f the reduced thickness f = ˜ T R ( p ; ˜ T A , ˜ T B ) ≡ (cid:18) ˜ T pA + ˜ T pB (cid:19) /p , (3.4)so named because it takes two thicknesses ˜ T A , ˜ T B and “reduces” them to athird thickness, similar to a reduced mass. This functional form—known asthe generalized mean—interpolates between the minimum and maximum of˜ T A , ˜ T B depending on the value of the dimensionless parameter p , and sim-plifies to the arithmetic, geometric, and harmonic means for certain values:˜ T R = max( ˜ T A , ˜ T B ) p → + ∞ , ( ˜ T A + ˜ T B ) / p = +1 , (arithmetic) q ˜ T A ˜ T B p = 0 , (geometric)2 ˜ T A ˜ T B / ( ˜ T A + ˜ T B ) p = − , (harmonic)min( ˜ T A , ˜ T B ) p → −∞ . (3.5) HAPTER 3. COMPUTATIONAL MODELS −2 −1 0 1 2 x [fm] T h i c k n e ss [ f m − ] Arithmetic: p = 1 Geometric: p = 0 Harmonic: p = − Participant × Beam view x Figure 3.1
Reduced thickness of a pair of nucleon partici-pants. The nucleons collide with a nonzero impact parameteralong the x -direction as shown in the upper right. The graydashed lines are one-dimensional cross sections of the partici-pant nucleon thickness functions ˜ T A , ˜ T B , and the colored linesare the reduced thickness ˜ T R for p = 1 , , − Physically, p interpolates among qualitatively different physical mechanismsfor entropy production. To see this, consider a pair of nucleon participantscolliding with some nonzero impact parameter, as shown in figure 3.1. For p = 1, the reduced thickness is equivalent to a Monte Carlo wounded nucleonmodel and deposits a blob of entropy for each nucleon, while for p = 0,the model deposits a single roughly symmetric blob at the midpoint of thecollision, and as p becomes negative, it suppresses entropy deposition alongthe direction of the impact parameter. Similar behavior was discussed inthe context of small collision systems in [161]. Note that the values 1 , , − p is a continuous parameter—and the scale-invariantconstraint (3.3) is always satisfied.We now detail the construction of the thickness functions ˜ T A,B ( x, y ),which combined with the definition of the reduced thickness completes thespecification of the model. The procedure is constructed from the groundup to handle a variety of collision systems; we begin with the simplest case.Consider a collision of two protons A, B with impact parameter b alongthe x -direction and nuclear densities ρ A,B = ρ proton ( x ± b/ , y, z ) , (3.6) HAPTER 3. COMPUTATIONAL MODELS R dz ρ proton either has a closed form or maybe evaluated numerically, so that the proton thickness functions can becalculated. The protons collide with probability [162] P coll = 1 − exp (cid:20) − σ gg Z dx dy Z dz ρ A Z dz ρ B (cid:21) , (3.7)where the integral in the exponential is the overlap integral of the protonthickness functions and σ gg is an effective parton-parton cross-section tunedso that the total proton-proton cross-section equals the experimental inelas-tic nucleon-nucleon cross-section σ NN .The collision probability is sampled once to determine if the protonscollide; assuming they do, we follow a procedure similar to [163] and assigneach proton a fluctuated thickness˜ T A,B ( x, y ) = w A,B Z dz ρ A,B ( x, y, z ) , (3.8)where w A,B are independent random weights sampled from a gamma distri-bution with unit mean, P k ( w ) = k k Γ( k ) w k − e − kw . (3.9)These gamma weights introduce additional multiplicity fluctuations in orderto reproduce the large fluctuations observed in experimental proton-protoncollisions. The shape parameter k may be tuned to optimally fit the data:small values (0 < k <
1) correspond to large multiplicity fluctuations, whilelarge values ( k (cid:29)
1) suppress fluctuations.With the projectile thickness functions in hand, the reduced thickness iscalculated to furnish the initial transverse entropy profile up to an overallnormalization factor, dS/dy | τ = τ ∝ ˜ T R ( p ; ˜ T A , ˜ T B ) . (3.10)Composite collision systems such as proton-nucleus and nucleus-nucleusare essentially treated as superpositions of proton-proton collisions. A setof nucleon positions is chosen for each projectile, typically by sampling anuncorrelated Woods-Saxon distribution or from more realistic correlated nu-clear configurations when available [124]. The collision probability (3.7) issampled for each pairwise interaction and those nucleons that collide with HAPTER 3. COMPUTATIONAL MODELS A then reads˜ T A = N part X i =1 w i Z dz ρ proton ( x − x i , y − y i , z − z i ) , (3.11)where w i and ( x i , y i , z i ) are the weights and position, respectively, of par-ticipant i in nucleus A . ˜ T B follows analogously.This completes the construction of the model, TRENTo (Reduced Thick-ness Event-by-event Nuclear Topology). In summary, the model depositsentropy proportional to the reduced thickness function (3.4), defined as thegeneralized mean of fluctuated participant thickness functions (3.11), witheach participant nucleon weighted by an independent gamma random num-ber (3.9). We now demonstrate TRENTo’s ability to simultaneously describe a widerange of collision systems. Note that the reduced thickness parameter p ,gamma fluctuation parameter k , and nucleon profile ρ proton are not rig-orously constrained here—see sections 5.2 and 5.3 for their quantitativeestimates—so the following results do not necessarily represent the best-fitof the model to data.We adopt a three-stage model for particle production similar to [163], inwhich the final multiplicity arises from a convolution of the initial entropydeposited by the collision, viscous entropy production during hydrodynamicevolution, and statistical hadronization at freeze-out. The average charged-particle multiplicity h N ch i after hydrodynamic evolution is to a good ap-proximation proportional to the total initial entropy [118] and hence to theintegrated reduced thickness via equation (3.10): h N ch i ∝ Z dx dy ˜ T R . (3.12)Then, assuming independent particle emission at freeze-out, the final num-ber of charged particles is Poisson distributed [164, 165], i.e. P ( N ch ) =Poisson( h N ch i ). The folding of the Poisson fluctuations with the gammaweights for each participant yields a negative binomial distribution [163],which has historically been used to fit proton-proton multiplicity fluctua-tions.To compare with experimental multiplicity distributions, we generate alarge ensemble of minimum-bias events, integrate their ˜ T R profiles, rescale HAPTER 3. COMPUTATIONAL MODELS N ch -4 -3 -2 -1 P ( N c h ) p = 1 p = 0 p = − ALICE 2.36 TeV NSD | η | < , p T correctedp+p × × × N ch -6 -4 -2 p = 1 p = 0 p = − ALICE 5.02 TeV | η | < , . < p T < . GeVp+Pb × × × N ch -6 -4 -2 p = 1 p = 0 p = − ALICE 2.76 TeV | η | < , . < p T < . GeVPb+Pb × × × Figure 3.2
Multiplicity distributions for proton-proton, proton-lead, and lead-lead collisions. The histograms are TRENTo results for reduced thickness param-eter p = − p = 0 (middle, blue), and p = 1 (bottom, green), withapproximate best-fit fluctuation parameters k and normalizations given in table 3.1.The shaded bands show the sensitivity from varying k by ± by an overall normalization constant, and sample a Poisson number for themultiplicity of each event. The left panel of figure 3.2 shows the N ch dis-tributions for proton-proton simulations with reduced thickness parameter p = 1, 0, −
1, and Gaussian beam-integrated proton density Z dz ρ proton = 12 πB exp (cid:18) − x + y B (cid:19) (3.13)with effective area B = (0 . . We tune the fluctuation parameter k foreach value of p to qualitatively fit the experimental proton-proton distri-bution [166], and additionally vary k by ±
30% to explore the sensitivity of
Table 3.1
Approximate best-fit fluctuation parameters k andnormalizations for each p value and collision system in figure 3.2. p k p+p norm p+Pb norm Pb+Pb norm+1 0.8 9.7 7.0 13.0 1.4 19. 17. 16. − HAPTER 3. COMPUTATIONAL MODELS Centrality % ε n ε Centrality % ε Arithmetic: p = 1 Geometric: p = 0 Harmonic: p = − Centrality %
Ratio
IP-Glasma
Figure 3.3
Left and middle plots: Eccentricity harmonics ε and ε as a functionof centrality for reduced thickness parameters p = 1, 0, − k by ±
30% from the values intable 3.1. Right plot: Ratio of the rms eccentricities p h ε i / p h ε i . against theallowed region (gray band) and the ratio computed by IP-Glasma (circles) [167].Note that the axes have different ranges in the ratio plot. the model to the gamma participant weights. For proton-lead and lead-leadcollisions [82] (middle and right panels), we use identical model parametersexcept for the overall normalization factor, which is allowed to vary indepen-dently across collision systems to account for differences in beam energy andkinematic cuts (annotated in the figure). The k values and normalizationsare given in table 3.1.The model is able to reproduce the experimental proton-proton distri-bution for each value of p , provided k is appropriately tuned. Varying thebest-fit k value (by ± p value also yields a reasonable fit to the shapes of the proton-lead and lead-lead distributions, although lead-lead appears to favor p ≈ p = 1 (wounded nucleon model) in proton-lead and lead-lead collisions (table 3.1) are not self-consistent, since proton-lead requires roughly half the normalization as lead-lead, even though theexperimental data were measured at a higher beam energy.Eccentricity harmonics ε n are calculated using the definition ε n e inφ = − R dx dy r n e inφ ˜ T R R dx dy r n ˜ T R . (3.14) HAPTER 3. COMPUTATIONAL MODELS ε and triangularity ε as a function of central-ity using the same lead-lead data as in figure 3.2. There is a clear trend ofincreasing eccentricity (particularly ε ) with decreasing p . This is a larger-scale manifestation of the behavior in figure 3.1: as p decreases, the gener-alized mean (3.4) attenuates entropy production in asymmetric regions ofthe collision, accentuating the elliptical overlap shape in non-central colli-sions and enhancing their eccentricity. Meanwhile, varying the fluctuationparameter k has limited effect.In addition, we perform the test proposed by [167], which uses flowdata and hydrodynamic calculations to determine an experimentally allowedband for the ratio of root-mean-square eccentricities q h ε i / q h ε i . as afunction of centrality. Among available initial condition models only IP-Glasma consistently falls within the allowed region. As shown in the rightpanel of figure 3.3, TRENTo with p = 0 (geometric mean) yields excellentagreement with the allowed band and is similar to IP-Glasma.As a final novel application, we return to the previously mentioned ultra-central uranium-uranium puzzle, where typical Glauber models are notablyinconsistent with experimental data. Unlike e.g. gold and lead, uraniumnuclei have a highly deformed prolate spheroidal shape, so uranium-uraniumcollisions may achieve maximal overlap via two distinct orientations: “tip-tip”, in which the long axes of the spheroids are aligned with the beamaxis and the overlap area is circular; or “side-side”, where the long axes areperpendicular to the beam axis and the overlap area is elliptical, as shown infigure 3.4. Hence side-side collisions will in general have larger initial-stateellipticity ε and final-state elliptic flow v than tip-tip.Side view Beam view ε N part N coll U tip-tip U U smaller equal largerU side-side U U larger equal smaller
Figure 3.4
Comparison of tip-tip and side-side uranium-uraniumcollisions. Schematics are shown from a side view and looking downthe beam axis, and the following quantities are compared: ellipticity ε , number of participating nucleons N part , and number of binarynucleon-nucleon collisions N coll . HAPTER 3. COMPUTATIONAL MODELS N ch / › N ch fi ε U+UAu+Au0–0.1% spectators N ch / › N ch fi U+UAu+Au0–1% spectators
TRENTO Glauber+NBD
Figure 3.5
Ellipticity ε as a function of normalized charged-particle multiplicity N ch / h N ch i in ultra-central uranium-uranium and gold-gold collisions at RHIC. Theleft and right plots show the top 0.1% and 1% of collisions selected by numberof spectators to mimic STAR’s experimental ZDC selection [158]. Blue pointswith error bars are binned TRENTo results with reduced thickness parameter p =0 and best-fit fluctuation parameter k = 1 .
4. Blue lines are linear fits within0 . < N ch / h N ch i < .
1. Gray lines represent the analogous Glauber+NBDslopes calculated in [158].
In the two-component Glauber model, tip-tip collisions produce more bi-nary nucleon-nucleon collisions than side-side, so tip-tip collisions have largercharged-particle multiplicity N ch . Therefore, the most central uranium-uranium events are dominated by tip-tip collisions with maximal N ch andsmall v , while side-side collisions have a smaller N ch and somewhat larger v . This predicted drop in elliptic flow as a function of N ch is known as the“knee” [168].Recent data by STAR on uranium-uranium collisions exhibits no evi-dence of a knee [158, 159], at odds with Glauber model predictions. It hasbeen proposed that fluctuations could wash out the knee [169], but a recentflow analysis showed that it would still be visible [160].The data therefore imply that multiplicity is independent of the numberof binary collisions, justifying the scale-invariant condition (3.3) postulatedduring the construction of the reduced thickness ansatz (3.4). Consequently,TRENTo predicts roughly the same number of charged particles in tip-tipand side-side uranium-uranium collisions. As shown in figure 3.5, the slope of ε as a function of N ch is approximately equal for uranium-uranium and gold-gold, in contrast to the Glauber model which predicts a much steeper slopefor uranium. Short of conducting a full hydrodynamic analysis, TRENTo HAPTER 3. COMPUTATIONAL MODELS
This subsection is adapted from:J. E. Bernhard et al., “Applying Bayesian parameter estimation to rel-ativistic heavy-ion collisions: simultaneous characterization of the initialstate and quark-gluon plasma medium”, Phys. Rev.
C94 , 024907 (2016),arXiv: .TRENTo is constructed to achieve maximal flexibility using a minimal num-ber of parameters and can mimic a wide range of existing initial conditionmodels. To demonstrate the efficacy of the generalized mean ansatz, equa-tion (3.4), we now show that the mapping can reproduce different theorycalculations using suitable values of the parameter p .Perhaps the simplest and oldest model of heavy-ion initial conditions isthe so called participant or wounded nucleon model, which deposits entropyfor each nucleon that engages in one or more inelastic collisions [172]. Inits Monte Carlo formulation [173–176], the wounded nucleon model may beexpressed in terms of participant thickness functions, equation (3.11), as s ∝ ˜ T A + ˜ T B . (3.15)Comparing to equation (3.5), we see that the wounded nucleon model isequivalent to the generalized mean ansatz with p = 1.More sophisticated calculations of the mapping f can be derived fromcolor glass condensate effective field theory. A common implementation ofa CGC based saturation picture is the KLN model [133, 177, 178], in whichentropy deposition at the QGP thermalization time can be determined fromthe produced gluon density, s ∝ N g , where dN g dy d r ⊥ ∼ Q s, min (cid:20) (cid:18) Q s, max Q s, min (cid:19)(cid:21) , (3.16)and Q s, max and Q s, min denote the larger and smaller values of the two satu-ration scales in opposite nuclei at any fixed position in the transverse plane[179]. In the original formulation of the KLN model, the two saturationscales are proportional to the local participant nucleon density in each nu-cleus, Q s,A ∝ ˜ T A , and the gluon density can be recast as s ∼ ˜ T min (cid:2) T max / ˜ T min ) (cid:3) (3.17) HAPTER 3. COMPUTATIONAL MODELS ~ T A [fm ¡ ] E n t r o p y d e n s i t y [ f m ¡ ] Gen. mean, p = ¡ : KLN ~ T A [fm ¡ ]Gen. mean, p = 0 EKRT ~ T A [fm ¡ ] f m ¡ f m ¡ ~ T B = f m ¡ Gen. mean, p = 1 Wounded nucleon
Figure 3.6
Profiles of the initial thermal distribution predicted by the KLN (left),EKRT (middle), and wounded nucleon (right) models (dashed black lines) comparedto a generalized mean with different values of the parameter p (solid blue lines).Staggered lines show different slices of the initial entropy density dS/ ( d r ⊥ dy ) asa function of the participant nucleon density ˜ T A for several values of ˜ T B = 1 , , − ]. The EKRT mapping is shown with model parameters K = 0 .
64 and β = 0 . to put it in a form which can be directly compared with the wounded nucleonmodel.Another saturation model which has attracted recent interest after itsuccessfully described an extensive list of experimental particle multiplic-ity and flow observables [103, 180] is the EKRT model, which combinescollinearly factorized pQCD minijet production with a simple conjecture forgluon saturation [181, 182]. The energy density predicted by the model aftera pre-thermal Bjorken free streaming stage is given by e ( τ , x, y ) ∼ K sat π p ( K sat , β ; T A , T B ) , (3.18)where the saturation momentum p sat depends on nuclear thickness func-tions T A and T B , as well as phenomenological model parameters K sat and β . Calculating the saturation momentum in the EKRT formalism is com-putationally intensive, and hence—in its Monte Carlo implementation—themodel parametrizes the saturation momentum p sat to facilitate efficient eventsampling [103]. The energy density in equation (3.18) can then be recast asan entropy density using the thermodynamic relation s ∼ e / to compareit with the previous models.Note that equation (3.18) is expressed as a function of nuclear thickness T which includes contributions from all nucleons in the nucleus, as opposed HAPTER 3. COMPUTATIONAL MODELS T . In order to express initial condition mappingsas functions of a common variable one could, e.g. relate ˜ T and T using ananalytic wounded nucleon model. The effect of this substitution on theEKRT model is small, as the mapping deposits zero entropy if nucleons arenon-overlapping, effectively removing them from the participant thicknessfunction. We thus replace T with ˜ T in the EKRT model and note thatsimilar results are obtained by recasting the wounded nucleon, KLN, andTRENTo models as functions of T using standard Glauber relations.Figure 3.6 shows one-dimensional slices of the entropy deposition map-ping predicted by the KLN, EKRT, and wounded nucleon models for typi-cal values of the participant nucleon density sampled in Pb+Pb collisions at √ s = 2 .
76 TeV. The vertically staggered lines in each panel show the changein deposited entropy density as a function of ˜ T A for several constant valuesof ˜ T B , where the dashed lines are the entropy density calculated using thevarious models and the solid lines show the generalized mean ansatz tunedto fit each model. The figure illustrates that the ansatz reproduces differ-ent initial condition calculations and quantifies differences among them interms of the generalized mean parameter p . The KLN model, for example,is well-described by p ∼ − .
67, the EKRT model corresponds to p ∼ p = 1. Smaller, more negativevalues of p pull the generalized mean toward a minimum function and hencecorrespond to models with more extreme gluon saturation effects.The three models considered in figure 3.6 are by no means an exhaustivelist of proposed initial condition models, see e.g. Refs. [176, 183–187]. No-tably absent, for instance, is the highly successful IP-Glasma model whichcombines IP-Sat CGC initial conditions with classical Yang-Mills dynamicsto describe the full pre-equilibrium evolution of produced glasma fields [131,136, 138]. The IP-Glasma model lacks a simple analytic form for initial en-ergy (or entropy) deposition at the QGP thermalization time and so cannotbe directly compared to the generalized mean ansatz. In lieu of such a com-parison, we examined the geometric properties of IP-Glasma and TRENTothrough their eccentricity harmonics ε n .We generated a large number of TRENTo events using entropy deposi-tion parameter p = 0, Gaussian nucleon width w = 0 . k = 1 .
6, which were previously shown to reproduce the ratio ofellipticity and triangularity in IP-Glasma [141]. We then free streamed [143,144] the events for τ = 0 . c to mimic the weakly coupled pre-equilibriumdynamics of IP-Glasma and match the evolution time of both models. Fi-nally, we calculated the eccentricity harmonics ε and ε weighted by energy HAPTER 3. COMPUTATIONAL MODELS Impact parameter b [fm] " n " " TRENTo + FS, p = 0 ± . IP-Glasma
Figure 3.7
Eccentricity harmonics ε and ε as a functionof impact parameter b for Pb+Pb collisions at √ s = 2 .
76 TeVcalculated from IP-Glasma and TRENTo initial conditions.IP-Glasma events are evaluated after τ = 0 . c classicalYang-Mills evolution [131]; TRENTo events after τ = 0 . c free streaming [143, 144] and using parameters p = 0 ± . k = 1 .
6, and nucleon width w = 0 . density e ( x, y ) according to the definition ε n e inφ = − R dx dy r n e inφ e ( x, y ) R dx dy e ( x, y ) , (3.19)where the energy density is the time-time component of the stress-energytensor after the free streaming phase, T . The resulting eccentricities, pic-tured in figure 3.7, are in good agreement for all but the most peripheralcollisions, where sub-nucleonic structure becomes important. This similar-ity suggests that TRENTo with p ∼ f . However, the resulting mapping cannot be encapsulatedby a single value of the parameter p , so we do not attempt to support orexclude the participant quark model in the present analysis. HAPTER 3. COMPUTATIONAL MODELS After publishing the preceding work, I implemented in TRENTo a minimumnucleon-nucleon distance parameter d min . As discussed in subsection 2.2.3,this is an unknown degree of freedom that could impact initial geometry.In the absence of a minimum distance, sampling uncorrelated nucleonpositions is straightforward: Independently sample the radial position ofeach nucleon from the Fermi distribution P ( r ) ∝ r
11 + exp (cid:0) r − Ra (cid:1) , (3.20)and sample the angles θ and φ isotropically.Now, the naïve way to impose a minimum distance is, after sampling eachnucleon’s coordinates, check its distance to all previously sampled nucleons,and if it falls within d min of any other nucleon, resample the coordinates.However, since the spherical volume element is dV = 4 πr dr , there is lessspace available at small r , so positions sampled with small r are more likelyto need resampling. This shifts density toward larger r , effectively modifyingthe target radial distribution, which is undesirable.To avoid this, one should pre-sample the radii for all nucleons beforeattempting to place any of them. Then, when choosing the full three-dimensional coordinates for each nucleon, sample only the angles, resam-pling as necessary to satisfy the minimum distance. This way, the radialdistribution is guaranteed not to change.With this algorithm, if d min is large it will occasionally be impossible toplace a nucleon no matter how many times the angles are resampled. Todecrease the likelihood of this happening, one should sort the pre-sampledradii in increasing order; this way, the nucleons with the smallest r , wherespace is at a premium, are placed first.Using this method, d min can be varied and estimated simultaneouslywith all other model parameters. Broadly speaking, pre-equilibrium models take the output of the initial con-dition model, compute the ensuing dynamics until the hydrodynamic start-ing time, and initialize the energy-momentum tensor T µν . The simplestmodel, and the present choice, is free streaming. HAPTER 3. COMPUTATIONAL MODELS This scheme assumes that the system consists of noninteracting, masslesspartons which stream freely (zero coupling) for a tunable time τ fs until asudden equilibration and switch to hydrodynamics (strong coupling). Wetherefore interpret the output of the initial condition model as the densityof partons in the transverse plane, n ( x, y ), at the initial time τ = 0 + . Thisis different from the previous interpretation—that the initial condition pro-vides the entropy density directly at the hydrodynamic starting time—butnot contradictory, since density has the same units as entropy density; weare effectively asserting that each particle carries some number of entropyunits. And for a model like TRENTo, which stipulates that entropy deposi-tion (or particle production) is purely eikonal, it arguably makes more senseto use its output immediately at τ rather than a later time.Since the partons are massless and noninteracting, they propagate alongstraight trajectories at the speed of light; at a later time τ > τ , the partonsat transverse point ( x, y ) were originally located on a ring of radius c ∆ τ centered at ( x, y ), where ∆ τ = τ − τ is the elapsed time. The energy-momentum tensor at position ( x, y ) and time τ is therefore proportional tothe integral of the density around the ring [143, 144]: T µν ( x, y ) = 1 τ Z dφ ˆ p µ ˆ p ν n ( x − ∆ τ cos φ, y − ∆ τ sin φ ) , (3.21)where ˆ p µ = p µ /p T is a transverse-momentum unit vector and the prefac-tor 1 /τ accounts for longitudinal expansion. Assuming longitudinal boostinvariance, we only need the tensor at midrapidity, in which case the unitvectors expand out toˆ p µ ˆ p ν = φ sin φ cos φ cos φ cos φ sin φ sin φ cos φ sin φ sin φ . (3.22)The result for T µν may also be derived by analytically solving the collision-less Boltzmann equation, p µ ∂ µ f ( x, p ) = 0 where f is the parton distributionfunction, and noticing that the result is independent of the original trans-verse momentum distribution for massless particles [144].At the switching time τ fs , we match the energy-momentum tensor to itshydrodynamic form T µν = e u µ u ν − ( P + Π)∆ µν + π µν . (3.23) HAPTER 3. COMPUTATIONAL MODELS T µν u ν = e u µ , (3.24)which is an eigenvalue equation whose physical solution is the one witha timelike eigenvector u µ . The equilibrium pressure P = P ( e ) can thenbe obtained via the equation of state, and the bulk pressure Π from thedifference with the total effective pressure, P + Π = −
13 ∆ µν T µν . (3.25)Lastly, the shear pressure tensor may be calculated as π µν = T µν − e u µ u ν + ( P + Π)∆ µν , (3.26)since everything on the right-hand side is now known. The corona
At time τ fs , the system is assumed to equilibrate and begin evolving hydro-dynamically. The hydrodynamic calculation then runs until the system coolsbelow a switching energy density e switch (usually parametrized as a temper-ature T switch , which can be converted to an energy density via the equationof state). Presumably, most of the system has energy density e > e switch at time τ fs , but the periphery of the collision inevitably has e < e switch .This low-density region, known as the “corona”, never enters the hydrody-namic calculation and is effectively neglected. Longer free streaming timesincrease the relative size of the corona, since the energy density decreases asthe system expands.In the present scenario, for lead-lead collisions at LHC energies with τ fs ∼ /c , the relative contribution of the corona is empirically negligible:less than 1% of the total energy for all but the most peripheral collisions.Thus, it is safe to neglect. However, for smaller collision systems (such asproton-lead), lower beam energies, or longer free streaming times, the coronacould become significant, and it may be necessary to consider its effects. Computational notes
My implementation of free streaming, written in Python, is available at https://github.com/Duke-QCD/freestream .Since the initial density is discretized onto a grid, it must be inter-polated to obtain a continuous function n ( x, y ) for integration in equa-tion (3.21). I use the bicubic interpolation provided by the Python class HAPTER 3. COMPUTATIONAL MODELS fitpack .Cubic interpolation faithfully captures the curvature of fluctuating initialconditions, but sometimes suffers from unphysical artifacts near where thedensity drops to zero, rapidly oscillating between small positive and negativevalues (clearly, density cannot be negative). Linear interpolation, on theother hand, is unable to capture the curvature but does not have the samedeficiencies. One way to remove the artifacts from cubic interpolation isto interpolate the density grid with both bilinear and bicubic algorithms,then if both return values greater than zero, use the result of the cubicinterpolation, otherwise use zero. Letting n and n be the linear and cubicinterpolating functions, this can be written n ( x, y ) = ( n ( x, y ) if n ( x, y ) > n ( x, y ) > , n ( x, y ) ≤ n ( x, y ) = 0 . (3.27)The Landau matching condition (3.24) can be solved as T µν u ν = e u µ by standard numerical eigensystem solvers. In most cases, the resultingeigenvectors must be renormalized so that u µ u µ = 1. The workhorse of computational heavy-ion collision models, hydrodynamicsis responsible for simulating the collective expansion of the hot and denseQGP medium and, via the equation of state, the transition to a hadron gas.In the present work, the implementation of choice is the Ohio State Univer-sity (2+1)-dimensional viscous hydrodynamics code, originally publishedunder the name VISH2+1 [92] and now updated to handle fluctuating initialconditions [155] and temperature-dependent shear and bulk viscosity [171].My customized version is available at https://github.com/jbernhard/osu-hydro .The boost-invariant approximation used in 2+1D hydrodynamics is adrawback for high-precision calculations, but the difference in midrapidityobservables is small compared to full 3+1D calculations [191, 192], and 2+1Dmodels run at least an order of magnitude faster. Given a finite amount ofcomputation time, many more events can be generated using a 2+1D model;from the perspective of parameter estimation, this reduction in statistical Two spatial dimensions plus time, using boost invariance for the third (longitudinal)dimension.
HAPTER 3. COMPUTATIONAL MODELS ∂ µ T µν = 0 , T µν = e u µ u ν − ( P + Π)∆ µν + π µν , (3.28)starting from initial profiles of the energy density e , flow velocity u µ , andviscous pressures π µν and Π supplied by the initial condition and pre-equilibrium models; the pressure P relates to the energy density via theequation of state. For the viscous pressures, the OSU code solves τ π ˙ π h µν i + π µν = 2 ησ µν − δ ππ π µν θ + φ π h µα π ν i α − τ ππ π h µα σ ν i α + λ π Π Π σ µν ,τ Π ˙Π + Π = − ζθ − δ ΠΠ Π θ + λ Π π π µν σ µν , (3.29)which differ from equations (2.26) by neglecting vorticity and some second-order terms, while retaining all first-order and shear-bulk coupling terms.The viscosity coefficients η and ζ are discussed in the next subsection; theremaining transport coefficients and relaxation times are fixed to the valuesderived in the limit of small masses [97]. In order to estimate the temperature-dependent specific shear and bulk vis-cosity, ( η/s )( T ) and ( ζ/s )( T ), we must parametrize them.The specific shear viscosity, as discussed in section 2.2.1, is expected toreach a minimum near the transition temperature T c . Above T c , I use amodified linear ansatz( η/s )( T ) = ( η/s ) min + ( η/s ) slope · ( T − T c ) · ( T /T c ) ( η/s ) crv (3.30)with three degrees of freedom: a minimum value at T c , a slope above T c , anda curvature parameter (crv for short), which has intuitive meaning wherezero curvature equates to a straight line and positive (negative) curvatureto the function being concave up (down). The left side of figure 3.8 showsthese degrees of freedom.In this parametrization, η/s reaches its minimum value precisely at T c ,fixed to the HotQCD value 154 MeV [6]. But this may not exactly be thecase; consider that the minimum η/s of other fluids can be located somewhatabove or below the critical temperature (depending on the pressure), as HAPTER 3. COMPUTATIONAL MODELS
150 200 250 300
Temperature [MeV] η / s Shear
150 200 250 300
Temperature [MeV] ζ / s Bulk
Figure 3.8
Degrees of freedom of the temperature-dependent specific shear andbulk viscosity parametrizations, equations (3.30) and (3.31). The parameters foreach curve are chosen for visual clarity and do not represent all possible variability,e.g. η/s may have a large slope with negative curvature, or ζ/s may have a tall andnarrow peak, neither of which are shown here. shown in figure 2.22. It would be reasonable to replace T c by a tunableparameter T in (3.30).A fourth parameter ( η/s ) hrg sets a constant η/s in the hadronic phase of the hydrodynamic model , i.e. in the narrow temperature window below T c but before converting the medium to particles and switching to Boltzmanntransport. In practice, the value of this parameter matters little since it con-trols such a small fraction of the hydrodynamic evolution, and in any case,most flow develops at higher temperatures. Note that ( η/s ) hrg is indepen-dent from ( η/s ) min , so η/s may be discontinuous at T c , a feature observedin other fluids as shown in figure 2.22.For the specific bulk viscosity, I use a three-parameter (unnormalized)Cauchy distribution ( ζ/s )( T ) = ( ζ/s ) max (cid:18) T − ( ζ/s ) T ( ζ/s ) width ) (cid:19) , (3.31)which is a symmetric peak with a tunable maximum, width, and location( T ), shown on the right of figure 3.8. This form is qualitatively similar tothe (1 / − c s ) dependence mentioned in section 2.2.1. HAPTER 3. COMPUTATIONAL MODELS The hydrodynamic equation of state (EoS) shall consist of a lattice calcu-lation for the high-temperature region and a hadron resonance gas (HRG)calculation at low temperatures. Similar to previous work [193], I constructthis “hybrid” EoS by connecting the HRG and lattice trace anomalies inan intermediate temperature range; the trace anomaly is the physical quan-tity computed directly on the lattice and may be integrated to obtain thepressure and other thermodynamic quantities.I use the lattice EoS recently calculated by the HotQCD Collaboration[6], which they parametrize as PT = 12 (cid:0) c t ( t − t )] (cid:1)(cid:18) p id + a n /t + b n /t + c n /t + d n /t a d /t + b d /t + c d /t + d d /t (cid:19) , (3.32)where t = T /T c , T c = 154 MeV, p id = 95 π /
180 is the ideal gas value of
P/T , and the fit coefficients are c t = 3 . , a n = − . , b n = 3 . , c n = 0 , d n = 0 . ,t = 0 . , a d = − . , b d = 0 . , c d = 0 , d d = − . . This form is intended for differentiation; in particular, the trace anomaly isΘ µµ T = e − PT = T ddT (cid:18) PT (cid:19) . (3.33)There is some uncertainty in the lattice EoS, but the impact on actualobservables is small: A recent analysis of systematic uncertainties, usingthe HotQCD and Wuppertal-Budapest equations of state in hydrodynamiccalculations, found ∼
1% differences in mean p T and ∼ v and v [194].The HRG trace anomaly may be computed from the energy density andpressure e = X sp g Z d p (2 π ) E f ( p ) , P = X sp g Z d p (2 π ) p E f ( p ) , (3.34)where the sums run over all species in the hadron gas; g and f are thedegeneracy and distribution function for each species. See the next sectionfor details on the hadron gas composition and particle distribution functions,which incorporate the effects of finite resonance width. HAPTER 3. COMPUTATIONAL MODELS
100 200 300 400
Temperature [MeV] ( e − P ) / T ← Connection range Trace anomaly
100 200 300 400
Temperature [MeV] c s Speed of sound
HRGLatticeHybrid
Figure 3.9
Trace anomaly and speed of sound for the hybrid equation of statecalculated using the HotQCD lattice EoS at high temperature and the HRG EoSat low temperature.
Procedure for constructing the hybrid EoS
1. Compute the HRG trace anomaly at an array of temperature points upto 165 MeV. This somewhat high temperature (above T c ) is necessaryto ensure continuity of the EoS across the switch from hydrodynamicsto Boltzmann transport—the EoS must exactly match the HRG cal-culation up to at least the maximum switching temperature, 165 MeVin the present work.2. Compute the lattice trace anomaly using equation (3.33) at an arrayof temperature points starting at 200 MeV.3. Connect the two curves between 165 and 200 MeV using a Kroghpolynomial, which ensures continuity of the functions and their firstseveral derivatives.4. Interpolate the trace anomaly across the full temperature range witha cubic spline.5. Numerically integrate the interpolating spline to obtain the pressure: P ( T ) T = P T + Z TT dT Θ µµ T , (3.35)with reference pressure P given by the HRG model at T = 50 MeV. HAPTER 3. COMPUTATIONAL MODELS s = ( e + P ) /T , and speed of sound c s = ∂P/∂e follow immediately.Figure 3.9 shows the result. An implementation of this procedure is includedwith my version of the OSU hydrodynamics code. While hydrodynamics excels at modeling the high-temperature QGP, mi-croscopic Boltzmann transport models are superior for the low-temperaturehadron gas (I will justify this claim in the next section). To switch to a mi-croscopic model, the continuous hydrodynamic medium must be convertedinto an ensemble of discrete particles. This process, “particlization”, is amodeling artifact, distinct from the physical processes of hadronization andfreeze-out, which is why such a neologism is necessary [195]. The physicalsystem is the same before and after particlization; only the modeled rep-resentation changes. In principle, there is a temperature window near theQCD crossover transition in which both hydrodynamics and microscopictransport are valid descriptions of the system, and it would be reasonableto particlize anywhere within this window.
Particlization is performed on an isothermal spacetime hypersurface definedby a switching temperature T switch , a variable model parameter presumablyclose to T c . This four-dimensional surface encloses the spacetime regionwhere T > T switch , which is modeled by hydrodynamics, and excludes theregion where
T < T switch , modeled by transport. On the switching surface,particles are emitted with momentum distributions given by the Cooper-Frye formula [196]
E dNd p = g (2 π ) Z σ f ( p ) p µ d σ µ , (3.36)where the integral runs over the surface σ ; the integration element d σ µ isa volume element of the four-dimensional surface whose magnitude is itssize and direction is normal to the surface. In thermal equilibrium, thedistribution function is a Bose-Einstein or Fermi-Dirac distribution f ( p ) = 1exp( p · u/T ) ∓ , (3.37) HAPTER 3. COMPUTATIONAL MODELS u is the velocity of the volume element; p · u is the energy, in the labframe, of a particle with momentum p in the frame of the volume element;and the upper sign corresponds to bosons, the lower to fermions. Rearrang-ing terms, it becomes apparent that the integrated yield is the total particleflux through the surface: N = Z σ d σ µ Z g d p (2 π ) f ( p ) p µ E , (3.38)where the inner integral is effectively a particle four-current [195]. In thesimple case of a single stationary volume element with zero normal vector, d σ µ = ( V, ), this reduces to something quite reasonable: N = V Z g d p (2 π ) f ( p ) = V n, (3.39)i.e. the product of the volume and the particle density.In computational models, the Cooper-Frye integral is replaced by a sumover discrete volume elements,
E dNd p = g (2 π ) X σ f ( p ) p µ ∆ σ µ , (3.40)with the elements and their normal vectors computed by a surface findingalgorithm such as Cornelius [195]. To produce an ensemble of particles,momenta are randomly sampled from this function by treating it as a prob-ability distribution. In doing so, it is standard practice to discard particleswith p µ ∆ σ µ <
0, meaning they are moving back into the hydrodynamicregion; this is a physical effect but is difficult to model realistically. Notealso that the Cooper-Frye formula provides the average number of emittedparticles, which is in general not an integer. A convenient way to convertthe average to a discrete number of particles is to interpret it as the meanof a Poisson distribution.I specify the complete sampling algorithm in subsection 3.4.4, after ad-dressing some other relevant aspects of particlization.
Particlization models have traditionally neglected the width of resonances,instead assigning every sampled resonance its pole mass. But it has beenknown for some time that accounting for finite width leads to increasedpion production, especially at low p T [197], and a recent detailed study ofthe ρ (770) resonance width confirmed the effect [198]. HAPTER 3. COMPUTATIONAL MODELS m is n = g Z d p (2 π ) f ( m , p ) , f ( m , p ) = 1 e √ m + p /T ± , (3.41)but if the particle is a resonance of finite width, then its mass probabilitydistribution P ( m ) must be integrated out of the distribution function: f ( p ) = Z dm P ( m ) f ( m, p ) . (3.42)Since mass is exponentially (rather than linearly) suppressed, the part ofthe distribution below m is enhanced more than the part above m issuppressed. The upshot is increased production of low-mass states relativeto high-mass, and depending on the specific form of P ( m ), a net change inoverall production.Given the precision goals of the present work, resonance width is tooimportant to neglect; therefore, I randomly sample the masses of all (severalhundred) resonances during particlization and allow the transport model tocalculate their scatterings and decays as part of the full ensemble of particles.I also account for finite width when calculating the HRG equation of state,as described in the previous section. For the mass distribution, I assume aBreit-Wigner distribution with a mass-dependent width: P ( m ) ∝ Γ( m )( m − m ) + Γ( m ) / , Γ( m ) = Γ s m − m min m − m min , (3.43)where m and Γ are the resonance’s Breit-Wigner mass and width, thethreshold mass m min is the total mass of the lightest decay products (e.g. m min = 2 m π for a resonance that can decay into a pair of pions), and themass-dependent width Γ( m ) is designed to be physically reasonable andsatisfy the constraints that Γ( m min ) = 0 and Γ( m ) = Γ . The distributionis normalized so that Z m max m min dm P ( m ) = 1 , m max = m + 4Γ . (3.44)Figure 3.10 shows the mass distributions for several resonances and theimpact on their densities. The ρ (770) and N (1535) have roughly symmetricmass distributions and their densities significantly increase, especially at lowmomentum. This is the general behavior of most species, but a minority ofresonances, such as the ∆(1232), have their pole mass not far above their HAPTER 3. COMPUTATIONAL MODELS m [GeV] P r o b a b ili t y ρ (770) ∆(1232) N (1535) p [GeV] p f ( p ) [ a r b . un i t s ] ↑ × ↑ × T = 150 MeV ρ (770) : +12% ∆(1232) : −9% N (1535) : +16%Finite widthZero width Figure 3.10
Left: Mass distributions for the ρ (770), ∆(1232), and N (1535) res-onances from equation (3.43). Right: Density distributions p f ( p ) including finitewidth (solid) and for zero width (dashed) at T = 150 MeV. Colors are the sameas on the left. The ∆(1232) and N (1535) distributions are scaled for visibility.Annotated are the relative changes in the particle densities from including finitewidth. threshold mass, leading to asymmetric mass distributions with more weightabove the pole mass, which reduces their total density.The Breit-Wigner distribution (3.43) is a simplifying assumption and isnot accurate for all resonances. But it is certainly closer to reality thanassigning every resonance its pole mass, and the chosen mass-dependentwidth ensures that the distribution is physically reasonable.It is difficult to predict the net effect of including finite resonance widthon stable hadron yields, spectra, and other observables. One likely conse-quence: Given the effects observed in figure 3.10, and the fact that nearlyall resonances decay to at least one pion, we can expect increased pion pro-duction relative to other species, especially at low p T . The f (500)The f (500) or σ meson is an unusual resonance with a controversial history[199]. It has an exceptionally small mass and large width, m = 475 ± = 550 ±
150 MeV in the 2017 Review of Particle Physics from theParticle Data Group (PDG) [29]. Since it is so light, it should be thermallyproduced in large quantities, and because it decays into pion pairs, shouldcontribute significantly to the total pion multiplicity.
HAPTER 3. COMPUTATIONAL MODELS
74I include the f (500) in the particlization routine, applying the sameBreit-Wigner distribution with mass-dependent width as for all other res-onances. This is not formally correct—the f (500) is known not to be aBreit-Wigner resonance—but it’s preferable to neglecting the resonance orusing only its pole mass. Note that, with the chosen mass-dependent widthand threshold mass m min = 2 m π ≈
280 MeV, the mass distribution is not atypical Breit-Wigner peak, but a highly asymmetric distribution with a longhigh-mass tail (like the ∆(1232) distribution in figure 3.10 but even moreextreme).Another issue is that the f (500) is unknown to many Boltzmann trans-port models, including the one used in this work. To circumvent this, Idecay each produced f (500) into a pion pair before initializing the trans-port model. This is physically justifiable since the resonance has such ashort mean lifetime: about 10 − seconds, or one-third fm /c , which is quickeven on the timescale of heavy-ion collisions.This scheme, while admittedly crude, should capture the basic physicsof producing some pions that would otherwise be missing. For the system to be physically self-consistent, the energy-momentum ten-sor T µν must be continuous across the transition from hydrodynamics toBoltzmann transport. After particlization, kinetic theory gives (assuming anoninteracting hadron gas) T µν = X sp g Z d p (2 π ) p µ p ν E f ( p ) , (3.45)where the sum runs over all species in the hadron gas; g and f are the degen-eracy and distribution function of each species. On the switching surface,the kinetic form must equal the hydrodynamic form T µν = e u µ u ν − ( P + Π)∆ µν + π µν , (3.46)in particular, the sampled particles must have the same energy density,thermal pressure, and viscous pressures as the hydrodynamic medium. Ex-amining the kinetic form (3.45), it is clear that the only way to achievecontinuity is to modify the distribution function f ( p ); if the hydrodynamicmedium is out of thermal equilibrium, so should be the system of particles.The standard modification to the distribution function is the additionof a small linear correction: f = f + δf , where f is the equilibrium Bose- HAPTER 3. COMPUTATIONAL MODELS δf = f (1 ± f ) τET (cid:20) η p i p j π ij + 1 ζ (cid:18) p − c s E (cid:19) Π (cid:21) , (3.47)where π ij is the shear tensor in the fluid rest frame and τ is a constant shearand bulk relaxation time for all species which gives the RTA its name. Butlinear corrections break down for large viscous pressure and/or momentum;eventually, δf dominates the equilibrium distribution, invalidating the as-sumption of a “small” correction and sometimes causing unphysical negativedensities ( f + δ f < p i → p i = p i + X j λ ij p j , λ ij = ( λ shear ) ij + λ bulk δ ij , (3.48)where λ ij is a linear transformation matrix, consisting of a traceless shearpart and a bulk part proportional to the identity matrix, chosen to satisfycontinuity of T µν [201]. This procedure lends itself naturally to computa-tional particlization models: Simply sample momentum vectors from theequilibrium distribution and then apply the transformation. I have adoptedthis general approach in this work. Shear corrections
In the limit of small shear pressure, the shear transformation is [201]( λ shear ) ij = τ η π ij , (3.49)where π ij is again the (spatial) shear pressure tensor in the local rest frame,and the ratio of the shear viscosity to the relaxation time in the noninter-acting hadron gas model is ητ = 115 T X sp g Z d p (2 π ) p E f (1 ± f ) . (3.50) Per convention, Latin indices are purely spatial; vectors and tensors like p i and π ij represent only spatial components. The temporal components of π µν are zero in the restframe, as required by orthogonality to the flow velocity, π µν u ν = 0, which together with u ν = (1 , ) in the rest frame implies π ν = 0. HAPTER 3. COMPUTATIONAL MODELS a /P (input) −0.10.00.10.20.30.40.5 O u t pu t s Shear only a /P Π /P ∆ e/e a /P a /P (input) −0.10.00.10.20.30.40.5 T = 150 MeVAll PDG resonances Shear and bulk
Figure 3.11
Test of the viscous correction method. The input value of a , definedin equation (3.52), is varied relative to the equilibrium pressure P and the followingoutput quantities are checked: a /P itself, the bulk pressure Π /P , the change inthe energy density ∆ e/e , and a /P defined in equation (3.54). Colored circlesare the test calculations and lines are the targets. Left: zero bulk pressure, right:Π = − . P . Inserting the transformed momentum vector p into the equilibrium distri-bution and expanding for small λ yields f ( p ) − f ( p ) ≈ f (1 ± f ) τET η p i p j π ij , (3.51)which is the same as the shear part of δf in equation (3.47) above, hence,this ansatz is equivalent to the δf correction for small shear pressure.If this procedure works as intended, then given an input π ij , the resultingsampled particles should actually have the specified π ij . In this vein, I havereproduced the test performed in [201], with the input π ij defined by a = π xx = − π yy = T xx − T yy , (3.52)and all other components set to zero. For each value of a , I sample a largenumber of thermal particles, transform their momentum vectors by ( λ shear ) ij as given in equation (3.49), and compute the energy-momentum tensor T µν = 1 V X parts p µ p ν E , (3.53)
HAPTER 3. COMPUTATIONAL MODELS V is the volume of thethermal source. From this, I calculate the output a (which should equalthe input), as well as the quantity a = 2 T zz − T xx − T yy √ , (3.54)which should be zero, and the energy density and pressure e = T tt = 1 V X parts E, P = T xx + T yy + T zz V X parts p E , (3.55)which should not deviate from their equilibrium values e and P . In general,the pressure may deviate, the difference being the bulk pressure Π, but thebulk pressure is zero for this test. The left panel of figure 3.11 shows theresults; a is reproduced faithfully, with some small deviations in the otherquantities for large a relative to the equilibrium pressure. This is expectedsince the procedure was derived in the limit of small shear pressure. Notethat the δf correction would also induce deviations for large shear pressurebecause, as mentioned, it sometimes causes negative densities which areimpossible to sample. The right panel of the figure is the same test butwith nonzero bulk pressure, which requires a separate correction that I willdescribe now. Bulk corrections
The form of the bulk transformation in equation (3.48) is λ bulk δ ij , whichtranslates to an overall scaling of the momentum: p = (1 + λ bulk ) p . Asrationale, consider that the total effective pressure of the system is the sumof the thermal pressure P and bulk pressure Π—as can be seen by how theyenter the hydrodynamic energy-momentum tensor (3.46)—and the total ki-netic pressure is P + Π = X sp g Z d p (2 π ) p E f ( p ) . (3.56)This relation may be satisfied for a given bulk pressure by replacing f ( p ) → f ( p ) = f ( p + λ bulk p ) and adjusting λ bulk . However, doing so would alsochange the energy density e = X sp g Z d p (2 π ) E f ( p ) , (3.57) HAPTER 3. COMPUTATIONAL MODELS z bulk , so thatthe complete replacement is f ( p ) → z bulk f ( p + λ bulk p ). The parameters λ bulk and z bulk together account for bulk corrections and are uniquely determinedby the requirement that the total pressure is reproduced without changingthe energy density.This parametric method, which I devised for use in computational par-ticlization routines, is not an approximation, but it does rely on some as-sumptions, namely: It modifies the momentum distributions in a simple way,only by scaling the magnitude of momentum; and it scales the density ofall particle species by the same factor, maintaining their equilibrium ratios.The physical interpretation is that bulk pressure implies a change in themomentum density of the system, and to conserve energy, the particle den-sity must be adjusted accordingly. Recall that the Navier-Stokes equationfor bulk viscosity is Π = − ζ ∇ · u , where ∇ · u is the fluid expansion rate; ifthe fluid is radially expanding (as is often the case in heavy-ion collisions),bulk viscosity acts as a kinetic “brake”, reducing the momentum of emit-ted particles and converting that kinetic energy into increased productionof low-momentum particles.The left panel of figure 3.12 verifies that the parametric method re-produces bulk pressure while preserving the energy density and shows thecorresponding modifications to the particle density and mean momentum,which are closely related to the parameters z bulk and λ bulk . The methodis accurate all the way down to Π = − P , meaning zero total pressure, atwhich point particles have zero momentum and all their energy is rest mass(this may not make much physical sense, but it works fine numerically).For large positive bulk pressure, the mean momentum diverges and the cor-rections must be truncated, which is why everything becomes flat aboveΠ /P ∼ .
7. This has negligible impact on heavy-ion collisions since veryfew volume elements have such large positive bulk pressure.The right panel of the figure compares the modified distribution functionfrom the parametric method and the RTA δf , equation (3.47). Calculating δf requires the ratio of the bulk viscosity to the relaxation time ζτ = 13 T X sp g Z d p (2 π ) m E (cid:18) c s E − p (cid:19) f (1 ± f ) (3.58)and the speed of sound c s = ∂P∂e = ∂P/∂T∂e/∂T = 13 P sp g R d p p f (1 ± f ) P sp g R d p E f (1 ± f ) . (3.59) HAPTER 3. COMPUTATIONAL MODELS −1.0 −0.5 0.0 0.5 1.0 Π /P −1012 R e l a t i v e c h a n g e ∆ n/n ∆ › p fi / › p fi Π /P ∆ e/e T = 150 MeVAll PDG resonances p [GeV] -8 -6 -4 -2 f ( p ) Π = 0 − . P − . P Pions, T = 150 MeVParametricRTA Figure 3.12
Left: Effect of bulk pressure on thermodynamic quantities and ver-ification of the parametric method. Shown are the changes in density, energy den-sity, pressure, and mean momentum as a function of bulk pressure relative to theequilibrium pressure. Colored circles are test calculations from sampled particlesand lines are the targets. Right: Effect of bulk pressure on the pion distributionfunction from the parametric method and the RTA.
Both methods generally decrease momentum with negative bulk pressure,but, importantly, the RTA distribution function goes (unphysically) negativefor even moderately large bulk pressure and momentum.Returning briefly to the previous figure 3.11: The right panel is a testof the complete viscous correction method with Π = − . P and variableshear pressure. As in the left panel, which has zero bulk pressure, there aresome deviations for large a input, but this is caused by the approximateshear correction method, not the parametric bulk method. I have developed a new computational particlization model, available at https://github.com/Duke-QCD/frzout , with online code documentationincluding additional information and numerical tests. The following sum-marizes the sampling algorithm.
Preliminary steps
1. Choose a list of hadron species consistent with the Boltzmann trans-port model.
HAPTER 3. COMPUTATIONAL MODELS η/τ ; for bulk,construct cubic interpolating splines that map Π to the parameters λ bulk and z bulk . These parameters are determined by the system ofequations P + Π = z bulk X sp g Z d p (2 π ) p E f ( p + λ bulk p ) ,e = z bulk X sp g Z d p (2 π ) E f ( p + λ bulk p ) , (3.60)which can be inverted numerically, but it would be too slow to do so forevery volume element, which would be necessary since each element ingeneral has a different bulk pressure. Steps to create the interpolatingsplines:(a) Compute the equilibrium particle density n , energy density e ,and pressure P .(b) For an array of λ bulk values from − n , energy density e , and pressure P . The z bulk necessary to preserve the equilibrium energy density is z bulk = h E i h E i = e /n e/n , (3.61)where h E i is the average energy per particle, and the resultingbulk pressure is Π = P e e − P . (3.62)(c) Interpolate the data points using Π as the input variable and thebulk parameters as the outputs. The interpolating functions canthen be evaluated quickly during the main sampling steps. Main sampling steps
Scott Pratt originally devised this algorithm [201]; I wrote new code imple-menting it and made some minor modifications.
HAPTER 3. COMPUTATIONAL MODELS σ µ is h dN i = p · ∆ σE d p (2 π ) g f ( p ) = p · ∆ σp · u d p (2 π ) g f ( p ) , (3.63)where in the second form p is the momentum in the rest frame of the volumeelement. Now multiplying and dividing by a volume V , this becomes h dN i = w ( p ) V d p (2 π ) g f ( p ) , w ( p ) = 1 V p · ∆ σp · u , (3.64)where w ( p ) is a particle emission probability. The volume V ensures w ( p ) ≤
1; its optimal value is V = max (cid:18) p · ∆ σp · u (cid:19) = u · ∆ σ + q ( u · ∆ σ ) − (∆ σ ) . (3.65)In view of these relations, the sampling algorithm is:1. Sample a particle four-momentum from a stationary thermal source ofvolume V . If the particle is a resonance, sample its mass in additionto the three-momentum.2. Apply the viscous correction transformation.3. Boost the momentum from the rest frame of the volume element, i.e.an inverse boost by four-velocity u .4. If p · ∆ σ <
0, reject the particle, otherwise accept the particle withprobability w ( p ).This process should be repeated for each volume element and each species.An efficient algorithm for achieving Poissonian particle production is:1. Initialize a variable S with the negative of an exponential randomnumber. Such a random number can be generated as S = log( U ),where U ∈ (0 ,
1] is a uniform random number.2. For each particle species in the hadron gas:(a) Add
V n to S , where n is the density of the species, so V n is theaverage number emitted from the volume. If the volume elementhas nonzero bulk pressure, determine the parameter z bulk andscale the density. HAPTER 3. COMPUTATIONAL MODELS
S <
0, continue to the next species, otherwise perform theabove sampling algorithm and then subtract an exponential ran-dom number from S . Continue sampling particles and subtract-ing from S until it again goes negative, then continue to the nextspecies.3. Repeat for each volume element.This works because the time between Poisson events has an exponentialdistribution.In boost-invariant hydrodynamics, the volume elements are ∆ σ µ = τ ∆ y ∆ σ µ , where τ is the proper time of the element and ∆ y is a rapid-ity range which must be chosen a priori . After accepting a particle in theabove algorithm, its longitudinal momentum only determines the differencebetween the spacetime and momentum rapidity y − η s = 12 log (cid:18) E + p z E − p z (cid:19) , (3.66)so some additional steps are required:1. Sample a momentum rapidity y uniformly in the range ∆ y . Boost theparticle’s momentum vector longitudinally so that it has rapidity y .2. From y and the difference y − η s , calculate the spacetime rapidity η s . Boost the particle’s position vector longitudinally so that it hasrapidity η s , namely t = τ cosh η s and z = τ sinh η s .There are many further subtleties which I omit here for brevity. See thecode documentation and comments for details. After particlization, a Boltzmann transport model simulates the microscopicdynamics of the hadronic system, including scatterings and decays, untilfreeze-out. As the name suggests, such models solve the Boltzmann equation df i ( x, p ) dt = C i ( x, p ) , (3.67)which stipulates that the time evolution of the distribution function f i forspecies i is driven by the collision kernel, or source term, C i , which accounts HAPTER 3. COMPUTATIONAL MODELS i , including collisions with other species, sothat the equations for each species are in general coupled.The most widely used implementation of Boltzmann transport, and thepresent choice, is UrQMD (Ultra-relativistic Quantum Molecular Dynam-ics) [202, 203]. UrQMD effectively solves the Boltzmann equation by prop-agating particles along classical (straight-line) trajectories, sampling theirstochastic binary collisions, and calculating resonance formation and de-cays. My version of UrQMD, tailored for use as a hadronic afterburner, i.e.as part of a multistage model following hydrodynamics and particlization,is available at https://github.com/jbernhard/urqmd-afterburner .There are other Boltzmann transport implementations, but since thephysics of hadronic scatterings and decays is well-understood, the priority isto use a stable, established code with a comprehensive set of hadronic reso-nances, which UrQMD satisfies. A more recent model, SMASH (SimulatingMany Accelerated Strongly-interacting Hadrons) [204], may ultimately re-place UrQMD as the standard Boltzmann transport model for heavy-ioncollisions, but at the time of this writing is not ready for production use. Microscopic transport models like UrQMD are ideal for modeling the late,hadronic stage of heavy-ion collisions. There is no assumption of thermalequilibrium, the feed down of resonances to stable hadrons is calculatedrealistically, and the various stages of freeze-out arise naturally from the mi-croscopic dynamics. Chemical freeze-out may occur earlier and at a highertemperature than kinetic freeze-out, as is generally understood to happen inreal collisions (see subsection 2.1.2). Different species may kinetically freeze-out separately, for example because they have different scattering cross sec-tions.These models also innately account for hadronic transport properties,obviating the need to specify transport coefficients such as η/s and ζ/s . Infact, the only free parameter is T switch , the particlization temperature. In the interest of computational tractability, the collision kernel usually in-cludes only binary collisions and 2 → n processes; hence, microscopic trans-port is a valid description of the system provided it is sufficiently dilute thatbinary scatterings are the dominant process and higher-order scatterings arerare. Hence, the system must have particle degrees of freedom and cannot HAPTER 3. COMPUTATIONAL MODELS
The final step in the modeling workflow is to compute observables, suchas multiplicities and anisotropic flow coefficients, for comparison with ex-perimental observations. I strive to replicate experimental data analysismethods as closely as possible.
I run minimum-bias events (no centrality or impact parameter cuts), sort theevents by charged-particle multiplicity dN ch /dη at midrapidity ( | η | < . dN ch /dη is not exactly the same asmost experiments, e.g. ALICE defines centrality by the energy deposited inits VZERO detectors [35], which are not at midrapidity. But this should notmake much difference, since these measures of particle or energy productionare strongly correlated. In any case, since the present hydrodynamic modelis boost-invariant, quantities away from midrapidity are fairly meaningless. After dividing the events into centrality bins, I compute observables from theparticle data output by the final stage of the model (Boltzmann transport);these virtual particles are analogous to their real counterparts recorded by anexperimental detector. I calculate quantities such as particle yields dN ch /dη and dN/dy , transverse energy E T , and mean transverse momentum h p T i bystraightforward counting and averaging; anisotropic flow cumulants v n { k } by the Q -cumulant method [60] (see discussion on page 27).It is always crucial to apply the same kinematic cuts as the experimentaldetector, for example ALICE measures flow cumulants using charged parti-cles with | η | < . . < p T < . | η | < . | y | < .
5, and extrapolated to zero p T [16], so no p T cut is necessary whencomputing them from the model. HAPTER 3. COMPUTATIONAL MODELS How many events should one generate with the model? It depends on theinherent statistical fluctuations in the desired observables: Yields and mean p T converge quickly; flow cumulants are noisier and therefore require moreevents to stabilize. I have found that ∼ × minimum-bias events achievesacceptable statistical noise for two-particle flow cumulants in 10% centralitybins. Four-particle cumulants need more—at least 10 . Since the hydrodynamic model usually takes much more time than the subse-quent particlization and Boltzmann transport models, it is standard practiceto run the particlization+transport combination multiple times per hydro-dynamic evolution. All particle data are then merged and used to computelow-noise observables.This strategy, known as “oversampling” in reference to sampling theCooper-Frye switching hypersurface, is advantageous because single eventsdon’t naturally produce enough particles to accurately measure their prop-erties, and by sampling each event several times, more information can beextracted—without incurring much more computational cost. Averagingover multiple samples certainly suppresses some event-by-event fluctuations,so one must take care that the observables of interest are not sensitive tothese fluctuations.To achieve a consistent statistical noise level across all events, I oversam-ple each event until a target number of particles are emitted, which generallymeans more samples for peripheral events than central. This is preferableto running a fixed number of samples, for then one would have to choose be-tween wasting time running too many samples for central events, or havingtoo few samples for peripheral events.
I have developed a workflow for generating large quantities of heavy-ion col-lision events, available at https://github.com/Duke-QCD/hic-eventgen .It runs the five modules described in this chapter, implements the consid-erations just mentioned in this section, and provides utilities for runningon high-performance computational systems, specifically the Open ScienceGrid (OSG) and the National Energy Research Scientific Computing Center(NERSC). See the online documentation for details.
Bayesian parameter estimation N ot coincidentally, the present situation conforms to the “generic setup”of the introduction (chapter 1): We have assorted experimental obser-vations of heavy-ion collisions (section 2.1), some related properties of QCDmatter that we wish to quantify (section 2.2), and a computational colli-sion model which takes those properties as input parameters and producessimulated observables analogous to the experimental data (chapter 3).In order to rigorously quantify the model parameters—and further, toclaim that they connect to genuine physical properties—the model must be areasonable representation of real collisions, evidenced by a global fit to a widevariety of observables. Complicating this endeavor is that each parameteris linked to multiple model observables, and vice versa; for example, thespecific shear viscosity η/s affects the anisotropic flow coefficients v n , butso too do the initial collision geometry and free-streaming time, which inturn also influence the transverse momentum distributions. In general, it issafe to assume that all parameters affect every observable to some extent.Undoubtedly, the only path to a global fit is a simultaneous treatment of allparameters and observables.Putting aside how to achieve such a fit, it is essential to realize thatparameters determined in this way are inherently uncertain. Notable—andunavoidable—sources of uncertainty include measurement errors in the ex-perimental data itself, the complex interplay among model parameters, anddiscrepancies between the model calculations and the data. Thus, the ob-jective is a quantitative estimate of each parameter, including the associateduncertainties.Bayesian statistics offers a natural framework for parameter estimationand uncertainty quantification, wherein the final result is a posterior prob-86 HAPTER 4. BAYESIAN PARAMETER ESTIMATION x = ( x , x , . . . , x n ) and denote the experimental data vector by y .We then define the prior P ( x ), a probability distribution encoding our initialknowledge of the parameters, and the likelihood P ( y | x ), a conditional prob-ability that quantifies the quality of the fit to data, accounting for all sourcesof uncertainty, given the parameters x . Next, we apply Bayes’ theorem toobtain the posterior distribution P ( x | y ) ∝ P ( y | x ) P ( x ) , (4.1)which encapsulates all our knowledge of the parameters given the prior andthe data. Usually, we are interested in the marginal distributions for eachparameter, calculated by marginalizing over (integrating out) all the rest,for example the marginal distribution for x is P ( x | y ) = Z dx · · · dx n P ( x | y ) . (4.2)From this, we can derive the desired estimate and uncertainty of x .In a prominent application of this methodology, the Laser Interferom-eter Gravitational-Wave Observatory (LIGO) Scientific Collaboration hasestimated properties of binary black hole and neutron star mergers fromgravitational wave observations [205–208]. Matching numerical relativitycalculations to the observed gravitational waveforms, they extracted modelparameters including the masses and spins of the progenitor objects andthe final object. Figure 4.1 shows the posterior distributions for the source-frame black hole masses in merger event GW150914 [207, 209], from whichthey derived m source1 /M (cid:12) = 36 +5 − and m source2 /M (cid:12) = 29 +4 − ( M (cid:12) is the solarmass), where the reported values are the posterior medians and the un-certainties are 90% credible intervals. This means that, e.g., 90% of theposterior density lies between 32 < m source1 /M (cid:12) <
41; based on all the avail-able information, there is a 90% chance that the true value of m source1 lieswithin this range.The figure also shows the joint probability distribution between the twomasses, obtained from a marginalization integral similar to (4.2), but inte-grating out all but two parameters, instead of all but one. From this visual-ization, we see that the estimates of the two masses are strongly correlated:Large m implies small m , and vice versa. This suggests that the total massis better constrained than the individual masses, and indeed, the reportedtotal is M source /M (cid:12) = 65 +5 − , which has less relative uncertainty than m HAPTER 4. BAYESIAN PARAMETER ESTIMATION entering our sensitive band [85,86] and could not haveformed from an asymptotically spin antialigned binary.We could exclude those systems if we believe the binary isnot precessing. However, we do not make this assumptionhere and instead accept that the models can only extractlimited spin information about a more general, precessingbinary.We also need to specify the prior ranges for the ampli-tude and phase error functions δ A k ð f ; ~ ϑ Þ and δϕ k ð f ; ~ ϑ Þ , seeEq. (5). The calibration during the time of observation ofGW150914 is characterized by a - σ statistical uncertaintyof no more than 10% in amplitude and 10° in phase [1,47].We use zero-mean Gaussian priors on the values of thespline at each node with widths corresponding to theuncertainties quoted above [48]. Calibration uncertaintiestherefore add 10 parameters per instrument to the modelused in the analysis. For validation purposes we alsoconsidered an independent method that assumes frequency-independent calibration errors [87], and obtained consistentresults. III. RESULTS
The results of the analysis using binary coalescencewaveforms are posterior PDFs for the parameters describ-ing the GW signal and the model evidence. A summary isprovided in Table I. For the model evidence, we quote(the logarithm of) the Bayes factor B s=n ¼ Z = Z n , whichis the evidence for a coherent signal hypothesis dividedby that for (Gaussian) noise [5]. At the leading order, theBayes factor and the optimal SNR ρ ¼ ½ P k h h Mk j h Mk i(cid:4) = arerelated by ln B s=n ≈ ρ = [88].Before discussing parameter estimates in detail, weconsider how the inference is affected by the choice ofthe compact-binary waveform model. From Table I, we seethat the posterior estimates for each parameter are broadlyconsistent across the two models, despite the fact thatthey are based on different analytical approaches and thatthey include different aspects of BBH spin dynamics. Themodels ’ logarithms of the Bayes factors, . (cid:2) . and . (cid:2) . , are also comparable for both models: the datado not allow us to conclusively prefer one model over theother [89]. Therefore, we use both for the Overall columnin Table I. We combine the posterior samples of bothdistributions with equal weight, in effect marginalizingover our choice of waveform model. These averaged resultsgive our best estimate for the parameters describingGW150914.In Table I, we also indicate how sensitive our results areto our choice of waveform. For each parameter, we givesystematic errors on the boundaries of the 90% credibleintervals due to the uncertainty in the waveform modelsconsidered in the analysis; the quoted values are the 90%range of a normal distribution estimated from the varianceof results from the different models. (If X were an edge of a credible interval, we quote systematic uncertainty (cid:2) . σ sys using the estimate σ sys ¼ ½ð X EOBNR − X Overall Þ þð X IMRPhenom − X Overall Þ (cid:4) = . For parameters with boundedranges, like the spins, the normal distributions shouldbe truncated. However, for transparency, we still quotethe 90% range of the uncut distributions. These numbersprovide estimates of the order of magnitude of the potentialsystematic error). Assuming a normally distributed error isthe least constraining choice [90] and gives a conservativeestimate. The uncertainty from waveform modeling is lesssignificant than the statistical uncertainty; therefore, we areconfident that the results are robust against this potentialsystematic error. We consider this point in detail later in theLetter.The analysis presented here yields an optimal coherentSNR of ρ ¼ . þ . − . . This value is higher than the onereported by the search [1,3] because it is obtained using afiner sampling of (a larger) parameter space.GW150914 ’ s source corresponds to a stellar-mass BBHwith individual source-frame masses m source ¼ þ − M ⊙ and m source ¼ þ − M ⊙ , as shown in Table I and Fig. 1.The two BHs are nearly equal mass. We bound the massratio to the range . ≤ q ≤ with 90% probability. Forcomparison, the highest observed neutron star mass is . (cid:2) . M ⊙ [91], and the conservative upper-limit for FIG. 1. Posterior PDFs for the source-frame component masses m source and m source . We use the convention that m source ≤ m source ,which produces the sharp cut in the two-dimensional distribution.In the one-dimensional marginalized distributions we show theOverall (solid black), IMRPhenom (blue), and EOBNR (red)PDFs; the dashed vertical lines mark the 90% credible intervalfor the Overall PDF. The two-dimensional plot shows thecontours of the 50% and 90% credible regions plotted over acolor-coded PDF. PRL week ending17 JUNE 2016
Figure 4.1
Posterior distributions for the source-framecomponent masses of black hole merger GW150914 [207,209]. The one-dimensional histograms are marginal distri-butions for the masses, where the colored lines correspondto different waveform models and the black line is the over-all (average) result, and the dashed lines indicate the 90%credible interval. The two-dimensional density plot is thejoint distribution between the two masses with credibleregion contours (the sharp cut is due to the convention m source2 ≤ m source1 ). and m . In effect, the ambiguity in the mass apportionment contributes tothe mutual uncertainty of both parameters. If we later determined that m is toward the lower end of its credible interval, we would then believe that m is on the large side. This mutual uncertainty is a typical characteristicof correlated parameter estimates.The one-dimensional marginal distributions in figure 4.1 are histograms,not smooth curves, because they were not actually obtained from directcalculation of marginalization integrals like equation (4.2). In general, it ismore computationally efficient and convenient to generate a large sampleof the posterior distribution through Markov chain Monte Carlo (MCMC) HAPTER 4. BAYESIAN PARAMETER ESTIMATION { x i } , with each x i = ( x i , x i , . . . , x ni ), thenthe sample of x is just { x i } .Generating every parameter sample x i entails a model evaluation—a seri-ous obstacle if the model is computationally expensive. This is certainly thecase for heavy-ion collisions: Calculating the centrality dependence of bulkobservables requires O (10 ) minimum-bias events, and at O (10 − ) hours perevent, this works out to O (10 ) hours per parameter sample. Assuming astatistically significant sample size O (10 ), the total computation time wouldbe O (10 ) hours, which is out of the question—even the largest NERSC al-locations provide “only” O (10 ) hours.To circumvent this obstacle, we use a model emulator to predict theoutput of the full model in much less time than an explicit calculation. Thestrategy, developed specifically for this type of scenario [210–213], proceedsby evaluating the model at a relatively small O (10 ) number of points inparameter space, training an emulator on the model input-output data, andthen using the emulator as a fast surrogate to the full model during MCMCsampling. This reduces the computation time requirement by several ordersof magnitude, more than making up for the disparity.The canonical choice for model emulators are Gaussian processes [214],statistical objects that can non-parametrically interpolate multidimensionalfunctions. When carefully constructed, Gaussian processes are sufficientlyflexible to emulate a wide variety of models, and since they provide theuncertainty of their predictions, are ideal for parameter estimation withquantitative uncertainties.Bayesian parameter estimation using Gaussian process emulators hasbeen successfully deployed in heavy-ion physics [122, 139, 140], includingmy own previously published work [119, 171, 215, 216], and in numerousother fields, such as galaxy formation history [217].In this chapter, I fully develop the parameter estimation procedure, rep-resented graphically in figure 4.2. Sections 4.1 and 4.2 address the modelinputs and outputs, respectively; I elaborate on the choice of input param-eters, their distribution in parameter space, and postprocessing of modelcalculations for emulation. In section 4.3, I discuss the theory of Gaussianprocesses and the practical details of building model emulators. In section4.4, I expand upon model calibration, including MCMC sampling, construc-tion of the posterior distribution, and uncertainty quantification. Lastly, insection 4.5, I point out my computer code implementing Bayesian parameterestimation for heavy-ion collisions. HAPTER 4. BAYESIAN PARAMETER ESTIMATION Input parameters
QGP properties
Model
Heavy-ion collisionspacetime evolution
Gaussian process emulator
Surrogate model
Bayesian calibration
Infer model parametersfrom data
Posterior distribution
Quantitative estimatesof each parameter
Experimental data
Heavy-ion collisionobservables
Figure 4.2
Overview of the parameter estimation process.
The goal of this section is to choose n model input parameters for estimation, x = ( x , x , . . . , x n ), and d points in parameter space at which to evaluatethe full model, arranged into a d × n design matrix X = ( x , x , . . . , x d ).These choices will have important downstream consequences for uncertaintyquantification and the performance of the Gaussian process emulator. The adage “as much as necessary, as little as possible” is sometimes invokedregarding antibiotics, meaning that they should be used to treat bacterialillness, but avoided to prevent antibiotic resistance. Similar considerationsapply here, although the potential repercussions are, fortunately, much lessdire.Any parameter that might have a meaningful impact on the model calcu-lation should be included in the design. Physical properties certainly satisfythis criterion, but all parameters need not have a direct physical connection. I use d for the number of design points because m shall be the number of modeloutputs. Mnemonic: d → d esign points, m → m odel outputs. HAPTER 4. BAYESIAN PARAMETER ESTIMATION do care about. We saw in theLIGO example, figure 4.1, how parameters can contribute to their mutualuncertainties through marginalization. Fixing a parameter to a nominalvalue—even a model-dependent nuisance parameter—can artificially biasthe results for other parameters.At the same time, we should not get carried away introducing frivolousparameters. That is why I say a meaningful impact, though what is mean-ingful is of course subjective.Sometimes, we may not know whether a parameter will affect the model.When in doubt, it is usually better to include such parameters in the designrather than risk bias. The primary drawback of adding parameters is that,as we shall see in the next subsection, more parameters require more designpoints, which means more computation time.In summary: As many parameters as necessary, as few as possible. Having chosen a set of parameters to estimate, we must now decide thenumber of design points and their locations in parameter space. The guidingmotive is to create an efficient scaffolding of parameter space for emulationusing as few design points as possible.First, we specify ranges, i.e. minimum and maximum values, for eachparameter. Effectively, this imposes a prior distribution which is zero outsidethe design range, a very strong assumption. The ranges should thereforeenclose any possibly reasonable values, erring on the side of generosity ratherthan risking truncation.Sometimes, choosing the ranges is a somewhat paradoxical problem,where part of the reason for performing parameter estimation is to determinereasonable ranges. One strategy I have used in this case is first performinga coarse-grained “pilot study”, that is, running a wide design range withfew design points and low statistics. Based on the resulting low-precisionposterior distribution, adjust the parameter ranges as necessary and re-runwith normal precision.Now, how many design points, and where? Figure 4.3 shows three pos-sible strategies. Factorial design, in which points are placed on a uniformlattice, is an obvious choice in low dimensions, but fails in high dimensions.A factorial design of k points in each of n dimensions has k n total points—fartoo many even for a modest k = 10 and n > HAPTER 4. BAYESIAN PARAMETER ESTIMATION x x Factorial x Random x Latin hypercube
Figure 4.3
Examples of factorial, random, and maximin Latin hypercube designsin two dimensions. Each has 7 = 49 points. Another simple design is purely random points. This is certainly morereasonable than factorial design in high dimensions, but still suboptimal, be-cause there’s no guarantee that random points fill the space. Often, purelyrandom samples leave large regions with no points, which will preclude ac-curate emulation in those empty regions.A common method for generating semi-random, space-filling designs isLatin hypercube sampling [218, 219], in which points are generated in an n -dimensional unit hypercube, [0 , n (which can then be scaled to the desiredranges), such that if each dimension is divided into equal subintervals, thereis exactly one point in each subinterval (like a Sudoku grid). For example,four points in two dimensions could be distributed like this:Such designs provide the desired efficient scaffolding because they uniformlyfill the space with relatively few points; the required number of points growsonly linearly with the number of dimensions. As a rule of thumb, 10 pointsper dimension yields acceptable emulation accuracy [220], although more isalways better if computation time permits. I usually aim for at least 20 perdimension. HAPTER 4. BAYESIAN PARAMETER ESTIMATION maximizes the minimum distance between points.A numerical library for generating Latin hypercube samples is publiclyavailable [221]. I have used it in my work, but I have not contributed to it.Comparing the example designs in figure 4.3, we see that the maximinLatin hypercube fills the space much more uniformly than the random de-sign, with smaller gaps and no points on top of each other. Note that, intwo dimensions, the factorial design may actually be the best choice, but asexplained above, it is not a viable option in higher dimensions.One final note: It is sometimes desirable to nonlinearly transform aparameter so that it affects the model smoothly across its range. Like ashower hot water knob, we want linear behavior as we turn the virtual knobof each parameter. In particular, this will facilitate training the Gaussianprocess emulator.
I use a 500 point maximin Latin hypercube design, repeated for Pb-Pbcollisions at 2.76 and 5.02 TeV, for 1000 total design points.
Initial condition
1. Normalization factor for the initial density profile (different normal-ization for each beam energy).2. TRENTo entropy deposition parameter p defined in equation (3.4).With a free-streaming stage, the initial condition provides the trans-verse density of partons, parametrized as n = Norm × (cid:18) ˜ T pA + ˜ T pB (cid:19) /p , (4.3)where ˜ T is a participant thickness function.3. Gaussian nucleon width w of the nucleon thickness function T p ( x, y ) = 12 πw exp (cid:18) − x + y w (cid:19) . (4.4) HAPTER 4. BAYESIAN PARAMETER ESTIMATION σ fluct = 1 / √ k ,where k is the shape parameter of the gamma distribution, equation(3.9), reproduced here: P k ( u ) = k k Γ( k ) u k − e − ku . (4.5)The fluctuated participant thickness functions are˜ T A = N part,A X i =1 u i T p ( x − x i , y − y i ) , (4.6)where ( x i , y i ) is the transverse position of nucleon participant i innucleus A , and the u i are sampled from the gamma distribution. Iuse the standard deviation σ fluct , instead of k itself, because it is moreintuitive and allows setting k to very large values ( σ fluct → k → ∞ ),which effectively disables fluctuations.5. Minimum distance between nucleons d min (subsection 3.1.4), trans-formed to the volume d . Pre-equilibrium
6. Free-streaming time τ fs (section 3.2). QGP medium η/s min, slope, and curvature, which set the temperature dependenceof the QGP specific shear viscosity in equation (3.30), reproduced here:( η/s )( T ) = ( η/s ) min + ( η/s ) slope · ( T − T c ) · ( T /T c ) ( η/s ) crv (4.7)10. Constant value of η/s in the hadronic phase of the hydrodynamic model (see discussion on page 67).11–13. ζ/s max, width, and location ( T ), which set the temperature depen-dence of the QGP specific bulk viscosity in equation (3.31), reproducedhere: ( ζ/s )( T ) = ( ζ/s ) max (cid:18) T − ( ζ/s ) T ( ζ/s ) width ) (cid:19) . (4.8)14. Particlization temperature T switch (section 3.4). HAPTER 4. BAYESIAN PARAMETER ESTIMATION Generically, the computational model takes a vector of n input parameters x = ( x , x , . . . , x n ) and produces a vector of m outputs y = ( y , y , . . . , y m ).For a heavy-ion collision model, each of the outputs y i is an observable in aparticular centrality class or kinematic bin ( p T or η ). If the outputs includethe centrality and/or kinematic dependence of several observables, the totalnumber m quickly becomes quite large.As we shall see in the next section, Gaussian processes (which we willuse to interpolate the model output) are scalar functions, i.e. they map avector input to a single output. The naïve way to handle all the modeloutputs is to use m independent Gaussian processes, but this could be com-putationally expensive, and it ignores correlations among the outputs. Sincephysical models generally produce many highly-correlated outputs, this isunsatisfactory.Instead, we transform the model outputs into a smaller number of un-correlated variables using principal component analysis (PCA), then treateach new variable independently. PCA [222] is a general procedure that defines an orthogonal linear transfor-mation from a set of correlated variables to a new set of linearly uncorrelatedvariables, aptly called principal components (PCs), which explain the maxi-mum possible variance of the original data. Figure 4.4 shows a typical PCAtransformation of two (randomly generated) correlated variables ( y , y ); itis effectively a rotation around the empirical mean into a different orthonor-mal basis. The first PC, z ≈ ( y + y ) / √
2, explains over 80% of the originalvariance—in other words, based on the observed correlation of y and y ,their sum contains most of the information about the individual variables.Meanwhile, the second PC, z ≈ ( y − y ) / √
2, is orthogonal to the first andaccounts for the remaining variance (i.e. information).In the present situation, the original variables are the m model outputs y = ( y , y , . . . , y m ), to be transformed into the principal components z =( z , z , . . . , z m ), where each z i is a linear combination of the y i . To constructthe PCA transformation, we first concatenate all the model outputs into an d × m matrix Y = ( y , y , . . . , y d ) whose rows correspond to design pointsand columns to model outputs. Each vector y i = ( y i , y i , . . . , y mi ) containsthe m model outputs at the i th design point, y ji being the j th model outputat design point i . We then standardize the data by centering and scaling HAPTER 4. BAYESIAN PARAMETER ESTIMATION −3 0 3 y −303 y PCA z (83%) z (17%) −3 0 3 z −303 z Figure 4.4
Principal component analysis (PCA) transformation of two correlatedvariables ( y , y ) into linearly uncorrelated variables ( z , z ). On the left, arrowsrepresent the principal component vectors, with labels including the fraction ofexplained variance. each column of Y to zero mean and unit variance. Zero mean is requiredsince PCA is a rotation around the empirical mean; unit variance is notexplicitly required, but the columns must all have the same units and similarmagnitude, and scaling to unit variance is a convenient way to achieve this.The transformation is now determined by the standardized data Y , suchthat the first principal component has the maximum possible variance (ex-plaining as much variance of Y as possible), the second component has max-imal variance while being orthogonal to the first, and so forth. This resultsin an orthonormal m × m matrix V which transforms the (standardized)model data as Z = Y V, (4.9)where Z = ( z , z , . . . , z d ) is another d × m matrix whose rows correspondto design points and columns to principal components, with the columnssorted from greatest variance to least—in contrast to Y , whose columnsall have unit variance. Analogous to the above notation, each vector z i =( z i , z i , . . . , z mi ), where z ji is the value of the j th PC at design point i . Thecolumns of V are orthonormal vectors v j , i.e. satisfying v j · v k = δ jk , eachcontaining the linear combination coefficients for PC j , so that z ji = v j · ˜ y i , (4.10) HAPTER 4. BAYESIAN PARAMETER ESTIMATION standardized model output.In numerical implementations [223], the PCA transformation is com-puted efficiently via the singular value decomposition (SVD) of the datamatrix Y . The SVD, a generalization of the eigendecomposition for non-square matrices, is the factorization Y = U Σ V T , (4.11)where U and V are orthogonal matrices containing the left and right singularvectors and Σ is diagonal containing the singular values. The matrix V ofthe right singular vectors is the PCA transformation matrix. Using theSVD, we can also see that PCA is related to the eigendecomposition of thesample covariance matrix as Y T Y = ( U Σ V T ) T U Σ V T = V Σ V T . (4.12)Hence, V contains the eigenvectors of the sample covariance matrix ( Y T Y )and Σ has the eigenvalues on the diagonal.Figure 4.5 shows some properties of a realistic application of PCA to thepresent heavy-ion collision model, using output from the design specified insubsection 4.1.3. The main left panel shows the linear combination coeffi-cients for the first three components, i.e. the values of v j , j = 1 , ,
3. Thefirst component, which by itself explains about half of the model’s variance,accounts for the mutual correlation of all the particle and energy productiondata, and their anti-correlation with the flow data. In this context, corre-lations refer to correlations across the parameter design space, for examplechanging a parameter that increases the charged-particle yield is likely toalso increase energy production and identified particle yields. The remainingcomponents, of which only the second and third are shown here, are orthog-onal to the first (and to each other) and describe various other correlationsand anti-correlations among the observables.The side right panel shows the convergence of the explained variance.Despite there being over 100 original observables, the first four principalcomponents explain about 95% of the total variance; the 99% threshold isattained with eight components. This trend is valuable for dimensionalityreduction.
Dimensionality reduction
If the original data are strongly correlated, which is often the case for phys-ical models, the first few principal components will usually explain most of
HAPTER 4. BAYESIAN PARAMETER ESTIMATION N ch E T N π N K N p › p πT fi › p KT fi › p pT fi δp T / › p T fi v v v −0.10.00.10.20.3 P C A c o e ff i c i e n t PC 1 (49%)PC 2 (23%)PC 3 (15%)
Number of PC C u m u l a t i v e e x p l a i n e d v a r i a n ce f r a c t i o n Figure 4.5
Application of PCA to heavy-ion collision model output for Pb-Pbcollisions at 2.76 TeV. Left: Linear combination coefficients for the observables la-beled along the top: charged-particle yield, transverse energy production, identifiedparticle yields, identified particle mean p T , mean p T fluctuations, and flow coeffi-cients (two-particle cumulants). Each point represents a centrality bin. The legendentries include the explained variance of each component. Right: Cumulative ex-plained variance fraction for up to 10 components. the original variance. Thus, we can use a smaller number of principal com-ponents k than the number of original model outputs m , sacrificing a smallamount of information in the process. Since the columns of the transfor-mation matrix V are sorted by their explained variance, we simply take thefirst k < m columns and transform the data as Z k = Y V k , (4.13)where V k is d × k containing the first k principal components. The inversetransformation is Y ’ Z k V T k , (4.14)where the equality is only approximate since we have discarded some infor-mation.PCA dimensionality reduction is particularly effective for data contain-ing statistical noise. Since noise is, by definition, uncorrelated with the truevariability of the model, PCA will naturally separate the true variability HAPTER 4. BAYESIAN PARAMETER ESTIMATION y = y in reality, with the differences caused entirely by statisticalfluctuations. Thus if we modeled only the first PC, we would account for allthe true variability. Caveats
PCA works best if the original data have a joint multivariate-normal distri-bution. It is not necessary for every model output to have a perfect normaldistribution, but they should roughly have a peak with tails. In fact, thisusually happens automatically when several parameters are varied, due tothe central limit theorem. Non-normal distributions can sometimes be mademore normal by applying a nonlinear transformation such as a Box-Coxpower transformation, but since this would later complicate propagation ofuncertainty, it should be avoided unless necessary.More important is that the model outputs are only linearly correlated;as a linear transformation, PCA can only remove linear correlations. Prac-tically speaking, this means that scatterplots of y i vs. y j should look ap-proximately elliptical, with no curved “S” or “C” shapes.Outliers will have an undue influence on the principal component direc-tions. But if outlier points are determined to be true model behavior, it maybe desirable to still include them in the analysis. They should nonethelessbe excluded from the data matrix Y when computing the SVD, then, afterdetermining the PCA transformation, the outlier points can be transformedas usual.One should always check these considerations before applying PCA to adataset. PCA is implemented in scikit-learn [223], a Python machine learninglibrary.1. Check model outputs for approximate normality, linear correlations,and outliers.2. Concatenate into the matrix Y = ( y , y , . . . , y d ) and standardize thecolumns to zero mean and unit variance.3. Compute the PCA transformation via the SVD (4.11). HAPTER 4. BAYESIAN PARAMETER ESTIMATION
Having evaluated the model at each of the parameter design points, the timehas come to construct an emulator to serve as a fast surrogate to the fullmodel, that is, to quickly predict the model output at any point in parameterspace. Gaussian processes are ideal for this purpose since they operate inarbitrarily high-dimensional space, require only minimal assumptions aboutthe model, and naturally quantify the uncertainty of their predictions. Theyare not the only valid emulation scheme, but exploring the alternatives isbeyond the scope of this work, and Gaussian processes are the de factochoice for parameter estimation with computationally expensive models.In the following subsections, I summarize the theory of Gaussian pro-cesses and discuss relevant practicalities of building model emulators. For acomplete treatment, see the seminal book
Gaussian Processes for MachineLearning by Rasmussen and Williams [214], especially chapters 2, 4, and 5.
The essential ingredients of a model emulator are: • A set of training points X t = ( x , x , . . . , x d ), where each x i is an n -dimensional input vector. • A corresponding set of model outputs y t = ( y , y , . . . , y d ), where each y i is the result of evaluating the model at x i .Given these training data, the emulator shall predict model outputs y p atnew points X p . Gaussian process (GP) emulators achieve this providedanother key ingredient: • A covariance function, which dictates the similarity between pairs ofoutputs ( y i , y j ).Given such a function, a GP predicts new model outputs y p by exploit-ing their covariance, i.e. similarity, with the training outputs y t . In fact,the predictions are probability distributions—specifically normal (Gaussian) HAPTER 4. BAYESIAN PARAMETER ESTIMATION
Input −2−1012 O u t pu t Random functions
InputConditioned ona few noiseless pointsMean predictionUncertaintyTraining data
InputConditioned onmany noisy points
Figure 4.6
Left: Random functions drawn from a Gaussian process. Center:Functions drawn from a GP conditioned on a few noiseless training points. Right:A function drawn from a GP conditioned on many noisy training points (only onesemitransparent line for visual clarity). In all plots, the gray dashed line and bandare the GP predictive mean and uncertainty (one standard deviation), respectively.All GPs have a squared exponential covariance function (4.17) with length scale ‘ = 0 .
6. On the right, the covariance function (4.21) also has a noise term withvariance σ n = 0 . distributions—for the likely values of y p , from which we can extract meanvalues and associated uncertainties.A brief word on notation: The subscripts t and p mean “training” and“predictive”, respectively, and I shall use them consistently throughout thissection. However, I am somewhat overloading the vector notation: Here, thevectors y t and y p contain multiple values of a single model output, whilein other contexts, y (without a subscript) is the vector of all the modeloutput variables, and y i (with an index subscript) is a single observation ofall the model outputs at design point i . The meaning of vector symbols willhopefully be clear from context.Before formalizing Gaussian process predictions, let us take a step back.A GP is, in one interpretation, a distribution over functions . Analogouslyto sampling random numbers from a probability distribution, we can drawrandom functions from a GP, demonstrated in the left panel of figure 4.6(I will explain precisely how to do this shortly). The colored lines are therandom functions, the dashed line is the GP mean (zero), and the band isthe standard deviation (one).After conditioning the GP on some training data, its mean and standarddeviation become functions of the input, as in the center panel. The mean HAPTER 4. BAYESIAN PARAMETER ESTIMATION y ∼ N ( µ , Σ) (4.15)for mean vector µ and covariance matrix Σ. This property also means thatany single variable has a (univariate) normal distribution y ∼ N ( µ, σ ) , (4.16)for mean µ and variance σ .Before drawing functions from a GP, we must specify a covariance func-tion, or kernel, k ( x , x ). A standard choice is the squared exponential (SE)covariance functioncov( y i , y j ) = k ( x i , x j ) = exp (cid:18) − | x i − x j | ‘ (cid:19) , (4.17)which we shall to refer to as the SE function, even though it’s obviously aGaussian function, to distinguish it from the “Gaussian” in “Gaussian pro-cess”; it’s also known as the radial basis function (RBF). Notice that thecovariance function describes the similarity between pairs of outputs , butis a function of the inputs . With this particular covariance function, out-puts from nearby input points are strongly correlated, while distant pointsbecome uncorrelated over a characteristic length scale ‘ .Now, to sample functions: We choose some input points X p and con- The input points are denoted by X p because they are technically predictive points,even though they are not actually predicting anything in this case. Soon, X t and X p willappear together and it will be important to distinguish between them. HAPTER 4. BAYESIAN PARAMETER ESTIMATION K pp , where this notation means a matrix fromapplying the covariance function to each pair of points in X p : K pp = k ( x p , x p ) k ( x p , x p ) · · · k ( x p , x p ) k ( x p , x p ) · · · ... ... . . . . (4.18)We then launch our favorite statistical software, generate random vectorsfrom the multivariate normal distribution y p ∼ N ( , K pp ) , (4.19)and plot the resulting vectors as smooth curves. The left panel of figure4.6 is the result of following this procedure, setting X p to an array of one-dimensional points from 0 to 4 and using the SE covariance function (4.17)with length scale ‘ = 0 . f ( x ) with a specified covariance structure k ( x , x ). The curves on the left of figure 4.6 are samples from the family offunctions, defined by the chosen SE covariance function (4.17), which varysmoothly over the chosen characteristic length scale.To use a GP as a model emulator, we assume that the training data( X t , y t ) are the inputs and outputs of a function f ( x ) from a GP. This isquite general, tantamount to assuming that there exists a covariance func-tion k ( x , x ) that describes the relationships between model outputs. The SEcovariance function used to this point is in fact appropriate for many phys-ical models, which tend to be well-behaved and smoothly varying. Otherfunctions allow control over the degree of smoothness. More sophisticatedcovariance functions can be constructed by adding together different kernelsas k = k + k + · · · , for example the sum of two SE functions with differentlength scales implies a covariance structure with both small- and large-scaletrends. If the model is periodic, a periodic covariance function would besuitable.After designating a covariance function, we condition a GP on the train-ing data ( X t , y t ), furnishing the predictive distribution for new model out-puts y p at input points X p , y p ∼ N ( µ , Σ) , µ = K pt K − tt y t , Σ = K pp − K pt K − tt K tp , (4.20) HAPTER 4. BAYESIAN PARAMETER ESTIMATION K pp defined in (4.18) and the other K ∗∗ matrices following analogously,for example K pt is the covariance matrix from applying the covariance func-tion to each pair of predictive and training points, i.e. its ij element is( K pt ) ij = k ( x pi , x tj ). The center panel of figure 4.6 shows the effects ofconditioning a GP on the plotted training points using the same SE covari-ance function as before; the dashed line is the predictive mean µ plottedas a smooth curve, the gray band is the mean plus or minus one predictivestandard deviation, and the colored lines are sampled functions. In general,conditioning a GP restricts its function space to functions that are consistentwith the training data; the plotted curves in the figure are several possiblefunctions that could have given rise to the training points.If the model calculations are non-deterministic—perhaps due to aver-aging over a finite sample—the training data will contain statistical noiseas y = f ( x ) + (cid:15) , where (cid:15) is a fluctuating noise term. We may account forthis by adding a noise kernel k ( x i , x j ) = σ n δ ij to the covariance function,which describes uncorrelated (independent for each training point) Gaussiannoise of variance σ n . Combining the noise kernel with the SE function, forexample, gives the total covariance function k ( x i , x j ) = exp (cid:18) − | x i − x j | ‘ (cid:19) + σ n δ ij . (4.21)The right panel of figure 4.6 shows the result of conditioning a GP on noisytraining data using this covariance function. Since the data are noisy, thepredictive mean does not pass through every point exactly, but rather be-haves more like a regression line, and the predictive standard deviation ac-counts for the noise. Functions drawn from the GP have random fluctuationswith variance σ n .We can gain more intuition for how GP emulators work by examiningthe conditional (predictive) distribution for a single output y p . Writing k p =( k ( x p , x ) , k ( x p , x ) , . . . , k ( x p , x d )) for the vector of covariances between thepredictive point and the training points, equation (4.20) reduces to y p ∼ N ( µ, σ ) ,µ = k T p K − tt y p ,σ = k ( x p , x p ) − k T p K − tt k p . (4.22)From this, we see that the mean µ is a linear combination of all the trainingpoints, with the relative contributions depending on the covariance func-tion. The variance σ consists of the variance at x p from the covariance HAPTER 4. BAYESIAN PARAMETER ESTIMATION y p , deter-mined by the covariance function; (4.20) is the posterior distribution, deter-mined by the covariance function in combination with the training data. Infigure 4.6, the left panel shows a prior GP with the SE covariance function(4.17) and the center shows a posterior GP after conditioning the prior onthe training data. The right panel shows another posterior GP, conditionedon noisy training data using the covariance function (4.21). Fundamentally, Gaussian processes map vector inputs to scalar outputs,but computational models often have many outputs. As detailed in theprevious section 4.2, we deal with this by transforming the model outputsusing principal component analysis (PCA) and building an independent GPemulator for each principal component. Given an input point x , we computethe predictive distributions for each PC, collect the mean predictions into avector z , and transform it into the desired model outputs y = V z , (4.23)where V is the PCA transformation matrix.Calculating the uncertainty on y is straightforward since it is related to z by a linear transformation; writing Σ z for the predictive covariance matrixof z , the covariance of y is Σ y = V Σ z V T . (4.24)The principal components are uncorrelated by construction, so the covari-ance matrix is diagonal, Σ z = diag( σ , . . . , σ m ) , (4.25)where each σ k is the predictive variance of the k th principal component, i.e.of the k th element of z . Hence, the covariance of y reduces to(Σ y ) ij = X k V ik σ k V jk . (4.26)Note that Σ y is in general not diagonal, meaning that the uncertainties onthe model outputs are correlated. These correlations manifest because we HAPTER 4. BAYESIAN PARAMETER ESTIMATION k < m principal components of the largestvariance. But the remaining components do contain a small amount of in-formation, so neglecting them contributes some uncertainty, which we musttake into account. Consider that neglecting a PC is equivalent to treating itas an unconditioned (prior) GP with zero mean and constant variance equalto the sample variance, therefore, we should take each neglected compo-nent’s sample variance as its predictive variance. In the diagonal covariancematrix Σ z (4.25), we set the first k variances ( σ , . . . , σ k ) according to theGP emulators, and set the remaining variances ( σ k +1 , . . . , σ m ) to the samplevariance of the respective principal component. To this point I have glossed over the free parameters that are often present incovariance functions, such as the characteristic length scale ‘ in the squaredexponential function. These hyperparameters are usually not known a priori and must be estimated from the data. The selection of hyperparameters,known as training, is typically accomplished by maximizing the likelihood L ( θ ) = 1 q (2 π ) d det K tt exp (cid:18) − y T t K − tt y t (cid:19) , (4.27)where θ is the vector of hyperparameters and K tt = K tt ( θ ) is the covariancematrix from applying the covariance function—which depends on θ —to thetraining points. The form of L ( θ ) is nothing but a multivariate normalprobability density, whose logarithmlog L ( θ ) = − y T t K − tt y t −
12 log(det K tt ) − d π (4.28)is preferable for numerical optimization, and more clearly separates intomeaningful components: The first term is a fit to the data, the second is acomplexity penalty which prevents overfitting, and the third is a normaliza-tion constant.To see how this works, let us consider an instructive example: Figure4.7 shows three GPs conditioned on the same noisy data, all using the SEcovariance function with a noise term (4.21), but with different values of the HAPTER 4. BAYESIAN PARAMETER ESTIMATION
Input −2−1012 O u t pu t Overfit ‘ = 0 . , σ n = 0 . InputUnderfit ‘ = 10 , σ n = 0 . InputMax likelihood ‘ = 2 . , σ n = 0 . GPActual function
Figure 4.7
Example of training a Gaussian process. The data points, whichare the same in all plots, were generated by evaluating the function plotted asthe dashed line and adding random Gaussian noise of variance σ = 0 .
09. Eachplot shows a GP conditioned on the noisy data using the SE covariance function(4.21) with variable length scale ‘ and noise term σ n . On the left and center,the hyperparameters were set manually; on the right, they were determined bynumerically maximizing the likelihood (4.28). hyperparameters θ = ( ‘, σ n ). On the left, a too-short length scale and too-small noise variance lead to an “overfit” model that passes through everypoint, mistakenly treating the noise as true variability, and thus offering nopredictive value. These hyperparameter values have a low likelihood dueto a high complexity penalty. In the center, the opposite: The model is“underfit” with a long length scale and large noise term, ascribing too muchof the true variability to noise. These values also have a low likelihood,this time because of a poor fit to the data. On the right, the maximumlikelihood hyperparameters strike a compromise, accurately capturing theactual underlying function and noise.In the present work, I use an anisotropic squared exponential covariancefunction k ( x i , x j ) = σ f exp (cid:20) − X k (cid:18) x ki − x kj ‘ k (cid:19) (cid:21) + σ n δ ij , (4.29)whose hyperparameters are the independent length scales ‘ k for each inputdimension (hence, anisotropic), overall variance scale σ f , and noise variance σ n . Using this covariance function essentially amounts to assuming thatthe model is well-behaved and smoothly varying, with no discontinuities,divergences, or other anomalous features. The noise variance allows for HAPTER 4. BAYESIAN PARAMETER ESTIMATION too smooth—a GPwith this covariance function is infinitely differentiable, which may not bethe case for some physical models. The Matérn class of covariance functionsattempts to resolve this by introducing a smoothness parameter while oth-erwise being similar to the SE function. I trained GPs to the model datausing once- and twice-differentiable Matérn covariance functions, which areboth somewhat less smooth than SE, but found no difference in practicalperformance. Thus, in the interest of simplicity, I use the SE covariancefunction.I determine the hyperparameters θ = ( σ f , ‘ , . . . , ‘ n , σ n ) by maximiz-ing the likelihood using a numerical optimization algorithm. To help pre-vent over or underfitting, I constrain the length scales to within an or-der of magnitude of the corresponding parameter’s design range, i.e. if∆ k = max( x k ) − min( x k ) is the design range of parameter x k , then theconstraint is 0 . < ‘ k / ∆ k <
10. As previously discussed, I train an indepen-dent GP on each principal component; the optimal hyperparameters are ingeneral different for each.Numerical optimizers sometimes converge to a local rather than globalmaximum. To ensure this is not the case, we can repeat the hyperparame-ter optimization several times starting from different initial values of θ , thentake the best result. However, if we do find several competing local max-ima, it may be a sign that we do not have enough information to uniquelydetermine the hyperparameters, i.e. there are too few training points. Inmy experience, using Latin hypercube designs with at least 20 points perdimension, the optimization algorithm converges to the same result almostevery time, regardless of the initial values. This lends confidence that thehyperparameters are well-determined by the data.If the hyperparameters were not well-determined, we could account forthat uncertainty by sampling them during the MCMC calibration phase ofthe analysis (next section, 4.4). Formally, we should always do this, sincethe hyperparameters are not known exactly, but it incurs significant com-putational cost, as the likelihood requires calculating the inverse covariancematrix, an O ( n ) operation. In any case, provided a sufficient number oftraining points, the actual emulator predictions will not depend strongly onthe hyperparameters, as long as they are not egregiously over or underfit.Still, it is reasonable to doubt whether a GP emulator with the maximumlikelihood hyperparameters truly captures the underlying model behavior.The ultimate test of emulator performance is whether it accurately predictsnew model calculations, which I will address in the next subsection. First, we HAPTER 4. BAYESIAN PARAMETER ESTIMATION −404 P C −404 P C −0.5 0.0 0.5 p −404 P C τ fs [fm /c ] η/s min Figure 4.8
Emulator diagnostic visualization. Each subplot shows the depen-dence of a principal component on a model parameter, as labeled on the axes. Thedots are the training data. The lines with bands are GP emulator predictions, withuncertainty, as a function of the given parameter over its full design range, holdingall other parameters fixed. The blue lines are with all other parameters fixed to themidpoint of their design range (50%), purple is 20%, and green is 80%. can perform some simple checks that the emulator is behaving reasonably, forexample plotting the dependence of model outputs on input parameters andverifying that the relationships align with expectations. Another diagnosticvisualization that I have found quite useful is shown in figure 4.8. Withoutrepeating the information in the caption, here are some characteristics wecan check: • Are the emulator predictions smooth and sensible? Changing a singleparameter can only affect the model so much; there should not be anyrapid oscillations or extreme behavior, which could be a sign of over-
HAPTER 4. BAYESIAN PARAMETER ESTIMATION • Are the predictions consistent with the training data? The 50% curveshould probably track through the middle of the cloud, while the 20%and 80% curves should be distinct (the values 20 and 80 are not special,the point is to probe closer to the corners of the design space). • The uncertainties should be much smaller than the spread of the train-ing data, which is due to varying all parameters simultaneously. Inother words, the predictive variance should be smaller than the totalvariance of the model. Equivalently, check that σ n (cid:28) σ f . • The uncertainty should usually increase for the higher order principalcomponents, since they describe more noise. This is why I have showncomponents 1, 3, and 10, to emphasize the increase of the uncertainty.Equivalently, check that σ n generally trends upward.The subplots in the figure are only a small subset of all the possible input-output combinations; I chose these representative instances to keep the figurea reasonable size. The most important test of emulator performance is if it faithfully predictsmodel calculations, that is, given an arbitrary input point x , the predictedmodel output y pred ( x ) should be close to the result of a full model calculation y calc ( x ). We should check a large sample of validation points to ensurestatistical significance.In the present work, I have a sample of model calculations from an earlierversion of the design that I will use for validation. However, it sometimesmay be too computationally expensive to run a separate validation sample.An alternative is cross-validation, a general technique in which the trainingdata is split into two sets, one for training and the other for validation.In k -fold cross-validation, the training data is partitioned into k equallysized subsets, then one subset is used for validation and the other k − k subsets, so that eventually alltraining points have been used for validation.The simplest validation test is a scatterplot of a calculated vs. predictedmodel output, for example in the main (left) panel of figure 4.9. It appearsthat the emulator is performing reasonably well, although it is difficult to sayprecisely how well from this plot alone. To quantify this, consider that since HAPTER 4. BAYESIAN PARAMETER ESTIMATION
200 400 600 800 1000 1200
Emulator prediction M o d e l c a l c u l a t i o n dN ch /dη −3−2−10123 N o r m a li ze d r e s i du a l s N o r m a l q u a n t il e s Figure 4.9
Validation of emulator predictions of a single model output. Left:Scatterplot of model calculations vs. emulator predictions of dN ch /dη in 20–30%centrality. The horizontal error bars are the standard deviation of the predictiveuncertainty, and the diagonal line is a reference for calculation = prediction. Center:Histogram of the normalized residuals, overlaid with a standard normal distribution N (0 ,
1) probability density. Right: Box plot of the normalized residuals comparedto normal distribution quantiles.
Gaussian processes predict probability distributions, they need not predictevery validation output exactly, but rather should predict the distribution of outputs. Specifically, GP predictions are normal distributions, therefore,it should be the case that for every output y , y pred − y calc σ pred ∼ N (0 , . (4.30)The left-hand side is a normalized residual: The difference of the predictivemean and the actual calculation, divided by the predictive uncertainty. Ifthe emulator is performing perfectly, these normalized residuals would havea standard zero-mean, unit-variance normal distribution. The center panelof figure 4.9 compares a histogram of the normalized residuals to the N (0 , HAPTER 4. BAYESIAN PARAMETER ESTIMATION N ch E T N π N K N p › p πT fi › p KT fi › p pT fi δp T / › p T fi v v v −2−1012 N o r m a li ze d r e s i du a l s R M S % e rr o r N o r m a l q u a n t il e s Figure 4.10
Validation of emulator predictions of all model outputs. The outputs(observables) are grouped by type, as labeled, with each box plot or dot correspond-ing to a centrality bin (most central on the left to most peripheral on the right).Top: Box plots of normalized residuals compared to normal distribution quantiles.Bottom: Root mean square (RMS) percentage predictive error for each observable.
We can use a box plot to validate the quantiles (or percentiles) of thenormalized residuals, as in the right the figure. The horizontal blue linemarks the median of the distribution, the box extends from the 25th to 75thquantile (the interquartile range), and the tails extend to the 10th and 90thquantiles. The gray reference lines and box indicate where these elementswould be located in the ideal case. From this, we see that the median issomewhat high, meaning that more validation points were overpredictedthan underpredicted, which we can also see qualitatively on the scatterplot.In general, the predictions are skewed high, but not unreasonably so.Comparing quantiles like this is a sensitive test, and the box plots arequite compact. Taking advantage of this, the top of figure 4.10 shows boxplots for all the model observables. To reiterate, in the ideal case, themedian line would coincide with the reference line at zero, the box wouldmatch the range indicated by the gray band, and the tails would extend tothe positive and negative reference lines. We observe overall very good per-formance, although many of the box tails extend a little too far, suggestingthat the uncertainty may be slightly underpredicted. This is likely because
HAPTER 4. BAYESIAN PARAMETER ESTIMATION y pred − y calc ) /y calc .Most observables are predicted to roughly 5% precision, which is quite goodconsidering that there are only 500 training points in a 14-dimensional space,so some interpolation uncertainty is inevitable. Some observables are alsointrinsically noisier than others, which is why the relative error increases formore peripheral bins, and why the error of v is greater than that of v ,which is greater than that of v . The error of the mean p T is smaller thanthe rest simply because it does not vary as much across the design space(imagine if an observable did not change at all across the design space, thenit would be easy to predict).The salient point to take away from this validation: The emulator ac-curately quantifies its own uncertainty, so it is safe to use for parameterestimation as long as we take that uncertainty into account. We are now prepared to calibrate the model to experimental data, therebyinferring quantitative estimates of the model parameters, including uncer-tainties. This is an inverse problem—we wish to learn about unknown modelinputs using data we have collected about the outputs—for which Bayesianinference offers a natural solution. In this framework, we extract parameterestimates from the posterior distribution for the model parameters P ( x |D ) ∝ P ( D| x ) P ( x ) , (4.31)where x are the parameters and D represents all the collected data, fromboth experiments and model calculations. In this relation, now familiar asBayes’ theorem, the left-hand side is the posterior distribution: the con-ditional probability of the parameters given the data. Written like so, asa proportionality, the posterior distribution is unnormalized, which is ac-ceptable for the present purposes since we are only concerned with relativeprobabilities. On the right, P ( D| x ) is the likelihood, the probability of ob-serving the data conditional on some assumed parameter values, and P ( x )is the prior distribution, which embodies our initial knowledge of the pa-rameters. HAPTER 4. BAYESIAN PARAMETER ESTIMATION
The prior distribution P ( x ) expresses any information we have about theparameters before observing the data. If we know little about the parame-ters, a uniform prior would be appropriate, P ( x ) = constant. In the presentmethod, we have designated a finite design range for each parameter, and theGaussian process emulator can only make predictions within those ranges,thus, we may choose a prior which is constant inside the hyperrectangulardesign region and zero outside, P ( x ) ∝ ( x i ) ≤ x i ≤ max( x i ) for all i, . (4.32)The prior being zero outside the design region is a very strong assumption:It means we believe it is impossible for the true value of any parameter to beoutside its design range. To ensure that plausible parameter combinationsare not excluded a priori , we ought to err on the side of too-wide designranges.We should not be fooled into thinking that a uniform prior is uninfor-mative or unbiased ; it does not amount to the absence of a prior. A uniformprior encodes a specific assumption: that any equally-sized volume of pa-rameter space is equally probable, regardless of location. For example, if weplace a uniform prior on parameter x from zero to one, we are assertinga belief that the true value of x is equally likely to fall within [0 , /
2] as[1 / ,
1] (or any other pair of equally-sized ranges). This may be reasonable,but we should not take it as a given.Further, uniform priors become nonuniform if the parameter is nonlin-early transformed. Continuing the above example, suppose that x entersthe model only as its square, then we might instead place a uniform prioron x , but that would encode a different assumption: that the true valueof x is equally likely to fall within [0 , / √
2] as [1 / √ , HAPTER 4. BAYESIAN PARAMETER ESTIMATION x and x have natural ranges of zero to one, but, based on physical consid-erations, it’s unlikely that both parameters are close to one, we could choosea prior that decreases when, e.g., x + x > The likelihood P ( D| x ) is the probability of observing the data given theparameters; it quantifies the compatibility of the model calculations, at aparticular parameter point x , with the experimental data. Here, the sym-bol D is shorthand for all the collected data, including the experimentalobservations, model calculations, and associated uncertainties.Before specifying the likelihood function, we define some terms. Let y e be the vector of experimental data, which is the result of observing thehypothetical “true” values y true e with some measurement error (cid:15) e . We writethis as y e = y true e + (cid:15) e , (cid:15) e ∼ N ( , Σ e ) , (4.33)where the second relation means that the error is distributed as a multivari-ate normal distribution with mean zero and covariance matrix Σ e , whichaccounts for all sources of experimental uncertainty, namely statistical andsystematic. Similarly, the model outputs y m for input parameters x are y m ( x ) = y ideal m ( x ) + (cid:15) m , (cid:15) m ∼ N ( , Σ m ) , (4.34)where the “ideal” model outputs represent the hypothetical calculations ofa perfect physical model to unlimited precision. Since we are using a modelemulator, y m ( x ) is an emulator prediction, and the model covariance matrixΣ m accounts for predictive uncertainty, model statistical uncertainty (e.g.from averaging over a finite sample), and model systematic uncertainty (e.g.from discretizing a continuous system onto a grid).Now, we assume that there exists some true values of the parameters x ? at which the ideal model calculations would match the true experimentaldata: y true e = y ideal m ( x ? ). Combining this with (4.33) and (4.34) gives y e = y m ( x ? ) + (cid:15) , (cid:15) ∼ N ( , Σ) , Σ = Σ e + Σ m , (4.35) HAPTER 4. BAYESIAN PARAMETER ESTIMATION P ( D| x ) = 1 p (2 π ) m det Σ exp (cid:26) −
12 [ y m ( x ) − y e ] T Σ − [ y m ( x ) − y e ] (cid:27) . (4.36)All that remains is to calculate the covariance matrix. We further breakdown the experimental part into statistical and systematic components,Σ e = Σ stat e + Σ sys e . (4.37)Statistical uncertainties are uncorrelated by definition, so the statistical co-variance matrix is diagonal,Σ stat e = diag h ( σ stat1 ) , ( σ stat2 ) , . . . , ( σ stat m ) i , (4.38)where σ stat i is the statistical uncertainty of experimental observable y i (the i th element of y e ). Systematic uncertainties are in general correlated, soΣ sys e is not diagonal. However, while experimental collaborations typicallyreport separate statistical and systematic uncertainties, they usually do notreport the systematic correlation structure, so we shall assume somethingreasonable. Quite generally, we can express the covariance between observ-ables ( y i , y j ) as Σ ij = cov( y i , y j ) = ρ ij σ i σ j , (4.39)where σ i is the uncertainty of y i and ρ ij is a correlation coefficient satisfying | ρ ij | ≤ , ρ ii = 1 (not a sum) , (4.40)with the following meaning ρ ij = 1 ( y i , y j ) are fully correlated , < ρ ij < ,ρ ij = 0 uncorrelated , − ≤ ρ ij < . (4.41)Indeed, Σ stat e can be cast in the form (4.39) with ρ stat ij = δ ij . For system-atic uncertainty, I assume that observables within a centrality dataset (e.g. dN ch /dη as a function of centrality) have correlation coefficients ρ sys ij = exp (cid:20) − (cid:18) c i − c j ‘ (cid:19) (cid:21) , (4.42) HAPTER 4. BAYESIAN PARAMETER ESTIMATION N ch E T N π N K N p › p πT fi › p KT fi › p pT fi δp T / › p T fi v v v Model (emulator) N ch E T N π N K N p › p πT fi › p KT fi › p pT fi δp T / › p T fi v v v N ch E T N π N K N p › p πT fi› p KT fi› p pT fi δp T / › p T fi v v v Experiment −1.0−0.50.00.51.0
Correlation
Figure 4.11
Visualizations of the model (emulator) and experimental correlationmatrices, whose elements are corr( y i , y j ) = cov( y i , y j ) / ( σ i σ j ). Observables aregrouped by type, as labeled on the axes, where each cell represents a centrality bin. where c i is the midpoint of the centrality bin for observable y i and ‘ is acorrelation length, which I set to ‘ = 1. I reduce the correlation by 20%,i.e. multiply ρ ij by 0.8, for pairs of observables in different datasets but ofthe same type, e.g. pion and kaon yield; and I assume that observables ofdifferent types are uncorrelated. All of this results in the block diagonal cor-relation matrix visualized in figure 4.11. Clearly, these are assumptions, butthe behavior is qualitatively correct and certainly preferable to neglectingsystematic error correlations.Model uncertainty consists of emulator predictive uncertainty, statisticalfluctuations, and systematic uncertainty:Σ m = Σ pred m + Σ stat m + Σ sys m . (4.43)In fact, the Gaussian process emulator accounts for both predictive andstatistical uncertainty since the GPs have estimated noise terms (subsection4.3.3), thus Σ m = Σ GP m + Σ sys m . (4.44) HAPTER 4. BAYESIAN PARAMETER ESTIMATION GP m = V Σ GP m,z V T , (4.45)where Σ GP m,z is the (diagonal) predictive covariance in principal componentspace and V is the PCA transformation matrix. The GP matrix dependson the position x in parameter space, but not strongly; figure 4.11 shows arepresentative correlation matrix from a random point in parameter space.Although it shares some qualitative features with the (assumed) experimen-tal correlation matrix, the emulator correlation structure is not assumed,it’s a direct consequence of the empirical correlations in the model outputdata.Model systematic uncertainty arises from non-random imperfections inthe computational model, such as grid discretization effects, uncertainty inthe hydrodynamic equation of state, and negative contributions to Cooper-Frye. It would be futile to attempt to enumerate every source of uncertaintyand compute a covariance matrix for each; instead, I define a simple param-eter σ sys m which is added in quadrature to the diagonal of Σ GP m,z in principalcomponent space, so that the complete model covariance matrix isΣ m = V h Σ GP m,z + ( σ sys m ) I i V T = Σ GP m + ( σ sys m ) V T V. (4.46)The natural range of this parameter is zero to one, relative to the overallvariance of the model: σ sys m = 0 means no systematic uncertainty, σ sys m =1 means that all the model’s variability is due to systematic uncertainty(which is obviously not the case). Since we do not know the “true” value of σ sys m , I leave it as a free parameter with a gamma distribution prior, P ( σ ) ∝ σ e − σ/s , s = 0 . , (4.47)which encodes that σ sys m is greater than zero but less than about 0.4. I willeventually marginalize over the posterior distribution for σ sys m , thereby ac-counting for our uncertainty in the uncertainty (not a typo). This treatment,while rudimentary, is preferable to neglecting model systematic uncertainty. Multiple collision systems
We may calibrate the model to data from multiple collision systems (or beamenergies) by calculating an independent likelihood for each system and thenthe joint likelihood as the product P ( D| x ) = Y s P ( D s | x s ) , (4.48) HAPTER 4. BAYESIAN PARAMETER ESTIMATION s is an index over systems; D s and x s are the data and parameters forsystem s . This factorized likelihood implicitly assumes that the uncertaintieson observables from different collision systems are uncorrelated.The parameters x s for each system are in general different: There couldbe parameters which are specific to a particular system, or parameters withpotentially different values for each system. Example: Suppose we are cali-brating to data from n s collision systems, and parameter x depends on thesystem, but the other parameters ( x , . . . , x n ) are common to all systems.Writing x ,s for the value of x for system s , the parameter vectors are x = ( x , , x , , . . . , x ,n s , x , . . . , x n ) , x s = ( x ,s , x , . . . , x n ) , (4.49)so that x s contains the parameters for system s and x contains the unionof all the parameters. In such cases, we calibrate all the parameters x ,distributing them to the appropriate system-specific likelihood functions.This entails constructing an independent Gaussian process emulator for eachsystem, each taking the system’s particular parameters x s and predictingits outputs y m,s . Markov chain Monte Carlo (MCMC) sampling is the key to computationalBayesian inference. A general class of algorithms for sampling probabilitydistributions, MCMC methods produce a representative sample of the pos-terior distribution by generating a random walk through parameter spaceweighted by the posterior probability. The sample (also called the chain) canthen be used to calculate marginal distributions, derive parameter estimates,and create visualizations.A simple, widely-used MCMC method is the Metropolis-Hastings algo-rithm [224, 225], which proceeds iteratively as follows: Given a position x i ,randomly choose a new proposal position x , then accept or reject x withprobability based on the ratio of the posterior probabilities at x i and x . Ifaccepted, set the next position to the proposal, x i +1 = x , otherwise repeatthe current position, x i +1 = x i . After repeating this many times, the distri-bution of the resulting positions { x , x , . . . , x n } approximates the posteriordistribution.In this work, I use the affine-invariant ensemble sampler [226], an MCMCalgorithm that uses a large ensemble of interdependent walkers. Ensemblesampling tends to performs well in most contexts and converges to the pos-terior distribution faster than Metropolis-Hastings sampling. Additionally, HAPTER 4. BAYESIAN PARAMETER ESTIMATION emcee [227].Since I have not personally developed the MCMC algorithm, I will notdescribe it in detail, but instead comment on some relevant practicalities.
Computing the posterior probability
Since the posterior typically varies over many orders of magnitude, it isnumerically preferable to operate on its logarithm,log P ( x |D ) = log P ( D| x ) + log P ( x ) + const , (4.50)where the additive constant is irrelevant in this context because only theratio of probabilities, i.e. the difference of the logs, enters MCMC sampling.The logs of the uniform prior (4.32) and likelihood (4.36) arelog P ( x ) = ( x i ) ≤ x i ≤ max( x i ) for all i, −∞ else , (4.51)log P ( D| x ) = − d T Σ − d −
12 log(det Σ) , d = y m ( x ) − y e , (4.52)where I have dropped normalization constants. Note that since the emulatorpredictive covariance is in general a function of x , the determinant of thecovariance matrix Σ is not constant and must be computed (if the covariancematrix were constant, we could safely neglect this term).The likelihood contains the inverse and determinant of the covariancematrix, which are both O ( n ) operations. Rather than evaluate the likeli-hood as written, it is numerically faster and more stable to use the Choleskydecomposition of the covariance matrix,Σ = LL T , (4.53)where L is a lower triangular matrix. This factorization is also an O ( n )operation, but allows us to avoid computing the inverse or determinant ex-plicitly. Given a Cholesky decomposition, numerical linear algebra librariescan efficiently solve the linear equation LL T α = d for α = Σ − d . (4.54)Since L is a triangular matrix, its determinant is simply the product of itsdiagonal entries, sodet Σ = det( LL T ) = det( L ) = Y i L ii . (4.55) HAPTER 4. BAYESIAN PARAMETER ESTIMATION P ( D| x ) = − d · α − X i log L ii . (4.56) Burn-in
It takes a number of MCMC steps for the chain to converge to the poste-rior distribution, so it is almost always necessary to discard the first part ofthe chain. This is called “burn-in”. After the burn-in phase, the chain (inprinciple) no longer depends on the starting position. The necessary num-ber of burn-in steps depends strongly on the specific problem and MCMCalgorithm, but is usually hundreds or thousands.
Number of walkers and steps
In ensemble sampling, a large number of walkers is usually necessary forsampling high-dimensional distributions. I use 1000 walkers as a defaultnumber, although that is likely overkill; a few hundred would probably sufficein most cases.I initialize the walkers at random positions in parameter space and runseveral hundred burn-in steps, perhaps up to 1000. Sometimes, walkers thatwere initialized in very low-probability regions may become stuck and takea very long time to burn-in. To accelerate this process, we can perform atwo-stage burn-in: Randomly initialize the walkers and run some burn-insteps, then resample the walker positions around the most probable positionssampled so far, and finally run some more burn-in steps.After burn-in, I run O (10 –10 ) steps to generate the posterior sample.This is enough to create smooth histogram visualizations but is overkillfor most other purposes, such as calculating medians or other summarystatistics. Keep in mind that the total number of samples is the number ofwalkers times the number of steps.The fraction of accepted proposal points is an important MCMC perfor-mance metric: If the acceptance fraction is very small (close to zero), thatindicates that the walkers are stuck; if the acceptance fraction is too large(close to one), that means the parameter space is being sampled completelyrandomly. In both cases, the MCMC sample will not be representative ofthe posterior distribution. I have typically observed acceptance fractionsaround 15–40%, with higher-dimensional distributions usually having lowerfractions. HAPTER 4. BAYESIAN PARAMETER ESTIMATION
Marginal distributions
A marginal distribution is a posterior distribution for a subset of the pa-rameters, obtained by marginalizing over (integrating out) all the rest; forexample the marginal distribution for x is P ( x |D ) = Z dx · · · dx n P ( x |D ) , (4.57)and the joint marginal distribution for ( x , x ) would be P ( x , x |D ) = Z dx · · · dx n P ( x |D ) . (4.58)Given an MCMC sample { x i } of the posterior distribution, marginalizationis trivial: • The values of x j from the MCMC sample, { x ji } , is a sample of themarginal distribution P ( x j |D ). • The values { ( x ji , x ki ) } is a sample of P ( x j , x k |D ).And so forth. Parameter uncertainties: credible intervals
We quantify the uncertainty on a parameter by a credible interval—a rangecontaining a certain fraction of the marginal distribution. For example, a90% credible interval contains 90% of the posterior density, and means thatthe true value of the parameter is expected to fall within the interval 90%of the time (recall, the gravitational wave posterior distribution figure 4.1showed 90% credible intervals). For a generic parameter x , let x l and x h bethe lower and upper endpoints of a credible interval containing a fraction0 < c <
1, then assuming the marginal distribution of x is unimodal, we canwrite Z x h x l dx P ( x |D ) = c Z x max x min dx P ( x |D ) . (4.59)More practically, we can extract credible intervals from an MCMC samplevia its percentiles, for example 0–90%, 1–91%, . . . , 10–100% are all 90%credible intervals. The narrowest interval containing the desired fraction iscalled the highest posterior density (HPD) interval. Given some samples { x , x , . . . , x n } , and again assuming a unimodal distribution, we can findthe HPD interval as follows: HAPTER 4. BAYESIAN PARAMETER ESTIMATION m = int( c × n ), the number of samples contained in an interval offraction c .2. Sort the samples in ascending order.3. Compute the widths of all n − m intervals containing the fraction c , { x m − x , x m +1 − x , . . . , x n − x n − m } .4. Choose the smallest interval.However, this algorithm is inefficient since it sorts all the samples, when weonly need the upper and lower ends to be sorted. To avoid this inefficiency,we can partition the samples on indices m − n and m , then sort only thesamples up to index m − n and after index m . The procedure is otherwiseidentical.A credible region is a generalization to multiple dimensions, e.g. an areaenclosing some fraction of a two-dimensional joint posterior distribution be-tween a pair of parameters (see again figure 4.1). Visualizations
The standard visualization of a posterior distribution is a triangle (or corner)plot: A triangular grid of subplots with the marginal distributions for eachparameter on the diagonal subplots and the joint distributions between eachpair of parameters on the off-diagonal subplots. Such visualizations com-pactly display the probability densities for all parameters and reveal cor-relations between parameters. Operationally, the marginal distributions onthe diagonal are histograms of MCMC samples, and the off-diagonal jointdistributions are two-dimensional histograms (density plots). In the nextchapter, figures 5.1, 5.2, 5.5, and 5.10 are triangle plots of actual posteriordistributions for heavy-ion collision parameters.Besides the distributions for the parameters themselves, it’s also usefulto visualize the model calculations compared to the experimental data. Inparticular, we can plot the model calculations from each design point over-laid on the data points, then after calibration, make a similar plot showingemulator predictions of the model output from random draws of the poste-rior MCMC sample. The first version, which effectively represents the prioron the model parameters, generally exhibits a wide spread around the datapoints, since there are several parameters varying across wide ranges. In thesecond version—the posterior—the emulator predictions should be tightlyclustered around the data, with the remaining spread arising from the finitewidth of the posterior distribution. Figures of this type: 5.3, 5.6, 5.11, 5.12.
HAPTER 4. BAYESIAN PARAMETER ESTIMATION
I have developed a complete parameter estimation code implementing themethods and strategies detailed in this chapter; it is the basis for my latestanalysis, the results of which are presented in section 5.3. The source codeis publicly available at https://github.com/jbernhard/hic-param-est with documentation at http://qcd.phy.duke.edu/hic-param-est . I en-courage interested readers to peruse the code and documentation, since inmany cases it is not obvious how to translate theoretical concepts into func-tioning code.The code makes use of several open-source Python libraries:
NumPy [228] and
SciPy [229] for general scientific computing, scikit-learn [223]for principal component analysis and Gaussian processes, emcee [227] forMCMC sampling, h5py [230] for data storage, matplotlib [231] for gen-erating plots.
Quantifying properties of hotand dense QCD matter O ver the past several years, I have conducted a series of case studiesapplying Bayesian parameter estimation to relativistic heavy-ion colli-sions, each time improving the analysis and advancing toward the ultimategoal: to quantitatively determine the properties of the quark-gluon plasma.This chapter is an exhibit of these case studies.The first two studies, which are published [119, 171], are somewhatlimited—primarily by earlier and less sophisticated versions of both thecomputational model of chapter 3 and the parameter estimation methodof chapter 4—but they nonetheless represent significant steps forward. Af-ter explaining the meaningful differences in these earlier iterations, I willdefer to the discussion I previously wrote.The third and final study is state of the art: It eliminates many of theshortcomings in the first two and realizes the most precise estimates of QGPproperties to date. These are new results, as of yet unpublished.The trio:I. A proof of conceptII. A more flexible approachIII. A precision extraction 125 HAPTER 5. QUANTIFYING QCD PROPERTIES
The purpose of this first study [119] is to begin developing the parameterestimation method and validate that it is a viable strategy in heavy-ionphysics. It succeeds in doing so, with the results quantitatively confirmingprior qualitative knowledge about the model parameters while revealing somepreviously unknown details. On a philosophical level, this is a positiveoutcome.
Initial conditions
This study precedes TRENTo, instead using two ex-isting initial condition models: the Monte Carlo Glauber model [129], awidely-used geometric model, and the Monte Carlo KLN model [134], animplementation of color-glass condensate (CGC) effective field theory. Theparameter estimation process is carried out separately for the two models.There is no pre-equilibrium free-streaming stage.
Hydrodynamics and particlization
The hydrodynamic model has con-stant shear viscosity η/s (no temperature dependence) and lacks bulk viscos-ity. The particlization routine was contributed by the Ohio State Universitygroup [155].
Parameters and observables
There is a modest set of five calibrationparameters:1. Initial condition normalization factor.2. A parameter specific to the initial condition model. Glauber: Thebinary collision fraction α , which controls how entropy is distributedto wounded nucleons and binary collisions. KLN: The saturation scaleexponent λ , a CGC parameter.3. QGP thermalization time (and hydrodynamic starting time) τ .4. Constant QGP specific shear viscosity η/s .5. Shear relaxation time τ π , controlled via the coefficient k π in the rela-tion τ π = 5 k π η/ ( sT ).Table 5.1 summarizes the parameters and their design ranges. HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.1
Input parameter ranges for the Glauber and KLNinitial condition models and for the hydrodynamic model.Parameter Description RangeGlauber Norm Overall normalization 20–60Glauber α Binary collision fraction 0.05–0.30KLN Norm Overall normalization 5–15KLN λ Saturation scale exponent 0.1–0.3 τ Thermalization time 0.2–1.0 fm η/s
Specific shear viscosity 0–0.3 k π Shear relaxation time coefficient 0.2–1.1
The observables are the centrality dependence of the average charged-particle multiplicity h N ch i and the flow cumulants v { } , v { } , with exper-imental data from the ALICE experiment, Pb-Pb collisions at √ s = 2 . Parameter estimation method
Most aspects of the parameter esti-mation method are similar or identical to chapter 4, including the Latin-hypercube parameter design, principal component analysis of the modeloutput, Gaussian process emulator, and MCMC algorithm. The primarydifference is much less sophisticated uncertainty quantification than in sub-section 4.4.2. The likelihood is evaluated in principal component space as P ( D| x ) ∝ exp (cid:26) −
12 [ z m ( x ) − z e ] T Σ − z [ z m ( x ) − z e ] (cid:27) , (5.1)where z e is the PCA transformation of the experimental data y e and z m ( x )contains the values of the principal components, predicted by the Gaus-sian processes, at parameter point x . The covariance matrix is diagonal inprincipal component space with a simple fractional uncertainty:Σ z = diag( σ z z e ) , σ z = 0 . . (5.2)This assumption precludes rigorous quantitative uncertainties on the modelparameters, but does not invalidate the overall results. (Editorial comment:This was a stopgap. As I wrote in the original publication, “The primarygoal of this study is to develop and test a model-to-data comparison frame-work; details such as the precise treatment of uncertainties can be improvedlater.”) HAPTER 5. QUANTIFYING QCD PROPERTIES
This subsection is adapted from:J. E. Bernhard et al., “Quantifying properties of hot and dense QCD matterthrough systematic model-to-data comparison”, Phys. Rev.
C91 , 054910(2015), arXiv: .The primary MCMC calibration results are presented in figures 5.1 and 5.2for the Glauber and KLN models, respectively. These are visualizations ofthe posterior probability distributions of the true parameters, including thedistribution of each individual parameter and all correlations. The diagonalhistograms show the marginal distributions for each parameter (all other pa-rameters integrated out); the lower-triangle plots are two-dimensional scat-ter histograms of joint distributions between pairs of parameters, wheredarker color denotes higher probability density; and the upper triangle hascontour plots of the same joint distributions, where the contour lines enclosethe 68%, 95%, and 99% credible regions.A wealth of information may be gained from these posterior visualiza-tions; the following highlights some important features.Focusing on the Glauber results in figure 5.1, we see the shear viscosity η/s (fourth diagonal plot) has a narrow approximately normal distributionlocated near the commonly quoted value 0.08. As expected, η/s is tightlyconstrained by experimental flow data. Going across the fourth row, we ob-serve nontrivial correlations among η/s and other parameters, for example, η/s and the hydrodynamic thermalization time τ are negatively correlated(fourth row, third column). As τ increases, the medium expands as a fluidfor less time, so less flow develops, and viscosity must decrease to compen-sate.Both τ and normalization (third and first diagonals) have broad distri-butions without strong peaks, and they are strongly-correlated (third row,first column). This is because the hydrodynamic model is boost-invariantand lacks any pre-equilibrium dynamics, so τ is effectively an inverse nor-malization factor. The joint distribution shows a narrow acceptable bandwhose shape is governed by the inverse relationship.The wounded nucleon / binary collision parameter α (second diagonal)has a roughly-normal distribution located near the typical value 0.12. Itis mainly related to the slope of multiplicity vs. centrality and hence has anontrivial correlation with normalization and τ , e.g. we can decrease thenormalization to the lower end of its distribution provided we also increase α to compensate. HAPTER 5. QUANTIFYING QCD PROPERTIES N o r m a li z a t i o n ® ¿ ´ = s
30 40 50 60
Normalization k ¼ ® ¿ ´=s k ¼ Figure 5.1
Posterior marginal and joint distributions of the calibration parame-ters for the Glauber model. On the diagonal are histograms of MCMC samples forthe respective parameters, on the lower triangle are two-dimensional scatter his-tograms of MCMC samples showing the correlation between pairs of parameters,and on the upper triangle are approximate contours for 68%, 95%, and 99% credibleregions along with a dot indicating the median.
HAPTER 5. QUANTIFYING QCD PROPERTIES N o r m a li z a t i o n ¸ ¿ ´ = s Normalization k ¼ ¸ ¿ ´=s k ¼ Figure 5.2
Same as figure 5.1 for the KLN model.
HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.2
Quantitative summary of posterior distributions. For each parameter,the previous estimate [232, 233], mean, median, and credible intervals are given.Credible intervals are computed from central percentiles, e.g. the 68% interval is16–84%. Credible intervalsParameter Prev. est. Mean Median 68% 95% 99% G l a ub e r Norm. 57 48.9 49.0 41.6–56.4 36.5–59.4 33.9–59.9 α τ η/s k π K L N Norm. 9.9 10.8 10.9 8.15–13.6 6.40–14.8 5.82–15.0 λ τ η/s k π Meanwhile, the shear stress relaxation time coefficient k π (fifth diag-onal) has an almost flat distribution and its joint distributions show nocorrelations. Evidently, this parameter does not influence flow coefficientsor multiplicity.The KLN results in figure 5.2 generally exhibit wider, less normal dis-tributions than Glauber. This could indicate an inferior fit to the data andsuggests that KLN is somewhat less flexible than Glauber, i.e. its overallbehavior is relatively insensitive to the specific values of input parameters.The shear viscosity η/s has a narrow, irregular distribution covering thecommon value 0.20. As with Glauber, η/s has a negative correlation with τ , there is a strong inverse relationship between normalization and τ , and k π has no effect. The KLN parameter λ has a flat marginal distribution, butthere are strongly excluded regions in the joint distributions with normal-ization and τ . This appears to be the same effect as observed with Glauber α , except the dependence on λ is significantly weaker.The posteriors may be validated by drawing samples from the calibrateddistributions and visualizing the corresponding emulator predictions: if themodel is correct and properly calibrated, the posterior samples will be closeto experimental measurements. Figure 5.3 confirms—for the most part—that the posteriors are indeed tightly clustered around the data points. Vi-sualizations such as this will always have some uncertainty since samples aredrawn from the full posterior, however, the posterior samples in the bottom HAPTER 5. QUANTIFYING QCD PROPERTIES G l a ub e r N ch ® v © ª v © ª Centrality % K L N Centrality %
Centrality %
Posterior G l a ub e r N ch ® v © ª v © ª Centrality % K L N Centrality %
Centrality %
Figure 5.3
Top two rows (prior): Model calculations from Glauber (blue) andKLN (green) initial conditions at each design point. Bottom two rows (posterior):Random samples of the calibrated posterior distributions for Glauber and KLN.From left to right: average charged-particle multiplicity h N ch i , elliptic flow two-particle cumulant v { } , and triangular flow two-particle cumulant v { } . Datapoints are experimental measurements from ALICE [82]. HAPTER 5. QUANTIFYING QCD PROPERTIES ´=s
Glauber 0.08 KLN 0.20
Figure 5.4
Comparison of posterior distributions of η/s forGlauber (blue) and KLN (green). These are the same histogramsas in figures 5.1 and 5.2, expanded and placed on the same axis.The vertical grey lines indicate the common values 0.08 for Glauberand 0.20 for KLN [232, 233]. of the figure are markedly narrower than the prior calculations in the top,in which the input parameters varied across their full ranges and were nottuned to match experiment.As shown in the posterior samples of figure 5.3, the Glauber model nearlyfits the centrality dependence of all the present observables ( h N ch i , v { } , v { } ). The v samples have a somewhat larger variance than the others, inpart due to the underlying noise in the model calculations and also because v is explicitly given a lower weight (recall that h N ch i : v { } : v { } areweighted 1.2 : 1.0 : 0.6).The KLN results in the bottom row tell a somewhat different story, asthey cannot fit all observables simultaneously. While the fit to h N ch i isexcellent, the ratio of v to v is simply too large and the model has nochoice but to compromise between the two, similar to previous KLN results[135]. The posterior biases more towards v than v due to the explicithigher weight on v .Figure 5.4 shows an expanded view of the η/s marginal distributionsfor Glauber and KLN. The Glauber distribution is approximately normalwith mean ∼ ∼ ∼ HAPTER 5. QUANTIFYING QCD PROPERTIES ∼ v , the additional constraint from v shifts the distribution to somewhat smaller values and causes the plateaushape: Rather than a strong peak, there is a range of values which all fitthe data roughly equally well.Table 5.2 quantitatively summarizes the posterior distributions for eachparameter including basic statistics, credible intervals, and comparisons toprevious estimates from earlier work with the same models [232, 233]. Allprevious estimates fall within 95% credible intervals, and most within 68%. With a markedly improved computational model, more parameters, andincreased constraining power from additional observables, this analysis [171]delivers new insights on the initial state of heavy-ion collisions and on QGPmedium properties, especially the temperature dependence of shear and bulkviscosity.Compared to
A proof of concept , the present model is much more flexi-ble, owing in large part to the parametric initial condition model TRENTo(section 3.1). This adaptability is of paramount importance to ensure faith-ful uncertainty quantification. Consider, for example, the posterior distri-butions for η/s in figure 5.4, obtained using the Glauber and KLN initialcondition models; the two distributions are almost entirely incompatible, de-spite the hydrodynamic model and the rest of the analysis being identical.This happened, in short, because the KLN model tends to produce moreelliptic initial geometry than Glauber, so requires a larger η/s to describeelliptic flow v .More generally, the choice of initial condition model can strongly affectthe estimates of η/s and other QGP medium properties. Since we do notknow the precise nature of the initial state, we should incorporate that uncer-tainty into our estimates of other model parameters. The TRENTo modelenables this by parametrically interpolating among a family of physicallyreasonable initial condition models, so that when we marginalize over itsparameters, we propagate any remaining uncertainty into all other param-eter estimates. Thus, by employing a flexible model, we can simultaneously characterize the initial state and QGP medium.Another way to view this: Choosing a specific initial condition modelis a strong prior, equivalent to asserting that particular model is the trueinitial condition. The posterior distribution will then reflect this prior. So HAPTER 5. QUANTIFYING QCD PROPERTIES η/s posterior distributions for Glauber and KLN are compatibleafter all—they are simply the consequences of different priors. On the otherhand, TRENTo is effectively a weak prior on the initial condition; as demon-strated in section 3.1, it can mimic the behavior of—and continuously inter-polate among—various particular initial condition models, including KLN,IP-Glasma, EKRT, and wounded nucleon.
Initial conditions
The TRENTo model is identical to the description insection 3.1, except there is no minimum nucleon distance parameter. Thereis no pre-equilibrium free-streaming stage.
Hydrodynamics and particlization
The hydrodynamic model has tem-perature-dependent shear and bulk viscosity, although the parametrizationsare somewhat different from section 3.3 (see below). The particlizationmodel lacks bulk viscous corrections (but does implement shear corrections).As stated in the original publication, “This precludes any quantitative con-clusions on bulk viscosity, since we are only allowing bulk viscosity to affectthe hydrodynamic evolution, not particlization. We will, however, be ableto determine whether ζ/s is nonzero.”
Parameters
There are nine model parameters for estimation, summarizedwith their ranges in table 5.3. Four control the parametric initial state:1. Initial condition normalization factor.2. TRENTo entropy deposition parameter p in the generalized meanansatz s ∝ ˜ T pA + ˜ T pB ! /p , (5.3)where ˜ T is a fluctuated participant thickness function˜ T ( x, y ) = N part X i =1 u i T p ( x − x i , y − y i ) , (5.4)with u i (a random fluctuation factor) and T p (the nucleon thicknessfunction) defined below.3. Multiplicity fluctuation parameter k . Nucleon fluctuation factors u i are sampled from a gamma distribution with unit mean and variance HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.3
Input parameter ranges for the initial conditionand hydrodynamic models.Parameter Description RangeNorm Overall normalization 100–250 p Entropy deposition parameter − k Multiplicity fluct. shape 0.8–2.2 w Gaussian nucleon width 0.4–1.0 fm η/s hrg Const. shear viscosity,
T < T c η/s min Shear viscosity at T c η/s slope Slope above T c − ζ/s norm Prefactor for ( ζ/s )( T ) 0–2 T switch Particlization temperature 135–165 MeV /k , whose probability density is P k ( u ) = k k Γ( k ) u k − e − ku . (5.5)4. Gaussian nucleon width w , which determines initial-state granularitythrough the nucleon thickness function T p ( x, y ) = 12 πw exp (cid:18) − x + y w (cid:19) . (5.6)The remaining five parameters are related to the QGP medium:5–7. The three parameters ( η/s hrg, min, and slope) that set the tempera-ture dependence of the specific shear viscosity in the piecewise linearparametrization( η/s )( T ) = ( ( η/s ) min + ( η/s ) slope ( T − T c ) T > T c ( η/s ) hrg T ≤ T c . (5.7)8. Normalization prefactor ( ζ/s ) norm for the temperature dependence of HAPTER 5. QUANTIFYING QCD PROPERTIES ζ/s )( T ) = ( ζ/s ) norm C + λ exp[( x − /σ ]+ λ exp[( x − /σ ] T < T a A + A x + A x T a ≤ T ≤ T b C + λ exp[ − ( x − /σ ]+ λ exp[ − ( x − /σ ] T > T b , (5.8)with x = T /T and coefficients C = 0 . , C = 0 . ,A = − . , A = 27 . , A = − . ,σ = 0 . , σ = 0 . , σ = 0 . , σ = 0 . ,λ = 0 . , λ = 0 . , λ = 0 . , λ = 0 . ,T = 0 .
18 GeV , T a = 0 . T , T b = 1 . T . Qualitatively, this form peaks near T = 180 MeV and falls off expo-nentially on either side.9. Particlization temperature T switch . Observables
Centrality dependence of identified particle yields dN/dy and mean transverse momenta h p T i , for charged pions, kaons, and protons,as well as two-particle anisotropic flow coefficients v n { } for n = 2, 3, 4.Table 5.4 summarizes the observables including kinematic cuts, centralityclasses, and experimental data, which are all from the ALICE experiment,Pb-Pb collisions at √ s NN = 2 .
76 TeV [15, 16].
Table 5.4
Experimental data to be compared with model calculations.Observable Particle species Kinematic cuts Centrality classes Ref.Yields dN/dy π ± , K ± , p ¯ p | y | < . h p T i π ± , K ± , p ¯ p | y | < . v n { } all charged | η | < n = 2, 3, 4 0 . < p T < . n = 2 only: 50–60, 60–70 Parameter estimation method
Nearly the same as in
A proof of concept (subsection 5.1.1), namely, the likelihood is P ( D| x ) ∝ exp (cid:26) −
12 [ z m ( x ) − z e ] T Σ − z [ z m ( x ) − z e ] (cid:27) , (5.9) HAPTER 5. QUANTIFYING QCD PROPERTIES z = diag( σ z z e ) , σ z = 0 . . (5.10)The sole difference is the uncertainty fraction: a more conservative 10%compared to 6% previously. This subsection is adapted from:J. E. Bernhard et al., “Applying Bayesian parameter estimation to rel-ativistic heavy-ion collisions: simultaneous characterization of the initialstate and quark-gluon plasma medium”, Phys. Rev.
C94 , 024907 (2016),arXiv: .The primary result of this study is the posterior distribution for the modelparameters, figure 5.5. In fact, this figure contains two posterior distribu-tions: one from calibrating to identified particle yields dN/dy (blue, lowertriangle), and the other from calibrating to charged particle yields dN ch /dη (red, upper triangle). We performed the alternate calibration to chargedparticles because the model could not simultaneously describe all identifiedparticle yields for any parameter values, as will be demonstrated shortly.In figure 5.5, the diagonal plots are marginal distributions for each modelparameter (all other parameters integrated out) from the calibrations toidentified (blue) and charged (red) particles, while the off-diagonals are jointdistributions showing correlations among pairs of parameters from the cali-brations to identified (blue, lower triangle) and charged (red, upper triangle)particles. Operationally, these are all histograms of MCMC samples.We discuss the posterior distributions in detail in the following subsec-tions. First, let us introduce several ancillary results.Table 5.5 contains quantitative estimates of each parameter extractedfrom the posterior distributions. The reported values are the medians of eachparameter’s distribution, and the uncertainties are highest posterior density(HPD) 90% credible intervals. Note that some estimates are influenced bylimited prior ranges, e.g. the lower bound of the nucleon width w .Figure 5.6 compares simulated observables (see table 5.4) to experimen-tal data. The top row has explicit model calculations at each of the 300design points; recall that all model parameters vary across their full ranges,leading to the large spread in computed observables. The bottom row showsemulator predictions of 100 random samples from the identified particle pos-terior distribution (these are visually indistinguishable for the charged par-ticle posterior). Here, the model has been calibrated to experiment, so its HAPTER 5. QUANTIFYING QCD PROPERTIES n o r m norm p k w [fm] ´=s min ´=s slope † ³=s norm n o r m T sw [GeV] -1.00.01.0 p p k k w [ f m ] w [ f m ] ´ = s m i n ´ = s m i n ´ = s s l o p e † ´ = s s l o p e † ³ = s n o r m ³ = s n o r m
100 130 160 norm T s w [ G e V ] -1.0 0.0 1.0 p k w [fm] ´=s min ´=s slope † ³=s norm T sw [GeV] T s w [ G e V ] Figure 5.5
Posterior distributions for the model parameters from calibrating toidentified particles yields (blue, lower triangle) and charged particles yields (red,upper triangle). The diagonal has marginal distributions for each parameter, whilethe off-diagonal contains joint distributions showing correlations among pairs ofparameters. † The units for η/s slope are [GeV − ]. HAPTER 5. QUANTIFYING QCD PROPERTIES T r a i n i n g d a t a ¼ § K § p ¹ p Yields dN=dy ¼ § K § p ¹ p Mean p T [GeV] v v v Flow cumulants v n © ª Centrality % P o s t e r i o r s a m p l e s ¼ § K § p ¹ p Centrality % ¼ § K § p ¹ p Centrality % v v v Figure 5.6
Simulated observables compared to experimental data from the AL-ICE experiment [15, 16]. Top row: explicit model calculations for each of the 300design points, bottom: emulator predictions of 100 random samples drawn fromthe posterior distribution. Left column: identified particle yields dN/dy , middle:mean transverse momenta h p T i , right: flow cumulants v n { } . calculations are clustered tightly around the data—although some uncer-tainty remains since the samples are drawn from a posterior distribution offinite width. Overall, the calibrated model provides an excellent simultane-ous fit to all observables except the pion/kaon yield ratio, which (althoughit is difficult to see on a log scale) deviates by roughly 10–30%. We addressthis deficiency in the following subsections. Initial condition parameters
The first four parameters are related to the initial condition model. Pro-ceeding in order:The normalization factor is not a physical parameter but nonethelessmust be tuned to fit overall particle production. Both calibrations producednarrow posterior distributions, with the identified particle result locatedslightly lower to compromise between pion and kaon yields. There are somemild correlations between the normalization and other parameters that affectparticle production.The TRENTo entropy deposition parameter p introduced in equation HAPTER 5. QUANTIFYING QCD PROPERTIES −1.0 −0.5 0.0 0.5 1.0 p KLN EKRT WN
Figure 5.7
Posterior distribution of the TRENTo entropy de-position parameter p introduced in equation (5.3). Approximate p -values are annotated for the KLN ( p ≈ − . ± . p ≈ . ± . p = 1) models. (5.3) has a remarkably narrow distribution, with the two calibrations inexcellent agreement. The estimated value is essentially zero with approxi-mate 90% uncertainty ± .
2, meaning that initial state entropy deposition isroughly proportional to the geometric mean of participant nuclear thicknessfunctions, s ∼ q ˜ T A ˜ T B . This confirms previous analysis of the TRENTomodel which demonstrated that p ≈ p and any other parameters, suggesting that its optimalvalue is mostly factorized from the rest of the model.Further, recall that the p parameter smoothly interpolates among differ-ent classes of initial condition models; figure 5.7 shows an expanded view ofthe posterior distribution along with the approximate p -values for the othermodels in figure 3.6. The EKRT model (and presumably IP-Glasma as well)lie squarely in the peak—this helps explain their success—while the KLNand wounded nucleon models are considerably outside.The distributions for the multiplicity fluctuation parameter k are quitebroad, indicating that it’s relatively unimportant for the present model andobservables. Indeed, these fluctuations are overwhelmed by nucleon positionfluctuations in large collision systems such as Pb+Pb.The Gaussian nucleon width w has fairly narrow distributions mostlywithin 0.4–0.6 fm. It appears we did not extend the initial range low enoughand so the posteriors are truncated; however we still resolve peaks at ∼ . ∼ .
49 fm for the identified and charged particle calibrations, respec-tively. Since the distributions are asymmetric, the median values are some-what higher than the modes. The quantitative estimates and uncertaintiesare in good agreement with the gluonic widths extracted from deep inelastic
HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.5
Estimated parameter values (medians) and uncertainties(90% credible intervals) from the posterior distributions calibratedto identified and charged particle yields (middle and right columns,respectively). The distribution for T switch based on charged particlesis essentially flat, so we do not report a quantitative estimate.Calibrated to:Parameter Identified ChargedNormalization 120 . +8 . − . . +11 . − . p − . +0 . − . . +0 . − . k . +0 . − . . +0 . − . w [fm] 0 . +0 . − . . +0 . − . η/s min 0 . +0 . − . . +0 . − . η/s slope [GeV − ] 0 . +0 . − . . +0 . − . ζ/s norm 1 . +0 . − . . +0 . − . T switch [GeV] 0 . +0 . − . — scattering data at HERA [235–237] and support the values used in EKRTand IP-Glasma studies [103, 131]. We also observe striking correlations be-tween the nucleon width and QGP viscosities—this is because decreasing thewidth leads to smaller scale structures and steeper gradients in the initialstate. So e.g. as the nucleon width decreases, average transverse momentumincreases, and bulk viscosity must increase to compensate. This explainsthe strong anti-correlation between w and ζ/s norm. QGP medium parameters
The shear viscosity parameters ( η/s ) min,slope set the temperature depen-dence of η/s according to the linear ansatz( η/s )( T ) = ( η/s ) min + ( η/s ) slope ( T − T c ) (5.11)for T > T c . The full parametrization, equation (5.7), also includes a constant( η/s ) hrg for T < T c ; this parameter was included in the calibration butyielded an essentially flat posterior distribution, implying that it has littleto no effect. This is not surprising, since hadronic viscosity is largely handledby UrQMD, not the hydrodynamic model. Therefore, we omit ( η/s ) hrg fromthe posterior distribution visualizations and tables.Examining the marginal distributions for η/s min and slope, we see a HAPTER 5. QUANTIFYING QCD PROPERTIES
Temperature [GeV] ´ = s KSS bound = ¼ Prior rangePosterior median90% CR
Figure 5.8
Estimated temperature dependence of the shearviscosity ( η/s )( T ) for T > T c = 0 .
154 GeV. The grayshaded region indicates the prior range for the linear ( η/s )( T )parametrization equation (5.11), the blue line is the medianfrom the posterior distribution, and the blue band is a 90%credible region. The horizontal gray line indicates the KSSbound η/s ≥ / π [99, 238, 239]. clear preference for ( η/s ) min (cid:46) .
15 and a slight disfavor of steep slopes; how-ever, the marginal distributions do not paint a complete picture. The jointdistribution shows a salient correlation between the two parameters, hence,while neither η/s min nor slope are strongly constrained independently, alinear combination is quite strongly constrained. Figure 5.8 visualizes thecomplete estimate of the temperature dependence of η/s via the medianmin and slope from the posterior (for identified particles) and a 90% credi-ble region. This visualization corroborates that the posterior for ( η/s )( T ) ismarkedly narrower than the prior and further reveals that the uncertainty issmallest at intermediate temperatures, T ∼ √ s NN = 2 .
76 TeV—perhaps it is where the system spends most ofits time and hence where most anisotropic flow develops, for instance—andthus the data provide a “handle” for η/s around 200 MeV. Data at otherbeam energies and other, more sensitive observables could provide additionalhandles at different temperatures, enabling a more precise estimate of thetemperature dependence of η/s .This result for ( η/s )( T ) supports several recent findings using other mod-els: a detailed study using the EKRT model [103] showed that a combi- HAPTER 5. QUANTIFYING QCD PROPERTIES η/s =0 .
095 reported [108] using the IP-Glasma model and the same bulk viscosityparametrization, equation (5.8). Finally, the present result remains compat-ible (within uncertainty) with the KSS bound η/s ≥ / π [99, 238, 239].One should interpret the estimate of ( η/s )( T ) depicted in figure 5.8 withcare. We asserted a somewhat restricted linear parametrization reaching aminimum at a fixed temperature, and evidently may not have extended theprior range for the slope high enough to bracket the posterior distribution;these assumptions, along with the flat 10% uncertainty [see equation (5.10)],surely affect the precise result. And in general, a credible region is not astrict constraint—the true function may lie partially or completely (howeverimprobably) outside the estimated region. Yet the overarching messageholds: we find the least uncertainty in η/s at intermediate temperatures,and estimate that its temperature dependence has at most a shallow positiveslope.For the ζ/s norm [the prefactor for the parametrization equation (5.8)],the calibrations yielded clearly peaked posterior distributions located slight-ly above one. Hence, the estimate is comfortably consistent with leavingthe parametrization unscaled, as in [108]. As noted in the previous subsec-tion, there is a strong anti-correlation between ζ/s norm and the nucleonwidth. We also observe a positive correlation with η/s min, which initiallyseems counterintuitive. This dependence arises via the nucleon width: in-creasing bulk viscosity requires decreasing the nucleon width, which in turnnecessitates increasing shear viscosity to damp out the excess anisotropy.Given the previously mentioned shortcomings in the current treatment ofbulk viscosity (neglecting bulk corrections at particlization, lack of a dy-namical pre-equilibrium phase), we refrain from making any quantitativestatements. What is clear, however, is that a nonzero bulk viscosity is nec-essary to simultaneously describe transverse momentum and flow data.The distributions for the particlization temperature T switch have by farthe most dramatic difference between the two calibrations. The posteriorfrom identified particle yields shows a sharp peak centered at T ≈
148 MeV,just below T c = 154 MeV; but with charged particle yields, the distribu-tion is nearly flat. This is because the final particle ratios—while somewhatmodified by scatterings and decays in the hadronic phase—are largely de-termined by the thermal ratios at the particlization temperature. So, when HAPTER 5. QUANTIFYING QCD PROPERTIES T switch is tightlyconstrained; on the other hand, lacking these data there is little else todetermine an optimal switching temperature. This reinforces the originalhybrid model postulate—that both hydro and Boltzmann transport modelspredict the same medium evolution within a temperature window [148, 150,151].Note that, while we do see a narrow peak for T switch , the model cannot si-multaneously fit pion, kaon, and proton yields; in particular, the pion/kaonratio is 10–30% low. The peak thus arises from a compromise between pionsand kaons—not an ideal fit—so we do not consider the quantitative valueof the peak to be particularly meaningful. This is a long-standing issue inhybrid models [241] and therefore likely indicates a more fundamental prob-lem with the particle production scheme rather than one with this specificmodel. Verification of high-probability parameters
As a final verification of emulator predictions and the model’s accuracy, wecalculated a large number of events using high-probability parameters andcompared the resulting observables to experiment. We chose two sets ofparameters based on the peaks of the posterior distributions, listed in table5.6. These values approximate the “most probable” parameters and thecorresponding model calculations should optimally fit the data.We evaluated O (10 ) minimum-bias events (no emulator) for each set ofparameters and computed observables, shown along with experimental datain figure 5.9. Solid lines represent calculations using parameters based onthe identified particle posterior while dashed lines are based on the chargedparticle posterior. Note that these calculations include a peripheral central-ity bin (70–80%) that was not used in parameter estimation. Table 5.6
High-probability parameters chosen based on the pos-terior distributions and used to generate figure 5.9. Pairs of val-ues separated by slashes are based on identified / charged particleyields, respectively. Single values are the same for both cases.Initial condition QGP mediumnorm 120. / 129. η/s min 0.08 p η/s slope 0.85 / 0.75 GeV − k ζ/s norm 1.25 / 1.10 w T switch HAPTER 5. QUANTIFYING QCD PROPERTIES ¼ § K § p ¹ pN ch £ solid: identifieddashed: charged Yields dN=dy , dN ch =d´ Centrality % M o d e l / E x p ¼ § K § p ¹ p Mean p T [GeV] Centrality % v v v Flow cumulants v n © ª Centrality %
Figure 5.9
Model calculations using the high-probability parameters listed in ta-ble 5.6. Solid lines are calculations using parameters based on the identified particleposterior, dashed lines are based on the charged particle posterior, and points aredata from the ALICE experiment [15, 16]. Top row: calculations of identified orcharged particle yields dN/dy or dN ch /dη (left), mean transverse momenta h p T i (middle), and flow cumulants v n { } (right) compared to data. Bottom: ratio ofmodel calculations to data, where the gray band indicates ± We observe an excellent overall fit; most calculations are within 10% ofexperimental data, the notable exceptions being the pion/kaon ratio (dis-cussed in the previous subsection) and central elliptic flow, both of whichare general problems within this class of models. Total charged particle pro-duction is nearly perfect—within 2% of experiment out to 80% centrality—indicating that the issues with identified particle ratios arise in the parti-clization and/or hadronic phases, not in initial entropy production. The v mismatch in the most central bin is a manifestation of the experimentalobservation that elliptic and triangular flow converge to nearly the samevalue in ultra-central collisions [15, 242], a phenomenon that hydrodynamicmodels have yet to explain [125, 243]. This final act represents the culmination of this work. Leveraging an ad-vanced computational model, Bayesian parameter estimation with rigorousuncertainty quantification, and diverse experimental data from two beamenergies, it lives up to its title.Building upon
A more flexible approach , the model is now more phys-
HAPTER 5. QUANTIFYING QCD PROPERTIES √ s = 2 .
76 and 5.02TeV, reduces the uncertainties on important model parameters, such as thetemperature-dependent shear and bulk viscosity. Those uncertainties arethe first in this series with true quantitative meaning, as they now accountfor all experimental and model errors.Before proceeding to the details and results, I should note that I pre-sented a preliminary version of this analysis at the Quark Matter 2017 con-ference [215, 216]. But with only minor differences compared to this finalversion, it does not warrant a separate discussion.
The model is exactly as described in chapter 3, with the five stages:1. TRENTo parametric initial conditions.2. Pre-equilibrium free streaming.3. Viscous relativistic 2+1D hydrodynamics, implemented by the OhioState University group.4. Particlization, performed by new implementation frzout .5. UrQMD for the hadronic phase.The model is identical at the two beam energies except for the inelasticnucleon cross section and the initial condition normalization factor. Theparameters to be estimated are listed below and summarized in table 5.7.
Initial condition parameters p . The initial density of par-tons is n = Norm × (cid:18) ˜ T pA + ˜ T pB (cid:19) /p , (5.12) HAPTER 5. QUANTIFYING QCD PROPERTIES T is a participant thickness function˜ T ( x, y ) = N part X i =1 u i T p ( x − x i , y − y i ) , (5.13)with the fluctuation factors u i and nucleon thickness function T p de-fined below.4. Gaussian nucleon width w of the nucleon thickness function T p ( x, y ) = 12 πw exp (cid:18) − x + y w (cid:19) . (5.14)5. Standard deviation of nucleon multiplicity fluctuations σ fluct = 1 / √ k ,where k is the shape parameter of the unit-mean gamma distribution P k ( u ) = k k Γ( k ) u k − e − ku . (5.15)6. Minimum distance between nucleons d min (subsection 3.1.4), trans-formed to the volume d . Pre-equilibrium parameter
7. Free-streaming time τ fs . QGP medium parameters η/s min, slope, and curvature (crv), which set the temperature depen-dence of the QGP specific shear viscosity for
T > T c as the modifiedlinear ansatz( η/s )( T ) = ( η/s ) min + ( η/s ) slope · ( T − T c ) · ( T /T c ) ( η/s ) crv . (5.16)11. Constant value of η/s in the hadronic phase ( T < T c ) of the hydrody-namic model (see discussion on page 67).12–14. ζ/s max, width, and location ( T ), which set the temperature de-pendence of the QGP specific bulk viscosity as the three-parameter(unnormalized) Cauchy distribution( ζ/s )( T ) = ( ζ/s ) max (cid:18) T − ( ζ/s ) T ( ζ/s ) width ) (cid:19) . (5.17)15. Particlization temperature T switch . HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.7
Model parameters to be estimated and their design ranges.Parameter Description RangeNorm Normalization factor 8–20 (2.76 TeV)10–25 (5.02 TeV) p Entropy deposition parameter − / / σ fluct Multiplicity fluct. std. dev. 0–2 w Gaussian nucleon width 0.4–1.0 fm d Minimum nucleon volume 0–1.7 fm τ fs Free streaming time 0–1.5 fm /cη/s hrg Const. shear viscosity,
T < T c η/s min Shear viscosity at T c η/s slope Slope above T c − η/s crv Curvature above T c − ζ/s max Maximum bulk viscosity 0–0.1 ζ/s width Peak width 0–0.1 GeV ζ/s T Peak location 150–200 MeV T switch Particlization temperature 135–165 MeV
All experimental data are from ALICE, Pb-Pb collisions at √ s = 2 .
76 and5.02 TeV. At the time of this writing, some datasets are not available at5.02 TeV, as noted below. The calibration observables are the centralitydependence of: • Charged-particle multiplicity dN ch /dη at midrapidity ( | η | < .
5) [14,20]. • Identified particle yields dN/dy of pions, kaons, and protons at midra-pidity ( | y | < .
5) (2.76 TeV only) [16]. • Transverse energy production dE T /dη at midrapidity ( | η | < .
6) (2.76TeV only) [21]. • Identified particle mean p T of pions, kaons, and protons at midrapidity( | y | < .
5) (2.76 TeV only) [16]. • Mean transverse momentum fluctuations δp T / h p T i (charged particles, | η | < .
8, 0 . < p T < . • Anisotropic flow cumulants v n { } from two-particle correlations, n =2, 3, 4 (charged particles, | η | < .
8, 0 . < p T < . HAPTER 5. QUANTIFYING QCD PROPERTIES
Mean transverse momentum fluctuations
The event-by-event fluctuations of the mean transverse momentum is a newcalibration observable in this analysis, included to provide more informa-tion on the p T distributions (beyond simply the mean). The dynamicalfluctuations of mean p T (as opposed to random statistical fluctuations) arequantified by the two-particle correlator [19]( δp T ) = DD(cid:0) p T,i − h p T i (cid:1)(cid:0) p T,j − h p T i (cid:1)EE , (5.18)where the outer double average runs over pairs of particles i, j in the sameevent and over events in a centrality class, and h p T i is the usual meantransverse momentum of the centrality class. This is typically normalizedby the mean p T to form the dimensionless ratio δp T / h p T i , i.e. the relativedynamical fluctuations.The expression (5.18) is numerically inconvenient since it involves a sumover pairs of particles. To recast it in a more favorable form, we first writeout the sums as( δp T ) = 1 P n ev k N pairs k n ev X k N k X i,j>i (cid:0) p T,i − h p T i (cid:1)(cid:0) p T,j − h p T i (cid:1) , (5.19)where index k runs over all n ev events in the centrality class, N k is thenumber of particles that satisfy the kinematic cuts in event k , and indices i, j run over all N pairs k = N k ( N k − / k . Now,in general, a sum over pairs can be expanded as N X i,j>i a i a j = 12 "(cid:18) N X i a i (cid:19) − N X i a i . (5.20)Applying this to (5.19) and collecting terms gives( δp T ) = 1 P n ev k N pairs k n ev X k " (cid:18) N k X i p T,i (cid:19) − N k X i p T,i (5.21)+ h p T i ( N k − N k X i p T,i + h p T i N pairs k , which involves only sums of p T and p T . HAPTER 5. QUANTIFYING QCD PROPERTIES
The method is exactly as described in chapter 4. A few specifics: • The parameter design is a 500 point Latin hypercube sample, repeatedat the two beam energies for 1000 total design points. • Model outputs at each beam energy are postprocessed separately, i.e.with independent PCA transformations. The first 10 principal com-ponents are used, accounting for about 99.6% of the total variance. • Independent Gaussian process emulators predict the model outputsfor each beam energy. Subsection 4.3.4 validates their performance. • The likelihood is computed as described in subsection 4.4.2, with ajoint likelihood P ( D| x ) = P ( D . | x . ) P ( D . | x . ) . (5.22)Here, the only difference between each energy’s parameter vector isthe normalization factor; all other parameters are the same. Figure 5.10 shows the now-familiar triangular visualization of the posteriordistribution, where the diagonal subplots are marginal distributions for eachparameter, and the off-diagonals are joint marginal distributions showingcorrelations between pairs of parameters. The annotations along the diago-nal, also listed in table 5.8, are quantitative parameter estimates, consistingof the posterior median and highest posterior density (HPD) 90% credibleinterval for each parameter. Note that for asymmetric distributions, themedian does not coincide with the mode (peak value).Before examining the posterior distribution in more detail, observe alsofigure 5.11, showing the model calculations at each design point compared toexperimental data, and 5.12, which is analogous but with emulator predic-tions at parameter points randomly drawn from the posterior distribution.As depicted in the second figure, the calibrated model accurately describesalmost all experimental data points—the notable exception being periph-eral mean p T fluctuations, which I will address later. Overall, the fit issuperior to the previous version (figure 5.6), with less tension among theidentified particle yields and improved centrality dependence of the mean p T and elliptic flow cumulant v { } . HAPTER 5. QUANTIFYING QCD PROPERTIES N o r m . T e V . +1 . − . N o r m . T e V . +1 . − . −0.50.00.5 p . +0 . − . σ f l u c t . +0 . − . w [ f m ] . +0 . − . d m i n [ f m ] . +0 . − . τ f s [ f m / c ] . +0 . − . η / s m i n . +0 . − . η / s s l o p e [ G e V − ] . +0 . − . −101 η / s c r v − . +0 . − . ζ / s m a x . +0 . − . ζ / s w i d t h [ G e V ] . +0 . − . ζ / s T [ G e V ] . +0 . − . T s w i t c h [ G e V ] . +0 . − . Norm σ m o d e l s y s Norm −0.5 0.0 0.5 p σ fluct w [fm] d min[fm] τ fs[fm /c ] η/s min η/s slope[GeV − ] −1 0 1 η/s crv ζ/s max ζ/s width[GeV] . . . ζ/s T [GeV] . . . T switch[GeV] σ model sys . +0 . − . Figure 5.10
Posterior distribution for the model parameters.Diagonal: marginal distributions for each parameter;off-diagonal: joint marginal distributions betweenpairs of parameters. The annotated estimatesare the posterior medians with 90%HPD credible intervals.
HAPTER 5. QUANTIFYING QCD PROPERTIES
Table 5.8
Posterior parameter estimates. Reported values are theposterior medians; uncertainties are the 90% HPD credible intervals.Initial condition / Pre-eq QGP mediumNorm 13 . +1 . − . (2.76 TeV) η/s min 0 . +0 . − . . +1 . − . (5.02 TeV) η/s slope 0 . +0 . − . GeV − p . +0 . − . η/s crv − . +0 . − . σ fluct . +0 . − . ζ/s max 0 . +0 . − . w . +0 . − . fm ζ/s width 0 . +0 . − . GeV d min . +0 . − . fm ζ/s T . +0 . − . GeV τ fs . +0 . − . fm /c T switch . +0 . − . GeV
Let us now consider each model parameter, proceeding in order alongthe diagonal. It will be useful to refer back to the posterior distributionfrom
A more flexible approach , figure 5.5.
Initial condition
The normalization factors are well-constrained by particle and energy pro-duction data. Interestingly, the factor at 5.02 TeV is about 30% larger thanat 2.76 TeV, even though experimental particle production only increasesby about 20% between the two energies [20]. This occurred because only dN ch /dη data are available at 5.02 TeV, while at 2.76 TeV there is also trans-verse energy and identified particle yields. Since there is still some tensionamong these observables—notice that the model samples of dN ch /dη areslightly low at 2.76 TeV—the normalization decreases.The TRENTo entropy deposition parameter p has a narrow, approxi-mately normal distribution centered at essentially zero, with half the uncer-tainty of the previous study, about ± .
08 compared to ± .
17. This stronglycorroborates that initial entropy deposition (or particle production) goes asthe geometric mean of participant nuclear thickness, s ∼ n ∼ q ˜ T A ˜ T B , (5.23)see equations (3.4) and (3.5). Although this does not tell us the physicalmechanism driving entropy deposition—and many possibly theories couldpredict such general behavior—it does rule out models that do not havethis approximate scaling. For example, consider figure 5.13, which shows HAPTER 5. QUANTIFYING QCD PROPERTIES d N c h / d η , d N / d y , d E T / d η [ G e V ] N ch E T πKp Pb-Pb 2.76 TeV N ch E T πKp Y i e l d s Pb-Pb 5.02 TeV › p T fi [ G e V ] πKp πKp M e a n p T δ p T / › p T fi M e a n p T f l u c t u a t i o n s Centrality % v n ' “ v v v Centrality % v v v F l o w c u m u l a n t s Figure 5.11
Model calculations at each design point. Experimental data arefrom ALICE, Pb-Pb collisions at √ s = 2 .
76 TeV (left column) [14–16, 19, 21] and5.02 TeV (right) [20, 22]. Some datasets are not available at 5.02 TeV.
HAPTER 5. QUANTIFYING QCD PROPERTIES d N c h / d η , d N / d y , d E T / d η [ G e V ] N ch E T πKp Pb-Pb 2.76 TeV N ch E T πKp Y i e l d s Pb-Pb 5.02 TeV › p T fi [ G e V ] πKp πKp M e a n p T δ p T / › p T fi M e a n p T f l u c t u a t i o n s Centrality % v n ' “ v v v Centrality % v v v F l o w c u m u l a n t s Figure 5.12
Emulator predictions at parameter points randomly drawn from theposterior distribution. Experimental data are the same as figure 5.11.
HAPTER 5. QUANTIFYING QCD PROPERTIES −1.0 −0.5 0.0 0.5 1.0 p KLN EKRT /IP-Glasma Woundednucleon . +0 . − . Figure 5.13
Marginal distribution of the TRENTo entropy deposition parameter p . Approximate p -values, established in subsection 3.1.3, are annotated for the KLN( p ≈ − . ± . p ≈ . ± . p = 1) models. an expanded view of the marginal distribution with the p -values of severalexisting models marked (compare to figure 5.7). Clearly, the KLN andwounded nucleon models are excluded by this analysis, while EKRT andIP-Glasma are substantiated.The standard deviation of nucleon multiplicity fluctuations has a strongpeak around σ fluct ∼
1, which corresponds to exponentially distributed fluc-tuations. But the most compelling feature is simply that there are lower andupper bounds, meaning that some fluctuations, but not too much, are neces-sary to describe the data. Of course, we already knew this from experimentalproton-proton collision multiplicity distributions (see figure 3.2), but it is re-markable that the parameter estimation framework can extract this informa-tion from Pb-Pb data alone. Note that the transformation from the gammadistribution shape parameter k to the standard deviation σ fluct = 1 / √ k fa-cilitated this inference, since zero fluctuations ( σ fluct = 0) corresponds to k → ∞ .It appears that the design range for the Gaussian nucleon width w wastruncated on the upper end, although we do resolve a peak at w ≈ . ∼ w < . w is the width HAPTER 5. QUANTIFYING QCD PROPERTIES w = 1 fm implies very large nucleons ofdiameter ∼ w → . d min enters the analysis as the vol-ume d , i.e. there is a uniform prior on d from 0 to 1.7 fm , and thevisualized distribution is over the volume (note the nonuniform axis tickmarks). The distribution is more or less flat, suggesting that d min does notinfluence the overall fit of the model to the present observables. However,there is no doubt that this parameter does affect the model: It modifiesthe initial eccentricity distributions and the final flow coefficients, and theemulator captures this dependence. We can see a hint of this in the jointdistribution between d min and σ fluct , which shows that increased multiplicityfluctuations correlate with increased minimum distance. The interpretation:A minimum distance prevents nucleons from piling up, but since only thebeam-integrated thickness matters, increasing fluctuations—which scale thethickness of each nucleon—easily negates this effect. Evidently, d min onlyweakly affects model calculations of the present observables, but it’s possiblethat calibrating to other data could reveal a nontrivial distribution for d min . Pre-equilibrium
The sole free parameter related to pre-equilibrium evolution is the free-streaming time τ fs , whose distribution has a peak at ∼ . ± . /c , consis-tent with the long-standing belief that hydrodynamic evolution begins early,around O (1 fm /c ). Although free streaming is not the most realistic model,the existence of a peak means that a brief weakly-coupled pre-equilibriumstage is necessary to describe to the data.Note that, in the present analysis, τ fs is required to be the same at bothbeam energies, which may not be the case. Future studies could seek toestimate independent values at different energies. QGP medium
The most salient QGP medium parameters are those controlling its trans-port coefficients, namely, the temperature dependence of the specific shearand bulk viscosity, ( η/s )( T ) and ( ζ/s )( T ), the determination of which is aprimary goal of this work.The marginal distribution for the minimum value of η/s is approximately HAPTER 5. QUANTIFYING QCD PROPERTIES
150 200 250 300
Temperature [MeV] η / s / π Shear viscosity
Posterior median90% credible region
150 200 250 300
Temperature [MeV] ζ / s Bulk viscosity
Figure 5.14
Estimated temperature dependence of the specific shear and bulkviscosity, ( η/s )( T ) and ( ζ/s )( T ). Lines are the parametrizations (5.16) and (5.17)with the parameters set to their posterior median values; shaded regions are 90%credible regions. The horizontal line in the shear viscosity plot indicates the con-jectured lower bound 1 / π [99, 238, 239]. normal with a peak at ( η/s ) min ≈ .
085 and 90% credible interval 0.05–0.11,strikingly close to the conjectured lower bound 1 / π ’ .
08 [99, 238, 239].This of course does not prove the conjecture, but it is sensible to concludethat the QGP created in heavy-ion collisions behaves as a nearly-ideal fluidnear the transition temperature.Regarding the other η/s parameters, there is a mild preference for asmall positive slope, although zero slope (i.e. constant η/s ) is not excluded.We observe an anti-correlation between the slope and minimum, similarto the previous study, and another anti-correlation between the slope andcurvature parameter, which itself has a broad marginal distribution withsomewhat more density at negative curvature. All of this points to η/s likely increasing slowly with temperature, perhaps with a negative secondderivative.Figure 5.14, left panel, visualizes the estimated temperature dependenceof η/s as the posterior median with a 90% credible region (compare to figure5.8, note the y -axis range is different). Similar to the previous study, thereis a marked narrowing of the uncertainty at intermediate temperatures, al-though the narrowest region is now somewhat lower, T ∼
175 MeV comparedto above 200 MeV before. It is difficult to say why the range moved, butregardless, this characteristic suggests that the data have their greatest re-
HAPTER 5. QUANTIFYING QCD PROPERTIES η/s is bestconstrained. I emphasize that nothing about the ( η/s )( T ) parametrizationwould impose such a narrowing—it arises naturally.It is possible that η/s does not reach its minimum value precisely at thetransition temperature T c , as the present parametrization requires. Futurework could add the location of the minimum as a degree of freedom andattempt to estimate it from the data.Moving on to bulk viscosity: The maximum value of ζ/s and the width ofthe peak both have skewed distributions, and their joint distribution showsthat they trade off, i.e. the peak can be tall or wide, but not both. Thisimplies that it is the integral of ( ζ/s )( T ) that matters, not its specific form.Meanwhile, the peak location ( T ) is not constrained, except for possiblyruling out a very narrow peak located at high temperature. The right panelof figure 5.14 shows the estimated temperature dependence of ζ/s , analogousto η/s .Given the excellent performance of the model and the uncertainty quan-tification framework in the present analysis—which properly accounts for ex-perimental statistical and systematic uncertainty and model uncertainty—we should take seriously the quantitative estimates shown in figure 5.14,especially their credible regions. Based on all the included information, andsubject to the assumptions of the model, there is a 90% chance that the trueQGP ( η/s )( T ) and ( ζ/s )( T ) curves lie within the pictured regions.Finally, the particlization temperature T switch has a narrow distributionlocated in the QCD crossover transition region. As established in the previ-ous study, T switch is determined primarily by identified particle yield ratios,but where there was previously a discrepancy between the pion and kaonyields, there is now much less tension. This is attributable to the inclusionof finite resonance mass width in the particlization model, which leads toincreased production of resonances that feed down to pions. Systematic uncertainty
The “ σ model sys” parameter is the model systematic uncertainty σ sys m intro-duced in subsection 4.4.2 to account for imperfections in the computationalmodel. As a reminder, it is defined relative to the overall variability of themodel, e.g. σ sys m = 0 . HAPTER 5. QUANTIFYING QCD PROPERTIES σ sys m . This reflects that, with large systematicerror, the specific values of the model parameters don’t matter as much. As a final verification of the calibrated model’s performance, I calculated alarge number of events using the maximum a posteriori ( MAP ) parameters,which are the mode of the posterior probability: x MAP = arg max x P ( x |D ) . (5.24)The MAP parameter values, determined by numerical optimization, arelisted in table 5.9. Table 5.9
Maximum a posteriori (MAP) parameters.Initial condition / Pre-eq QGP mediumNorm 13.94 (2.76 TeV) η/s min 0.08118.38 (5.02 TeV) η/s slope 1.11 GeV − p η/s crv − . σ fluct ζ/s max 0.052 w ζ/s width 0.022 GeV d min ζ/s T τ fs /c T switch Using the
MAP events, I computed the usual calibration observableslisted in subsection 5.3.2, which should approximate a “best-fit” of the modelto data. Further, if the calibrated model is a realistic representation of re-ality, it should be able to describe other observables that were not used forcalibration, and that potentially contain more information about the physi-cal system. To check this, I computed several higher-order flow observablesthat are too noisy for calibration, but are stable given the larger quantityof
MAP events.I emphasize that the following is a secondary result of the analysis; theprimary result is the full posterior distribution, which a single model cal-culation cannot capture. Ideally, one would perform model calculations ata number of parameter points sampled from the posterior distribution, butdoing so would require a prohibitive amount of computation time. About 1 × events, compared to 4 × for the design points. Starting the optimization algorithm from the posterior median.
HAPTER 5. QUANTIFYING QCD PROPERTIES d N c h / d η , d N / d y , d E T / d η [ G e V ] N ch E T πKp Yields v n ' k “ v ' “ v ' “ v ' “ v ' “ Flow cumulants2.76 TeV5.02 TeV ±10%0 20 40 60 800.91.01.1 R a t i o ±10%0 20 40 60 800.91.01.1 R a t i o › p T fi [ G e V ] πKp Mean p T δ p T / › p T fi Mean p T fluctuations ±10%0 20 40 60 80 Centrality % R a t i o ±10%0 20 40 60 80 Centrality % R a t i o Figure 5.15
Model calculations using the MAP parameters listed in table 5.9.Solid lines are calculations at √ s = 2 .
76 TeV; dashed 5.02 TeV. Filled symbols areALICE data at 2.76 TeV [14–16, 19, 21]; empty 5.02 TeV (where available) [20, 22].The ratio axes show the ratio of the model calculations to data, where the grayband indicates ± HAPTER 5. QUANTIFYING QCD PROPERTIES
MAP point compared to experimental data. The upper right “flow cu-mulants” panel also shows the four-particle elliptic flow v { } , which wasnot a calibration observable. The overall fit is superb, with almost everydata point described within 10%. Arguably the worst fit is to the mean p T fluctuations, where the model calculations do not increase rapidly enoughas a function of centrality. We can likely attribute this to the lack of nu-cleon substructure in the initial condition model; a model with quark and/orgluon constituents would have smaller hotspots, creating larger relative p T fluctuations in peripheral collisions [244].As an additional cross-check observable, I computed symmetric cumu-lants SC( m, n ), which quantify the correlations between event-by-event fluc-tuations of flow harmonics v m and v n [23, 62]. They are defined as thefour-particle observableSC( m, n ) = hh cos[ m ( φ − φ ) + n ( φ − φ )] ii− hh cos[( m ( φ − φ )] iihh cos[ n ( φ − φ )] ii≈ h v m v n i − h v m ih v n i , (5.25)where the double average is over particles and events, as usual for flowcumulants (see discussion on page 26), and the second equality is only ap-proximate due to nonflow effects. Since the two-particle correlations for v m and v n are subtracted, SC( m, n ) is zero if v m and v n are uncorrelated. Em-pirically, symmetric cumulants calculated from hydrodynamic models arehighly sensitive to the temperature dependence of η/s [23].Symmetric cumulants may be computed using Q -vectors; the single-event two-particle correlation is h cos[ n ( φ − φ )] i = 1 P M, ( | Q n | − M ) (5.26)and the four-particle mixed-harmonic correlation is [23] h cos[ m ( φ − φ ) + n ( φ − φ )] i = 1 P M, n | Q m | | Q n | − < [ Q m + n Q ∗ m Q ∗ n ] − < [ Q m Q ∗ m − n Q ∗ n ] + | Q m + n | + | Q m − n | − ( M − | Q m | + | Q n | ) + M ( M − o , (5.27)where P M,k is the number of k -particle permutations, namely P M, = M ( M − ,P M, = M ( M − M − M − . (5.28) HAPTER 5. QUANTIFYING QCD PROPERTIES S C ( m , n ) Central
Minimum bias
SC(4 , , Centrality % −0.050.000.050.100.15 S C ( m , n ) / › v m fi› v n fi Centrality % −0.250.000.250.500.751.00
SC(4 , / › v fi› v fi SC(3 , / › v fi› v fi Figure 5.16
Model calculations of symmetric cumulants using the MAP param-eters. Solid lines are calculations at 2.76 TeV; dashed at 5.02 TeV (a prediction, asthe data are not available). Data are from ALICE (2.76 TeV only) [23].
The double averages in (5.25) are then obtained by averaging over events ina centrality class, weighting the single-event correlations by P M,k .Figure 5.16 shows model calculations of SC(4 ,
2) and SC(3 , m, n ) / h v m ih v n i , compared toexperimental data [23]. Considering that this is a sensitive observable andit did not enter into the calibration, the model provides a good description,with the correct signs and qualitative centrality trends. But given this loneresult, we can only speculate why the fit is imperfect or how it could beimproved (unfortunately, a very large number of events, O (10 ), is requiredto compute symmetric cumulants with reasonable statistical noise). HAPTER 5. QUANTIFYING QCD PROPERTIES
Although
A precision extraction accomplished many of the salient goals ofthis work, there is always room for improvement. The following is a non-exhaustive list of possible enhancements; a wish list.
RHIC data
In addition to the two LHC beam energies, we can calibrateon data from gold-gold collisions at √ s = 200 GeV at the Relativistic Heavy-ion Collider, which should enhance constraining power. In principle, thereis no reason not to do this; it is a matter of running the events and takingcare to compute all observables correctly. Nucleon substructure
Implementing nucleon substructure in the initialcondition model would permit calibration to data from small collision sys-tems such as proton-lead, and could improve the performance of the modelin peripheral nucleus-nucleus collisions. In fact, this is already in progress,using an extension of TRENTo that models nucleons as superpositions ofseveral smaller constituents [245].
Full three-dimensional calculations
Moving to full three-dimensional(not boost-invariant) initial conditions and hydrodynamics would enablecalibration to new observables, such as particle rapidity distributions andrapidity-dependent flow. A recent study [246] applied Bayesian parameterestimation to constrain a 3D initial condition model, but did so withouthydrodynamics, by mapping initial-state quantities directly to final-stateobservables. This shortcut was necessary to avoid the great computationalcost of 3+1D hydrodynamics, which indeed will be difficult to overcome. Onepossible solution is to run hydrodynamic calculations on graphics processingunits (GPUs); such an implementation was recently developed [247]. GPUscan calculate single events much faster than CPUs, but the relative dearthof GPU computing resources inhibits running on a large scale.
Beam energy dependence
The only model parameter that I allowed tovary as a function of beam energy was the initial condition normalizationfactor. But in principle, all parameters related to the initial condition orpre-equilibrium stages could be functions of energy, such as the nucleon sizeand free-streaming time.
HAPTER 5. QUANTIFYING QCD PROPERTIES
Systematic uncertainty correlations
I assumed a particular correla-tion structure for experimental systematic uncertainties, as described in sub-section 4.4.2, specifically equation (4.42). This could certainly be improved,ideally with input from experimentalists. It would also be interesting to testhow much of an impact this has on parameter estimates.
Discrepancy model
The complete formulation of Bayesian model cali-bration includes a discrepancy term which accounts for deviations betweenthe model and reality [210, 212, 248]. Adding this in, the relation betweenthe model calculations and experimental data (4.35) becomes y e = y m ( x ) + δ + (cid:15) , (5.29)where δ is the discrepancy term, usually decomposed into some kind ofbasis functions. The physical model parameters are then calibrated simul-taneously with the discrepancy.In practice, the simplified model systematic error parameter σ sys m , intro-duced in equation (4.46), certainly subsumes some model discrepancy, butan explicit treatment of discrepancy would be preferable. Sensitivity analysis
A category of techniques for quantifying the rela-tionships between model inputs and outputs, sensitivity analysis providespertinent information such as which input parameters have the strongest ef-fect on the outputs, which observables constrain each parameter, and whichobservables would benefit most from reduced uncertainty. See, for example,sensitivity analysis applied to heavy-ion collisions [140] and galaxy formation[249].
The Bayesian parameter estimation method is not specific to the modeland data used in this work; it can be applied to other types of physicalmodels and experimental data which describe different aspects of heavy-ion collisions, enabling inferences on new physical properties. In particular,while this work focused on bulk properties and observables, there is alreadyprogress on quantifying properties related to hard processes, for example,a recent Bayesian analysis estimated the heavy-quark diffusion coefficient[250], and the recently created
Jetscape
Collaboration [251] is applyingsimilar techniques to jets in heavy-ion collisions.
Conclusion Q uark-gluon plasma is one of the most exotic substances ever created,and one of the most extraordinarily difficult to characterize. Producedas tiny fluid-like droplets in ultra-relativistic heavy-ion collisions, it almostinstantly disintegrates into particles—the only observable evidence of theQGP’s existence; the remnants, in essence, of a long-past explosion.But all is not lost. Although we can only observe the final state of heavy-ion collisions, we can infer the time evolution by matching the output ofdynamical model calculations to corresponding experimental observations.By encoding physical properties as model input parameters and tuning theparameters so that the model optimally describes the data, we can estimatethe fundamental properties of the QGP and related characteristics of thecollision.This idea is not new, but prior to this work, its execution in heavy-ionphysics had been limited. Most previous studies considered only a sin-gle model parameter and observable, and reported rough estimates lackingmeaningful uncertainties. This is not to disparage earlier model-to-dataanalysis—it was integral to the progression of the field and informed manyaspects of the present work—but the reality is that heavy-ion collision mod-els have multiple interrelated parameters and there are a wide variety of ex-perimental observables. If we are to claim rigorous, quantitative estimatesof QGP properties, we must account for all relevant sources of uncertaintyand demand that the model describe as much data as possible.In this dissertation, I have overcome previous limitations and producedthe first estimates of QGP properties with well-defined quantitative uncer-tainties. I developed a complete framework for applying Bayesian parameterestimation methods to heavy-ion collisions, calibrated a dynamical collision166 HAPTER 6. CONCLUSION η/s )( T ) and ( ζ/s )( T ). The final result is shown in figure 5.14,reproduced here:
150 200 250 300
Temperature [MeV] η / s / π Shear viscosity
Posterior median90% credible region
150 200 250 300
Temperature [MeV] ζ / s Bulk viscosity
The illustrated credible regions indicate quantitative 90% uncertainties, ac-counting for experimental and model errors and subject to the assumptionsof the model. The estimated minimum value of the specific shear viscosity, η/s = 0 . +0 . − . , is conspicuously close to the conjectured lower bound1 / π ∼ . η/s )( T ) and ( ζ/s )( T ), we understand the preci-sion of our knowledge. Before this work, it was not possible to constructmeaningful probability regions for temperature-dependent QGP transportcoefficients.These coefficients are fundamental physical properties; their measure-ment a long-standing primary goal of heavy-ion physics. Countless publica-tions and presentations have studied and constrained η/s and ζ/s . Whitepapers have explicitly stated determination of transport coefficients as aprincipal objective.In addition to transport properties, I simultaneously estimated charac-teristics of the initial state of heavy-ion collisions, including the scaling ofinitial state entropy deposition, the effective size of nucleons, and the dura-tion of the pre-equilibrium stage that precedes QGP formation. This was HAPTER 6. CONCLUSION do not have , andthe posterior will be artificially narrow as a result. Bayesian inference makesexplicit how posterior results depend on prior knowledge.It is quite astonishing that we are capable of creating quark-gluon plasmaand characterizing it with any precision. In all likelihood, QGP does notpresently exist anywhere else in the natural universe. Only by collidingnuclei at ultra-relativistic speeds can we compress and heat matter enoughto overcome the strong force and liberate quarks and gluons. In the firstmoments after the Big Bang, similar temperature and density may havecreated a single large QGP from which everything originated. We are, quitepossibly, studying the source material of the universe itself.There is, of course, more to be done. The estimate of the minimum valueof η/s , while more precise than previous results, still has 30% uncertainty,which would not be considered particularly precise for many other measure-ments. In section 5.4, I outlined some possible improvements to both thecomputational model and parameter estimation method which could reduceuncertainty and provide insights on new physical properties. Other exten-sions of the analysis may inform pivotal decisions such as which experimentsto run and which observables to measure.Finally, the Bayesian parameter estimation method is not specific to themodel used in this work. There are entire other classes of models and datarelated to different physical phenomena in heavy-ion collisions. Work isalready underway applying the developed methodology in these areas.Hopefully, this is only the beginning. eveloped software
Physics models and analysis tools that I have developed in my research
Original code trento
Initial condition modelRelevant section: 3.1Source code: https://github.com/Duke-QCD/trento
Documentation: http://qcd.phy.duke.edu/trento freestream
Pre-equilibrium free streamingRelevant section: 3.2Source code: https://github.com/Duke-QCD/freestream
Documentation: https://github.com/Duke-QCD/freestream frzout
Particlization model (Cooper-Frye sampler)Relevant section: 3.4Source code: https://github.com/Duke-QCD/frzout
Documentation: http://qcd.phy.duke.edu/frzout hic-eventgen
Heavy-ion collision event generatorRelevant section: 3.6Source code: https://github.com/Duke-QCD/hic-eventgen
Documentation: https://github.com/Duke-QCD/hic-eventgen hic-param-est
Implementation of Bayesian parameter estimationRelevant chapter: 4Source code: https://github.com/jbernhard/hic-param-est
Documentation: http://qcd.phy.duke.edu/hic-param-est
EVELOPED SOFTWARE
Adapted and modified code osu-hydro
The Ohio State University viscous hydrodynamics codeOriginal source: https://u.osu.edu/vishnu
Relevant section: 3.3Source code: https://github.com/jbernhard/osu-hydro
Documentation: https://github.com/jbernhard/osu-hydro urqmd-afterburner
UrQMD tailored for use as a hadronic afterburnerOriginal source: https://urqmd.org
Relevant section: 3.5Source code: https://github.com/jbernhard/urqmd-afterburner
Documentation: https://github.com/jbernhard/urqmd-afterburner cknowledgments
This work was supported by: • U.S. Department of Energy (DOE) grant number DE-FG02-05ER41367. • National Science Foundation (NSF) grant number ACI-1550225.Computation time was provided by: • The National Energy Research Scientific Computing Center (NERSC),the primary scientific computing facility for the Office of Science in theDOE. • The Open Science Grid (OSG), funded by DOE and NSF.Open-source programming libraries used in this work: • NumPy [228] • SciPy [229] • scikit-learn [223] • h5py [230] • matplotlib [231] • emcee [227] 171 eferences [1] D. R. Williams, Sun Fact Sheet , (Dec. 16, 2016) https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html .[2] J. C. Collins and M. J. Perry, “Superdense Matter: Neutrons OrAsymptotically Free Quarks?”, Phys. Rev. Lett. , 1353 (1975).[3] E. V. Shuryak, “Quantum Chromodynamics and the Theory ofSuperdense Matter”, Phys. Rept. , 71–158 (1980).[4] G. F. Chapline, M. H. Johnson, E. Teller, and M. S. Weiss, “Highlyexcited nuclear matter”, Phys. Rev. D8 , 4302–4308 (1973).[5] J. D. Bjorken, “Highly Relativistic Nucleus-Nucleus Collisions: TheCentral Rapidity Region”, Phys. Rev. D27 , 140–151 (1983).[6] A. Bazavov et al., “Equation of state in ( 2+1 )-flavor QCD”, Phys.Rev.
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