Hot and Dense QCD Matter

nraveling the Mysteries of the Strongly Interacting Quark-Gluon-Plasma

ommunity White Paper onDetermination
the Future of Relativistic
Heavy-Ion
Physics
inthe
USof QGP Parametersfrom a Global Bayesian Analysis
Steffen A. Bass

http://www.facebook.com/DukeQCD@Steffen_Bass

Phase Diagram of QCD MatterPhase Diagram of QCD Matter
Ordinary Matter:• phases determined by (electromagnetic) interaction between
molecules• apply heat & pressure to study
phase-diagram• calculate via derivatives ofpartition function

Phase Diagram of QCD Matter
Ordinary Matter:• phases determined by (electromagnetic) interaction between
molecules• apply heat & pressure to study
phase-diagram• calculate via derivatives ofpartition function

Phases of QCD matter:•heat & compress QCD matter:‣collide heavy atomic nuclei•numerical simulations:‣solve partition function (Lattice
Field Theory)

Phase Diagram of QCD Matter
Ordinary Matter:• phases determined by (electromagnetic) interaction between
molecules• apply heat & pressure to study
phase-diagram• calculate via derivatives ofpartition function

Phases of QCD matter:•heat & compress QCD matter:‣collide heavy atomic nuclei•numerical simulations:‣solve partition function (Lattice
Field Theory)

Equation of State for an ideal QGP:

(ultra-relativistic gas of massless bosons)!LGT predicts a phase-transition
to a state of deconfined nearlymassless quarks and gluons!QCD becomes simple at hightemperature and/or density

The Early Universe: Quark-Gluon-Plasma•a few microseconds after the
Big Bang the entire Universe
was in a QGP state•compressing & heating
nuclear matter allows toinvestigate the history of the
Universe•the only means of recreating
temperatures and densitiesof the early Universe is bycolliding beams of ultrarelativistic heavy-ionsProperties of QCD: Transport Coefficientsshear and bulk viscosity are defined as the coefficients in the expansion of the stress tensor in termsof the velocity fields:

Tik = ⇤ui uk + P ( ik + ui uk ) ⇥ ⇤i uk + ⇤k ui⇥

2
ik ⇤ · u +⌅ ik ⇤ · u3

η/s from Lattice QCD:
The confines of the Euklidian Formulation:•extracting η/s formally requires taking thezero momentum limit in an infinite spatial
volume, which is numerically notpossible…

•preliminary estimates:T

1.58 TC

2.32 TCη/s

0.2-0.25

0.25-0.5

A. Nakamura & S. Sakai: Phys. Rev. Lett. 94 (2005) 072305
Harvey B. Meyer: Phys. Rev. D79 (2009) 011502
Harvey B. Meyer: arXiv:0809.5202 [hep-lat]Properties of QCD: Transport Coefficientsshear and bulk viscosity are defined as the coefficients in the expansion of the stress tensor in termsof the velocity fields:

Tik = ⇤ui uk + P ( ik + ui uk ) ⇥ ⇤i uk + ⇤k ui⇥

2
ik ⇤ · u +⌅ ik ⇤ · u3

η/s from Lattice QCD:
The confines of the Euklidian Formulation:•extracting η/s formally requires taking thezero momentum limit in an infinite spatial
volume, which is numerically notpossible…

•preliminary estimates:T

1.58 TC

2.32 TCη/s

0.2-0.25

0.25-0.5

A. Nakamura & S. Sakai: Phys. Rev. Lett. 94 (2005) 072305
Harvey B. Meyer: Phys. Rev. D79 (2009) 011502
Harvey B. Meyer: arXiv:0809.5202 [hep-lat]

The determination of the QCD transport coefficients is one of the key goals of the global relativistic
heavy-ion effort!

Standing on the Shoulders of Giants
Jonah E. Bernhard, J. Scott Moreland & Steffen A. Bass,
to appear in Nature Physics on August 12th 2019
Pc/2

Water

Pc

10

2Pc

η/s

Pc/2

He

Pc
2Pc

Helium1

0.1

Quark-gluon plasma
0.0

0.5

1.0

1.5

2.0

2.5

T/Tc

H 2O

• more than a decade of hard work by multiple
research groups• cooperation between theory & experiment• significant investment by the funding agencies

An Effort by the Heavy-Ion Community
Important experimental and theoretical developments0
Huovinen et al. (2001) Phys.Lett.B503, 58
Hydro
Hydro
Hydro

2

PHENIX π
PHENIX K
PHENIX p

0.2

2000

0.1

2004

2006

fluctuations importantfor v2 analysis in small systems

2002

0
0

1
2
Transverse Momentum p [GeV]3

T

Early success of hydrodynamics missing physics
of lattice QCD equation of state and viscosity.

2

Anisotropy Parameter v

0.2

η/s=0.08 Glauberη/s=0.16 CGC IC

η/s=0.16 Glauber

0.1

2008
0
0

1
2
Transverse Momentum p [GeV]

2

2000

pT dependence

ideal hydroidentified particle flow

2002
LO pQCD

rapidity
dependence

AdS/CFT limit

2004

analysis improved
errors reduced

T

2010

n

v2
v3
v4
v5

0.2

PHENIX v2
PHENIX v3
PHENIX v4

2012

20-30%
η/s=0.08

viscous hydrofirst flow results fromviscous fluid-dynamics

2008

v3

2010

vn

Schenke et al. (2012) Phys.Rev.C85, 024901

0.3

2006

reliable QCD equation of statefrom the lattice included

3

Bounds on shear viscosity but large uncertainties
from initial conditions.

Anisotropy Parameter v

1.5

STAR charged hadrons, corrected
η/s=0.08 CGC IC

vn correlations

2012

P(vn )

0.1

0
0

1
2
Transverse Momentum p [GeV]3kinetic theory
lattice QCD
AdS/CFT limit
viscous hydro+ flow data2014

η/s near Tc

2022

2014

T

approximate range ofmaximal initial temperaturesprobed by RHIC

4

by LHC

3
2

e
enc

end

1
s
pos

0

RHIC BES-II upgrade required toHigher moments constrain viscosity and fluctuating
initial conditions better, but temperature dependenceof η/s is not yet determined.

pQCD

2012 response of the US relativistic
heavy-ion community to the request
for comments by the NSAC
subcommittee, that was tasked torecommend optimizations to the US
Nuclear Science Program over thefollowing five years.

We are well on our way deliver onthese goals!

1

Luzum, Romatschke (2008) Phys.Rev.C78, 034915

0.3

η/s

Goal: by 2022 determine thetemperature dependence of η/s andζ/s as well as relaxation times andother QGP transport coefficients ofinterest (e.g. q-hat and e-hat)

0.5experimental techniquesdeveloped

v2 systematics developed

0.3

Anisotropy Parameter v

2012 RHIC community White Paper
identified key developments andlaid out milestones for thedetermination of QGP properties:

Increasing precisionof key observable

0

0.1

0.2

ible

0.3

per
tem

ep
re d

atu

AdS/CFT limit
0.4

0.5

0.6

0.7

T[GeV]
To determine (η/s)(T) different initial
temperatures need to be accessible.
Only possible with combined data from
LHC and RHIC beam energy scan.

• measure lower
bound to ~10%
• determine (η/s)(T)dependence• measure bulk viscosity
and relaxation times• constrain initial stateand its fluctuations

2022
0

0.5

1

1.5

2

Telescopes for the Early Universe:
Heavy-Ion Collider Facilities

Heating & Compressing QCD Matter
The only way to heat & compress QCD matter under controlledlaboratory conditions is by colliding two heavy atomic nuclei!

Probes of the Early Universe
ALICE experiment at CERN:•1000+ scientists from 105+ institutions•dimensions: 26m long, 16m high, 16m wide
•weight: 10.000 tons
two other experiments: CMS, ATLASHeavy-Ion Collision Data

typical Pb+Pb Collision at the LHC:
• thousands of particle tracks• challenge: reconstruction of final state tocharacterize matter created in collision

Transport Theory:
Connecting Data to Knowledge

Transport Theory
microscopic transport models based on the
Boltzmann Equation:• transport of a system of microscopic particles• all interactions are based on binary scattering

p
+
t E

r

f1 (p, r, t) =C(p, r, t)
processes

diffusive transport models basedon the Langevin Equation:• transport of a system of microscopic particles in a thermal medium• interactions contain a drag term related to the properties of themedium and a noise term representing random collisionsp(t +t) = p(t)2Tv·

t+ (t) t(viscous) relativistic fluid dynamics:• transport of macroscopic degrees of freedom• based on conservation laws:

Tik

=

+

ui uk + P ( ik + ui uk )2
i uk +
k ui
ik
3
·u
ik

·u

(plus an additional 9 eqns. for dissipative flows)

hybrid transport models:•combine microscopic & macroscopic degrees of freedom•current state of the art for RHIC modeling

Each transport model relies on roughly a dozen physics parameters to describe the time-evolution of thecollision and its final state. These physics parameters act as a representation of the information we wish to
extract from RHIC & LHC.

Computational Modeling
3+1D Hydro + Boltzmann HybridComputational Modeling
3+1D Hydro + Boltzmann Hybrid

Probing the QGP in Relativistic Heavy-Ion Collisions

Probing the QGP in Relativistic Heavy-Ion Collisions

1x 10-23 s

10 x 10-23 s

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

1x 10-23 s

10 x 10-23 s

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

Quark-Gluon-Plasma

1x 10-23 s

10 x 10-23 s

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

Quark-Gluon-Plasma

1x 10-23 s

hadronic final stateinteractions

10 x 10-23 s

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

Quark-Gluon-Plasma

1x 10-23 s

hadronic final stateinteractions

10 x 10-23 s

measurable (stable)
particles in detector

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

Quark-Gluon-Plasma

10 x 10-23 s

1x 10-23 s

non-equilibrium
early time dynamics

hadronic final stateinteractions

viscous fluid
dynamics

hadronic transportmeasurable (stable)
particles in detector

30 x 10-23 s

Probing the QGP in Relativistic Heavy-Ion Collisions
nuclei at 99.99%
speed of light

Quark-Gluon-Plasmameasurable (stable)
particles in detector

10 x 10-23 s

1x 10-23 s

non-equilibrium
early time dynamics

hadronic final stateinteractions

viscous fluid
dynamics

hadronic transport

Principal Challenges of Probing the QGP with Heavy-Ion Collisions:• time-scale of the collision process: 10-24 seconds! [too short to resolve]• characteristic length scale: 10-15 meters! [too small to resolve]• confinement: quarks & gluons form bound states, experiments don’t observe them directly‣computational models are need to connect the experiments to QGP properties!

30 x 10-23 s

Knowledge Extraction fromRelativistic Heavy-Ion Collisions

Probing QCD in Heavy-Ion Collisions
Data:

Model:
initial conditions, τ0, η/s, ζ/s, ….

Glauber 20-25%

102pT>0.5 GeV, |η|<2.5

pT>0.5 GeV, |η|<2.5

Pb+Pb

ATLAS Pb+Pb

ATLAS Pb+Pb

sNN=2.76 TeV

sNN=2.76 TeV

sNN=2.76 TeV

Lint = 7 µb-1

Lint = 7 µb-1

ATLAS
10

Lint = 7 µb-1

10

0oGel
ATLAS

4

p(v )

2

p(v 3)1p(v ).L1 20-25%

P(v2 )

pT>0.5 GeV, |η|<2.5

10

1
10-1 centrality:

-2

10

0

0-1%
5-10%
20-25%
30-35%
40-45%
60-65%

10-1
0.1

0.2

v2

0

centrality:
0-1%
5-10%
20-25%
30-35%
40-45%

centrality:
0-1%
5-10%
20-25%
30-35%
40-45%

0.05

0.1

v3

1
0

0.01

extracted QGP properties: η/s, …
0.02

0.03

v4

0.04

0.05

0.00

0.05

0.10

v2

0.15

0.20

0.00

0.05

0.10

v2

0.15

0.20

Determining the QGP Properties via aModel to Data Comparison
Model Parameter:
eqn. of state
shear viscosity
initial statepre-equilibrium dynamics
thermalization time
quark/hadron chemistry
particlization/freeze-out

experimental data:
π/K/P spectra
yields vs. centrality & beam
elliptic flow
HBT
charge correlations & BFs
density correlations

Determining the QGP Properties via aModel to Data Comparison
Model Parameter:
eqn. of state
shear viscosity
initial statepre-equilibrium dynamics
thermalization time
quark/hadron chemistry
particlization/freeze-out

experimental data:
π/K/P spectra
yields vs. centrality & beam
elliptic flow
HBT
charge correlations & BFs
density correlations

Determining the QGP Properties via aModel to Data Comparison
Model Parameter:
eqn. of state
shear viscosity
initial statepre-equilibrium dynamics
thermalization time
quark/hadron chemistry

experimental data:
π/K/P spectra
yields vs. centrality & beam
elliptic flow
HBT
charge correlations & BFs
density correlations

particlization/freeze-out• large number of interconnected parameters w/ non-factorizable data dependencies• data have correlated uncertainties• develop novel optimization techniques: Bayesian Statistics and MCMC methods• transport models require too much CPU: need new techniques based on emulators• general problem, not restricted to RHIC Physics→collaboration with Statistical Sciences

Bayesian Analysis
Each computational model relies on a set of physics parameters to describe the dynamics and properties of thesystem. These physics parameters act as a representation of the information we wish to extract from RHIC & LHC.

Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

estim

ate

or c

alcu

late

para

met
ers

Physics Model:• Trento• iEbE-VISHNU

Experimental Data
• ALICE flow & spectra

vabl
r
e
s
b
o
te
calcula

es &

to da
e
r
a
p
com

ta

Bayesian Analysis
Each computational model relies on a set of physics parameters to describe the dynamics and properties of thesystem. These physics parameters act as a representation of the information we wish to extract from RHIC & LHC.

Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

Physics Model:
Bayesian analysis

Experimental Data
• ALICE flow & spectra• Trento• iEbE-VISHNU

Bayesian Analysis
Each computational model relies on a set of physics parameters to describe the dynamics and properties of thesystem. These physics parameters act as a representation of the information we wish to extract from RHIC & LHC.

Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

Physics Model:
Bayesian analysis• Trento• iEbE-VISHNU

Experimental Data
• ALICE flow & spectra• Bayesian analysis allows us to simultaneously calibrate all model parameters via a model-to-data comparison• determine parameter values such that the model best describes experimental observables• extract the probability distributions of all parameters

Example: Gravitational Waves
LIGO gravitational wave signal:

Bayesian analysis of GR model of merging
black holes of masses m1 and m2 that is
capable of reproducing LIGO data:Setup of a Bayesian Statistical Analysis
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Components of the Bayesian Analysis

Methodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parametersPhysics Model: Trento + iEbE-VISHNU
hadronic phaseand freeze-out

QGP andhydrodynamic expansion
initial stateparameterized
initial QGP state
hadronization

iEbE-VISHnew:

Trento:y [fm]

-

8

8

4

4

0

0

−4

−4

−8
−8

- EbE 2+1D viscous RFD- describes QGP dynamics &hadronization- EoS from Lattice QCD- temperature-dependentshear and bulk viscosity as
input

parameterized initialcondition model basedon phenomenologicalconcepts for entropy
deposition to a QGP−40x [fm]

4

8

−8
−8

−4

0
x [fm]

4

8

UrQMD:- non-equilibrium
evolution of aninteracting hadron gas- hadron gas shear & bulk
viscosities are implicitly
contained in calculation

Methodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Calibration Parameters• the calibration parameters are the model parametersthat codify the physical properties of the system that
we wish to characterize with the analysis

Calibration Parametersy [fm]• the calibration parameters are the model parametersthat codify the physical properties of the system that
we wish to characterize with the analysis

Trento initial condition:• p: attenuation parameter - entropy deposition• k: governs fluctuation in nuclear thickness• w: Gaussian nucleon width
8

8

4

4

0

0

−4

−4

−8
−8

−4

0
x [fm]

4

8

−8
−8

Pb+Pb @ LHC−40x [fm]

48

Calibration Parameters• the calibration parameters are the model parametersthat codify the physical properties of the system that
we wish to characterize with the analysis

Trento initial condition:• p: attenuation parameter - entropy deposition• k: governs fluctuation in nuclear thickness• w: Gaussian nucleon widthy [fm]• hydro to micro switching temperature Tsw

8

8

4

4

0

0

−4

−4

−8
−8

−4

0
x [fm]

4

8

−8
−8

Pb+Pb @ LHC−40x [fm]

48

Calibration Parameters• the calibration parameters are the model parametersthat codify the physical properties of the system that
we wish to characterize with the analysis

Trento initial condition:• p: attenuation parameter - entropy deposition• k: governs fluctuation in nuclear thickness• w: Gaussian nucleon widthy [fm]• hydro to micro switching temperature Tsw

8

8

4

4

0

0

−4

−4

−8
−8

temperature dependent shear viscosity:
𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)β× v

parameters:• intercept:(η/s)min at TC• slope: (η/s)slope• curvature: β

r u

u

−4

0
x [fm]

4

8

−8
−8

Pb+Pb @ LHC−40x [fm]

48

Calibration Parameters• the calibration parameters are the model parametersthat codify the physical properties of the system that
we wish to characterize with the analysis

Trento initial condition:• p: attenuation parameter - entropy deposition• k: governs fluctuation in nuclear thickness• w: Gaussian nucleon widthy [fm]• hydro to micro switching temperature Tsw

8

8

4

4

0

0

−4

−4

−8
−8

temperature dependent shear viscosity:

u

4

8

−8
−8

−4

0
x [fm]

4

8

ζ/s(T)=(ζ/s)max / [1+(T-(ζ/s)peak)2/Γ2]
parameters:• magnitude (ζ/s)max• width: Γ• peak position:(ζ/s)peak

Ö v

× v

r u

0
x [fm]

temperature dependent bulk viscosity:

𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)βparameters:• intercept:(η/s)min at TC• slope: (η/s)slope• curvature: β−4

Pb+Pb @ LHC

r uuMethodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

T

0.6
0.5
0.4
0.3
0.2
0.1

(a)
Full: positive, Open: negative

0
10

102

π+, π103dN ch/dη

〈pT〉 (GeV/ c )the entire success of the analysis depends
on the quality of the exp. data!

2
1.8

ALICE, Pb-Pb, sNN = 2.76 TeV

1.6STAR, Au-Au, sNN = 200 GeV

1.4

PHENIX, Au-Au, sNN = 200 GeV

1.2
1

ALICE, Pb-Pb, sNN = 2.76 TeV

0.8
0.6

STAR, Au-Au, sNN = 200 GeV

0.4

(c)

0.2

sNN = 200 GeV
Full: PHENIX,
positive, Au-Au,
Open: negative

p, p

102

103

0
10

dN ch/dη
〈pT〉 (GeV/ c )Data:• ALICE v2, v3 & v4 flow cumulants• identified & charged particle yields• identified particle mean pT• 2 beam energies:
2.76 & 5.02 TeV〈p 〉 (GeV/ c )Training Data

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10

(b)
Full: positive, Open: negative
102

+

K ,K

-

103

dN ch/dηMethodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Exploring the Model Parameter-SpaceExploring the Model Parameter-Spacebrute force analysis:• 14 model parameters• 9 centrality bins
• 20 bins per parameter• need to evaluate model at 9 ×2014 points• fluctuating initial conditions: 𝒪(104) events per point →1018 events• assume 1 cpu hour per event: 1018 cpu-hours!
• 2 billion years 100% use of TITAN @ ORNL (Cray XK7 w/ 560,640 cores)• then start MCMC to find point that optimally describes data…

Exploring the Model Parameter-Spacebrute force analysis:• 14 model parameters• 9 centrality bins
• 20 bins per parameter• need to evaluate model at 9 ×2014 points• fluctuating initial conditions: 𝒪(104) events per point →1018 events• assume 1 cpu hour per event: 1018 cpu-hours!
• 2 billion years 100% use of TITAN @ ORNL (Cray XK7 w/ 560,640 cores)• then start MCMC to find point that optimally describes data…

Need to find techniques that cut down the cpu needed by at least a factor of
1010: Gaussian Process Emulators

Exploring the Model Parameter-Space
2

Random functions

Outputbrute force
analysis:
Gaussian
process:
1
14 model function:
parameters• •stochastic
inputs bins
to normally distributed outputs•maps
9 centrality0• •specified
by meanand covariance functions
20 bins perparameter−1• need to evaluate model at 9 ×2014 points
−2 18
GP
as
a
model
emulator:4• fluctuating initial conditions: 𝒪(10 ) events per point →10 events‣ narrow near training points, wide in gaps

Output•• non-parametricinterpolationof physics
model
Conditioned on training data (dots)
assume 1 cpu hour
per event:1018 cpu-hours!
2•• predicts
distributions
for model
output2 billionprobability
years 100%use of TITAN@ ORNL(Cray XK7 w/ 560,640 cores)
1anystart
given
input value• atthen
MCMC
to find point that optimally describes data…
0• needs to be conditioned on training data (Latin−1
Dashed line: mean
hypercube points)
Band: 2σ uncertainty−2
Coloredlines: sampled
functions
Need
to
find
techniques
that
cut
down
the
cpu
needed
by
at
least
a
factorof• fast surrogate to actual model
0
1
2
3
4
10
10 : Gaussian Process Emulators
Input

5

Computer Experiment Design
Example:• Latin-hypercube projection for 𝜂/s parameters× v oLatin hypercube:• algorithm for generating semi-randomized, spacefilling points (here: maximin Latin hypercube)• avoids large gaps and tight clusters• all parameters varied simultaneously• needs only m≥10n points, with
n: number of model parameters

× vlm

Computer Experiment Design

this design:• n=15 model parameters• 9 centrality bins, 2 energies• Latin hypercube with m=500 points

Example:• Latin-hypercube projection for 𝜂/s parameters× v oLatin hypercube:• algorithm for generating semi-randomized, spacefilling points (here: maximin Latin hypercube)• avoids large gaps and tight clusters• all parameters varied simultaneously• needs only m≥10n points, with
n: number of model parameters• 𝒪(104) events per point, for a total of approx.
35,000,000 events• use Gaussian Process Emulators to interpolate
between points

× vlm

Computer Experiment ExecutionComputer Experiment Execution
Edison @ NERSC:• Cray XC30: 5586 nodes w/ 24 cores each• 2 hyperthreads per core
• 2.57 Petaflops/s
Duke QCD workflow:• 1000 nodes per job: running on 48K cores
simultaneously• entire model design with 30M events can becomputed in 1 day

Computer Experiment Execution
Edison @ NERSC:• Cray XC30: 5586 nodes w/ 24 cores each• 2 hyperthreads per core
• 2.57 Petaflops/s
Duke QCD workflow:• 1000 nodes per job: running on 48K cores
simultaneously• entire model design with 30M events can becomputed in 1 day

Calibration
Vector of input parameters: x=[p,k,w,(𝜂/s)min,(𝜂/s)slope,(𝜁/s)norm,Tsw,…]• assume true parameters x★ exist

find probability distribution for x★

Bayes’ Theorem: P(x★ |X,Y,yexp)• P(x★ |X,Y,yexp) = posterior
probability of x★ given observations (X,Y,yexp)• X: training data design points• Y: model output on X

∝ P(X,Y,y | x )P(x )
exp

★

★

• P(x★) = priorinitial knowledge of x★
• P(X,Y,yexp| x★) = likelihood
probability of observing (X,Y,yexp) given proposed x★

Markov-Chain Monte-Carlo:• random walk through parameter space weighted by posterior• large number of samples
chain equilibrates to posterior distribution

Likelihood and Uncertainty Quantification:• flat prior within design range, zero outside• posterior ~ likelihood within design range, zero outside• Σmodel conservatively estimated as 5%

Likelihood ∝ exp[-1/2 (y-yexp)⊤Σ-1(y-yexp)]• covariance matrix Σ = Σexperiment + Σmodel• Σexperiment=stat(diagonal) + sys(non-diagonal)Prior vs. Posterior
Prior: model calculations evenly distributed over full design space
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Prior: model
calculations
evenlyfor
distributed
over full design
space
Posterior:
emulator
predictions
highest likelihood
parameter
values
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Analysis Results

Methodology: Jonah E. Bernhard, J. Scott Moreland, Steffen A. Bass, Jia Liu, Ulrich Heinz: PRC94 (2016) 024907, arXiv:1605.03954
Results: Jonah E. Bernhard, PhD thesis arXiv:1804.06469

Methodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Calibrated Posterior Distribution• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Calibrated Posterior Distribution• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Tsw⩽TcCalibrated Posterior Distributionp≈0: IP-Glasma & EKRT type scaling

Tsw⩽Tc• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Calibrated Posterior Distributionp≈0: IP-Glasma & EKRT type scaling• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameterstemperature-dependent viscosities:

Tsw⩽TcCalibrated Posterior Distributionp≈0: IP-Glasma & EKRT type scaling• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameterstemperature-dependent viscosities:

Tsw⩽TcTemperature Dependence of Shear & Bulk Viscosities
temperature dependent shear viscosity:• analysis favors small value and shallow rise
• results do not fully constrain temperaturedependence:

𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)β• inverse correlation between (η/s)slope slope andintercept (η/s)min• insufficient data to obtain sharply peaked
likelihood distributions for (η/s)slope and curvature
β independently• current analysis most sensitive to T< 0.23 GeV‣RHIC data may disambiguate further

temperature dependent bulk viscosity:• setup of analysis allows for vanishingvalue of bulk viscosity• significant non-zero value near TC favored,
confirming the presence / need for bulk viscosity
caveat of current analysis:• bulk-viscous corrections are implemented using
relaxation-time approximation & regulated toprevent negative particle densities

ζ/s(T)=(ζ/s)max / [1+(T-(ζ/s)peak)2/Γ2]

Constraining the Initial Statep quantifies the attenuation of entropy production in theoff-diagonal regions of dS/dyTR(p;TA,TB):‣analysis strongly favors eccentricity∝• -1 < p < 0: features seen in• p=1: woundedsaturation models:

nucleon model• p = 0: IP-Glasma & EKRT
• p ≈ -0.65: KLN

scaling and entropy deposition seenin the EKRT & IP-Glasma models‣wounded nucleon and KLN models
disfavored‣no conclusion yet on 2 component
WN+BC model• still need to corroborate scale offluctuations being probed
note: no pre-equilibrium flow in the current Trento analysis, may account for larger 𝜀nTrento vs. IP-Glasma:
EccentrLcLtLeV

0ultiplicity

0.7

.L1
,3-GlaVPa
1part
7rento p 0, N 1.4

ε2
0.4
dS/dy

0.5

4

10

εn

0.4
0.3

ε3

0.2

0.2

2

10

0.1
0.0
0

3

10

εn

0.6

(ccentricity

0.6

7rento (no pre-eT.)
I3-GlaVPa (0.4 IP/c)

Au-Au @ 200 GeV
2

4

6

8

b [IP@

10

12

14

0

20

40

60

Centrality %3E+3E @ 2.76 7eV
80

100

0.0
0

10

20

30

40

Centrality %50

60

Precision Scienceor“Smoke & Mirrors”?

Validation• generate a separate Latin hypercube validation design with 50 points• evaluate the full physics model at each validation point• compare physics model output to that of the previously conditioned GP emulators:

0v

7

centrality:

7b1 7 7à 7

a@7b1 7 r$ à

7b1 7 • note that since GPEs are stochastic functions, only ~68% of predictions need to fall within 1 standard deviation

Verification: Explicit Model Calculation• explicit physics model calculations (no emulator)
with parameter values set to the maximum of theposterior probability distributions yield excellent
agreement with data!

• description of data to within ±10% accuracy

Non-Calibrated Observables
The robustness and quality of the Physics Model can be tested by making predictions on observables
not used during calibration using highest likelihood parameter values.
Example: correlations between event-by-event fluctuations of flow harmonics
SC(m,n) = ⟨v2mv2n⟩ - ⟨v2m⟩ ⟨v2n⟩o

1 mu

1o b bombmbl l 0b(( ru 7b1 bomm

SC(m,n) are sensitive to:• initial conditions• evolution model• QGP transportcoefficients• excellent agreement ofmodel prediction to data!

ALICE: PRL 117 (2016) 182301, 1604.07663

mu b

mu b

Closure Test
Need to verify that analysis can recover “true” values for the parameters: run physics model with chosen
set of parameters, generate “fake data” from model output and then conduct analysis on that fake data
to test if the input parameters can be recovered!

• both, smooth functions as well as peaked functions, canbe reproduced well within the 90% CR• note: due to reduction of information when going frommodel output to observables & model/GP uncertainties
one should not expect a one-to-one reconstruction• bulk analysis is mostly sensitive to area under bulk peak,
not peak position, height & width independently

Summary I: Key Physics Results
Trento initial condition:εn• analysis strongly favors
eccentricity scaling andentropy deposition of
EKRT & IP-Glasma
model

EccentrLcLtLeV
0.7.L1,3-GlaVPa
1part
7rento p 0, N 1.4

0.6
0.5
0.4
0.3
0.2

• Glauber and KLN
models strongly
disfavored

0.1
0.0
0

Au-Au @ 200 GeV
2

4

6

8

b [IP@

1012

14

Summary I: Key Physics Results
Trento initial condition:εn• analysis strongly favors
eccentricity scaling andentropy deposition of
EKRT & IP-Glasma
modeltemperature dependent shear viscosity:

EccentrLcLtLeV• analysis favors small value and shallow rise
• slope vs. curvature needs disambiguation

0.7

.L1
,3-GlaVPa
1part
7rento p 0, N 1.4

0.6
0.5

𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)β

0.4
0.3
0.2

• Glauber and KLN
models strongly
disfavored

0.1
0.0
0

Au-Au @ 200 GeV
2

4

6

8

b [IP@

1012

14

Summary I: Key Physics Results
Trento initial condition:εn• analysis strongly favors
eccentricity scaling andentropy deposition of
EKRT & IP-Glasma
modeltemperature dependent shear viscosity:

EccentrLcLtLeV• analysis favors small value and shallow rise
• slope vs. curvature needs disambiguation

0.7

.L1
,3-GlaVPa
1part
7rento p 0, N 1.4

0.6
0.5

𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)β

0.4
0.3
0.2

• Glauber and KLN
models strongly
disfavored

0.1
0.0
0

Au-Au @ 200 GeV
2

4

6

8

10

12

b [IP@

temperature dependent bulk viscosity:• non-zero value near TC favored• ambiguities exist for peak hight vs. widthζ/s(T)=(ζ/s)max / [1+(T-(ζ/s)peak)2/Γ2]

14

Summary I: Key Physics Results
Trento initial condition:εn• analysis strongly favors
eccentricity scaling andentropy deposition of
EKRT & IP-Glasma
modeltemperature dependent shear viscosity:

EccentrLcLtLeV• analysis favors small value and shallow rise
• slope vs. curvature needs disambiguation

0.7

.L1
,3-GlaVPa
1part
7rento p 0, N 1.4

0.6
0.5

𝜂/s(T) = (𝜂/s)min + (𝜂/s)slope × (T-TC)×(T/TC)β

0.4
0.3
0.2

• Glauber and KLN
models strongly
disfavored

0.1
0.0
0

Au-Au @ 200 GeV
2

4

6

8

10

12

14

b [IP@

temperature dependent bulk viscosity:• non-zero value near TC favored• ambiguities exist for peak hight vs. widthζ/s(T)=(ζ/s)max / [1+(T-(ζ/s)peak)2/Γ2]hydro to micro switching temperature Tsw• strong likelihood for a value of Tsw just around TC• indicative of the non-equilibrium nature anddynamical breakup of the hadronic system

Summary II: Methodology
Model Parameters - System Properties• initial state• temperature-dependent viscosities• hydro to micro switching temperature

calculate events on Latin hypercube

Physics Model:• Trento• iEbE-VISHNU

Gaussian Process Emulator

Experimental Data

• non-parametric interpolation• fast surrogate to full Physics Model• ALICE flow & spectra

MCMC(Markov-Chain Monte-Carlo)

after many steps, MCMC equilibrates to• random walk through parameter spaceweighted by posterior probability

Posterior Distribution
Bayes’ Theorem∝likelihood × priorposterior• prior: initial knowledge of parameters• likelihood: probability of observing exp.
data, given proposed parameters• diagonals: probability distribution of eachparameter, integrating out all others• off-diagonals: pairwise distributions showing
dependence between parameters

Outlook & Future Directions
current analysis focus was on the properties of bulk QCD matter and utilized only
LHC data on soft hadrons. The analysis needs to be extended to:• include data from lower beam energies‣ necessary for determination of the temperature and 𝜇B dependence of transportcoefficients• include asymmetric collision systems (p+A, d+A, 3He+A, A+B)
‣ generate improved understanding of the initial state• include hard probes (jets and heavy quark observables)‣ consistent determination of jet and heavy flavor transport coefficients• include other physics models‣ analysis is model agnostic, allows for quantitative comparison among different models
and verification/falsification of models/conceptual approaches
this work has been madepossible through support byPast & Present Collaborators & Sponsors
Duke QCD Group:• Jonah Bernhard (now Lowe’s Corporate)• J. Scott Moreland• Weiyao Ke• Yingru Xu• Jean-Francois Paquet
Duke Dept. of Statistical Sciences:• Robert E. Wolpert• Jake Coleman
Ohio State Nuclear Theory:• Ulrich W. Heinz• Jia Liu (now SAP)
• Chun Shen (now BNL)
U. of Wyoming Dept. of Statistics:• Snehalata Huzurbazar• Peter W. Marcy (now LANL)This work was made possible
through support by:
US Dept. of Energy
National Science
Foundation
Open Science Grid
NERSC
SAMSI

Pioneering work by the MADAI Collaboration, led by Scott E. Pratt, MSU (2009-2014)

Resources
Trento:• J. Scott Moreland, Jonah E. Bernhard & Steffen A. Bass: Phys. Rev. C 92, 011901(R)• https://github.com/Duke-QCD/trento
iEbE-VISHNU:• Chun Shen, Zhi Qiu, Huichao Song, Jonah Bernhard, Steffen A. Bass & Ulrich Heinz:
Computer Physics Communications in print, arXiv:1409.8164• http://u.osu.edu/vishnu/
UrQMD:• Steffen A. Bass et al. Prog. Part. Nucl. Phys. 41 (1998) 225-370 , arXiv:nucl-th/9803035• Marcus Bleicher et al. J.Phys. G25 (1999) 1859-1896 , arXiv:hep-ph/9909407• http://urqmd.org
MADAI Collaboration:• Visualization and Bayesian Analysis packages• https://madai-public.cs.unc.edu
Duke Bayesian Analysis Package:• https://github.com/jbernhard/mtd

The EndTime Evolution of a Heavy-Ion Collision
initial state

hadronic phaseand freeze-out

QGP andhydrodynamic expansion

pre-equilibrium• Initial State:hadronization• QGP and hydrodynamic expansion:- fluctuates event-by-event- classical color-field dynamics• Pre-equilibrium:- rapid change-over from glue-field dominatedinitial state to thermalized QGP- time scale: 0.15 to 2 fm/c in duration- build-up of transverse velocity fields?

- proceeds via 3D viscous RFD- EoS from Lattice QCD• hadronic phase & freeze-out- interacting hadron gas- separation of chemical and kinetic
freeze-out

Constraining the IS of Heavy-Ion Collisions
initial state

hadronic phaseand freeze-out

QGP andhydrodynamic expansion

pre-equilibrium• treatment of QGP evolution andhadronic freeze-out is well establishedand largely understood

hadronization• QGP and hydrodynamic expansion:- proceeds via 3D viscous RFD- EoS from Lattice QCD• major success: first extraction of QGP
properties such as η/s• major challenges:• quantify uncertainties in extracted QGP
properties• temperature dependence of transportcoefficients• hadronic phase & freeze-out- interacting hadron gas- separation of chemical and kinetic
freeze-out

Constraining the IS of Heavy-Ion Collisions
initial state

hadronic phaseand freeze-out

QGP andhydrodynamic expansion

pre-equilibrium• Initial State:- fluctuates event-by-event- classical color-field dynamics• Pre-equilibrium:- rapid change-over from glue-field dominatedinitial state to thermalized QGP- time scale: 0.15 to 2 fm/c in duration- build-up of transverse velocity fields?

hadronization• physics of initial state and preequilibrium dynamics are stillconceptually challenging with many
open questions• what processes drive the system
towards equilibration?
• on what timescale?
• …

• a major source of uncertainty forthe extraction of QGP properties

Constraining the IS of Heavy-Ion Collisions
hadronic phaseand freeze-out

QGP andhydrodynamic expansion

initial stateparameterized
initial QGP statepre-equilibrium

parameterized initial QGP
state:- based on simple-

phenomenological ideas for
entropy deposition
constrained by global model todata fitprovides guidance to ab-initio IS
models on features needed todescribe the data

hadronization• QGP and hydrodynamic expansion:- proceeds via 3D viscous RFD- EoS from Lattice QCD• hadronic phase & freeze-out- interacting hadron gas- separation of chemical and kinetic
freeze-out

Initial Condition Model: Trento• effective, parametric, description of entropy production prior to thermalization• based on reduced thickness* T as ansatz for dS/dy:
R

dS/dy |⌧ =⌧0 / TR (p; TA , TB ) ⌘• determine participant nucleons in A, B
by sampling for each nucleon pair:Pcoll = 1

TA =Nuclear Thickness*:

exp

dz

i

gg

dx dy

nucleon (x✓pTA + TB

2dz

xi , ymodel parameters:
attenuation parameter: pfluctuation parameter: kwidth of nucleon: w
overall normalization: Cnorm
model output:
event by event spatial
entropy density distribution
at mid-rapidity atthermalization time τ0

p ◆1/p

dz

A

•
•
•
•

B

zi )

yi , z•i• sum is over participant nucleons with positions sampled from an uncorrelated Woods-Saxon
distribution or correlated nuclear configurations when availablek
introduce
fluctuationsvia𝛾,
sampled
from
a
gamma
k
•
k 1
Pk ( ) =e ki(1) determine participants:(2) construct thickness functions:
TA
5

TB

2

x +y
2w2

8

◆
2

0

0

dS/dy

dS/dy

4

5

0

8

b=0.2 fm

4

4

0

0

−4

−4

−8
−8

−8
8
−8

(3) calculate entropy deposition:
Generalized Mean p=1
TA
TB
TR

3
−2

5

TTAB

1
dz ⇢proton =
exp
2
2⇡w

✓

Thickness [fm ]•nucleon density ⍴nucleon modeled as Gaussian intransverse planeZ(k)y [fm]

distribution with unit mean:

2

1

5

5

5
0

−10

−5

0

x [fm]

5

10

−4

0
x [fm]

4

Multivariate Output
Scaling of analysis with # of observables:• independent emulators for each output?
• neglects correlations among outputs• what if # of outputs scales to 100?

‣ training of individual GPE’s may become unfeasible
and unnecessary in case of strong correlations

Multivariate Output
Scaling of analysis with # of observables:• independent emulators for each output?
• neglects correlations among outputs• what if # of outputs scales to 100?

Principal Components:• linear combinations of model output• orthogonal and uncorrelated
emulate each PC‣ training of individual GPE’s may become unfeasible
and unnecessary in case of strong correlations

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Multivariate Output
Scaling of analysis with # of observables:• independent emulators for each output?
• neglects correlations among outputs• what if # of outputs scales to 100?

Principal Components:• linear combinations of model output• orthogonal and uncorrelated
emulate each PC‣ training of individual GPE’s may become unfeasible
and unnecessary in case of strong correlations

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Next steps:•sub-nucleon degrees of freedom•forward/backward rapidity

W. Ke, J.S. Moreland, J.E. Bernhard & S.A. Bass: Phys. Rev. C96 (2017) 044912, arXiv:1610.08490
J.S. Moreland, J.E. Bernhard & S.A. Bass, arXiv1808.02106

Nucleon Substructure
Original Trento model:• sample nucleon positions from spherical ordeformed Woods-Saxon distributions• solid angles resampled to preserve minimum
distance dmin
• Gaussian nucleons of width w• works very well for large nuclei
208Pb nucleus

Caveat:• spherical protons do not allow for proper
eccentricities in p+A or small/asymmetric collisionsystemsNucleon Substructure
Original Trento model:• sample nucleon positions from spherical ordeformed Woods-Saxon distributions• solid angles resampled to preserve minimum
distance dmin
• Gaussian nucleons of width w• works very well for large nuclei
208Pb nucleus

Trento with nucleon substructure:• trade Gaussian nucleons for lumpy nucleons• additional parameters:• sampling radius of constituent positions• constituent Gaussian width• number of constituents in each nucleon

sampling radius:

constituent width:

Caveat:• spherical protons do not allow for proper
eccentricities in p+A or small/asymmetric collisionsystems# of constituents:

Simultaneous Calibration on AA and pA
prior

posterior• ALICE & CMS data for AA & pA
at 5.02 TeV• calibration on 15 parameters,
for initial state, shear and bulk
viscosities• restriction on 1 energy to keep
computational effort reasonable• generally larger uncertainties inposterior, due to less data thanin the AA calibrations for 2energies…

Key results: initial state
IP-Glasma & EKRT eccentricity
scaling for initial state confirmed
no strong preference fora particular constituent #as long as n>3

constituent width & samplingradius are well constrained to• r = 0.99 ± 0.16
• w = 0.47 ± 0.18

Key results: viscosities• shear and bulk viscosities are fullycompatible with previous calibration
on Pb+Pb @ 2.76 TeV & 5.02 TeV• uncertainty bands are larger in AA +pA analysis due to focus on single
beam energy• for bulk properties, multiple beamenergies are more important thaninclusion of small systems

Other Examples: Equation of State
Example: determine the EoS of QGP matter from experimental measurements
what equation of state would the physics model choose to best describe the experimental data?
create set of QCD Equations of State (aka the prior)run physics model with each EoS
use comparison with RHIC/LHC data to determine which Equations of State are consistent with data (i.e. the posterior)posterior is very similar to Lattice EoS!!
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Other Examples: Heavy Quarks
Extraction of the Heavy Quark Diffusion Coefficient• calibration on heavy quark v2 and RAA
67A5 ; 0 , 0-10%

0.10

0.05
0.00

3

4

5

6

7

−0.05
1

pT [GeV@

1.0

AL,CE ; 0 , ; ∗ , ; ∗ +5 < pT < 8 GeV

0.8

0.4
0.2

0.05

Au+Au@200 GeV

2

3

4

5

6

Au+Au@200 GeV

100 200 300 400

t part

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4

5

6

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pT [GeV@

AL,CE D 0 , D ∗ , D ∗ +8 < pT < 16 GeV

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AL,CE ; 0 , ; ∗ , ; ∗ +30-50%

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0.6
0.4

0.0
0

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pT [GeV@

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0.1

0.2
3E+3E@2.76 7eV

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1

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Au+Au@200 GeV

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0

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0.10

pAA

pAA

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v1

pAA

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1.0

67A5 ; 0 , 10-40%

0.15

0.6

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2

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0.20

0.8

0.2

0.25

0.25

0.0
3E+3E@2.76 7eV

100 200 300 400

t part−0.1
2

3E+3E@2.76 7eV4

6

8 10 12 14

pT [GeV@Y. Xu, J.E. Bernhard, S.A. Bass, M. Nahrgang & S. Cao: Phys. Rev C97 (2018) 014907

Ds 2πT

1.0

16
priRr
14
AuAu, 90% C5
123b3b, 90% C5
Au+3b, 90% C5
10
8
6
4
2
0
0.5 1.0 1.5 2.0 2.5 3.0

T/Tc• combining RHIC and LHC data yields significant
improvement for the extraction of Ds(T)