arXiv:1605.03954 [nucl-th] Precision extrac on of QGP proper es with quan fied uncertain es Part II: methodology and results Jonah E. Bernhard, Steffen A. Bass INT workshop: Bayesian methods in nuclear physics Wednesday, June 15, 2016 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 1 / 25 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 2 / 25 

Heavy-ion collision models 1. Ini al condi ons t = 0+ Entropy deposi on 2. (Pre-equilibrium) t < 1 fm/c Early- me dynamics and thermaliza on 3. Hydrodynamics 1 < t < 10 fm/c Hot and dense quark-gluon plasma 4. Hadronic phase 10 < t < 100 fm/c Expanding and cooling gas J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 3 / 25 

Ini al condi on models Provide ini al entropy density for hydrodynamics ↓ Many different theore cal and phenomenological approaches ↓ Affects es mates of QGP proper es! J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 4 / 25 

Ini al condi on models Provide ini al entropy density for hydrodynamics ↓ Many different theore cal and phenomenological approaches ↓ Affects es mates of QGP proper es! Alterna ve: parametric models ↓ Mimic theory calcula ons ↓ Simultaneously characterize ini al condi ons and QGP medium J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 4 / 25 

TRENTo: parametric IC model Ansatz Entropy density propor onal to generalized mean of local nuclear density s ∝ ( Tp A + Tp B 2 )1/p p ∈ (−∞, ∞) = tunable parameter p = +1 p = 0 p = −1 TA + TB 2 √ TA TB 2TA TB TA + TB −5 0 5 −5 0 5 y [fm] −5 0 5 x [fm] −5 0 5 J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 5 / 25 

TRENTo: parametric IC model s ∝ ( Tp A + Tp B 2 )1/p Compare to geometry of other models Lines = TRENTo with different p values 0 2 4 6 8 10 12 14 Impact parameter b [fm] 0.0 0.2 0.4 0.6 Ellipticity ε2 KLN IP-Glasma Wounded nucleon Mimics and interpolates other models! −5 0 5 −5 0 5 y [fm] −5 0 5 x [fm] −5 0 5 J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 5 / 25 

Hydrodynamics Viscous rela vis c hydrodynamics Energy and momentum conserva on + dissipa ve correc ons Equa on of state from la ce QCD (HotQCD collabora on) Transport coefficients: Shear viscosity (linear increase in QGP phase) (η/s)(T) = (η/s)min + (η/s)slope(T − Tc), Tc = 154 MeV Bulk viscosity (peak near 180 MeV, exponen al decrease) (ζ/s)(T) = (ζ/s)norm × f(T) J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 6 / 25 

Hadronic phase Ultra-rela vis c quantum molecular dynamics (UrQMD) Switch from hydrodynamics to par cles at Tswitch Temperature window where both models are valid? Solves Boltzmann equa on with Monte Carlo methods Simulates sca erings and decays Non-equilibrium breakup and freeze-out J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 7 / 25 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 8 / 25 

Input parameters Ini al condi on parameters Normaliza on factor Entropy deposi on p Gaussian nucleon width w Mul plicity fluctua on k QGP medium parameters η/s min and slope ζ/s norm Hydro → par cles Tswitch La n hypercube design 300 semi-random, space-filling parameter points 0.0 0.1 0.2 0.3 η/s min 0.0 0.5 1.0 1.5 2.0 η/s slope J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 9 / 25 

Observables Pion, kaon, and proton yields dN/dy Overall par cle produc on and species ra os Mean transverse momentum ⟨pT⟩ Magnitude of radial expansion Anisotropic flow coefficients vn Azimuthal momentum anisotropy ) c (GeV/ T p 0 0.5 1 1.5 2 2.5 3 ] -2 ) c ) [(GeV/ y d T p /(d N 2 ) d T p π 1/(2 ev N 1/ -3 10 -2 10 -1 10 1 10 2 10 3 10 4 10 5 10 6 10 π Range of combined fit 0-5% 80-90% positive negative combined fit individual fit (a) ) c (GeV/ T p 0 0.5 1 1.5 2 2.5 3 ] -2 ) c ) [(GeV/ y d T p /(d N 2 ) d T p π 1/(2 ev N 1/ -4 10 -3 10 -2 10 -1 10 1 10 2 10 3 10 4 10 5 10 K Range of combined fit 0-5% 80-90% positive negative combined fit individual fit (b) ) c (GeV/ T p 0 1 2 3 4 5 ] -2 ) c ) [(GeV/ y d T p /(d N 2 ) d T p π 1/(2 ev N 1/ -5 10 -4 10 -3 10 -2 10 -1 10 1 10 2 10 3 10 4 10 p Range of combined fit 0-5% 80-90% positive negative combined fit individual fit (c) 0 10 20 30 40 50 60 70 80 n v 0 0.05 0.1 (a) | > 1} ! " {2, | 2 v | > 1} ! " {2, | 3 v | > 1} ! " {2, | 4 v {4} 3 v RP # 3/ v 2 2 # 3/ v $ 100 n % / n v 0.1 0.2 0.3 0.4 (b) {2} CGC 2 % / | > 1} ! " {2, | 2 v {2} CGC 3 % / | > 1} ! " {2, | 3 v {2} W 2 % / | > 1} ! " {2, | 2 v {2} W 3 % / | > 1} ! " {2, | 3 v All experimental data from the ALICE collabora on at the LHC Pb-Pb collisions at √ s = 2.76 TeV J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 10 / 25 

Training data 0 10 20 30 40 50 60 70 Centrality % 100 101 102 103 dN/dy π ± K ± p ̄ p Identified particle yields 0 10 20 30 40 50 60 70 Centrality % 0.0 0.3 0.6 0.9 1.2 1.5 1.8 pT [GeV] π ± K ± p ̄ p Identified particle mean pT 0 10 20 30 40 50 60 70 Centrality % 0.00 0.02 0.04 0.06 0.08 0.10 0.12 vn {2} v2 v3 v4 Flow cumulants Model calcula ons at each design point To be used as training data for emulator J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 11 / 25 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 12 / 25 

Gaussian process emulator Gaussian process: Stochas c func on: maps inputs to normally-distributed outputs Specified by mean and covariance func ons As a model emulator: Non-parametric interpola on Predicts probability distribu ons Narrow near training points, wide in gaps Fast surrogate to actual model −2 −1 0 1 2 Output Random functions 0 1 2 3 4 5 Input −2 −1 0 1 2 Output Dashed line: mean Band: 2σ uncertainty Colored lines: sampled functions Conditioned on training data (dots) J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 13 / 25 

Mul variate output Many highly correlated outputs → principal component analysis PCs = eigenvectors of sample covariance matrix Y⊺Y = UΛU⊺ Transform data into orthogonal, uncorrelated linear combina ons Z = √ m YU Emulate each PC independently 0 500 1000 1500 2000 dNπ ± /dy 0.00 0.03 0.06 0.09 0.12 v2 {2} 20–30% 68% 32% J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 14 / 25 

Mul variate output Many highly correlated outputs → principal component analysis PCs = eigenvectors of sample covariance matrix Y⊺Y = UΛU⊺ Transform data into orthogonal, uncorrelated linear combina ons Z = √ m YU Emulate each PC independently 68 outputs → 8 PCs 1 2 3 4 5 6 7 8 Number of PC 0.7 0.8 0.9 1.0 Explained variance J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 14 / 25 

Valida on Independent 50-point valida on design Run full model and predict with emulator 0 1000 2000 3000 Predicted dNπ ± /dy 0 1000 2000 3000 Observed 0–5% 30–40% 0.4 0.5 0.6 0.7 Predicted ­ pT ® π ± 0.4 0.5 0.6 0.7 0.00 0.03 0.06 0.09 0.12 Predicted v2 {2} 0.00 0.03 0.06 0.09 0.12 J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 15 / 25 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 16 / 25 

Calibra on Assume true parameters x⋆ exist → find posterior distribu on P(x⋆|X, Y, yexp) ∝ P(X, Y, yexp|x⋆) P(x⋆) given design X, training data Y, experimental data yexp Flat prior Likelihood (in PC space): P(X, Z, zexp|x⋆) ∝ exp { − 1 2 (z⋆ − zexp)⊺Σ−1 z (z⋆ − zexp) } with flat 10% uncertainty on PCs Σz = diag(σ2 z zexp), σz = 0.10 J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 17 / 25 

MCMC Markov chain Monte Carlo Random walk through parameter space weighted by posterior Large number of samples → chain equilibrates to posterior distribu on This study Emulator serves as stand-in for full model Affine-invariant ensemble sampler: many interdependent walkers 1000 walkers, 106 burn-in steps, 107 produc on steps J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 18 / 25 

Overview Input parameters QGP proper es Model heavy-ion collision space me evolu on Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribu on quan ta ve es mates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 19 / 25 

Training data 0 10 20 30 40 50 60 70 Centrality % 100 101 102 103 dN/dy π ± K ± p ̄ p Identified particle yields 0 10 20 30 40 50 60 70 Centrality % 0.0 0.3 0.6 0.9 1.2 1.5 1.8 pT [GeV] π ± K ± p ̄ p Identified particle mean pT 0 10 20 30 40 50 60 70 Centrality % 0.00 0.02 0.04 0.06 0.08 0.10 0.12 vn {2} v2 v3 v4 Flow cumulants Model calcula ons at each design point J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 20 / 25 

Posterior samples 0 10 20 30 40 50 60 70 Centrality % 100 101 102 103 dN/dy π ± K ± p ̄ p Identified particle yields 0 10 20 30 40 50 60 70 Centrality % 0.0 0.3 0.6 0.9 1.2 1.5 1.8 pT [GeV] π ± K ± p ̄ p Identified particle mean pT 0 10 20 30 40 50 60 70 Centrality % 0.00 0.02 0.04 0.06 0.08 0.10 0.12 vn {2} v2 v3 v4 Flow cumulants Model calcula ons at each design point ↓ Emulator predic ons from calibrated posterior J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 20 / 25 

100 130 160 norm 120. +8. −8. −1 0 1 p −0. 02+0. 16 −0. 18 0.8 1.5 2.2 k 1. 7+0. 5 −0. 5 0.4 0.7 1.0 w 0. 48+0. 10 −0. 07 0.00 0.15 0.30 η/s min 0. 07+0. 05 −0. 04 0 1 2 η/s slope 0. 93+0. 65 −0. 92 0 1 2 ζ/s norm 1. 2+0. 2 −0. 3 100 130 160 norm 0.14 0.15 0.16 Tswitch −1 0 1 p 0.8 1.5 2.2 k 0.4 0.7 1.0 w 0.00 0.15 0.30 η/s min 0 1 2 η/s slope 0 1 2 ζ/s norm 0.14 0.15 0.16 Tswitch 0. 148+0. 002 −0. 002 Posterior distribu on Es mated values: medians Uncertain es: 90% credible intervals 

Constraining ini al condi ons TRENTo ansatz: s ∝ ( Tp A + Tp B 2 )1/p −1.0 −0.5 0.0 0.5 1.0 p KLN EKRT / IP-Glasma Wounded nucleon Mimics other models: 0 2 4 6 8 10 12 14 Impact parameter b [fm] 0.0 0.2 0.4 0.6 Ellipticity ε2 KLN IP-Glasma Wounded nucleon Entropy deposi on approx. propor onal to geometric mean of nuclear density: s ∼ √ TA TB Confirms success / failure of exis ng models J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 22 / 25 

100 130 160 norm 120. +8. −8. −1 0 1 p −0. 02+0. 16 −0. 18 0.8 1.5 2.2 k 1. 7+0. 5 −0. 5 0.4 0.7 1.0 w 0. 48+0. 10 −0. 07 0.00 0.15 0.30 η/s min 0. 07+0. 05 −0. 04 0 1 2 η/s slope 0. 93+0. 65 −0. 92 0 1 2 ζ/s norm 1. 2+0. 2 −0. 3 100 130 160 norm 0.14 0.15 0.16 Tswitch −1 0 1 p 0.8 1.5 2.2 k 0.4 0.7 1.0 w 0.00 0.15 0.30 η/s min 0 1 2 η/s slope 0 1 2 ζ/s norm 0.14 0.15 0.16 Tswitch 0. 148+0. 002 −0. 002 Posterior distribu on Es mated values: medians Uncertain es: 90% credible intervals 

Es mate of (η/s)(T) 0.15 0.20 0.25 0.30 Temperature [GeV] 0.0 0.2 0.4 0.6 η/s KSS bound 1/4π Prior range Posterior median 90% CR First systema c, quan ta ve es mate of T-dependent η/s “Handle” near 200 MeV → need mul ple beam energies! J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 23 / 25 

100 130 160 norm 120. +8. −8. −1 0 1 p −0. 02+0. 16 −0. 18 0.8 1.5 2.2 k 1. 7+0. 5 −0. 5 0.4 0.7 1.0 w 0. 48+0. 10 −0. 07 0.00 0.15 0.30 η/s min 0. 07+0. 05 −0. 04 0 1 2 η/s slope 0. 93+0. 65 −0. 92 0 1 2 ζ/s norm 1. 2+0. 2 −0. 3 100 130 160 norm 0.14 0.15 0.16 Tswitch −1 0 1 p 0.8 1.5 2.2 k 0.4 0.7 1.0 w 0.00 0.15 0.30 η/s min 0 1 2 η/s slope 0 1 2 ζ/s norm 0.14 0.15 0.16 Tswitch 0. 148+0. 002 −0. 002 Posterior distribu on Es mated values: medians Uncertain es: 90% credible intervals 

100 130 160 norm norm p k w η/s min η/s slope ζ/s norm norm Tswitch −1 0 1 p p 0.8 1.5 2.2 k k 0.4 0.7 1.0 w w 0.00 0.15 0.30 η/s min η/s min 0 1 2 η/s slope η/s slope 0 1 2 ζ/s norm ζ/s norm 100 130 160 norm 0.14 0.15 0.16 Tswitch −1 0 1 p 0.8 1.5 2.2 k 0.4 0.7 1.0 w 0.00 0.15 0.30 η/s min 0 1 2 η/s slope 0 1 2 ζ/s norm 0.14 0.15 0.16 Tswitch Tswitch Iden fied par cles Charged par cles 

Most probable parameters norm 120. / 129. η/s min 0.08 p 0.0 η/s slope 0.85 / 0.75 GeV−1 k 1.5 / 1.6 ζ/s norm 1.25 / 1.10 w 0.43 / 0.49 fm Tswitch 0.148 GeV 0 10 20 30 40 50 60 70 80 100 101 102 103 104 dN/dy, dNch /dη×5 π ± K ± p ̄ p Nch Yields 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.0 0.3 0.6 0.9 1.2 1.5 pT [GeV] π ± K ± p ̄ p Identified particle mean pT 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.00 0.02 0.04 0.06 0.08 0.10 vn {2} v2 v3 v4 Flow cumulants 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 24 / 25 

Most probable parameters norm 120. / 129. η/s min 0.08 p 0.0 η/s slope 0.85 / 0.75 GeV−1 k 1.5 / 1.6 ζ/s norm 1.25 / 1.10 w 0.43 / 0.49 fm Tswitch 0.148 GeV 0 10 20 30 40 50 60 70 80 100 101 102 103 104 dN/dy, dNch /dη×5 π ± K ± p ̄ p Nch Yields 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.0 0.3 0.6 0.9 1.2 1.5 pT [GeV] π ± K ± p ̄ p Identified particle mean pT 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.00 0.02 0.04 0.06 0.08 0.10 vn {2} v2 v3 v4 Flow cumulants 0 10 20 30 40 50 60 70 80 Centrality % 0.8 0.9 1.0 1.1 1.2 Ratio J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 24 / 25 

Summary TRENTo parametric ini al condi ons, viscous rela vis c hydrodynamics, hadronic a erburner (UrQMD) Excellent simultaneous fit to experimental data Es mated ini al condi on and QGP medium proper es Entropy deposi on ∼ geometric mean of nuclear density Rela on between η/s min and slope, handle near 200 MeV Finite bulk viscosity Tswitch constrained by par cle ra os only Addi onal beam energies (200 GeV, 2.76 TeV, 5.02 TeV) Improve treatment of uncertainty J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 25 / 25 

Gaussian processes Defini on A Gaussian process is a collec on of random variables, any finite number of which have a joint Gaussian distribu on. Stochas c func on: x → y x = n-dimensional input vector y = normally distributed output Specified by Mean func on μ(x) Covariance func on σ(x, x′), e.g.: σ(x, x′) = exp ( − |x − x′|2 2ℓ2 ) J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 1 / 2 

Condi oning a Gaussian process Given training input points X and observed training outputs y at X the predic ve distribu on at arbitrary test points X∗ is the mul variate-normal distribu on y∗ ∼ N(μ, Σ), μ = σ(X∗, X)σ(X, X)−1y, Σ = σ(X∗, X∗) − σ(X∗, X)σ(X, X)−1σ(X, X∗). J. E. Bernhard, S. A. Bass (Duke) Precision extrac on of QGP proper es (1605.03954) 2 / 2