Characterization of the initial state and QGP medium from a combined Bayesian analysis of LHC data at 2.76 and 5.02 TeV

Transcript

1. Characterization of the initial state and QGP medium from a

   combined Bayesian analysis of LHC data at . and . TeV Jonah E. Bernhard J. Scott Moreland, Ste en A. Bass

2. “ The next 5–10 years of the US relativistic heavy-ion

   program will deliver...the quantita- tive determination of the transport coefficients of the Quark Gluon Plasma, such as the tem- perature dependent shear-viscosity to entropy- density ratio (η/s)(T) ... ”

3. Overview Input parameters QGP properties Model heavy-ion collision spacetime evolution

   Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 2 / 22

4. Overview Input parameters QGP properties Model heavy-ion collision spacetime evolution

   Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 2 / 22

5. Model 1. Initial conditions t = 0+ • Entropy deposition

   2. Pre-equilibrium t < 1 fm/c • Early-time dynamics and thermalization 3. Hydrodynamics 1 < t < 10 fm/c • Hot and dense QGP 4. Particlization and hadronic phase 10 < t < 100 fm/c • Conversion to particles • Expanding and cooling gas J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 3 / 22

6. TRENTo: parametric initial condition model Ansatz: entropy density proportional to

   generalized mean of local nuclear density s ∝ Tp A + Tp B 2 1/p p ∈ (−∞, ∞) = tunable parameter; varying p mimics other models: • p = 1 =⇒ s ∝ TA + TB wounded nucleon model • p = 0 =⇒ s ∝ TA TB similar to IP-Glasma, EKRT • Previous work: p = 0.0 ± 0.2 PRC 92 011901 [1412.4708] PRC 94 024907 [1605.03954] See talk by S. Moreland, Wed. 10:40 −5 0 5 −5 0 5 y [fm] −5 0 5 x [fm] −5 0 5 J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 4 / 22

7. Pre-equilibrium Free-streaming approximation • Expanding, noninteracting gas of massless partons

   • Sudden thermalization and switch to hydrodynamics at tunable time τfs • Smooths out initial density, increases radial flow velocity PRC 80 034902 [0812.3393] PRC 92 064906 [1504.02160] J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 5 / 22

8. Hydrodynamics OSU VISH2+1 PRC 77 064901 [0712.3715] CPC 199 61

   [1409.8164] • Boost-invariant viscous hydrodynamics • Hybrid equation of state • HRG EOS at low temperature • HOTQCD lattice EOS at high temperature PRD 90 094503 [1407.6387] • Temperature-dependent shear and bulk viscosities J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 6 / 22

9. Temperature-dependent viscosities Shear Tunable minimum at Tc, slope, and curvature

   (η/s)(T) = (η/s)min + (η/s)slope(T − Tc) × T Tc (η/s)crv 0.15 0.20 0.25 0.30 Temperature [GeV] 0.0 0.2 0.4 η/s Bulk Cauchy distribution with peak at Tc, tunable height and width (ζ/s)(T) = (ζ/s)max 1 + T − Tc (ζ/s)width 2 0.08 0.12 0.16 0.20 Temperature [GeV] 0.00 0.04 0.08 ζ/s J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 7 / 22

10. Particlization and hadronic phase Convert hydrodynamic medium → particles at

    tunable Tswitch • Particle species and momenta sampled from thermal hadron resonance gas (Cooper-Frye) • Novel implementation of shear and bulk viscous corrections based on relaxation-time approximation PRC 82 044901 [1003.0413] PRC 85 044909 [1109.5181] PRC 83 044910 [1012.5927] • Masses of unstable resonances sampled from Breit-Wigner distribution Hadronic scatterings and decays: UrQMD J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 8 / 22

11. Overview Input parameters QGP properties Model heavy-ion collision spacetime evolution

    Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 8 / 22

12. Input parameters Initial condition • TRENTo entropy deposition p •

    Multiplicity fluctuation σfluct • Gaussian nucleon width w Pre-equilibrium • Free streaming time τfs QGP medium • η/s min, slope, curvature • ζ/s max, width • Tswitch (hydro to UrQMD) Latin hypercube design 500 semi-random, space-filling parameter points; ∼3 × 104 min-bias events per point 0.0 0.1 0.2 η/s min 0 1 2 3 η/s slope [GeV−1] J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 9 / 22

13. Observables All experimental data from the ALICE collaboration at the

    LHC Pb-Pb collisions at √ s = 2.76 and 5.02 TeV Centrality dependence of: • Charged particle yields dNch/dη PRL 106 032301 [1012.1657], PRL 116 222302 [1512.06104] • Identified particle (π, K, p) yields dN/dy and mean transverse momenta pT (2.76 TeV only) PRC 88 044910 [1303.0737] • Anisotropic flow cumulants vn{2} PRL 116 132302 [1602.01119] 〉 part N 〈 0 100 200 300 400 〉 η /d ch N d 〈 〉 part N 〈 2 4 6 8 10 ALICE = 5.02 TeV 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 η | ) 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) Centrality percentile 0 10 20 30 40 50 60 70 80 n v 0.05 0.1 0.15 5.02 TeV |>1} η ∆ {2, | 2 v |>1} η ∆ {2, | 3 v |>1} η ∆ {2, | 4 v {4} 2 v {6} 2 v {8} 2 v 2.76 TeV |>1} η ∆ {2, | 2 v |>1} η ∆ {2, | 3 v |>1} η ∆ {2, | 4 v {4} 2 v 5.02 TeV, Ref.[27] |>1} η ∆ {2, | 2 v |>1} η ∆ {2, | 3 v ALICE Pb-Pb Hydrodynamics (a) Centrality percentile 0 10 20 30 40 50 60 70 80 Ratio 1 1.1 1.2 /s(T), param1 η /s = 0.20 η (b) Hydrodynamics, Ref.[25] 2 v 3 v 4 v Centrality percentile 0 10 20 30 40 50 60 70 80 Ratio 1 1.1 1.2 (c) J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 10 / 22

14. Training data Model calculations at each design point 0 10

    20 30 40 50 60 70 80 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p Yields 0 10 20 30 40 50 60 70 80 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p Mean pT 0 10 20 30 40 50 60 70 80 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 2.76 TeV Flow cumulants 0 10 20 30 40 50 60 70 80 Centrality % 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 5.02 TeV J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 11 / 22

15. Overview Input parameters QGP properties Model heavy-ion collision spacetime evolution

    Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 11 / 22

16. Gaussian process emulator Gaussian process: • Stochastic function: maps inputs

    to normally-distributed outputs • Specified by mean and covariance functions As a model emulator: • Non-parametric interpolation • Predicts probability distributions • 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 Conditioned on data Mean prediction Uncertainty Training data J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 12 / 22

17. Bayesian model calibration Bayes’ theorem posterior ∝ likelihood × prior

    Prior = flat in design space Likelihood ∝ exp −1 2 (y − yexp) Σ−1(y − yexp) • Σ = covariance matrix = Σexperiment + Σmodel • Σexperiment = stat (diagonal) + sys (non-diagonal) • Σmodel conservatively estimated as 5% (to be improved) Markov chain Monte Carlo Construct posterior distribution by MCMC sampling (weighted random walk through parameter space) J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 13 / 22

18. Training data Model calculations at each design point 0 10

    20 30 40 50 60 70 80 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p Yields 0 10 20 30 40 50 60 70 80 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p Mean pT 0 10 20 30 40 50 60 70 80 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 2.76 TeV Flow cumulants 0 10 20 30 40 50 60 70 80 Centrality % 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 5.02 TeV J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 14 / 22

19. Posterior samples Emulator predictions from calibrated posterior 0 10 20

    30 40 50 60 70 80 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p Yields 0 10 20 30 40 50 60 70 80 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p Mean pT 0 10 20 30 40 50 60 70 80 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 2.76 TeV Flow cumulants 0 10 20 30 40 50 60 70 80 Centrality % 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.0 0.5 1.0 1.5 2.0 pT [GeV] π ± K ± p ̄ p 0 10 20 30 40 50 60 70 80 Centrality % 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 5.02 TeV J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 14 / 22

20. Overview Input parameters QGP properties Model heavy-ion collision spacetime evolution

    Gaussian process emulator surrogate model MCMC calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data LHC Pb-Pb collisions J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 14 / 22

21. −0.5 0.0 0.5 p 0. 03+0. 08 −0. 08 0

    1 2 σ fluct 1. 1+0. 3 −0. 2 0.30 0.65 1.00 w [fm] 0. 89+0. 11 −0. 12 0.0 0.5 1.0 τ fs [fm/c] 0. 59+0. 41 −0. 41 0.00 0.15 0.30 η/s min 0. 06+0. 03 −0. 03 0.0 1.5 3.0 η/s slope [GeV−1] 2. 0+1. 0 −0. 8 −1 0 1 η/s crv 0. 05+0. 95 −0. 73 0.00 0.05 0.10 ζ/s max 0. 015+0. 025 −0. 015 0.000 0.025 0.050 ζ/s width [GeV] 0. 02+0. 02 −0. 02 −0.5 0.0 0.5 p 0.130 0.145 0.160 T switch [GeV] 0 1 2 σ fluct 0.30 0.65 1.00 w [fm] 0.0 0.5 1.0 τ fs [fm/c] 0.00 0.15 0.30 η/s min 0.0 1.5 3.0 η/s slope [GeV−1] −1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.000 0.025 0.050 ζ/s width [GeV] 0.130 0.145 0.160 T switch [GeV] 0. 155+0. 005 −0. 006 Posterior distribution Diagonals: prob. dists. of each param. Off-diagonals: correlations b/w pairs Estimated values: medians Uncertainties: 90% credible intervals

22. Initial entropy deposition −1.0 −0.5 0.0 0.5 1.0 p KLN

    EKRT / IP-Glasma Wounded nucleon 0. 03+0. 08 −0. 08 s ∝ Tp A + Tp B 2 1/p • Entropy deposition ∼ geometric mean of local nuclear density TA TB • Uncertainty halved from previous work • Corroborates eccentricity scaling of IP-Glasma and EKRT J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 16 / 22

23. Shear viscosity (η/s)(T) = (η/s)min + (η/s)slope(T − Tc) ×

    T Tc (η/s)crv 0.00 0.15 0.30 η/s min 0. 06+0. 03 −0. 03 0.0 1.5 3.0 η/s slope [GeV−1] 2. 0+1. 0 −0. 8 0.00 0.15 0.30 η/s min −1 0 1 η/s crv 0.0 1.5 3.0 η/s slope [GeV−1] −1 0 1 η/s crv 0. 05+0. 95 −0. 73 0.15 0.20 0.25 0.30 Temperature [GeV] 0.0 0.2 0.4 η/s KSS bound 1/4π Prior range Posterior median 90% credible region • Zero η/s excluded; min consistent with AdS/CFT • Constant η/s excluded • Best constrained T 0.23 GeV • RHIC data could disambiguate slope and curvature J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 17 / 22

24. Bulk viscosity (ζ/s)(T) = (ζ/s)max 1 + T − Tc

    (ζ/s)width 2 0.00 0.05 0.10 ζ/s max 0. 015+0. 025 −0. 015 0.00 0.05 0.10 ζ/s max 0.000 0.025 0.050 ζ/s width [GeV] 0.000 0.025 0.050 ζ/s width [GeV] 0. 02+0. 02 −0. 02 0.08 0.12 0.16 0.20 Temperature [GeV] 0.00 0.04 0.08 ζ/s Prior range • Can be “tall” or “wide”, but not both • Short and wide (green) slightly favored See also talk by G. Denicol, Wed. 17:30 J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 18 / 22

25. −0.5 0.0 0.5 p 0. 03+0. 08 −0. 08 0

    1 2 σ fluct 1. 1+0. 3 −0. 2 0.30 0.65 1.00 w [fm] 0. 89+0. 11 −0. 12 0.0 0.5 1.0 τ fs [fm/c] 0. 59+0. 41 −0. 41 0.00 0.15 0.30 η/s min 0. 06+0. 03 −0. 03 0.0 1.5 3.0 η/s slope [GeV−1] 2. 0+1. 0 −0. 8 −1 0 1 η/s crv 0. 05+0. 95 −0. 73 0.00 0.05 0.10 ζ/s max 0. 015+0. 025 −0. 015 0.000 0.025 0.050 ζ/s width [GeV] 0. 02+0. 02 −0. 02 −0.5 0.0 0.5 p 0.130 0.145 0.160 T switch [GeV] 0 1 2 σ fluct 0.30 0.65 1.00 w [fm] 0.0 0.5 1.0 τ fs [fm/c] 0.00 0.15 0.30 η/s min 0.0 1.5 3.0 η/s slope [GeV−1] −1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.000 0.025 0.050 ζ/s width [GeV] 0.130 0.145 0.160 T switch [GeV] 0. 155+0. 005 −0. 006 Posterior distribution Much more information here! But what can the model do at a “best-fit” point?

26. 0 10 20 30 40 50 60 70 80 100

    101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p Yields ±10% 0 10 20 30 40 50 60 70 80 0.8 1.0 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.0 0.5 1.0 1.5 pT [GeV] π ± K ± p ̄ p Mean pT ±10% 0 10 20 30 40 50 60 70 80 0.8 1.0 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 2.76 TeV Flow cumulants ±10% 0 10 20 30 40 50 60 70 80 0.8 1.0 1.2 Ratio 0 10 20 30 40 50 60 70 80 100 101 102 103 104 dNch /dη, dN/dy Nch π ± K ± p ̄ p ±10% 0 10 20 30 40 50 60 70 80 Centrality % 0.8 1.0 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.0 0.5 1.0 1.5 pT [GeV] π ± K ± p ̄ p ±10% 0.0 0.2 0.4 0.6 0.8 1.0 Centrality % 0.8 1.0 1.2 Ratio 0 10 20 30 40 50 60 70 80 0.00 0.04 0.08 0.12 vn {2} v2 v3 v4 Pb+Pb 5.02 TeV ±10% 0 10 20 30 40 50 60 70 80 Centrality % 0.8 1.0 1.2 Ratio TRENTo p = 0 σfluct = 1 w = 0.9 fm τfs = 0.6 fm/c Tswitch = 150 MeV η/s min = 0.06, slope = 2.2 GeV−1, crv = −0.4 ζ/s max = 0.015, width = 0.01 GeV

27. Flow correlations Correlation between event-by-event fluctuations of the magnitudes of

    flow harmonics m and n: SC(m, n) = v2 m v2 n − v2 m v2 n 0 2 4 6 8 10 Centrality % −0.8 −0.4 0.0 0.4 0.8 SC(m, n) 1e−7 Most central collisions 2.76 TeV 5.02 TeV (prediction) 0 10 20 30 40 50 60 70 Centrality % −2 −1 0 1 2 1e−6 Minimum bias SC(4, 2) SC(3, 2) *Model calculation at best-fit point only Data: ALICE, PRL 117 182301 [1604.07663] J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 20 / 22

28. More ow observables 0 1 2 3 4 5 Centrality

    % 0.02 0.03 0.04 vn {2} v2 v3 Central two-particle cumulants 0 10 20 30 40 50 60 70 Centrality % 0.04 0.06 0.08 0.10 v2 {4} Four-particle cumulants Model 2.76 TeV Model 5.02 TeV ALICE 2.76 TeV ALICE 5.02 TeV *Model calculation at best-fit point only Data: ALICE, PRL 107 032301 [1105.3865] PRL 116 132302 [1602.01119] J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 21 / 22

29. Summary and outlook • Global, multi-parameter fit of event-by-event collision

    model to diverse experimental data at 2.76 and 5.02 TeV • Constrained initial state entropy deposition, fluctuations, and granularity • Estimated temperature dependence of QGP shear and bulk viscosities • Excluded both zero and constant η/s • Found preference for short, wide ζ/s peak • Include RHIC data to further constrain transport coefficients • Improve uncertainty quantification This project is open source! https://github.com/jbernhard/qm2017 J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 22 / 22

30. 10.0 17.5 25.0 Norm 2.76 TeV 14. 4+1. 2 −1.

    3 12 21 30 Norm 5.02 TeV 20. 8+1. 7 −1. 8 −0.5 0.0 0.5 p 0. 03+0. 08 −0. 08 0 1 2 σ fluct 1. 1+0. 3 −0. 2 0.30 0.65 1.00 w [fm] 0. 89+0. 11 −0. 12 0.0 0.5 1.0 τ fs [fm/c] 0. 59+0. 41 −0. 41 0.00 0.15 0.30 η/s min 0. 06+0. 03 −0. 03 0.0 1.5 3.0 η/s slope [GeV−1] 2. 0+1. 0 −0. 8 −1 0 1 η/s crv 0. 05+0. 95 −0. 73 0.00 0.05 0.10 ζ/s max 0. 015+0. 025 −0. 015 0.000 0.025 0.050 ζ/s width [GeV] 0. 02+0. 02 −0. 02 10.0 17.5 25.0 Norm 2.76 TeV 0.130 0.145 0.160 T switch [GeV] 12 21 30 Norm 5.02 TeV −0.5 0.0 0.5 p 0 1 2 σ fluct 0.30 0.65 1.00 w [fm] 0.0 0.5 1.0 τ fs [fm/c] 0.00 0.15 0.30 η/s min 0.0 1.5 3.0 η/s slope [GeV−1] −1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.000 0.025 0.050 ζ/s width [GeV] 0.130 0.145 0.160 T switch [GeV] 0. 155+0. 005 −0. 006 Posterior distribution with normalization factors

31. Uncertainty quanti cation Given a set of experimental uncertainties {σi},

    what is the covariance matrix Σ? In general: Σij = cijσiσj where cij = correlation coefficient between observations i, j • Statistical (uncorrelated) uncertainty cij = δij =⇒ Σstat = diag(σ2 i ) • Systematic uncertainty: assume Gaussian correlation function cij = exp − 1 2 xi − xj 100 2 where xi is the centrality % midpoint of observation i J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 2 / 4

32. Conditioning a Gaussian process Given • training input points X

    • observed training outputs y at X • covariance function σ the predictive distribution at arbitrary test points X∗ is the multivariate-normal distribution 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 (Duke U.) Bayesian characterization of the initial state and QGP medium 3 / 4

33. Multivariate output Many highly correlated outputs → principal component analysis

    PCs = eigenvectors of sample covariance matrix Y Y = UΛU Transform data into orthogonal, uncorrelated linear combinations Z = √ m YU Emulate each PC independently Example transformation of two observables 0 400 800 1200 dNπ ± /dy 0.00 0.04 0.08 0.12 v2 {2} 74% 26% J. E. Bernhard (Duke U.) Bayesian characterization of the initial state and QGP medium 4 / 4