Initial Conditions of Bulk Matter in Ultrarelativistic Nuclear Collisions

Ph.D. Defense

Transcript

1. Initial Conditions of Bulk Matter in Ultrarelativistic Nuclear Collisions J. Scott Moreland Ph.D. defense | Duke University | 27 March 2019 Support provided by the Department of Energy National Nuclear Security Administration Stewardship Science Graduate Fellowship (DOE NNSA SSGF) under grant DE-FC52-08NA28752, and by the U.S. Department of Energy (DOE) under grant DE-FG02-05ER41367. Computing resources provided by the Open Science Grid (OSG) and by the National Energy Research Scientific Computing Center (NERSC).
2. QCD predicts new phase of deconfined nuclear matter Atomic nuclei Hadron resonance gas Quark-gluon plasma Increasing temperature Quantum chromodynamics (QCD) predicts that hadrons melt at Tc ∼ 150 MeV to form a deconfined soup of quarks and gluons dubbed the quark-gluon plasma (QGP). J. Scott Moreland (Duke) 1 / 44
3. Motivation: What are the properties of QGP? How and under what conditions is it formed in nature? How does it recom- bine to form colorless hadrons? Equation of state? Relations between thermal quantities, e.g. P = P( ) Transport properties? Shear/bulk viscosity, probe energy loss, etc J. Scott Moreland (Duke) 2 / 44
4. Objective: Understand the QCD phase diagram Naturally, the properties of a substance depend on the conditions of its environment QCD phase diagram Describes the phases of nuclear matter as a function of the matter’s temperature T and baryon chemical potential µB . Big picture questions • What are the phases of nuclear matter? • Where are they located in the phase diagram? • What are the matter’s intrinsic properties at each point in the phase diagram? When 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 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 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 Search for the Critical Point: “A Landmark Study” Quark-Gluon Plasma The Phases of QCD Temperature Hadron Gas Early Universe Future FAIR Experiments Future LHC Experiments Nuclear Matter Vacuum Color Superconductor Critical Point Current RHIC Experiments RHIC Energy Scan Crossover Baryon Chemical Potential ~170 MeV 0 MeV 900 MeV 0 MeV Neutron Stars 1st order phase tran sition 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. Frontiers of Nuclear Science, A Long Range Plan, arXiv:0809.3137 J. Scott Moreland (Duke) 3 / 44
5. Producing and studying QGP in the laboratory Collide ions at top energy: compress and heat matter e empera- ndergo nge mal e, water vapor 00° C; if the pres- say. The por for ed as a ph of tem- curve d and bstance, exotic iagram ables, as nt by id- Beyond nction and us. The he 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 ing further experiments in which nuclear al Point: “A Landmark Study” Quark-Gluon Plasma The Phases of QCD Temperature Hadron Gas Early Universe Future FAIR Experiments Future LHC Experiments Nuclear Matter Vacuum Color Superconductor Critical Point Current RHIC Experiments RHIC Energy Scan Crossover Baryon Chemical Potential ~170 MeV 0 MeV 900 MeV 0 MeV Neutron Stars 1st order phase tran sition 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. Frontiers of Nuclear Science, A Long Range Plan, arXiv:0809.3137 Relativistic Heavy-Ion Collider (RHIC) Large Hadron Collider (LHC) J. Scott Moreland (Duke) 4 / 44
6. QGP produced by experiments is not directly detectable Model ? Experiment sees Experiment controls Hidden process projec�les and beam energy Final par�cle proper�es Fig. by Zhi Qiu ALICE Pb-Pb J. Scott Moreland (Duke) 5 / 44
7. Established model: QGP expands hydrodynamically time kinetic freezeout Boltzmann transport (red glyphs) hydrodynamics (heatmap) initial conditions pre-collision Figure credit Zhi Qiu Initial conditions Starting point of the simulation Minimum requirements: energy density e, flow velocity uµ, initial viscous corrections πµν and Π Each quantity is a three-dimensional field at hydrodynamic starting time τ0 Common simplification: system invariant under limited boosts along the beam direction (boost invariance)—reduces three spatial dimensions to two J. Scott Moreland (Duke) 6 / 44
8. Established model: QGP expands hydrodynamically time kinetic freezeout Boltzmann transport (red glyphs) hydrodynamics (heatmap) initial conditions pre-collision Figure credit Zhi Qiu Hydrodynamics Expands local stress-energy tensor in gradients of the fluid velocity uµ Tµν = (e +P)uµuν −Pgµν +πµν −∆µνΠ Solves energy-momentum conservation: ∂µ Tµν = 0 Shear correction πµν = 2η∆µναβ∂α uβ Bulk correction Π = −ζ∂µ uµ η Resists shear strain ζ Resists expansion/compression J. Scott Moreland (Duke) 6 / 44
9. Established model: QGP expands hydrodynamically time kinetic freezeout Boltzmann transport (red glyphs) hydrodynamics (heatmap) initial conditions pre-collision Figure credit Zhi Qiu Boltzmann transport As system expands and cools, matter begins to break-up and interactions become infrequent Fluid discretized into particles and modeled using non-equilibrium Boltzmann transport: dfi (x, p) dt = Ci (x, p) Kinetic freezeout Last interactions cease, and particles freestream to the “detector” J. Scott Moreland (Duke) 6 / 44
10. Using hydro simulations to extract the QGP shear viscosity Hydrodynamics converts initial spatial anisotropy into final momentum anisotropy Initial spatial anisotropy Energy density Spatial harmonics n = 2 n = 3 n = 4 εn einΦ = − dx dy rn einφρ(x, y) dx dy rn ρ(x, y) Final momentum anisotropy Particle yield 0 π/2 π 3π/2 2π φ dN/dφ v2 v3 v4 Flow harmonics dN/dφ ∝ 1 + 2 ∞ n=1 vn cos[n(φ − Ψn )] Conversion efficiency governed by the QGP specific shear viscosity: vn εn ↔ η s J. Scott Moreland (Duke) 7 / 44
11. Using hydro simulations to extract the QGP shear viscosity 1. Choose your favorite initial condition (IC) model 2. Run hydrodynamic simulations and calculate flow anisotropy vn 3. Tune simulation’s specific shear viscosity η/s to fit flow anisotropy 0 100 200 300 400 N Part 0 0.02 0.04 0.06 0.08 0.1 v 2 PHOBOS Glauber η/s=10-4 η/s=0.08 η/s=0.16 0 100 200 300 400 N Part 0 0.02 0.04 0.06 0.08 0.1 v 2 PHOBOS CGC η/s=10-4 η/s=0.08 η/s=0.16 η/s=0.24 Different ICs different η/s! Luzum and Romatschke, PRC.79.039903 J. Scott Moreland (Duke) 8 / 44
12. This work: a new initial condition model for data-driven inference J. S. Moreland, J. E. Bernhard, and S. A. Bass, “Alternative ansatz to wounded nucleon and binary collision scaling in high-energy nuclear collisions”, Phys. Rev. C92, 011901 (2015) arXiv:1412.4708 [nucl-th]
13. Heuristic approaches Bottom-up (ab initio) approach Theory 1 Theory 2 Theory 3 Theory 4 ◎ Experimental data • Color Glass Condensate EFT • pQCD minijet production & gluon saturation • AdS-CFT holography • Color flux tubes and string fragmentation Top-down (data-driven) approach Plausible theory space ◎ Experimental data parameter p • Construct plausible “meta-model” • Parametrize uncertain d.o.f. • Place reasonable prior on parameter • Update prior using global model-to-data comparison (Bayesian approach) J. Scott Moreland (Duke) 10 / 44
14. Nuclear structure Nucleon positions Sampled nuclear density is modeled as sum of localized nucleon densities ρA (x) = A i=1 ρp (x − xi ), ρp (x) = 1 (2πw2)3/2 exp − |x|2 2w2 . Sample nucleon positions from Fermi dist. ρ(r) = ρ0 1 + exp r−R a , with optional polar-angle deformation R → R[1 + β2 Y 0 2 (θ) + β4 Y 0 4 (θ)]. J. Scott Moreland (Duke) 11 / 44
15. Inelastic cross sections Inelastic nucleon participants Inelastic nucleon-nucleon collision prob: Pcoll (b) = 1 − exp[−σgg Tpp (b)], Tpp (b) = d2x⊥ Tp (x⊥ )Tp (x⊥ − b) Tp (x⊥ ) = dz ρp (x⊥ , z) Ə Ɛ Ƒ ƒ Ɠ Ɣ 0Œ=lœ ƏĺƏƏ ƏĺƑƔ ƏĺƔƏ ƏĺƕƔ ƐĺƏƏ 1oѴѴ Ő0ő bm;Ѵ mm ƷѵĺƓ=lƑ 0Ѵ-1h7bvh ‰ƷƏĺƓŒ=lœ ‰ƷƏĺƕŒ=lœ ‰ƷƏĺƖŒ=lœ J. Scott Moreland (Duke) 12 / 44
16. Participant thickness functions Participant thickness functions ˜ TA, ˜ TB Transverse density of participant matter: ˜ TA,B(x⊥) = Npart i=1 γi Tp(x⊥ − x⊥i ± b/2) Random weight γi sampled from Gamma distribution with unit mean and variance 1/k. These ad hoc fluctuations are necessary to describe large minimum-bias proton-proton multiplicity fluctuations. J. Scott Moreland (Duke) 13 / 44
17. Midrapidity energy and entropy deposition as a mapping ˜ TA ˜ TB Before τ0 dη dx dy ˜ TA ˜ TB After In ultrarelativistic limit, energy and entropy densities are local mappings e0 = fe( ˜ TA, ˜ TB), s0 = fs( ˜ TA, ˜ TB). Scalar functions fe and fs must obey basic physical constraints J. Scott Moreland (Duke) 14 / 44
18. Parametrizations in the literature Two-component ansatz Perhaps the oldest ansatz is participant nucleon scaling e0 s0 ∝ npart A + npart B 2 = ˜ TA + ˜ TB 2 Couldn’t describe multiplicity distribution, so a binary collision term ncoll AB ∝ ˜ TA ˜ TB was added e0 s0 ∝ (1 − α) npart A + npart B 2 + α ncoll AB Parameter α is tuned to fit the data (right) 101 102 103 dNch /dη Pb-Pb, sNN =2.76 TeV dNch /η∝ Npart dNch /η∝ Ncoll Two-component: α=0.08 ALICE, |η| <0.5 0 10 20 30 40 50 60 70 80 Centrality % 0.5 1.0 1.5 Ratio Two-component model fit to Pb-Pb data J. Scott Moreland (Duke) 15 / 44
19. Two-component parametrization fails to describe the data 0 0.02 0.04 0.06 ) 2 p(v -1 10 1 10 centrality: 0-1% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.36 2 ∈ MC-KLN 0.31 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L 0 0.05 0.1 0.15 -1 10 1 10 centrality: 5-10% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.46 2 ∈ MC-KLN 0.30 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L 0 0.05 0.1 0.15 0.2 0.25 -2 10 -1 10 1 10 centrality: 20-25% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.40 2 ∈ MC-KLN 0.29 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L 2 v 0 0.1 0.2 ) 2 p(v -2 10 -1 10 1 10 centrality: 30-35% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.36 2 ∈ MC-KLN 0.28 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L 2 v 0 0.1 0.2 0.3 -2 10 -1 10 1 10 centrality: 40-45% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.31 2 ∈ MC-KLN 0.26 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L 2 v 0 0.1 0.2 0.3 -3 10 -2 10 -1 10 1 10 centrality: 55-60% |<2.5 η >0.5GeV,| T data p 2 ∈ Glauber 0.24 2 ∈ MC-KLN 0.21 ATLAS Pb+Pb =2.76 TeV NN s -1 b µ = 7 int L ATLAS collaboration JHEP11(2013)183 Two-component ansatz (MC-Glauber) fails to describe the elliptic flow data v2 . Model predicts wrong shape for the event-by-event flow distribution P(v2 ). J. Scott Moreland (Duke) 16 / 44
20. Alternative ansatz to participant and binary collision scaling Generalized mean ansatz Replace arithmetic mean (participant scaling) with larger family of means: e0 s0 ∝ Mp ( ˜ TA, ˜ TB ) ≡ ˜ Tp A + ˜ Tp B 2 1/p Mp (x, y) =                  max(x, y) p → +∞, (x + y)/2 p = +1, √ x y p = 0, 2 xy/(x + y) p = −1, min(x, y) p → −∞. Parameter p inferred from data (right) 101 102 103 dNch /dη Pb-Pb, sNN =2.76 TeV Gen. mean: p=1 Gen. mean: p=0 Gen. mean: p= − 1 ALICE, |η| <0.5 0 10 20 30 40 50 60 70 80 Centrality % 0.75 1.00 1.25 Ratio Generalized mean ansatz fit to Pb-Pb data J. Scott Moreland (Duke) 17 / 44
21. Midrapidity energy and entropy deposition Reduced thickness TR = Mp ( ˜ TA, ˜ TB ) Depicted: geometric mean p = 0 Generalized mean ansatz: e0 s0 = Norm × Mp( ˜ TA, ˜ TB), Mp(x, y) ≡ xp + yp 2 1/p Each mean predicts unique geometry: Harmonic p = −1 Geometric p = 0 Arithmetic p = +1 J. Scott Moreland (Duke) 18 / 44
22. Model namesake Previous steps complete the specification of the TRENTo model: Reduced Thickness Event-by-event Nuclear Topology1 Variable model parameters 1. Overall energy (or entropy) normalization 2. Generalized mean parameter p 3. Inverse variance of the nucleon fluctuation gamma weights k 4. Gaussian nucleon width w [fm] 5. Inter-nucleon minimum distance dmin [fm] 1Yes, this is an abuse of terminology J. Scott Moreland (Duke) 19 / 44
23. Multiplicity distributions 0 10 20 30 40 50 Nch 10-4 10-3 10-2 10-1 100 101 P(Nch) p=1 p=0 p= − 1 ALICE 2.36 TeV NSD |η| <1, pT corrected p+p ×102 ×101 ×100 0 50 100 150 200 Nch 10-6 10-4 10-2 100 102 p=1 p=0 p= − 1 ALICE 5.02 TeV |η| <1, 0.2<pT <3.0 GeV p+Pb ×104 ×102 ×100 0 1000 2000 3000 Nch 10-6 10-4 10-2 100 102 p=1 p=0 p= − 1 ALICE 2.76 TeV |η| <1, 0.2<pT <3.0 GeV Pb+Pb ×104 ×102 ×100 • Figure: TRENTo charged-particle distributions P(Nch ) compared to ALICE data • TRENTo yield calculated assuming entropy deposition s0 ∝ Mp ( ˜ TA, ˜ TB ). For ideal hydrodynamics, dNch /dη ∝ dS/dη. • Only the geometric mean (p = 0) describes all three systems J. Scott Moreland (Duke) 20 / 44
24. Eccentricity harmonics 0 30 60 90 Centrality % 0.0 0.2 0.4 0.6 εn ε2 0 30 60 90 Centrality % 0.0 0.2 0.4 0.6 ε3 Arithmetic: p=1 Geometric: p=0 Harmonic: p= − 1 0 10 20 30 40 Centrality % 0.3 0.6 0.9 1.2 Ratio IP-Glasma • Left and middle: second and third eccentricity harmonics ε2 and ε3 for the Pythagorean means. • Far right: ratio of second and third eccentricities strongly constrained by heavy-ion flow data (gray band) calculated by Retinskaya et al. PRC.89.014902. • TRENTo nicely agrees with flow constraints using geometric mean (p = 0). J. Scott Moreland (Duke) 21 / 44
25. Application: estimating initial condition and QGP medium properties J. E. Bernhard, J. S. Moreland, and S. A. Bass, “Bayesian estimation of the specific shear and bulk viscosity of the quark-gluon plasma”, Nature Physics (submitted), (2019) This study is the capstone of J. Bernhard’s dissertation on Bayesian parameter estimation for heavy-ion collisions. I am not the author of the Bayesian parameter estimation framework.
26. Heavy-ion collision model Figure: H. Petersen, MADAI TrENTo initial conditions parametric energy deposition PRC.92.011901 freestream pre-flow infinitely weak coupling limit PRC.91.064906, PRC.80.034902 VISH2+1 viscous hydro 14-mom. approx w/ shear & bulk PRC.77.064901, J.CPC.2015.08.039 frzout sampler non-RTA shear & bulk corrections J.E. Bernhard thesis UrQMD Boltzmann cascade hadronic afterburner, simulate scatterings and decays PPNP.98.00058, JPG.25.9.308 Final observables calculated as similar as possible to experiment J. Scott Moreland (Duke) 23 / 44
27. Model parameters (quantities of interest) Initial conditions 1–2. Overall normalizations at 2.76 and 5.02 TeV 3. Generalized mean p for initial energy deposition Harmonic Geometric Arithmetic 4. Std. dev. of nucleon-nucleon multiplicity fluctuations σfluct 5. Gaussian nucleon width w 6. Inter-nucleon minimum distance dmin QGP medium 7. Pre-equilibrium free streaming time τfs 8–10. three parameters (η/s min, slope, and crv) for the temperature dependence of the specific shear viscosity (η/s)(T) 11–13. three parameters (ζ/s max, width, and loc) for the temperature dependence of the specific bulk viscosity (ζ/s)(T) 150 200 250 300 Temperature [MeV] 0.0 0.2 0.4 0.6 η/s 150 200 250 300 Temperature [MeV] 0.00 0.02 0.04 0.06 0.08 0.10 ζ/s 14. Particlization temperature Tswitch J. Scott Moreland (Duke) 24 / 44
28. Experimental calibration data Pb-Pb collision data at √ sNN = 2.76 and 5.02 TeV taken from the ALICE experiment Observable Label Particles Reference Integrated Charged particle yield dNch/dη charged [1012.1657] [1512.06104] Identified particle yield dN/dy π, K, p [1303.0737] Transverse energy yield dET /dη [1603.04775] Azimuthal Two-particle flow cumulants vn{k} charged [1105.3865] [1602.01119] Radial Mean transverse momentum pT π, K, p [1303.0737] Transverse momentum fluctuations δpT / p charged [1407.5530] J. Scott Moreland (Duke) 25 / 44
29. The challenge of rigorous model-to-data comparison Parameter Observable shear viscosity bulk viscosity pre-equilibrium flow nucleon width hadronization temp p+p fluctuations charged particle yields identified particle yields transverse energy anisotropic flow cumulants mean transverse momentum transverse momentum fluctuations Testing a single set of parameters requires ∼1000 CPU hours ...and evaluating fourteen different parameters five times each requires 514 × 103 ≈ 6 × 1012 CPU hours. That’s roughly 7 × 108 computer years! J. Scott Moreland (Duke) 26 / 44
30. Solution: Bayesian parameter estimation Input parameters IC and QGP properties Physics theory initial stages, hydro, and Boltzmann transport Computer model minimum bias event-by- event simulations Gaussian process emulator surrogate model Markov chain Monte Carlo calibrate model to data Posterior distribution quantitative estimates of each parameter Experimental data yields, mean pT , flows J. Scott Moreland (Duke) 27 / 44 Jonah Bernhard, Duke Ph.D. dissertation “Bayesian parameter estimation for heavy-ion collisions” Extensible and adaptable framework
31. Bayes’ theorem Definition posterior P(x|E) ∝ prior P(x) × likelihood P(E|x) Prior: flat within design range, zero otherwise Likelihood: multivariate Gaussian form P(E|x) = 1 (2π)mdetΣ exp − 1 2 [ym(x) − ye]T Σ−1(x)[ym(x) − ye] Use MCMC to draw samples from unnormalized posterior distribution. Histogram samples to visualize the posterior. J. Scott Moreland (Duke) 28 / 44
32. 0 20 40 60 80 100 101 102 103 104 105 dNch /dη, dN/dy, dET/dη [GeV] Nch ET π K p Pb-Pb 2.76 TeV 0 20 40 60 80 100 101 102 103 104 105 Nch ET π K p Yields Pb-Pb 5.02 TeV 0 20 40 60 80 0.0 0.5 1.0 1.5 pT [GeV] π K p 0 20 40 60 80 0.0 0.5 1.0 1.5 π K p Mean pT 0 20 40 60 80 0.00 0.01 0.02 0.03 0.04 δpT/ pT 0 20 40 60 80 0.00 0.01 0.02 0.03 0.04 Mean pT fluctuations 0 20 40 60 80 Centrality % 0.00 0.05 0.10 vn 2 v2 v3 v4 0 20 40 60 80 Centrality % 0.00 0.05 0.10 v2 v3 v4 Flow cumulants Training data (prior) Bayesian analysis by J. Bernhard 8 14 20 Norm 2.76 TeV 14.0+5.2 −5.6 10.0 17.5 25.0 Norm 5.02 TeV 17.5+6.0 −7.5 0.5 0.0 0.5 p 0.000+0.449 −0.451 0 1 2 σ fluct 1.00+1.00 −0.80 0.4 0.7 1.0 w [fm] 0.70+0.24 −0.30 0.000 2.456 4.913 d min3 2.46+2.39 −2.03 0.00 0.75 1.50 τ fs [fm/c] 0.75+0.74 −0.61 0.0 0.1 0.2 η/s min 0.10+0.08 −0.10 0 4 8 η/s slope [GeV−1] 4.00+3.35 −3.84 1 0 1 η/s crv −0.000+0.813 −0.987 0.00 0.05 0.10 ζ/s max 0.05+0.05 −0.04 0.00 0.05 0.10 ζ/s width [GeV] 0.050+0.040 −0.050 0.150 0.175 0.200 ζ/s T0 [GeV] 0.175+0.022 −0.023 0.135 0.150 0.165 T switch [GeV] 0.150+0.013 −0.014 8 14 20 Norm 2.76 TeV 0.0 0.2 0.4 σ model sys 10.0 17.5 25.0 Norm 5.02 TeV 0.5 0.0 0.5 p 0 1 2 σ fluct 0.4 0.7 1.0 w [fm] 0.000 2.456 4.913 d min3 0.00 0.75 1.50 τ fs [fm/c] 0.0 0.1 0.2 η/s min 0 4 8 η/s slope [GeV−1] 1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.00 0.05 0.10 ζ/s width [GeV] 0.150 0.175 0.200 ζ/s T0 [GeV] 0.135 0.150 0.165 T switch [GeV] 0.0 0.2 0.4 σ model sys 0.13+0.14 −0.11 Parameter values (prior)
33. 0 20 40 60 80 100 101 102 103 104 105 dNch /dη, dN/dy, dET/dη [GeV] Nch ET π K p Pb-Pb 2.76 TeV 0 20 40 60 80 100 101 102 103 104 105 Nch ET π K p Yields Pb-Pb 5.02 TeV 0 20 40 60 80 0.0 0.5 1.0 1.5 pT [GeV] π K p 0 20 40 60 80 0.0 0.5 1.0 1.5 π K p Mean pT 0 20 40 60 80 0.00 0.01 0.02 0.03 0.04 δpT/ pT 0 20 40 60 80 0.00 0.01 0.02 0.03 0.04 Mean pT fluctuations 0 20 40 60 80 Centrality % 0.00 0.05 0.10 vn 2 v2 v3 v4 0 20 40 60 80 Centrality % 0.00 0.05 0.10 v2 v3 v4 Flow cumulants Emulator samples (posterior) Bayesian analysis by J. Bernhard 8 14 20 Norm 2.76 TeV 13.9+1.2 −1.1 10.0 17.5 25.0 Norm 5.02 TeV 18.5+1.8 −1.7 0.5 0.0 0.5 p 0.006+0.078 −0.078 0 1 2 σ fluct 0.90+0.24 −0.27 0.4 0.7 1.0 w [fm] 0.96+0.04 −0.05 0.000 2.456 4.913 d min3 2.08+2.10 −2.08 0.00 0.75 1.50 τ fs [fm/c] 1.16+0.29 −0.25 0.0 0.1 0.2 η/s min 0.085+0.026 −0.025 0 4 8 η/s slope [GeV−1] 0.83+0.83 −0.83 1 0 1 η/s crv −0.37+0.79 −0.63 0.00 0.05 0.10 ζ/s max 0.037+0.040 −0.022 0.00 0.05 0.10 ζ/s width [GeV] 0.029+0.045 −0.026 0.150 0.175 0.200 ζ/s T0 [GeV] 0.177+0.023 −0.021 0.135 0.150 0.165 T switch [GeV] 0.152+0.003 −0.003 8 14 20 Norm 2.76 TeV 0.0 0.2 0.4 σ model sys 10.0 17.5 25.0 Norm 5.02 TeV 0.5 0.0 0.5 p 0 1 2 σ fluct 0.4 0.7 1.0 w [fm] 0.000 2.456 4.913 d min3 0.00 0.75 1.50 τ fs [fm/c] 0.0 0.1 0.2 η/s min 0 4 8 η/s slope [GeV−1] 1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.00 0.05 0.10 ζ/s width [GeV] 0.150 0.175 0.200 ζ/s T0 [GeV] 0.135 0.150 0.165 T switch [GeV] 0.0 0.2 0.4 σ model sys 0.10+0.09 −0.08 Parameter values (posterior)
34. 0 20 40 60 80 101 102 103 104 105 dNch /dη, dN/dy, dET/dη [GeV] Nch ET π K p Yields 0 20 40 60 80 0.00 0.05 0.10 0.15 vn k v2 2 v2 4 v3 2 v4 2 Flow cumulants 2.76 TeV 5.02 TeV ±10% 0 20 40 60 80 0.9 1.0 1.1 Ratio ±10% 0 20 40 60 80 0.9 1.0 1.1 Ratio 0 20 40 60 80 0.0 0.5 1.0 1.5 pT [GeV] π K p Mean pT 0 20 40 60 80 0.00 0.02 0.04 δpT/ pT Mean pT fluctuations ±10% 0 20 40 60 80 Centrality % 0.9 1.0 1.1 Ratio ±10% 0 20 40 60 80 Centrality % 0.9 1.0 1.1 Ratio Pb-Pb, 2.76 & 5.02 TeV (“best fit”) Bayesian analysis by J. Bernhard 0 20 40 60 80 101 102 103 104 105 dNch /dη, dN/dy, dET/dη [GeV] N×25 ch E×5 T π K p Yields 0 20 40 60 80 0.00 0.05 0.10 0.15 vn k v2 2 v3 2 v4 2 Flow cumulants ±10% 0 20 40 60 80 0.8 1.0 1.2 Ratio ±10% 0 20 40 60 80 0.8 1.0 1.2 Ratio 0 20 40 60 80 0.0 0.5 1.0 1.5 pT [GeV] π K p Mean pT 0 20 40 60 80 0.00 0.02 0.04 δpT/ pT Mean pT fluctuations ±10% 0 20 40 60 80 Centrality % 0.8 1.0 1.2 Ratio ±10% 0 20 40 60 80 Centrality % 0.8 1.0 1.2 Ratio Xe-Xe, 5.44 TeV (prediction) Predictions by S. Moreland Same parameters
35. Results: initial condition and QGP medium properties Posterior energy deposition Posterior QGP viscosities 1.0 0.5 0.0 0.5 1.0 p KLN EKRT / IP-Glasma Wounded nucleon 150 200 250 300 Temperature [MeV] 0.0 0.1 0.2 0.3 0.4 η/s 1/4π Shear viscosity Posterior median 90% credible region 150 200 250 300 Temperature [MeV] 0.00 0.02 0.04 0.06 0.08 ζ/s Bulk viscosity Evidently, initial energy deposition is well described by geometric mean scaling: e0 ∝ ˜ TA ˜ TB Recall, ˜ TA, ˜ TB are the transverse densities of participant nuclear matter Properly accounting for IC uncertainties (left) allowed for first quantitative estimate of (η/s)(T) and (ζ/s)(T) with meaningful uncertainties (above) These uncertainty bands would be significantly underestimated if we used a bottom-up ab initio initial condition model! J. Scott Moreland (Duke) 31 / 44
36. Application: investigating hydrodynamic flow in small collision systems J. S. Moreland, J. E. Bernhard, and S. A. Bass, “Bayesian calibration of a hybrid nuclear collision model using p-Pb and Pb-Pb data from the LHC”, Phys. Rev. C (submitted), (2019) arXiv:1808.02106 [nucl-th]
37. Recent discovery: collectivity in small collision systems! η ∆ -4 -2 0 2 4 (radians) φ ∆ 0 2 4 φ ∆ d η ∆ d pair N 2 d trig N 1 2.4 2.6 2.8 < 260 offline trk N ≤ = 2.76 TeV, 220 NN s (a) CMS PbPb < 3 GeV/c trig T 1 < p < 3 GeV/c assoc T 1 < p η ∆ -4 -2 0 2 4 (radians) φ ∆ 0 2 4 φ ∆ d η ∆ d pair N 2 d trig N 1 3.1 3.2 3.3 3.4 < 260 offline trk N ≤ = 5.02 TeV, 220 NN s (b) CMS pPb < 3 GeV/c trig T 1 < p < 3 GeV/c assoc T 1 < p Pb Pb Pb p • Hydrodynamic description of heavy-ion collisions is now well established. Primary “proof” is remarkable global description of bulk observables, e.g. see previous figure. • The framework surely breaks down at some small length scale. After all, it is an effective theory. A priori, hydro was not expected to describe collisions involving protons and light ions. However, data seems to suggest otherwise! • The exact same collectivity signals used to justify flow in heavy-ion collisions now seen in p-p and p-A collisions. Natural question: Can we apply the same theory machinery, but down to a smaller length scale? J. Scott Moreland (Duke) 33 / 44
38. The “spherical cow” problem • Energy density profile produced by light-ion collisions, e.g. p-p and p-Pb, is sensitive to each nucleon’s sampled density profile. • So far I’ve assumed the nucleon is a Gaussian “sphere” for simplicity. • Geometric mean of two Gaussians is also a Gaussian. p=1 p=0 p= − 1 proton-proton collisions → • Flow anisotropy is sensitive to initial state eccentricity, so let’s relax the Gaussian proton assumption. J. Scott Moreland (Duke) 34 / 44
39. Nucleon substructure (lumpiness) Lump dispersion 1 fm .8 fm .6 fm .4 fm Lump width Number of lumps Free parameters: • Width of the Gaussian distribution used to sample constituent lump radial positions • Number of Gaussian constituent lumps inside each nucleon • Width of the Gaussian constituent lumps Absent from this work: • Lump spatial correlations
40. Qualitative effect of nucleon substructure Transverse lead density 10 fm no substructure substructure Transverse proton density 2 fm no substructure substructure Pictured above: transverse nuclear densities T(x⊥ ) = dz ρ(x⊥ , z) J. Scott Moreland (Duke) 36 / 44
41. Model parameters (quantities of interest) Initial conditions 1. Single normalization at 5.02 TeV 2. Generalized mean p for initial energy deposition Harmonic Geometric Arithmetic 3. Std. dev. of nucleon-nucleon multiplicity fluctuations σfluct 4. Gaussian width of constituent radial position distribution r 5. Gaussian constituent number nc 6. Gaussian width of each constituent v 7. Inter-nucleon minimum distance dmin QGP medium 8. Pre-equilibrium free streaming time τfs 9–11. three parameters (η/s min, slope, and crv) for the temperature dependence of the specific shear viscosity (η/s)(T) 12–14. three parameters (ζ/s max, width, and loc) for the temperature dependence of the specific bulk viscosity (ζ/s)(T) 150 200 250 300 Temperature [MeV] 0.0 0.2 0.4 0.6 η/s 150 200 250 300 Temperature [MeV] 0.00 0.02 0.04 0.06 0.08 0.10 ζ/s 15. Particlization temperature Tswitch J. Scott Moreland (Duke) 37 / 44
42. Experimental calibration data p-Pb and Pb-Pb collision data at √ sNN = 5.02 TeV taken from ALICE and CMS experiments Collision system Observable Rapidity cut Momentum cut Ref. Pb-Pb, 5.02 TeV Yield dNch/dη |η| < 0.5 — 1512.06104 Flow cumulants vn{2}, |η| < 0.8, |∆η| > 1 0.2 < pT < 5.0 GeV 1602.01119 n = 2, 3, 4 p-Pb, 5.02 TeV Yield dNch/dη |η| < 1.4 — 1412.6828 Mean transverse momentum pT |η| < 0.3 0.15 < pT < 10 GeV 1307.1094 Flow cumulants vn{2}, |η| < 2.4, |∆η| > 2 0.3 < pT < 3.0 GeV 1305.0609 n = 2, 3 J. Scott Moreland (Duke) 38 / 44
43. 0 20 40 60 Centrality % 100 101 102 103 104 dNch /dη Nch p-Pb 5.02 TeV 0 20 40 60 80 Centrality % Nch Yields Pb-Pb 5.02 TeV 2 4 6 nch / nch 0.0 0.5 1.0 1.5 pT [GeV] ch 0 20 40 60 80 Centrality % Mean pT 2 4 6 noffline trk / noffline trk 0.00 0.05 0.10 0.15 vn 2 v2 v3 0 20 40 60 80 Centrality % v2 v3 v4 Flow cumulants Training data (prior) Bayesian analysis by S. Moreland 9.0 18.5 28.0 Norm [GeV] 18.5+7.6 −9.5 1 0 1 p −0.000+0.997 −0.802 0 1 2 σ fluct 1.00+0.99 −0.81 0.0 0.6 1.2 r [fm] 0.60+0.51 −0.57 1 5 9 nc 5.0+3.8 −3.4 0.2 0.7 1.2 v [fm] 0.70+0.41 −0.49 0.000 2.456 4.913 d min3 2.46+2.18 −2.24 0.1 0.8 1.5 τ fs [fm/c] 0.80+0.67 −0.59 0.0 0.1 0.2 η/s min 0.10+0.08 −0.10 0 4 8 η/s slope [GeV−1] 4.00+3.36 −3.84 1 0 1 η/s crv 0.001+0.997 −0.802 0.00 0.05 0.10 ζ/s max 0.05+0.05 −0.04 0.00 0.05 0.10 ζ/s width [GeV] 0.05+0.05 −0.04 0.150 0.175 0.200 ζ/s T0 [GeV] 0.175+0.021 −0.024 9.0 18.5 28.0 Norm [GeV] 0.135 0.150 0.165 T switch [GeV] 1 0 1 p 0 1 2 σ fluct 0.0 0.6 1.2 r [fm] 1 5 9 nc 0.2 0.7 1.2 v [fm] 0.000 2.456 4.913 d min3 0.1 0.8 1.5 τ fs [fm/c] 0.0 0.1 0.2 η/s min 0 4 8 η/s slope [GeV−1] 1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.00 0.05 0.10 ζ/s width [GeV] 0.150 0.175 0.200 ζ/s T0 [GeV] 0.135 0.150 0.165 T switch [GeV] 0.150+0.012 −0.015 Parameter values (prior)
44. 0 20 40 60 Centrality % 100 101 102 103 104 dNch /dη Nch p-Pb 5.02 TeV 0 20 40 60 80 Centrality % Nch Yields Pb-Pb 5.02 TeV 2 4 6 nch / nch 0.0 0.5 1.0 1.5 pT [GeV] ch 0 20 40 60 80 Centrality % Mean pT 2 4 6 noffline trk / noffline trk 0.00 0.05 0.10 0.15 vn 2 v2 v3 0 20 40 60 80 Centrality % v2 v3 v4 Flow cumulants Emulator samples (posterior) Bayesian analysis by S. Moreland 9.0 18.5 28.0 Norm [GeV] 20.0+2.6 −2.5 1 0 1 p 0.002+0.157 −0.180 0 1 2 σ fluct 0.91+0.32 −0.33 0.0 0.6 1.2 r [fm] 0.88+0.26 −0.23 1 5 9 nc 6.0+3.0 −3.4 0.2 0.7 1.2 v [fm] 0.52+0.28 −0.20 0.000 2.456 4.913 d min3 1.39+2.16 −1.39 0.1 0.8 1.5 τ fs [fm/c] 0.47+0.55 −0.37 0.0 0.1 0.2 η/s min 0.08+0.07 −0.07 0 4 8 η/s slope [GeV−1] 1.24+1.46 −1.24 1 0 1 η/s crv −0.09+0.80 −0.91 0.00 0.05 0.10 ζ/s max 0.026+0.032 −0.026 0.00 0.05 0.10 ζ/s width [GeV] 0.035+0.043 −0.035 0.150 0.175 0.200 ζ/s T0 [GeV] 0.174+0.020 −0.024 9.0 18.5 28.0 Norm [GeV] 0.135 0.150 0.165 T switch [GeV] 1 0 1 p 0 1 2 σ fluct 0.0 0.6 1.2 r [fm] 1 5 9 nc 0.2 0.7 1.2 v [fm] 0.000 2.456 4.913 d min3 0.1 0.8 1.5 τ fs [fm/c] 0.0 0.1 0.2 η/s min 0 4 8 η/s slope [GeV−1] 1 0 1 η/s crv 0.00 0.05 0.10 ζ/s max 0.00 0.05 0.10 ζ/s width [GeV] 0.150 0.175 0.200 ζ/s T0 [GeV] 0.135 0.150 0.165 T switch [GeV] 0.149+0.013 −0.014 Parameter values (posterior)
45. 10-1 100 101 102 103 104 105 dNch /dη, dN/dy, dET/dη [GeV] N×25 ch E×5 T π K p p-Pb 5.02 TeV N×25 ch E×5 T π K p Yields Pb-Pb 5.02 TeV 0 10 20 30 40 50 Centrality % 0.8 1.0 1.2 Ratio 0 20 40 60 80 Centrality % 0.0 0.5 1.0 1.5 pT [GeV] ch π K p ch π K p Mean pT 1 2 3 4 5 6 Nch / Nch 0.8 1.0 1.2 Ratio 0 20 40 60 80 Centrality % 0.00 0.05 0.10 vn k v2 2 v2 4 v3 2 v2 2 v2 4 v3 2 v4 2 Flow cumulants 1 2 3 4 5 6 Noffline trk / Noffline trk 0.8 1.0 1.2 Ratio 0 20 40 60 80 Centrality % p-Pb & Pb-Pb, 5.02 TeV (“best fit”) Bayesian analysis by S. Moreland Unified hydrodynamic description of small & large systems? p-Pb Pb-Pb Nucleon substructure is essential to describe both systems simultaneously
46. Initial condition and QGP medium properties Posterior energy deposition Posterior QGP viscosities 1.0 0.5 0.0 0.5 1.0 p KLN EKRT / IP-Glasma Wounded nucleon Bayesian calibration on: p-Pb, Pb-Pb 5.02 TeV Pb-Pb 2.76, 5.02 TeV 150 200 250 300 Temperature [MeV] 0.0 0.1 0.2 0.3 0.4 η/s 1/4π 1/4π Shear viscosity 90% credible region Posterior median 150 200 250 300 Temperature [MeV] 0.00 0.02 0.04 0.06 0.08 ζ/s Bulk viscosity Bayesian calibration on: p-Pb, Pb-Pb 5.02 TeV Pb-Pb 2.76, 5.02 TeV Uncertainties are larger, but nevertheless consistent with geometric mean scaling e0 ∝ ( ˜ TA ˜ TB )1/2 as before. Note, this was not guaranteed! If model was wildly wrong, unlikely that similar scaling would hold. Again, viscosity estimates self-consistent with Pb-Pb calibration to 2.76 and 5.02 TeV data without substructure. J. Scott Moreland (Duke) 41 / 44
47. Nucleon substructure properties 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Constituent sampling radius r [fm] 0.2 0.4 0.6 0.8 1.0 1.2 Constituent width v [fm] 1 3 5 7 9 Constituent number nc Sampling radius r varies constituent dispersion: position distribution → P(x) ∝ exp(−|x|2/2r2) Width v varies each constituent’s size: density → ρ(x) ∝ exp(−|x|2/2v2) Parameter nc varies number of constituents (lumps) inside each nucleon. Evidently nc > 1 favored, but no strong preference for specific number. Measure proton size and shape with ultrarelativistic nuclear collisions? J. Scott Moreland (Duke) 42 / 44
48. Conclusion • Developed the TRENTo model for ultrarelativistic nuclear collisions • Applied model to extract QGP transport coefficients from combined p-Pb, Pb-Pb analysis • Demonstrated hydrodynamics is able to simultaneously describe bulk observables in small & large collision systems Harmonic Geometric Arithmetic 150 200 250 300 Temperature [MeV] 0.0 0.1 0.2 0.3 0.4 η/s 1/4π 1/4π Shear viscosity 90% credible region Posterior median 150 200 250 300 Temperature [MeV] 0.00 0.02 0.04 0.06 0.08 ζ/s Bulk viscosity Bayesian calibration on: p-Pb, Pb-Pb 5.02 TeV Pb-Pb 2.76, 5.02 TeV J. Scott Moreland (Duke) 43 / 44
49. Backup slides
50. Insight from first-principles theory Lattice QCD QCD is solvable in thermal equilibrium using lattice discretization techniques • Method limited to zero baryon chemical potential µB = 0 • Provides QCD equation of state relating energy density, entropy density, pressure, etc. at fixed temperature and baryon chemical potential • Lattice calc. indicate phase transition to QGP is smooth crossover at Tc ∼ 150 MeV for µB = 0 (ε-3p)/T4 p/T4 s/4T4 0 1 2 3 4 130 170 210 250 290 330 370 T [MeV] stout HISQ HotQCD and Wuppertal-Budapest lattice QCD equations of state, 10.1103/PhysRevD.90.094503 J. Scott Moreland (Duke) 1 / 8
51. Israel Stewart equations & Cooper-Frye formula Second-order Israel-Stewart equations in the 14-momentum approximation: τΠΠ + ˙ Π = −ζθ − δΠΠΠθ + λΠππµνσµν, (1a) τπ ˙ π µν + πµν = 2ησµν − δπππµνθ + φ7π µ α πν α − τπππ µ α σν α + λπΠΠσµν. (1b) Fluid discretized into particles using the Cooper-Frye formula: E dNi d3p = gi (2π)3 Σ fi (x, p) pµ d3σµ, (2) where i is an index over species, fi the particle species’ distribution function, and d3σµ a volume element (located at spacetime position x) of the isothermal hypersurface Σ defined by Tswitch . J. Scott Moreland (Duke) 2 / 8
52. Lattice QCD details Quarks defined at lattice sites. Gluons defined on links connecting lattice sites (right). Start by calculating vacuum-to-vacuum transition amplitude Z = DAa µ (x)D ¯ Ψ(x)DΨ(x)ei d4xL[Aa µ ,¯ Ψ,Ψ] Replace time t everywhere with imaginary time, t → iτ Z = DAa µ (x, τ)D ¯ Ψ(x, τ)DΨ(x, τ)e− β 0 dτ d3xLE [Aa µ ,¯ Ψ,Ψ] Associate with the partition function Z = trˆ ρ where ˆ ρ = e−β ˆ H is the density operator for grand canonical ensemble in therm equilibrium J. Scott Moreland (Duke) 3 / 8
53. ALICE detector • Systems: p-p, p-Pb, Pb-Pb, Xe-Xe • Detector layers from inner to outer: first tracking system, then EM and hadronic calorimeters, and finally muon tracking system. • Magnetic field bends tracks of charged particles for momentum and charge determination • Complemented by 18 detectors to specify mass, velocity, and electric sign. • Inner tracking system: (1) silicon pixel, drift, and strip detectors, (2) time projection chamber, and (3) transition radiation detector. • Particle identification: charged hadrons specified by mass & charge. Mass deduced from momentum and velocity. Momentum and charge sign from curvature of track in TPC. Velocity from time-of-flight and radiation characteristics. J. Scott Moreland (Duke) 4 / 8
54. Pre-equilibrium evolution Arrows: fluid velocity weighted by energy density Free streaming: • Massless non-interacting parton gas matched to viscous hydrodynamics PRC.91.064906, PRC.80.034902. • Parametrize initial energy density e0 • Initialize hydro with non-zero uµ and πµν $bl; Ə bm= o†rѴbm] =u;;v|u;-l _‹7uo =v J. Scott Moreland (Duke) 5 / 8
55. Compare parametrization to existing IC models 0 1 2 3 Entropy density [fm−3] 1 fm−2 2 fm−2 TB = 3 fm−2 Gen. mean, p = 1 WN 0 1 2 3 Entropy density [fm−3] Gen. mean, p = 0 EKRT 0 1 2 3 4 Participant thickness TA [fm−2] 0 1 2 3 Entropy density [fm−3] Gen. mean, p = − 0. 67 KLN • Wounded nucleon model dS d2x⊥ τ0 dη ∝ TA + TB ∗T denotes participant thickness • EKRT model PRC 93, 024907 (2016) after brief free streaming phase dET d2x⊥ τ0 dη ∼ Ksat π p3 sat (Ksat, β; TA, TB ) • KLN model PRC 75, 034905 (2007) dNg d2x⊥ τ0 dη ∼ Q2 s,min 2 + log Q2 s,max Q2 s,min J. Scott Moreland (Duke) 6 / 8
56. TRENTo proton shapes Plausible p-p geometries considered in Ref. 10.1103/PhysRevC.87.064906 Actual p-p geometries predicted by the arithmetic, geometric, and harmonic means p=1 p=0 p= − 1 J. Scott Moreland (Duke) 7 / 8
57. TRENTo event-by-event flow distributions Right: TRENTo, IP-Glasma, and AMPT initialized hydrodynamic simulations for P(v2 ) compared to experimental data from ALICE and ATLAS. 0 0.5 1 1.5 2 2.5 〉 2 v 〈 )* 2 P(v 3 − 10 1 0 0.5 1 1.5 2 2.5 〉 2 v 〈 )* 2 P(v 3 − 10 1 |<0.8 η Pb-Pb 5.02 TeV, | <3 GeV/c T ALICE: 0.2<p 〉 2 v 〈 / 2 v 0 0.5 1 1.5 2 2.5 〉 2 v 〈 )* 2 P(v 3 − 10 1 Pb-Pb 5.02 TeV IP-Glasma + MUSIC AMPT-IC + iEBE-VISHNU TRENTo-IC + iEBE-VISHNU 0 0.5 1 1.5 2 2.5 1 5-10% 0 0.5 1 1.5 2 2.5 1 25-30% 〉 2 v 〈 / 2 v 0 0.5 1 1.5 2 2.5 1 |<2.5 η Pb-Pb 2.76 TeV, | <1 GeV/c T ATLAS: 0.5<p >0.5 GeV/c T ATLAS: p 45-50% 10.1007/JHEP07(2018)103 J. Scott Moreland (Duke) 8 / 8