Probing nucleon substructure with Bayesian parameter estimation J. S. Moreland, J. E. Bernhard, W. Ke, S. A. Bass—Duke U. BNL RIKEN Seminar, May 5, 2017 

Lattice predicts existence of a quark-gluon plasma Lattice QCD calculations find a pseudo-critical phase transition temperature T ≈ 155 MeV, where hadrons melt to form a deconfined soup of quarks and gluons dubbed a quark-gluon plasma (QGP) T ~ 155 MeV Baryon Density μ [GeV] Temperature T [MeV] critical point? quark-gluon plasma early universe hadron gas nuclear collisions J. S. Moreland (Duke U.) Nucleon substructure 1 / 50 

What are the quark-gluon plasma bulk properties? How and under what conditions is it formed in a nuclear collision? How does it recombine 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. S. Moreland (Duke U.) Nucleon substructure 2 / 50 

Quark-gluon plasma is not directly detectable What the experiment sees... Quark-gluon plasma Measure extrinsic final-state properties • particle yields • mean particle momenta • angular particle correlations ~ 10-14 m extent ~ 10-23 s lifetime Study intrinsic QGP properties • equation of state • transition temperature • viscosity J. S. Moreland (Duke U.) Nucleon substructure 3 / 50 

Transport models connect experiment and theory 1. Initial conditions describe Tµν at time τ = 0+ 2. Pre-equilibrium dynamics rapidly drive the system to hydrodynamic applicability 3. Relativistic viscous hydrodynamics solves ∂µ Tµν = 0, converts spatial anisotropy into momentum anisotropy 4. QGP medium is particlized near phase transition temperature 5. Hadronic afterburner simulates subsequent collisions and decays time: 0 fm/c 20 fm/c 

Transport models connect experiment and theory 1. Initial conditions describe Tµν at time τ = 0+ 2. Pre-equilibrium dynamics rapidly drive the system to hydrodynamic applicability 3. Relativistic viscous hydrodynamics solves ∂µ Tµν = 0, converts spatial anisotropy into momentum anisotropy 4. QGP medium is particlized near phase transition temperature 5. Hadronic afterburner simulates subsequent collisions and decays This talk: QGP initial conditions 

Model-to-data comparison: the inverse problem Theory • QCD equation of state • relativistic hydrodynamics • hadronic cross sections Measured observables Model Data Simulated observables tune free model parameters Procedure given model with some unknown parameters f(x 1 , x 2 , ..., x n ) optimize the parameters x 1 , x 2 , ..., x n to fit an observed data set y 1 , y 2 , ..., y m extract new physics insight J. S. Moreland (Duke U.) Nucleon substructure 5 / 50 

Part I: Constructing parametric QGP initial conditions at midrapidity Part II: Bayesian parameter estimation Part III: Adding nucleon substructure 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

QGP initial conditions: sampling nuclei Correlated nucleus algorithm: J. Bernhard 1. Pre-sample nucleon positions from Woods-Saxon dist. for nucleus 2. Place nucleons in order of increasing radii 3. For each nucleon, sample random θ and φ 4. Resample angles if any two nucleons are too close |xi − xj | < dmin Woods-Saxon Density ρ(r) = ρ0 1 + e(r−R0)/a Coeff. R0 and a for various nuclei from Atomic Data and Nuclear Data Tables Vol. 59, Issue 2, 185-381 J. S. Moreland (Duke U.) Nucleon substructure 7 / 50 

Correlated vs uncorrelated nuclei Uncorrelated Pb208 dmin = 0 fm Correlated Pb208 dmin = 1.5 fm J. S. Moreland (Duke U.) Nucleon substructure 8 / 50 

U238 Pb208 Au197 Cu62 He3 d p Collision systems at RHIC and the LHC J. S. Moreland (Duke U.) Nucleon substructure 9 / 50 

Collision geometry nucleus A nucleus B dS/dη|η=0 TA TB T(x, y) = dz ρA,B(x ± b, y, z) dS dη η=0 = f (TA , TB ) J. S. Moreland (Duke U.) Nucleon substructure 10 / 50 

Entropy deposition at midrapidity Collisions are stochastic. Entropy deposition mapping only defined on subset of participant matter dS dη η=0 = f (Tpart A , Tpart B ), where transverse participant density is given by Tpart(x, y) = Npart i=1 Tp(x − xi , y − yi ), and Tp is the proton thickness, typically modeled by a Gaussian. J. S. Moreland (Duke U.) Nucleon substructure 11 / 50 

Entropy deposition scaling with nuclear density? Two general approaches: Ab initio theory calculations Data-driven model inference Derive Validate Refine Examples: M.V. model AdS-CFT pQCD & saturation Flexible model Bayesian parameter estimation Experiment Constrained model Examples in other fields: LIGO black hole masses Cosmological Standard Model J. S. Moreland (Duke U.) Nucleon substructure 12 / 50 

TRENTo parametric initial condition model Sample nucleon positions from Woods-Saxon density with minimum distance criteria |xi − xj | > dmin J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

TRENTo parametric initial condition model Nuclei collide at some random impact parameter b dP(b) = 2πb db J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

TRENTo parametric initial condition model Determine nucleons which participate inelastically Pcoll(b) = 1 − exp[−σgg Tpp(b)], σinel NN = 2πb db Pcoll(b). J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

TRENTo parametric initial condition model Construct participant thickness functions ˜ T(x) = Npart i=1 γi 1 2πw2 exp − (x − xi )2 2w2 J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

TRENTo parametric initial condition model Construct participant thickness functions ˜ T(x) = Npart i=1 γi 1 2πw2 exp − (x − xi )2 2w2 J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

TRENTo parametric initial condition model Convert local overlap density → entropy deposition dS d2r dy y=0 ∝ ˜ Tp A + ˜ Tp B 2 1/p generalized mean ansatz J. S. Moreland (Duke U.) Nucleon substructure 13 / 50 

Motivation for the generalized mean ansatz Generalized mean ansatz: dS d2r dy ∝ Tp A + Tp B 2 1/p 8 6 4 2 0 2 4 6 8 Š [fm] 0 2 4 Thickness [fm 2] Pb+Pb 2.76 TeV $min <$<$max 1

Motivation for the generalized mean ansatz Generalized mean ansatz: dS d2r dy ∝ Tp A + Tp B 2 1/p 8 6 4 2 0 2 4 6 8 Š [fm] 0 2 4 Thickness [fm 2] Pb+Pb 2.76 TeV $min <$<$max 1

Motivation for the generalized mean ansatz Generalized mean ansatz: dS d2r dy ∝ Tp A + Tp B 2 1/p 8 6 4 2 0 2 4 6 8 Š [fm] 0 2 4 Thickness [fm 2] Pb+Pb 2.76 TeV $min <$<$max 1

Motivation for the generalized mean ansatz Generalized mean ansatz: dS d2r dy ∝ Tp A + Tp B 2 1/p 8 6 4 2 0 2 4 6 8 Š [fm] 0 2 4 Thickness [fm 2] Pb+Pb 2.76 TeV $min <$<$max 1

Parametrization mimics existing theory calculations 0 1 2 3 Entropy density [fm 3] 1 fm 2 2 fm 2 $ =3 fm 2 Gen. mean, r=1 WN 0 1 2 3 Entropy density [fm 3] Gen. mean, r=0 EKRT 0 1 2 3 4 Participant thickness $ [fm 2] 0 1 2 3 Entropy density [fm 3] Gen. mean, r= 0.67 KLN • Wounded nucleon model dS dy d2r⊥ ∝ TA + TB ∗T denotes participant thickness • EKRT model PRC 93, 024907 (2016) after brief free streaming phase dET dy d2r⊥ ∼ Ksat π p3 sat (Ksat, β; TA, TB ) • KLN model PRC 75, 034905 (2007) dNg dy d2r⊥ ∼ Q2 s,min 2 + log Q2 s,max Q2 s,min J. S. Moreland (Duke U.) Nucleon substructure 15 / 50 

Parametrization mimics existing theory calculations 0 1 2 3 Entropy density [fm 3] 1 fm 2 2 fm 2 $ =3 fm 2 Gen. mean, r=1 WN 0 1 2 3 Entropy density [fm 3] Gen. mean, r=0 EKRT 0 1 2 3 4 Participant thickness $ [fm 2] 0 1 2 3 Entropy density [fm 3] Gen. mean, r= 0.67 KLN • Wounded nucleon model dS dy d2r⊥ ∝ TA + TB ∗T denotes participant thickness • EKRT model PRC 93, 024907 (2016) after brief free streaming phase dET dy d2r⊥ ∼ Ksat π p3 sat (Ksat, β; TA, TB ) • KLN model PRC 75, 034905 (2007) dNg dy d2r⊥ ∼ Q2 s,min 2 + log Q2 s,max Q2 s,min 0 2 4 6 8 10 12 14 Impact parameter b [fm] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 "n "2 "3 TRENTo + FS, p = 0 ± 0.1 IP-Glasma No simple analytic form for IP-Glasma model. However, initial entropy density yield and eccentricity harmonics (above) agree closely with a generalized mean described by p ≈ 0. J. S. Moreland (Duke U.) Nucleon substructure 15 / 50 

Brief recap TRENTo model: • Predicts entropy deposition at midrapidity in a relativistic nuclear collision (includes several free parameters). • Parametric construction spans a large subspace of reasonable ab initio theory calculations. Application: • Answer the what without the why. How does entropy (energy) deposition scale with nuclear density? • What can we learn about nuclear matter at extreme energy density given the correct initial conditions? J. S. Moreland (Duke U.) Nucleon substructure 16 / 50 

Embed IC in computationally intensive model Cannot directly observe initial state physics (τ ≤ 3 · 10−24 s), only final hadrons. Must simulate full spacetime evolution of a relativistic heavy-ion collision. Hydrodynamic standard model Initial conditions → viscous hydrodynamics → microscopic hadronic afterburner time: 0 fm/c 20 fm/c Fig: H. Petersen, MADAI collaboration J. S. Moreland (Duke U.) Nucleon substructure 17 / 50 

Challenge of rigorous model-to-data comparison Parameter Observable shear viscosity bulk viscosity pre-equilibrium flow nucleon width hadronization temp p+p fluctuations identified yields identified mean pT flow cumulants mode mixing observables event plane decorrelations HBT interferometry Testing a single set of parameters requires O(104) hydro events ...and evaluating eight different parameters five times each requires 58 × 104 ≈ 109 hydro events. That’s roughly 105 computer years! J. S. Moreland (Duke U.) Nucleon substructure 18 / 50 

Part I: Constructing parametric QGP initial conditions at midrapidity Part II: Bayesian parameter estimation Part III: Adding nucleon substructure 

Solution: Bayesian parameter estimation Thesis work by Jonah Bernhard Quantifying properties of hot and dense QCD matter through systematic model-to-data comparison, Bernhard, et. al. PRC 91 (2015). Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and quark-gluon plasma medium, Bernhard, Moreland, Bass, Liu, and Heinz, PRC 94 (2016). Methodology based on Computer model calibration using high-dimensional output, Higdon, Gattiker, Williams, and Rightley, J.Amer.Stat.Assoc. 103, 570 (2008). Determining properties of matter created in ultrarelativistic heavy-ion collisions, Novak, et. al, PRC 89 (2014). Used prominently in Observation of gravitational waves from a binary black hole merger, LIGO and Virgo collaborations, PRL 116 (2016). Planck 2013 results. Cosmological parameters, Planck collaboration, Astron.Astrophys. 571 (2014). J. S. Moreland (Duke U.) Nucleon substructure 20 / 50 

Overview: Bayesian parameter estimation 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. S. Moreland (Duke U.) Nucleon substructure 21 / 50 

Details: heavy-ion collision model Fig: Zhi Qiu • TRENTo initial conditions Parametric entropy deposition Moreland, Bernhard, Bass, PRC 92, 011901 (2015) • Pre-equilibrium freestreaming Initial (infinitely) weak coupling phase Liu, Chen, Heinz, PRC 91, 064906 (2015) • HotQCD equation of state Lattice QCD (2+1)-flavor EoS Bazavov, et. al. PRD 90, 094503 (2014) • iEBE-VISHNU hydrodynamics Boost-invariant shear+bulk viscous hydrodynamics Shen, et. al, Comp. Phys. Comm. 199, 61 (2016) • UrQMD hadronic afterburner Simulates hadronic rescattering and resonance decay Bass et. al, Prog. Part. Nucl. Phys. 41, 255 (1998) time J. S. Moreland (Duke U.) Nucleon substructure 22 / 50 

Experimental calibration data 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. S. Moreland (Duke U.) Nucleon substructure 23 / 50 

Model 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. S. Moreland (Duke U.) Nucleon substructure 24 / 50 

Evaluating the model 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. S. Moreland (Duke U.) Nucleon substructure 25 / 50 

Training the 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. S. Moreland (Duke U.) Nucleon substructure 26 / 50 

Markov-chain Monte Carlo Perform random walk through parameter space, weighted by the Bayesian posterior probability: 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) Posterior dist. determined from equilibriated walker density J. S. Moreland (Duke U.) Nucleon substructure 27 / 50 

Model calibration 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 Model calculations at each of the 500 design points J. S. Moreland (Duke U.) Nucleon substructure 28 / 50 

Model calibration 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 One-hundred random samples drawn from the posterior J. S. Moreland (Duke U.) Nucleon substructure 28 / 50 

−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 

−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 −1.0 −0.5 0.0 0.5 1.0 p KLN EKRT / IP-Glasma Wounded nucleon 0. 03+0. 08 −0. 08 Entropy deposition scales with geometric mean of nuclear density dS/dy ≈ √ ρA ρB 

Geometric mean scaling in the literature AdS-CFT holography Pre-equillibrium radial flow from central shock-wave collisions in AdS5, Paul Romatschke, J. Drew Hogg, JHEP 1304 (2013) 048. “We find that the early-time radial flow buildup is identical to that expected from ideal hydrodynamics with an entropy den- sity proportional to the square root of the product of the matter densities in the individual nuclei.” Color flux-tube model Collision process with ν soft gluon exchanges akin to random- walk in color space. Strength of effective color charge Q ∝ √ ν. Particle production (entropy) scales like dN/dy ∝ √ ρA ρB . J. S. Moreland (Duke U.) Nucleon substructure 30 / 50 

What’s the best the model can do? 

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 

Checking non-calibrated ow observables Correlations 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) Data: ALICE, PRL 117 182301 [1604.07663] J. S. Moreland (Duke U.) Nucleon substructure 32 / 50 

PRC 94, 024907 [1605.03954] Success of saturation models + hydro Hydro models with saturation IC, e.g. TRENTo, IP-Glasma and EKRT, provide excellent description of bulk observables in A+A collisions 100 101 102 103 104 π± K± p¹ p Nch × 5 solid: identified dashed: charged Yields dN/dy, dNch /dη 0 10 20 30 40 50 60 70 Centrality % 0.8 1.0 1.2 Model/Exp 0.0 0.4 0.8 1.2 π± K± p¹ p Mean pT [GeV] 0 10 20 30 40 50 60 70 Centrality % 0.8 1.0 1.2 0.00 0.03 0.06 0.09 v2 v3 v4 Flow cumulants vn {2} 0 10 20 30 40 50 60 70 Centrality % 0.8 1.0 1.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 20 40 60 80 100 〈v n 〉 centrality percentile 0.5 GeV < pT < 1GeV η/s=0.18 filled - ATLAS open - IP-Glasma+MUSIC 〈v2 〉 〈v3 〉 〈v4 〉 PRL 113, 102301 [1405.3605] 0 10 20 30 40 50 60 70 80 centrality [%] 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 vn 2 (a) pT =[0.2 5.0] GeV LHC 2.76 TeV Pb+Pb η/s=0.20 η/s=param1 η/s=param2 η/s=param3 η/s=param4 ALICE vn 2 PRC 93, 024907 [1505.02677] 

Aside: agreement validates scaling, not narrative! ɳ How do we proceed if two models with different narratives predict similar initial conditions? How can we disambiguate model scaling behavior and the underlying theoretical framework? These questions aside: there is strong evidence that saturation-like effects govern particle production in relativistic nuclear collisions and their magnitude is well constrained. This work: dS d2r dy y=0 ≈ √ ρA ρB Purely observational inference! Need not be exact, but robust to moderate perturbations. J. S. Moreland (Duke U.) Nucleon substructure 34 / 50 

Part I: Constructing parametric QGP initial conditions at midrapidity Part II: Bayesian parameter estimation Part III: Adding nucleon substructure 

Flow-like signatures observed in small systems Naively, hydrodynamic behavior was only expected in heavy-ion collisions, not in proton-proton and proton-lead collisions. “When does hydrodynamics turn on/off?” 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 φ ∆ d pair 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 Long-range rapidity correlations characteristic of flow J. S. Moreland (Duke U.) Nucleon substructure 36 / 50 

What if we apply A+A models to small systems? PHENIX submitted to PRC [1609.02894] (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 0-5% p+Au 200 GeV 2 PHENIX v AMPT SONIC superSONIC IPGlasma+Hydro (a) PHENIX (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 0-5% d+Au 200 GeV (b) (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 He+Au 200 GeV 3 0-5% (c) IP-Glasma + hydro model could not reproduce experimental multiparticle correlations in small systems... what’s wrong? • Perhaps hydro isn’t valid, incorporate initial CGC correlations • Saturation IC + hydro correct picture, small systems require additional nucleon substructure Note: SONIC model pictured above has not demonstrated same level of agreement as IP-Glasma + hydro in Pb+Pb collisions J. S. Moreland (Duke U.) Nucleon substructure 37 / 50 

What if we apply A+A models to small systems? PHENIX submitted to PRC [1609.02894] (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 0-5% p+Au 200 GeV 2 PHENIX v AMPT SONIC superSONIC IPGlasma+Hydro (a) PHENIX (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 0-5% d+Au 200 GeV (b) (GeV/c) T p 0.5 1 1.5 2 2.5 3 2 v 0 0.05 0.1 0.15 0.2 0.25 He+Au 200 GeV 3 0-5% (c) IP-Glasma + hydro model could not reproduce experimental multiparticle correlations in small systems... what’s wrong? • Perhaps hydro isn’t valid, incorporate initial CGC correlations • Saturation IC + hydro correct picture, small systems require additional nucleon substructure Note: SONIC model pictured above has not demonstrated same level of agreement as IP-Glasma + hydro in Pb+Pb collisions J. S. Moreland (Duke U.) Nucleon substructure 37 / 50 

"Eccentric protons?" Schenke, Venugopalan PRL • Data highly constrains functional form of entropy deposition. • Cannot modify mapping without spoiling bulk A+A observables, but we can add fine structure to the inputs (thickness functions) Optical nucleus Nucleus w/ nucleons Historical analogue: nucleon hot spots necessary for triangular flow J. S. Moreland (Duke U.) Nucleon substructure 38 / 50 

"Eccentric protons?" Schenke, Venugopalan PRL • Data highly constrains functional form of entropy deposition. • Cannot modify mapping without spoiling bulk A+A observables, but we can add fine structure to the inputs (thickness functions) Optical proton Proton w/ partons Possibly similar picture for partons inside the nucleon? J. S. Moreland (Duke U.) Nucleon substructure 38 / 50 

Theory perspective: proton shape uctuations Several important contributions: Shapes of the proton, Gerald A. Miller, PRC 68 (2003) 022201 Spin-dependent proton shape fluctuations Revealing proton shape fluctuations with incoherent diffraction at high energy, Mäntysaari, Schenke, PRD 94, 034042 Evidence of strong proton shape fluctuations from incoherent diffraction, Mäntysaari, Schenke, PRL 117, 052301 Work to implement proton shape fluctuations within IP-Glasma. Shape parameters constrained by J/Ψ differential cross section. −1 0 1 y[fm] −1 0 1 x[fm] −1 0 1 y[fm] −1 0 1 x[fm] 0.00 0.05 0.10 0.15 0.20 −1 0 1 y[fm] −1 0 1 x[fm] −1 0 1 y[fm] −1 0 1 x[fm] 0.00 0.05 0.10 Various lumpy proton shapes input to IP-Sat J. S. Moreland (Duke U.) Nucleon substructure 39 / 50 

This work: • Implement parametric nucleon substructure within the TRENTo initial condition model • Estimate new substructure parameters using Bayesian methodology J. S. Moreland (Duke U.) Nucleon substructure 40 / 50 

TRENTo model with nucleon substructure Nucleon dof Parton dof • Nucleon width [fm] • Nucleon width [fm] • Parton width [fm] • Parton number J. S. Moreland (Duke U.) Nucleon substructure 41 / 50 

TRENTo model with nucleon substructure Nucleon dof Parton dof Gaussian nucleons Tp (x) = 1 2πw2 exp − x2 2w2 Sum of Gaussian partons Tp (x) = Npartons i=1 1 2πv2 exp − |x − xi |2 2v2 P(|xi |) ∼ exp[−0.5 r2/(w2 − v2)] J. S. Moreland (Duke U.) Nucleon substructure 41 / 50 

TRENTo model with nucleon substructure Nucleon dof Parton dof Sample proton-proton inelastic cross section Tpp(b) = d2x Tp(x + b) Tp(x) Pcoll(b) = 1 − exp[−σgg Tpp(b)] σinel NN = 2πb db Pcoll(b) J. S. Moreland (Duke U.) Nucleon substructure 41 / 50 

TRENTo model with nucleon substructure Nucleon dof Parton dof Entropy density scales with generalized mean of participant nucleon density, e.g. dS d2r dy y=0 ∝ TATB geometric mean J. S. Moreland (Duke U.) Nucleon substructure 41 / 50 

Substructure e ect on nuclear thickness functions Parton width [fm] Lead nucleus 5 0 5 3 partons 20 partons width 0.2 fm 5 0 5 5 0 5 5 0 5 width 0.3 fm x [fm] y [fm] Parton number Proton 1 0 1 3 partons 20 partons width 0.2 fm 1 0 1 1 0 1 1 0 1 width 0.3 fm x [fm] y [fm] Parton number nucleon width fixed, w = 0.5 fm J. S. Moreland (Duke U.) Nucleon substructure 42 / 50 

Substructure e ect on initial entropy density TRENTo: Pb+Pb, √ sNN = 2.76 TeV, p = 0, w = 0.88 fm Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƐƏ Ɛ ƐƏƏ ƐƏƐ ƐƏƑ r-u|om‰b7|_ˆƷƏĺƓ=l Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ ƐƏ Ƒ ƐƏ Ɛ ƐƏƏ ƐƏƐ ƐƏƑ r-u|omm†l0;umƷƒ 7"ņ7‹Ň‹ƷƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ Ƒ Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ ƒ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ Əĺƕ ƏĺѶ r-u|om‰b7|_Œ=lœ Ɠ ѵ Ѷ ƐƏ ƐƑ ƐƓ Ɛѵ ƐѶ ƑƏ r-u|omm†l0;u J. S. Moreland (Duke U.) Nucleon substructure 43 / 50 

Substructure e ect on initial entropy density TRENTo: p+Pb, √ sNN = 2.76 TeV, p = 0, w = 0.88 fm Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƐƏ Ɛ ƐƏƏ r-u|om‰b7|_ˆƷƏĺƓ=l Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƏĺƐƔ ƏĺƑƏ ƏĺƑƔ ƏĺƒƏ ƏĺƒƔ ƏĺƓƏ ƐƏ Ƒ ƐƏ Ɛ ƐƏƏ r-u|omm†l0;umƷƒ 7"ņ7‹Ň‹ƷƏ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ Ƒ Ə ƑƔ ƔƏ ƕƔ ƐƏƏ ;m|u-Ѵb|‹ѷ ƏĺƐƔ ƏĺƑƏ ƏĺƑƔ ƏĺƒƏ ƏĺƒƔ ƏĺƓƏ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ ƒ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ Əĺѵ Əĺƕ ƏĺѶ r-u|om‰b7|_Œ=lœ Ɠ ѵ Ѷ ƐƏ ƐƑ ƐƓ Ɛѵ ƐѶ ƑƏ r-u|omm†l0;u J. S. Moreland (Duke U.) Nucleon substructure 43 / 50 

Bayesian analysis with nucleon substructure Disclaimer, hydrodynamic description may not be suitable for small collision systems! ɔ Temporarily defer applicability questions to explore feasibility—would hydro “even work”? Work flow: 1. Vary generalized mean, nucleon width, parton width, parton number, and QGP medium parameters 2. Calibrate to simultaneously fit p+Pb and Pb+Pb particle yields, mean pT and flow harmonics at midrapidity 3. Test quantitative accuracy of hydro in small systems. What are the preferred substructure parameters? J. S. Moreland (Duke U.) Nucleon substructure 44 / 50 

Adding proton-lead observables 0 25 50 75 Centrality % 0 10 20 30 40 50 7ch /7 2 4 ch / ch 0.0 0.2 0.4 0.6 0.8 1.0 r$ 2.5 5.0 7.5 offline trk / offline trk 0.00 0.02 0.04 0.06 ˆm {2} ˆ2 ˆ3 Data: PRC 901 (2015) 064905, PLB 727 (2013) 371-380, PLB 724 (2013) 213 • Calibrate on yields, mean pT and flows (same as Pb+Pb) • Flows vn {2} are dicey in small systems. Use CMS data with peripheral substraction to remove non-flow contributions • Rescale Nch and Ntrk by mean values to that the ensure observable vector can be calculated at all design points J. S. Moreland (Duke U.) Nucleon substructure 45 / 50 

Stay tuned! Under construction... 0.0 0.5 1.0 norm 0.0 0.5 1.0 r 0.0 0.5 1.0 h 0.0 0.5 1.0 partons 0.0 0.5 1.0 ‰ 0.0 0.5 1.0 ˆ 0.0 0.5 1.0 norm 0.0 0.5 1.0 /vlbm 0.0 0.5 1.0 r 0.0 0.5 1.0 h 0.0 0.5 1.0 partons 0.0 0.5 1.0 ‰ 0.0 0.5 1.0 ˆ 0.0 0.5 1.0 /vlbm ? J. S. Moreland (Duke U.) Nucleon substructure 46 / 50 

Summary and Outlook Presented • Yields, mean pT and flows strongly constrain local entropy deposition in Pb+Pb collisions. • Bayesian analysis supports the approximate scaling. dS d2r dy y=0 ≈ √ ρA ρB • I’ve shown a natural framework to extend parametric initial conditions to include nucleon substructure. To do • Hydro model for small systems requires further justication. • Initial state correlations exist, how big are they? • Fire up the super computing cluster! Currently working to simultaneously calibrate on p+Pb and P+Pb observables. J. S. Moreland (Duke U.) Nucleon substructure 47 / 50 

Backup Slides 

Nucleon substructure—a new degree of freedom PRC 87 064906 [1304.3403v3] Schematic (a) (b) (c) TRENTo p=1 p=0 p= 1 

Nucleon substructure—a new degree of freedom PRC 87 064906 [1304.3403v3] Schematic (a) (b) (c) TRENTo p=1 p=0 p= 1 

Understanding the model and data discrepancy 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Š [fm] 0.0 0.2 0.4 0.6 0.8 1.0 Thickness [fm 2] Arithmetic: r=1 Geometric: r=0 Harmonic: r= 1 Participant× 0.3 Beam view x J. S. Moreland (Duke U.) Nucleon substructure 50 / 50