Heavy-Quark Kinetics in the Quark-Gluon Plasma

Heavy-Quark Kinetics in the Quark-Gluon Plasma Hendrik van Hees Justus-Liebig Universität Gießen May 3, 28 with M. Mannarelli, V. Greco, L. Ravagli and R. Rapp Institut für Theoretische Physik JUSTUS-LIEBIG- UNIVERSITÄT GIESSEN Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 1 / 32

Outline 1 Heavy-quark interactions in the sqgp Heavy-quark observables in heavy-ion collisions Heavy-quark diffusion: The Fokker-Planck Equation Elastic pqcd heavy-quark scattering Non-perturbative interactions: effective resonance model 2 Non-photonic electrons at RHIC 3 Microscopic model for non-pert. HQ interactions Static heavy-quark potentials from lattice QCD T-matrix approach 4 Resonance-Recombination Model 5 Summary and Outlook Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 2 / 32

Heavy-Ion collisions in a Nutshell Theory of strong interactions: Quantum Chromo Dynamics, QCD At high enough densities/temperatures: hadrons dissolve into a Quark-Gluon Plasma (QGP) hope to create QGP in Heavy-Ion Collisions at RHIC (and LHC) RHIC: collide gold nuclei with energy of 2 GeV per nucleon: Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 3 / 32

Evidence for QGP from heavy-ion observables particle p T spectra show hydrodynamical behavior collective flow of matter in local thermal equilibrium nuclear modification factor degree of thermalization R AA (p T ) = dn AA/dp T N coll dn pp /dp T no QGP R AA = 1; observed: R AA < 1 (suppression) at high p T in non-central collisions: anisotropic collective flow initially reaction zone of elliptic shape pressure gradients: p x > p y measure of flow anisotropy: p 2 x p 2 y v 2 = p 2 x + p 2 = cos(2φ p ) y Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 4 / 32

Heavy Quarks in Heavy-Ion collisions c q g hard production of HQs described by PDF s + pqcd (PYTHIA) c,b quark sqgp HQ rescattering in QGP: Langevin simulation drag and diffusion coefficients from microscopic model for HQ interactions in the sqgp q c Hadronization to D,B mesons via quark coalescence + fragmentation V. Greco, C. M. Ko, R. Rapp, PLB 595, 22 (24) K ν e e ± semileptonic decay non-photonic electron observables Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 5 / 32

correct equil. lim. Einstein relation: B 1 (t, p) = T (t)e p A(t, p) Hendrik Relativistic van Hees (JLU Gießen) Langevin simulation Heavy-Quark Kinetics for flowing the QGP medium May 3, 28 6 / 32 Heavy-Quark diffusion Fokker Planck Equation f(t, p) t = p i [ p i A(t, p) + ] B ij (t, p) f(t, p) p j drag (friction) and diffusion coefficients p i A(t, p) = p i p i B ij (t, p) = 1 (pi p 2 i)(p j p j) ( = B (t, p) δ ij p ) ip j p 2 + B 1 (t, p) p ip j p 2 transport coefficients defined via M X( p ) = 1 1 d 3 q d 3 q d 3 p γ c 2E p (2π) 3 2E q (2π) 3 2E q (2π) 3 2E p M 2 (2π) 4 δ (4) (p + q p q ) ˆf( q)x( p )

Meaning of Fokker-Planck coefficients non-relativistic equation with constant A = γ and B = D 1 = D f t = γ p ( pf) + D 2 f p 2 Green s function: { } γ 3/2 G(t, p; p ) = 2πD[1 exp( 2γt)] { exp γ [ p p exp( γt)] 2 } 2D 1 exp( 2γt) Gaussian with p(t) = p exp( γt), p 2 (t) p(t) 2 = 3D γ [1 exp( 2γt)] = t 6Dt γ: friction (drag) coefficient; D: diffusion coefficient equilibrium limit for t : D = mt γ (Einstein s dissipation-fluctuation relation) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 7 / 32

Relativistic Langevin process Fokker-Planck equation equivalent to stochastic differential equation Langevin process: friction force + Gaussian random force in the (local) rest frame of the heat bath d x = p E p dt, d p = A p dt + 2dt[ B P + B 1 P ] w w: normal-distributed random variable dependent on realization of stochastic process to guarantee correct equilibrium limit: Use Hänggi-Klimontovich calculus, i.e., use B /1 (t, p + d p) for constant coefficients: Einstein dissipation-fluctuation relation B = B 1 = E p T A. to implement flow of the medium use Lorentz boost to change into local heat-bath frame use update rule in heat-bath frame boost back into lab frame Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 8 / 32

Elastic pqcd processes Lowest-order matrix elements [Combridge 79] Debye-screening mass for t-channel gluon exch. µ g = gt, α s =.4 not sufficient to understand RHIC data on non-photonic electrons Hendrik [Moore, van Hees Teaney (JLU 25] Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 9 / 32

Non-perturbative interactions: effective resonance model General idea: Survival of D- and B-meson like resonances above T c Chiral symmetry SU V (2) SU A (2) in light-quark sector of QCD 2 D = [( µ Φ i )( µ Φ i ) m 2 DΦ i Φ i] + massive (pseudo-)vectors D L () i=1 Φ i : two doublets: pseudo-scalar ( D D ) and scalar Φ i : two doublets: vector ( D D ) and pseudo-vector L () qc = qi/ q + c(i/ m c )c q: light-quark doublet ( ) u d c: singlet Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 1 / 32

Interactions Interactions determined by chiral symmetry For transversality of vector mesons: heavy-quark effective theory vertices ( L int = G S q 1 + /v 2 Φ 1c v + q 1 + /v ) 2 iγ5 Φ 2 c v + h.c. G V ( q 1 + /v 2 γµ Φ 1µc v + q 1 + /v 2 iγµ γ 5 Φ 2µc v + h.c. v: four velocity of heavy quark in HQET: spin symmetry G S = G V ) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 11 / 32

Resonance Scattering elastic heavy-light-(anti-)quark scattering q c q c D, D, D s c s q c u D, D, D s q D- and B-meson like resonances in sqgp q D, D, D s D, D, D s parameters m D = 2 GeV, Γ D =.4....75 GeV m B = 5 GeV, Γ B =.4....75 GeV k c k Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 12 / 32

Cross sections σ [mb] 12 1 8 6 4 Res. s-channel Res. u-channel pqcd qc scatt. pqcd gc scatt. 2 1.5 2 2.5 3 3.5 4 s [GeV] total pqcd and resonance cross sections: comparable in size BUT pqcd forward peaked resonance isotropic resonance scattering more effective for friction and diffusion Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 13 / 32

Transport coefficients: pqcd vs. resonance scattering three-momentum dependence γ [1/fm].2.15.1.5 T=2 MeV resonances Γ=.3 GeV resonances Γ=.4 GeV resonances Γ=.5 GeV pqcd: α s =.3 pqcd: α s =.4 pqcd: α s =.5 D [GeV/fm].7.6.5.4.3.2.1 T=2 MeV resonances Γ=.3 GeV resonances Γ=.4 GeV resonances Γ=.5 GeV pqcd: α s =.3 pqcd: α s =.4 pqcd: α s =.5.5 1 1.5 p[gev].5 1 1.5 p[gev] resonance contributions factor 2... 3 higher than pqcd! Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 14 / 32

Transport coefficients: pqcd vs. resonance scattering Temperature dependence γ [1/fm].5.4.3.2 resonances: Γ=.4 GeV pqcd: α s =.4 total D [GeV/fm].4.3.2 resonances: Γ=.4 GeV pqcd: α s =.4 total.1.1.2.25.3.35.4 T [GeV].2.25.3.35.4 T [GeV] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 15 / 32

Time evolution of the fire ball Elliptic fire-ball parameterization fitted to hydrodynamical flow pattern [Kolb ] V (t) = π(z + v z t)a(t)b(t), a, b: half-axes of ellipse, v a,b = v [1 exp( αt)] v[1 exp( βt)] Isentropic expansion: S = const (fixed from N ch ) QGP Equation of state: s = S V (t) = 4π2 9 T 3 (16 + 1.5n f ), n f = 2.5 obtain T (t) A(t, p), B (t, p) and B 1 = T EA for semicentral collisions (b = 7 fm): T = 34 MeV, QGP lifetime 5 fm/c. simulate FP equation as relativistic Langevin process Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 16 / 32

Initial conditions need initial p T -spectra of charm and bottom quarks (modified) PYTHIA to describe exp. D meson spectra, assuming δ-function fragmentation exp. non-photonic single-e ± spectra: Fix bottom/charm ratio 1/(2 πp T ) d 2 N/dp T dy [a.u.] 1-2 1-3 1-4 1-5 1-6 d+au s NN =2 GeV STAR D STAR prelim. D * ( 2.5) c-quark (mod. PYTHIA) c-quark (CompHEP) 1-7 1 2 3 4 5 6 7 8 p T [GeV] 1/(2πp T ) dn/dp T [a.u.] 1-3 1-4 1-5 1-6 1-7 1-8 1-9 1-1 e ± σ bb /σ cc = 4.9 x1-3 STAR (prel, pp) STAR (prel, d+au/7.5) STAR (pp) D e B e sum 1-11 1 2 3 4 5 6 7 8 p T [GeV] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 17 / 32

Spectra and elliptic flow for heavy quarks 1.5 c, reso (Γ=.4-.75 GeV) c, pqcd, α s =.4 b, reso (Γ=.4-.75 GeV) R AA 1.4 1.2 1 D (2πT) = 1.5 D (2πT) = 3 D (2πT) = 6 D (2πT) = 12 D (2πT) = 24 α α s = α s = 1.7 αs=.5 αs=.35.25 s = R AA 1.5 Au-Au s=2 GeV (b=7 fm) 1 2 3 4 5 p T [GeV] µ D = gt, α s = g 2 /(4π) =.4 resonances c-quark thermalization without upscaling of cross sections Fireball parametrization consistent with hydro.8.6.4.2 [Moore, Teaney 4].5 1 1.5 2 2.5 3 3.5 4 4.5 5 p T (GeV) µ D = 1.5T fixed spatial diff. coefficient: D = D s = T ma 2πT D 3 2α 2 s Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 18 / 32

Spectra and elliptic flow for heavy quarks v 2 [%] 2 15 1 5 c, reso (Γ=.4-.75 GeV) c, pqcd, α s =.4 b, reso (Γ=.4-.75 GeV) Au-Au s=2 GeV (b=7 fm) 1 2 3 4 5 p T [GeV] (p T ) v 2.22.2.18.16.14.12.1.8.6.4.2 LO QCD [Moore, Teaney 4].5 1 1.5 2 2.5 3 3.5 4 4.5 5 p T (GeV) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 19 / 32

Observables: p T -spectra (R AA ), v 2 Hadronization: Coalescence with light quarks + fragmentation c c, b b conserved single electrons from decay of D- and B-mesons 2 1.5 PHENIX prel (2-4%) STAR (1-4%) Au-Au s=2 GeV (b=7 fm) c+b reso c+b pqcd c reso 2 15 PHENIX prel (min bias) Au-Au s=2 GeV (b=7 fm) c+b reso c+b pqcd R AA 1 v 2 [%] 1 5.5 1 2 3 4 5 6 7 8 p T [GeV] -5 1 2 3 4 5 p T [GeV] Without further adjustments: data quite well described [HvH, V. Greco, R. Rapp, Phys. Rev. C 73, 34913 (26)] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 2 / 32

Observables: p T -spectra (R AA ), v 2 Hadronization: Fragmentation only single electrons from decay of D- and B-mesons R AA 2 1.5 1 PHENIX prel (2-4%) c+b reso STAR (1-4%) c+b pqcd c reso Au-Au s=2 GeV (b=7 fm) v 2 [%] 2 15 1 5 PHENIX prel (min bias) Au-Au s=2 GeV (b=7 fm) c+b reso c+b pqcd.5 1 2 3 4 5 6 7 8 p T [GeV] -5 1 2 3 4 5 p T [GeV] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 2 / 32

Observables: p T -spectra (R AA ), v 2 Central Collisions single electrons from decay of D- and B-mesons Coalescence+Fragmentation Fragmentation only R AA 1.5 1 central Au-Au s=2 GeV c+b reso c+b pqcd c reso PHENIX prel (-1%) STAR (-5%) R AA 1.5 1 central Au-Au s=2 GeV c+b reso c+b pqcd c reso PHENIX prel (-1%) STAR (-5%).5.5 1 2 3 4 5 6 7 8 p T [GeV] 1 2 3 4 5 6 7 8 p T [GeV] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 2 / 32

Comparison to newer data R AA 1.8 1.6 (a) 1% central Armesto et al. (I) 1.4 1.2 1 van Hees et al. (II) 3/(2πT) Moore & 12/(2πT) Teaney (III).8.6.4 HF v 2.2.2.15.1.5 Au+Au @ s NN = 2 GeV (b) minimum bias π R AA, p T π v 2, p T e ± R, e ± vhf AA 2 > 4 GeV/c > 2 GeV/c PHENIX Collaboration PRL 98 17231 (27) 1 2 3 4 5 6 7 8 9 p [GeV/c] T Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 21 / 32

Microscopic model: Static potentials from lattice QCD V lqcd Kaczmarek et al color-singlet free energy from lattice use internal energy U 1 (r, T ) = F 1 (r, T ) T F 1(r, T ), T V 1 (r, T ) = U 1 (r, T ) U 1 (r, T ) Casimir scaling for other color channels [Nakamura et al 5; Döring et al 7] V 3 = 1 2 V 1, V 6 = 1 4 V 1, V 8 = 1 8 V 1 Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 22 / 32

T-matrix Brueckner many-body approach for elastic Qq, Q q scattering q, q c T = V + V T Σ = Σ glu + T reduction scheme: 4D Bethe-Salpeter 3D Lipmann-Schwinger S- and P waves same scheme for light quarks (self consistent!) Relation to invariant matrix elements M(s) 2 ( d a Ta,l= (s) 2 + 3 T a,l=1 (s) 2 ) cos θ cm q Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 23 / 32

T-matrix -Im T (GeV -2 ) 12 1 8 6 4 2 s wave color singlet T=1.1 T c T=1.2 T c T=1.3 T c T=1.4 T c T=1.5 T c T=1.6 T c T=1.7 T c T=1.8 T c 1 2 3 4 E cm (GeV) -Im T (GeV -2 ) 3 2 1 s wave color triplet T=1.1 T c T=1.2 T c T=1.3 T c T=1.4 T c T=1.5 T c T=1.6 T c T=1.7 T c T=1.8 T c 1 2 3 4 E cm (GeV) resonance formation at lower temperatures T T c melting of resonances at higher T! sqgp P wave smaller resonances near T c : natural connection to quark coalescence [Ravagli, Rapp 7] model-independent assessment of elastic Qq, Q q scattering problems: uncertainties in extracting potential from lqcd in-medium potential V vs. F? Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 24 / 32

Transport coefficients.15.1 T-matrix: 1.1 T c T-matrix: 1.4 T c T-matrix: 1.8 T c pqcd: 1.1 T c pqcd: 1.4 T c pqcd: 1.8 T c 4 3 pqcd+t-mat pqcd Α (1/fm) 2πT D s 2.5 1 1 2 3 4 5 p (GeV).2.22.24.26.28.3.32.34 T (GeV) from non-pert. interactions reach A non-pert 1/(7 fm/c) 4A pqcd A decreases with higher temperature higher density (over)compensated by melting of resonances! spatial diffusion coefficient increases with temperature D s = T ma Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 25 / 32

Non-photonic electrons at RHIC same model for bottom quark coalescence+fragmentation D/B e + X R AA v 2 (%) 1.6 1.4 1.2 1.8.6.4.2 T-matrix pqcd, α s =.4 Au-Au s=2 GeV (central).5 1 2 3 4 5 p T (GeV) 15 T-matrix pqcd, α s =.4.15 1 5 Au-Au s=2 GeV b=7 fm R AA v 2 1.5 1.1.5-1% central Au+Au s=2 AGeV minimum bias theory [Wo] frag. only [Wo] theory [SZ] PHENIX STAR PHENIX PHENIX QM8 1 2 3 4 5 p T (GeV) 1 2 3 4 5 p T [GeV] coalescence crucial for explanation of data increases both, R AA and v 2 momentum kick from light quarks! resonance formation towards T c coalescence natural [Ravagli, Rapp 7] Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 26 / 32

Properties of the sqgp measure for coupling strength in plasma: η/s relation to spatial diffusion coefficient η s 1 2 T D s (AdS/CFT), η s 1 5 T D s (wqgp) η/s 2 1.5 1.5 T-mat + pqcd reso + pqcd pqcd KSS bound -.5.2.25.3.35.4 T (GeV) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 27 / 32

Resonance-Recombination Model successes of quark-coalescence models in HI phenomenology high baryon/meson ration in heavy-ion compared to pp collisions compared Constituent-quark number scaling of v 2 v 2,had (p T ) = n q v 2,q (p T /n q ) experiment: CQNS better for KE T than p t problems with naive coalescence models violates conservation laws (energy, momentum!) violates 2 nd theorem of thermodynamics (entropy) Resonance structures close to T c transport process with q q(qq) R Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 28 / 32

Resonance-Recombination Model t f M(t, p) = Γ f M (t, p) + g(p) f (eq) M γ (p) = γ p p Γ g(p) d 3 p 1 d 3 p 2 g(p) = (2π) 6 d 3 xf q (x, p 1 )f q (x, p 2 )σ(s)v rel δ (3) (p p 1 p 2 ) 4π (Γm) 2 σ(s) = g σ kcm 2 (s m 2 ) 2 + (Γm) 2 1 1 1 T=18 MeV, β =.55 E dn/d 3 p (GeV -2 ) 1-1 1-2 1-3 1-4 output 1-5 blast wave 1 2 3 4 5 p T (GeV) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 29 / 32

Meson spectra q- q input: Langevin simulation meson output: resonance-recombination model E dn/d 3 p (GeV -2 ) 1 1 1 1-1 1-2 1-3 φ T=18 MeV 1-4 PHENIX STAR 1-5 STAR (new) 1 2 3 4 5 p T (GeV) E dn/d 3 p (GeV -2 ) 1-3 1-4 1-5 1-6 J/ψ T=18 MeV 1-7 PHENIX 1 2 3 4 5 p T (GeV) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 3 / 32

Constituent-quark number scaling usual coalescence models: factorization ansatz f q (p, x, ϕ) = f q (p, x)[1 + 2v q 2 (p T ) cos(2ϕ)] CQNS usually not robust with more realistic parametrizations of v 2 here: q input from Langevin simulation.1.2.8.15.6 v 2 v 2.1.4.2 charm strange light 1 2 3 4 5 KE T (GeV).5 J/ψ φ D 1 2 3 4 5 KE T (GeV) Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 31 / 32

Summary and Outlook Summary Heavy quarks in the sqgp non-perturbative interactions mechanism for strong coupling: resonance formation at T T c lqcd potentials parameter free res. melt at higher temperatures consistency betw. R AA and v 2! also provides natural mechanism for quark coalescence resonance-recombination model problems extraction of V from lattice data potential approach at finite T : F, V or combination? Outlook include inelastic heavy-quark processes (gluon-radiation processes) other heavy-quark observables like charmonium suppression/regeneration Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics in the QGP May 3, 28 32 / 32