Heavy Quark Dynamics in Heavy Ion Collisions

Heavy Ion Pub 2010/08/17 Heavy quarks in heavy ion collisions Yukinao Akamatsu (Univ. of Tokyo) Heavy Ion Pub 1

Outline 1. Quark-gluon plasma and heavy ion collision 2. Heavy quark diffusion in heavy ion collision 3. Heavy quark diffusion from spectral function 4. Summary Heavy Ion Pub 2

1. Quark-gluon plasma and heavy ion collision Quantum Chromo Dynamics (QCD) ( ij i D i L = ) 1 Asymptotic freedom Confinement a a m F F ij j 4 QCD Phase Diagram RHIC, LHC QGP (quark-gluon plasma) SB CSC (color superconductivity) (chiral symmetry breaking) Heavy Ion Pub 3

Heavy ion collision and hydrodynamic model = Dynamics of light particles (u,d,s,g) 0 T = + ( ) T e p u u pg hadron gas time QGP fluid collision axis z 0 Au Au Perfect fluidity, Local thermal equilibrium Early thermalization, Hadronization Heavy Ion Pub 4

2. Heavy quark diffusion in heavy ion collision Ref : Akamatsu, Hatsuda, Hirano, PRC79( 09),054907, PRC80( 09), 031901(R) Charm and bottom in QGP T/Mc,b<<1 impurity probe hadron gas time QGP fluid collision axis z 0 Au Au Heavy Ion Pub 5

Previous works on charm and bottom in QGP Radiative energy loss Resonance scattering Collisional energy loss Based on weak coupling or model calculation How about strong coupling model ? Heavy Ion Pub 6

Model of heavy quark in medium Relativistic Langevin equation Fokker-Planck equation by Ito discretization 2 p T p x = + ( ) p p t t + ( , , ) P t x t E M 1 p p p x = + ( ) ( ) ( , , ) p p P t = p p x t 2 E 2 ( ) p p ( ) exp P = = + 2 ( ) ( ) p T M p t 2 ( ) p t Generalized fluctuation-dissipation theorem ( p Peq exp ) + 2 2 M T 3 ( p ) ( ET ) 2 d p p T + = = + ( ) ( ) ( ) p p E T 2 ( ) 2 d M Heavy Ion Pub 7

Drag coefficient Weak coupling (pQCD) ~ 2 . 0 (leading order) Poor convergence (Caron-Huot 08) Strong coupling (SYM by AdS/CFT sQGP) [ for na ve perturbation] YM g 4 N=4 SYM theory 2 g 2 N 2 d p v T 2 = = YM 2 ( , ) gYM N N T p dt M 2 1 v (Gubser 06, Herzog et al. 06, Teaney 06) Translation to sQGP = 1 . 2 5 . 0 (Gubser 07) Heavy Ion Pub 8

Heavy Quark Langevin + Hydro Model 0 fm . Little Bang generated by PYTHIA 0.6 fm Initial Condition (pp + Glauber) Local temperature and flow Brownian Motion Full 3D hydrodynamics QGP T(x), u(x) (Hirano 06) Heavy Quark Spectra _ c(b) D(B) e-+ e+ etc (independent fragmentation) O(10)fm Electron Spectra + . Experiment (PHENIX, STAR 07) time Heavy Ion Pub 9

Notes in our model Initial condition <decayed electron in pp> <HQ in pp> available only spectral shape above pT ~ 3GeV Reliable at high pT No nuclear matter effects in initial condition No quark coalescence effects in hadronization Where to stop in mixed phase at 1storder P.T. 3 choices (no/half/full mixed phase) Heavy Ion Pub 10

Numerical Results Nuclear Modification Factor 1 / dN dp = + A A T R AA / N dN dp + coll p p T Experimental result =1-3 AdS/CFT =2.1 0.5 Different freezeouts at 1storder P.T. Heavy Ion Pub 11 Bottom dominant

Elliptic Flow = 2 ( ) cos v T p 2 Poor statistics, but at least consistent with =1-3. (Still preliminary, PHENIX at Run7: v2~0.05-0.1 for pT~3-5GeV) Heavy Ion Pub 12

Including Recombination Model Strong medium effect at freezeout can explain large v2. Heavy Ion Pub 13

Degree of HQ Thermalization ~ 3 [ 4 fm ] St Stay time M Relaxation time HQ 2 T 22 72 6.7 21 2.2 7.2 thermalized not thermalized Experimental result =1-3 charm : nearly thermalized, bottom : not thermalized Heavy Ion Pub 14

Azimuthal Correlation Back to back correlation of a heavy quark pair diffusion Loss of correlation in decay products from D & B e(mid)- (fwd) correlation : one peak no contribution from vector meson decay IAA : quantitative measure 1 dN = max assoc ( ) d AA N d min trig ZYAM = / I AA AA pp e- azimuthal correlation: sensitive probe for heavy quark thermalization rate Heavy Ion Pub 15

e-h correlation (mid) : two peaks A sensitive probe but not clean Effects we ignore : Hadronic interaction of associates Medium response to HQ propagation Fictitious correlation due to bulk v2 Relative angle range for IAA Near side : -0.5 0.5 Away side : 0.5 1.5 Heavy Ion Pub 16

Outlook for updates FONLL initial condition Systematic study using Bag EoS and Lattice EoS for hydro Heavy Ion Pub 17

3. Heavy quark diffusion from spectral function Goal Non-perturbative definition of drag parameter Ref : Petreczky and Teaney 06 Caron-Huot and Moore 08 Caron-Huot, Laine and Moore 09 Burnier, Laine, Langelage, Mether 10 Heavy Ion Pub 18

Response and spectral function 1 ( ) 00 4 0 0 ( ) [ ( ), 0 ( )] exp k d x J x J ik x 2 Linear response theory = 0 3 0 0 ( ) ' ( ) ' t [ ( ), ( ' )] ( ) ' x J x i dt d r t J x J x A 0 Switch off external field A0 at t=0 = t ( ) ( ) ( e ) A x t A r 0 0 Response at t>0 k i 0 3 0 izt r ( , ) ( ) e (Im ) 0 J k z d r dt J x z 0 = ) k + i , 0 00 ( , ) ( ) J k z k = hydrodynamics Re ( A Heavy Ion Pub 0 19

Spectral function from hydrodynamics Conservation law = 0 J Constitutive equation 1 = + 0 J J D J (Introduction of relaxation time ) t susceptibi lity Response at t>0 1 T 1 ( iz ) ( 2 ) iz A k J r 3 0 0 ( ) ) 0 ( d r J = 0 0 ( , ) J k z + 2 2 Dk Dk z Dk = 00 ( , ) k + D 2 2 2 2 ( ) 3 + i = = ii ( , 0 ) k 2 2 1 Heavy Ion Pub 20

Spectral function from Fokker-Planck equation (Non-relativistic) Fokker-Planck equation p + 1 ( , , ) P p r t t M = + ( , , ) p P p r t 2 p p ( ) = + = , 2 p p t MT Definition of current 0 3 ( ) ( , , ) J x d r P p r t p 3 ( ) ( , , ) J x d r P p r t M Heavy Ion Pub 21

Conservation law = 0 J Constitutive equation p p i j = 3 ( ) ( , , ) ( ) J x d p P p r t J x t i M p p ( ) T x T i j = 0 0 ( ) ( ) J x J x ij ij M M M ( equilibriu ) local m T T = = + 0 0 ( ) ( ) ( ) ( ) ( ) J x J x J x J x J x t M M ( 2 ) k 2 k T M / 1 , T M D = 00 ( , ) k ( ) 2 + M 2 2 2 T M 1 ( ) = + 0 J J D J 3 T i t = = ii ( , 0 ) k ( )2 + 1 Heavy Ion Pub 22

Kubo formula for the drag parameter 6 2 1 T normal Kubo formula = = ii lim ( , 0 ) k 0 i M Weak + coupling M 3 ( 2 / 2 ) 3 T MT T = = ii ( , 0 ) ( ) k 2 ( / 2 : ) g MT M i g 0 4 M Very tiny peak for very heavy quark Very narrow and steep for weak coupling Euclidean correlator SPF (Petreczky and Teaney 06) Lattice calculation for Euclidean correlator is found to be insensitive to diffusion constant . cannot extract Heavy Ion Pub 23

Alternative definition of the drag parameter ( 2 ) MT 2 2 / 2 3 MT i = = 3 M 2 ii ( , 0 ) M k MT + ( 2 / 2 ) 2 1 1 = = = 4 lim ( , ) 0 ( ), ) 0 ( k d x E x E Q Q i Q i Q E E 3 6 T Q 0 i + + E Q g E Q Q g E Q Q c c No sharp or low peak expected because EQ is not even an approximately conserved quantity. M : too heavy to change velocity, therefore spatial current is conserved. g 0: spatial current will not undergo scattering, therefore conserved. Leading order perturbation Wilson line=1 N 4 3 Hard Thermal Loop 2 m 4 m C g T T T f = + + + (ln ) (ln ) H N c 18 2 3 16 d p D D 00 = = 2 , HTL ( , 0 ) s p G p = = 64718 . 0 4 , 3 / C 3 9 2 ( ) H Heavy Ion Pub 24 Caron-Huot and Moore 08

Integrating out heavy quark field Real time contour Corresponding Euclidean correlator Tr Tr Heavy Ion Pub 25

A way to lattice QCD E E ) 0 ; Tr ( ; ) ( ) ( ) 0 ( W i i i W i 2 T eucl W eucl ( ) G ) 0 ; 3 Tr ( i cosh( 2 ) d = = ( ) ( E E i ) mink eucl sinh 2 0 1 = lim ( ) Caron-Huot, Laine and Moore 09 0 Lattice calculation around Tc may be a noisy measurement because of the Polyakov in the denominator of G( ). Heavy Ion Pub 26

4. Summary Heavy quark drag parameter ~2T2/M is extracted from phenomenological study. Drag parameter near Tc is interesting and may be relevant to heavy quark hadronization. Heavy quark (momentum) diffusion constant is defined non-perturbatively. In near future, lattice determination may be possible. Heavy Ion Pub 27