A Nearly Perfect Ink !? Theoretical Challenges from RHIC Dublin - 29 July 2005 LATTICE 2005 Berndt Mueller (Duke University)

A perfect ink… Is brilliantly dark and opaque Yet flows smoothly and easily A painful challenge to fountain pen designers A delightful challenge to physicists:  Are the two requirements really compatible? Hint:

The road to the quark-gluon plasma STAR …Is hexagonal and 2.4 miles long

Two wealths Challenges A wealth of … D A T A

Cornerstone results from RHIC Anisotropic transverse flow “Jet” suppression Baryon/meson enhancement

Azimuthal anisotropy v 2 pypy Coordinate space: initial asymmetry Momentum space: final asymmetry pxpx x y Semiperipheral collisions Signals early equilibration (t eq  0.6 fm/c)

Jet quenching in Au+Au PHENIX Data: Identified  0 d+Au Au+Au No quenching Quenching!

Baryons vs. mesons baryonsmesons R CP p T (GeV/c)  behaves like meson ? (also  -meson)

Overview QCD Thermodynamics  What are the dynamical degrees of freedom?  Is there a critical point, and where is it? Thermalization  How can it be so fast? Transport in a thermal medium  Viscosity, energy loss, collective modes Hadronization  Recombination vs. fragmentation

QCD phase diagram BB Hadronic matter Critical endpoint ? Plasma Nuclei Chiral symmetry broken Chiral symmetry restored Color superconductor Neutron stars T 1 st order line ? Quark- Gluon RHIC

Space-time picture  eq Pre-equil. phase Bjorken formula

Stages of a r.h.i. collision Initial collision ≈ break-up of the coherent gluon field (“color glass condensate”) Pre-equilibrium: the most puzzling stage Equilibrium (T > T c ): hydrodynamic expansion in longitudinal and transverse directions Hadronization: are there theoretically accessible domains in p T ? Hadronic stage (T < T c ): Boltzmann transport of the hadronic resonance gas

Initial state: Gluon saturation ~ 1/Q 2 Details of space-time picture depend on gauge! Gribov, Levin, Ryskin ’83 Blaizot, A. Mueller ’87 McLerran, Venugopalan ‘94 “Color glass condensate (CGC)”

CGC dynamics Initial state occupation numbers ~ 1/  s  1 → classical fields generated by random color sources on light cone: Boost invariance → Hamiltonian gauge field dynamics in transverse plane (x ,  ). 2-dim lattice simulation shows rapid equipartitioning of energy (  eq ~ Q s -1 ). Krasnitz, Nara & Venugopalan; Lappi (hep-ph/ ) Challenge: (3+1) dim. simulation without boost invariance

Saturation and dN/d  Kharzeev & Levin Pseudorapidity  = - ln tan  Rapidity y = ln tanh(E+p L )/(E-p L ) Assume nucleus is “black” for all gluons with k T  Q s : Q s (x) → Q s (y) with x = Q s e -|Y-y|. Also predicts beam energy dependence of dN/dy. Challenge: How much entropy is produced by simple decoherence, how much during the subsequent full equilibration?

The(rmalization) mystery Experiment demands:  th  0.6 fm/c “Bottom up” scenario ( Baier, A. Mueller, Schiff, Son ) :  “Hard” gluons with k T ~ Q s are released from the CGC;  Released gluons collide and radiate thermal gluons;  Thermalization time  th ~ [  s 13/5 Q s ] -1 ≈ 2-3 fm/c Perturbative dynamics among gluons does not lead to rapid thermalization. Quasi-abelian instability ( Mrowczynski; Arnold et al; Rebhan et al ):  Non-isotropic gluon distributions induce exponentially growing field modes at soft scale k ~ gQ s ;  These coherent fields deflect and isotropize the “hard” gluons.

After thermalization…... matter is described by (relativistic) hydrodynamics ! Requires f  L and small shear viscosity . HTL pert. theory (n f =3): Dimensionless quantity  /s. Classical transp. th.:  ≈ 1.5  T f, s ≈ 4  →  /s ≈ 0.4T f. (Baym…; Arnold, Moore & Yaffe)

Azimuthal anisotropy v 2 pypy Coordinate space: initial asymmetry Momentum space: final asymmetry pxpx x y Semiperipheral collisions Signals early equilibration (t eq  0.6 fm/c)

 – How small can it be? D. Teaney Boost invariant hydro with T 0  0 ~ 1 requires  /s ~ 0.1. N=4 SUSY Yang-Mills theory (g  1):  /s = 1/4  (Kovtun, Son, Starinets). Absolute lower bound on  /s ?  /s = 1/4  implies f ≈ (5 T) -1 ≈ 0.3 d Challenge: (3+1) dim. relativistic viscous fluid dynamics QGP(T ≈T c ) = sQGP

First attempts Nakamura & Sakai SU(3) YM Related (warm-up?) problem: EM conductivity  ≈  q 2   f /2T. n f = 3 QGP →  /  ≈ 20 T 2. Caveat: f (glue)  f (quarks) Method: spectral function repr. of G  and G ret, fit of spectral fct. to G . Challenge: Calculate  /s for real QCD

QCD equation of state 20% Challenge: Devise method for determining from data Challenge: Identify the degrees of freedom as function of T Is the (s)QGP a gaseous, liquid, or solid plasma ? Challenge: QCD e.o.s. with light domain wall quarks

A possible method Eliminate T from  and s : Lower limit on requires lower limit on s and upper limit on . BM & K. Rajagopal, hep-ph/

Measuring  and s Entropy is related to produced particle number and is conserved in the expansion of the (nearly) ideal fluid: dN/dy → S → s = S/V. Energy density is more difficult to determine:  Energy contained in transverse degrees of freedom is not conserved during hydrodynamical expansion.  Focus in the past has been on obtaining a lower limit on  ; here we need an upper limit.  New aspect at RHIC: parton energy loss. dE/dx is telling us something important – but what exactly?

Entropy Two approaches: 1) Use inferred particle numbers at chemical freeze-out from statistical model fits of hadron yields; 2) Use measured hadron yields and HBT system size parameters as kinetic freeze-out (Pratt & Pal). Method 2 is closer to data, but requires more assumptions. Good news: results agree within errors:  dS/dy = 5100 ± 400 for Au+Au (6% central, 200 GeV/NN) → s = (dS/dy)/(  R 2  0 ) = 33 ± 3 fm -3

Jet quenching in Au+Au PHENIX Data: Identified  0 d+Au Au+Au No quenching Quenching!

High-energy parton loses energy by rescattering in dense, hot medium. q q Radiative energy loss: “Jet quenching” = parton energy loss qq g Scattering centers = color charges L Density of scattering centers Range of color force Scattering power of the QCD medium:

Energy loss at RHIC Data suggest large energy loss parameter: RHIC Eskola, Honkanen, Salgado & Wiedemann p T = 4.5–10 GeV Dainese, Loizides, Paic

The Baier plot Plotted against , is the same for a  gas and for a perturbative QGP. Suggests that is really a measure of the energy density.  Data suggest that may be larger than compatible with Baier plot.  Nonperturbat. calculation is needed. Pion gas QGP Cold nuclear matter ˆ q ˆ q ˆ q RHIC data sQGP Challenge: Realistic calculation of gluon radiation in medium

Eikonal formalism quark   xx x - + Gluon radiation: x  = 0 xx Kovner, Wiedemann

Eikonal form. II Challenge: Compute  F +i (x)F + i (0)  for x 2 = 0 on the lattice Not unlike calculation of gluon structure function, maybe moments are calculable using euclidean techniques.

Where does E loss go? STARp+pAu+Au Lost energy of away-side jet is redistributed to rather large angles! Trigger jetAway-side jet

Wakes in the QGP Mach cone requires collective mode with  (k) < k : J. Ruppert and B. Müller, Phys. Lett. B 618 (2005) 123 Colorless sound ? Colored sound = longitudinal gluons ? Transverse gluons Angular distribution depends on energy fraction in collective mode and propagation velocity T. Renk & J. Ruppert

Baryons vs. mesons baryonsmesons R CP p T (GeV/c)  behaves like meson ? (also  -meson)

Hadronization mechanisms Fragmentation Recombination

Recombination wins… … for a thermal source Baryons compete with mesons Fragmentation dominates for a power-law tail

Quark number scaling of v 2 In the recombination regime, meson and baryon v 2 can be obtained from the quark v 2 :

Hadronization RHIC data (Runs 4 and 5) will provide wealth of data on: Identified hadron spectra up to much higher p T (~10 GeV/c); Elliptic flow v 2 up to higher p T with particle ID; Identified di-hadron correlations; Spectra and v 2 for D-mesons…. D D recombinationfragmentation Challenge: Unified framework treating recombination as special case of QCD fragmentation “in medium”. A. Majumder & X.N. Wang, nucl-th/

Some other challenges Where is the QCD critical point in ( ,T)? What is the nature of the QGP in T c < T < 2T c ? How well is QCD below T c described by a weakly interacting resonance gas? Thermal photon spectral function  (m 2,T). Are there collective modes with  (k) < k ? Can lattice simulations help understand the dynamics of bulk (thermal) hadronization?

Don’t be afraid… “Errors are the doors to discovery” James Joyce

Back-up slides

Hard-soft dynamics Nonabelian Vlasov equations generalizing “hard-thermal loop” effective theory. Can be defined on (spatial 3-D) lattice with particles described as test charges or by multipole expansion (Hu & Müller, Moore, Bödeker, Rummukainen). Poss. problem: short-distance lattice modes have wrong  (k). k ~ gQ s k ~ Q s Challenge: Full (3+1) dim. simulation of hard-soft dynamics

Reco: Thermal quarks Quark distribution function at “freeze-out” Relativistic formulation using hadron light-cone frame: For a thermal distribution, the hadron wavefunctions can be integrated out, eliminating the model dependence of predictions. Remains exactly true even if higher Fock states are included!

Heavy quarks Heavy quarks (c, b) provide a hard scale via their mass. Three ways to make use of this:  Color screening of (Q-Qbar) bound states;  Energy loss of “slow” heavy quarks;  D-, B-mesons as probes of collective flow. RHIC program: c-quarks and J/  ; LH”I”C program: b-quarks and .  RHIC data for J/  are forthcoming (Runs 4 & 5).

J/  suppression ? V qq is screened at scale (gT) -1  heavy quark bound states dissolve above some T d. Karsch et al. Color singlet free energy Quenched lattice simulations, with analytic continuation to real time, suggest T d  2T c ! S. Datta et al. (PRD 69, ) Challenge: Compute J/  spectral function in unquenched QCD

C-Cbar kinetics J/   LHC RHIC J/ ,  can be ionized by thermal gluons. If resonances persist above T c, J/  and  can be formed by recombination in the medium: J/  may be enhanced at LHC! Challenge: Multiple scattering theory of heavy quarks in a thermal medium Analogous to the multiple scattering theory for high-p T partons, but using methods (NRQCD etc.) appropriate for heavy quarks.