String theory and heavy ion collisions Hong Liu Massachusetts Institute of Technology As someone who thinks the world mostly in 10 dimensional terms, I want to thank the organizers for this Opportunity to speak at this conference and to come back to four dimensions for a while. HL, Krishna Rajagopal, Urs A. Wiedemann hep-ph/0605178, PRL in press hep-ph/0607062, submitted to PRL and to appear

String theory and heavy ion collisions Son and Kapusta’s talks The AdS/CFT computation of the shear viscosity: could ``explain’’ the perfect fluid observed at RHIC possibly a universal lower bound Here, I would like to convince you this is likely to be the first chapter of a long story. On Tuesday you heard the talk by Son on the AdS/CFT computation of the shear viscosity in supersymmetric YM theories, the significance of which was also discussed by various speakers including Kapusta. Here I would like to convince you that this beautiful story is only the beginning, the first chapter of a long story of the marriage between string theory and heavy ion collisions. In later chapters, many more experimental results could be explained, and predictions can be made. In later chapters, many more experimental results could be explained, and predictions can be made.

Chapter n: AdS/CFT and Jet quenching Today I will focus on one of the chapters.

Parton energy loss in QGP The dominant effect of the medium on a high energy parton is medium-induced Bremsstrahlung. Baier, Dokshitzer, Mueller, Peigne, Schiff (1996): In the high energy limit the dominant effect of the medium produced in heavy ion collisions on a hard parton is medium induced Bresstrahlung, by emitting High energy gluons. The energy loss can be described by a pocket formula, which depends on a transport coefficient qhat. L here is the length of the medium. qhat characterizes the ability of the medium to “quench” jets. In physical terms qhat can be described as the transverse momentum square transfer per unit distance as the parton travels through the medium. : reflects the ability of the medium to “quench” jets. : Debye mass : mean free path

Wanted: a first principle computation of Experimental results can be explained by values of qhat between 5-15 Gev^2/fm. The challenge for theorists is to see whether such a big value can arise from a first-principle calculation. : 5-15 GeV2/fm

New theoretical techniques needed! The main theoretical techniques for dealing with strongly coupled problems are lattice calculations. Lattice techniques are not well adapted to calculate transport coefficients, or dynamical processes of any sort. Here we need new ideas and new techniques, since it is rather hard for standard lattice techniques to do such a calculation.

AdS/CFT correspondence Maldacena (1997), Gubser, Klebanov,Polyakov; Witten (1998) N = 4 Super-Yang-Mills theory in 4d with SU(NC) A string theory in 5d AdS Finite temperature Black hole in AdS5 Large NC and strong coupling limit Classical gravity limit As already mentioned by other speakers, here we can take advantage of some recent development in string theory. I will only have time to highlight some features which are relevant for my discussion below. AdS/CFT is an equivalence between a SYM theory and a string theory in AdS spacetime. AdS spacetime is a homogenous space with a negative cosmological constant. It is very important that string theory lives one dimensional higher. You can also put the gauge theory at finite T, which is then dual to a putting a BH in AdS. BH is a thermo system with a Hawking T. In this relation one identifies the T of gauge theory with the Hawking T. The large color and large coupling limit of the SYM is described by the classical limit of string theory. A consequence of this AdS/CFT correspondence is that …... Can be translate into a computation in classical gravity. YM observables at infinite NC and infinite coupling can be computed using classical gravity Apply to both dynamical and thermodynamic observables.

Strategy Need a non-perturbative definition of Compute in strongly coupled Super-Yang-Mills theory using AdS/CFT We will first need a suitable non-perturbative definition of qhat. Similar strategy was used to compute the shear viscosity.

N = 4 Super-Yang-Mills theory is NOT QCD Caution: N = 4 Super-Yang-Mills theory is NOT QCD Later: Using sQGP of N = 4 SYM to understand sQGP of RHIC may NOT be far-fetched. Now: Accumulate data points

: a non-perturbative formulation Hard: weakly coupled Let us first carry out the first step. What we need is a convenient formalism to describe the rescatterings of a hard parton with the medium. Soft: likely strongly coupled : multiple rescatterings of hard particles with the medium

Soft scatterings Zakharov (1997); Wiedemann (2000) Amplitude for a particle propagating in the medium High energy limit (eikonal approximation): Soft scatterings are captured by Light like Wilson lines.

A non-perturbative definition of Wiedemann (2000) L: conjugate to the pT Light-like Wilson loop: : length of the medium Assuming: Thermal average (Hard to calculate using lattice) Nonperturbative definition of

Wilson loop from AdS/CFT Maldacena (1998); Rey and Yee (1998) Recipe: Our (3+1)-dim world, Wilson loop C in our world : area of string worldsheet with boundary C The minimal surface can be considered as the surface spread by the open string connecting the quark-anti-quark pair running along the Wilson loop. horizon Black hole in AdS spacetime: radial coordinate r, horizon: r=r0 constant r surface: (3+1)-dim Minkowski spacetime

Finding S(C) Wilson loop can be considered as the spacetime trajectories of a quark and antiquark pair. Key: open string connecting the quark pair can venture into the radial dimension. Finding S (C) : finding the shape of the string hanging from the spatial infinity of a black hole. Not more difficult than finding the Catenary !

Shape of the string r=r0 = The string hangs down from infinity and touches the horizon. Interactions between the quark and the medium Interaction of the string with the horizon of a black hole.

of N=4 SYM theory BDMPS transport coefficient reads: It is not proportional to number of scattering centers Take: Experimental estimates: 5-15 GeV2/fm

Jet quenching in a wind We have assumed that the medium is static, in realistic situations the medium itself can be moving: : the angle between the velocities of the quark and the medium HL,Rajagopal Wiedemann : velocity of the medium Example:

Summary Is the agreement meaningful? In QGP of QCD, the energy loss of a high energy parton can be described perturbatively up to a non-perturbative jet-quenching parameter. We calculate the parameter in N=4 SYM (not necessarily full energy loss of SYM) It appears to be close to the experimental value. Is the agreement meaningful?

Is agreement meaningful? N=4 SYM theory Conformal no asymptotic freedom, no confinement supersymmetric no chiral condensate no dynamical quarks, 6 scalar and 4 Weyl fermionic fields in the adjoint representation. Physics near vacuum and at very high energy is very different from that of QCD

Is agreement meaningful? (continued) N=4 SYM at finite T QCD at T ~TC -3 TC conformal no asymptotic freedom, no confinement supersymmetric (badly broken ) no chiral condensate no dynamical quarks, 6 scalars and 4 fermions in the adjoint representation. near conformal (lattice) not intrinsic properties of sQGP not present may be taken care of by proper normalization

Maybe the agreement is not an accident after all ! Take: Remind you the shear viscosity story Experimental estimates: 5-15 GeV2/fm Caveat: AdS/CFT calculation is in the infinite NC and infinite coupling limit

for other theories General conformal field theories (CFT) with a gravity dual: (large N and strong coupling) aCFT : central charge Theories near conformal: corrections small Buchel Finite coupling and NC corrections: hard Armesto, Edelstein and Mas R-charge chemical potentials: Lin, Matsuo, Avramis, Sfetsos, Armesto,Edelstein, Mas, ……. corrections mall when chemical potential is small

Drag force for heavy quarks in N=4 SYM Herzog, Karch, Kovtun, Kozcaz, Yaffe; Gubser, ……. Drag force for a heavy quark moving in the medium: Casalderrey-Solana, Teaney Fluctuation-dissipation theorem Diffusion coefficient: It is possible to analyze the energy flow pattern (indications of conical flow) Friess,Gubser Michalogiorgakis, Pufu Note: D can not be used to find Fluctuation-dissipation theorem assumes the quark is in equilibrium with the medium: does not apply to high energy jet

Chapter n+1: Quarkonium suppression: predictions for LHC or RHIC II

Quarkonium suppression at high PT HL,Rajagopal,Wiedemann Techniques discussed above can also be used to calculate screening length between a quark pair. Static quarks: great success from lattice calculation Heavy quarks produced in heavy ion collisions typically move relative to the medium: hard to do using lattice. Heavy quark mesons with larger velocity disassociate at a lower temperature: effect may be significant at RHIC II or LHC See Urs Wiedemann’s talk AdS/CFT: (for conformal theory)

Conclusions: a nice honeymoon AdS/CFT provides powerful tools to understand dynamics of strong coupled gauge theories. Expect many more chapters to be written for the marriage between string theory and physics of QCD in extreme conditions.