1. 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/ , PRL in press
hep-ph/ , submitted to PRL
and to appear

2. 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.

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

4. 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

5. 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.
: GeV2/fm

6. 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.

7. 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.

8. 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.

9. 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

10. : 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

11. 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.

12. 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

13. 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

14. 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 !

15. 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.

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

17. 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:

18. 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?

19. 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

20. 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

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

22. 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

23. 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

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

25. 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)

26. 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.