1. Holographic description of heavy-ions collisions
Irina Aref’eva
Steklov Mathematical Institute, Moscow
7th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics
September 2012, Belgrade, Serbia

2. Outlook
Physical picture of formation of Quark-Gluon Plasma in heavy-ions collisions
Holography description of QGP in equilibrium
Holography description of heavy-ions collisions
New formula for multiplicity
Further directions

3. Quark-Gluon Plasma (QGP): a new state of matter
QGP is a state of matter formed from deconfined quarks, antiquarks, and gluons at high temperature
QCD: asymptotic freedom, quark confinement
nuclear
matter
Deconfined
phase
T increases, or
density increases
Нагревать= warm up,
150MeV= K
ЧИСЛА!
Mu_b baryon chemical potential
170 MeV
300 MeV

4. Experiments: Heavy Ions collisions produced a medium
HIC are studied in several experiments:
started in the 1990's at the Brookhaven Alternating
Gradient Synchrotron (AGS),
the CERN Super Proton Synchrotron (SPS)
the Brookhaven Relativistic Heavy-Ion Collider (RHIC)
the LHC collider at CERN.
Text from v2.pdf
There are strong experimental evidences that RHIC or LHC have created some medium which behaves collectively:
modification of particle spectra (compared to p+p)
jet quenching
high p_T-suppression of hadrons
elliptic flow
suppression of quarkonium production 4
Study of this medium is also related with study of Early Universe

5. QGP in Heavy Ion Collision and Early Universe
One of the fundamental questions in physics is: what happens to matter at extreme densities and temperatures as may have existed in the first microseconds after the Big Bang
The aim of heavy-ion physics is to create such a state of matter in the laboratory.
Evolution of the Early Universe
170 MeV
1K=10 ^{-13}GeV
Evolution of a Heavy Ion Collision

6. The Standard Model of Nature
1. A Gauge Theory with a light Higgs for electro-weak and strong interactions.
2. General Relativity with a small Λ for gravity.

7. Theory. Heavy Ions collisions
Macroscopic: thermodynamics, hydrodynamics, kinetic theory, …
Fermi(1950); Pomeranchuk(1952); Landau(1953); Heisenberg(1952),
Rozental, Chernavskij (UFN,1954);
Landau, Bilenkij (UFN,1955); Feinberg (1972);
Cooper,…(1975); Bjorken(1983);
1950-th the hydrodynamic model to high energy collisions of protons
Text from v2.pdf
Kolb, Heinz (2003); Janik, Peschansky (2006); Shuryak(,,,,, 2009);
Peigne, Smilga (UFN, 2009)
Dremin, Leonidov (UFN, 2010); Muller, Schafer(2011),….
Microscopic: QCD, holographic (AdS/CFT)
Application of holographic approach to formation of QGP is the subject of this talk 7

8. pp collisions vs heavy ions collisionsPb Pb
PbPb – lead-lead

9. QGP in heavy-ions collisions
There are strong experimental evidences that RHIC or LHC have created some medium:
Elliptic flow
Text and picture from Son-cern2008
more pressure
along the small axis 9

10. Multiplicity: Landau’s/Hologhrapic formula vs experimental data
Landau formula
Plot from: Alice
Collaboration
PRL, 2010
dNch/dη ≈ 1600 ± 76 (syst)
≈ 30,000 particles in total, ≈ 400 times pp !
Energy density ε > 3 x RHIC (fixed τ0,)
Temperature + 30%
Plot from: ATLAS
Collaboration
Simplest
Hologhrapic
Model
Modified
Hologhrapic
Model

11. Multiplicity
s1/2NN dependence of the charged particle dNch/ dh | h =0 per colliding nucleon pair Npart/2 from a variety of measurements in p+p and p+-p and central A+A collisions, 0-5% centrality ALICE and CMS
The curves show different expectations for the s1/2NN dependence in A+A collisions:
results of a Landau hydrodynamics calculation (dotted line) ;
an s0.15 extrapolation of RHIC and SPS data (dashed line);
a logarithmic extrapolation of RHIC and SPS data (solid line)

12. Multiplicity in Landau model
BACKUP
Multiplicity in Landau model
Thermodynamic methods to investigating the process of high-energy collision.
E. Fermi, Prog. Theor. Phys. 5, 570 (1950)
Pomeranchuk, Dokl. Akad. Nauk SSSR, 78, 889 (1951)
L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953)
To determine the total number of particles it is necessary to compute the entropy in the first moment of collision
Key[i:] ideal: E. Fermi, Prog. Theor. Phys. 5, 570 (1950)
L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953)
R. Hagedorn, CERN-Report (1971).
Main eq. is dE=TdS-pdV
Dividing on dV we get eq. for densities
s=dS/dV, etc.
V~m0/E means the Lorentz contraction

13. QGP as a strongly coupled fluid
Conclusion from the RHIC and LHC experiments: appearance of QGP (not a weakly coupled gas of quarks and gluons, but a strongly coupled fluid).
This makes perturbative methods inapplicable
The lattice formulation of QCD does not work, since we have to study real-time phenomena.
This has provided a motivation to try to understand the dynamics of QGP through the gauge/string duality

14. “Holographic description of quark-gluon plasma”
Holographic description of quark-gluon plasma in equilibrium
Holography description of quark-gluon plasma formation in heavy-ions collisions

15. Holography and duality.
Holography: ‘t Hooft, 1993; Susskind, 1994
Strings
Maldacena 1997:
D=4
Gauge Theory
D=5
D=4
D=3
D=4
‘t Hooft, “dimensional reduction in quantum gravity” gr-qc/
Susskind, “The World as a hologram” hep-th/
String theory
on AdS5xS5
Gravity
in AdS5
Duality:
D=4 Gauge theory
N=4 YM with SU(Nc)
Group symmetry
bosonic part SO(2,4) x SU(4) <-> isometries of AdS5xS5
conformal symmetry and R-symmetry

16. Dual description of QGP as a part of Gauge/string duality
There is not yet exist a gravity dual construction for QCD.
Differences between N = 4 SYM and QCD are less significant, when quarks and gluons are in the deconfined phase (because of the conformal symmetry at the quantum level N = 4 SYM theory does not exhibit confinement.)
Lattice calculations show that QCD exhibits a quasi-conformal behavior at temperatures T >300 MeV and the equation of state can be approximated by e = 3 p (a traceless conformal energy-momentum tensor).
The above observations, have motivated to use the AdS/CFT correspondence as a tool to get non-perturbative dynamics of QGP.
There is the considerable success in description of the static QGP.
Review: Solana, Liu, Mateos, Rajagopal, Wiedemann,

17. AdS/CFT correspondence in Euclidean space. T=0
The correspondence between the strongly coupled N = 4 SYM theory in D=4 and classical (super)gravity on AdS5xS5
Simplest case:
Witten;Gubser, Klebanov,
Polyakov,1998
Renormalization
IA, I.Volovich, PLB, 1998

18. AdS/CFT correspondence in Minkowski space.
incoming wave at

19. AdS/CFT correspondence in Euclidean space. T=0
denotes Euclidean time ordering
+ requirement of regularity at horizon 19

20. AdS/CFT correspondence. T=0
QFT at finite temperature (D=4)
Classical gravity with black hole (D=5)
-- retarded temperature Green function in 4-dim Minkowski
F -- kernel of the action for the (scalar) field in AdS5
Son, Starinets, 2002
QFT with T
Bogoliubov-Tyablikov Green function

21. AdS/CFT correspondence. T=0
Gravity: AdS5 with black hole
fk(z) is the solution to the mode equation:
with unit boundary conditions:

22. AdS/CFT correspondence. T=0
Gravity: AdS5 with black hole

23. D=2 CFT at T=0 and BTZ black hole
Euclidean finite temperature correlates of the local operator with conformal dimension
Conformal map from the infinite plane to the cylinder with circumference L=1/T
(*)

24. D=2 CFT at T=0 and BTZ black hole
Gravity calculations

25. D=2 CFT at T=0 and BTZ black hole
On shell
Boundaries:
(**)

26. D=2 CFT at T=0 and BTZ black hole
tIdentification of the prefactors
Retarded Green’s (**) taken at
are equal to the expression given by eq. (*)

27. BACKUP
Relation between Bogoliubov-Tyablikov GR and Matsubara GE Green functions
Bogoliubov-Tyablikov
Matsubara
denotes Euclidean time ordering

28. Dual description of QGP as a part of gauge/string duality
In the phenomenological hydrodynamical (Landau or Bjorken models of QGP),
the plasma is characterized by a space-time profile of the energy-momentum tensor
AdS/CFT: operators in the gauge theory correspond to fields in SUGRA .
In the case of the energy-momentum tensor, the corresponding field is the 5D metric.

29. Dual description of QGP as a part of gauge/string duality
Static uniform plasma
Janik, 05
REF.-exact
Solution:
Performing a change of coordinates
The standard AdS
static black hole

30. Conclusion Black Hole in AdS5  static QGP in 4-dim
Today:
Black Hole in AdS5  static QGP in 4-dim
Formula for correlates
Static uniform plasma  BH AdS
Tomorrow:
2008-now
formation of QGP  formation of BH in AdS