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 9 - 19 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 nuclear matter Deconfined phase T increases, or density increases 300 MeV 170 MeV QCD: asymptotic freedom, quark confinement

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. 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 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 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, … 1950-th the hydrodynamic model to high energy collisions of protons Fermi(1950); Pomeranchuk(1952); Landau(1953); Heisenberg(1952), Rozental, Chernavskij (UFN,1954); Landau, Bilenkij (UFN,1955); Feinberg (1972); Cooper,…(1975); Bjorken(1983); 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

8. pp collisions vs heavy ions collisions Pb

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

10. Plot from: Alice Collaboration PRL, 2010 Modified Hologhrapic Model Multiplicity: Landau’s/Hologhrapic formula vs experimental data Landau formula Simplest Hologhrapic Model Plot from: ATLAS Collaboration 1108.6027

11. s 1/2 NN dependence of the charged particle dN ch / dh | h =0 per colliding nucleon pair N part /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 s 1/2 NN dependence in A+A collisions: results of a Landau hydrodynamics calculation (dotted line) ; an s 0.15 extrapolation of RHIC and SPS data (dashed line); a logarithmic extrapolation of RHIC and SPS data (solid line) Multiplicity

12. 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 BACKUP

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. Duality: D=4 Gauge theory N=4 YM with SU(Nc ) Holography: ‘t Hooft, 1993; Susskind, 1994 D=4 String theory on AdS 5 xS 5 Gravity in AdS 5 D=5 Group symmetry Strings Maldacena 1997: D=4 Gauge Theory bosonic part SO(2,4) x SU(4) isometries of AdS5xS5 conformal symmetry and R-symmetry D=4 D=3

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, 1101.0618

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 AdS 5 xS 5 Simplest case: Renormalization Witten;Gubser, Klebanov, Polyakov,1998 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 g:

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

21. AdS/CFT correspondence. T=0 Gravity: AdS 5 with black hole f k (z) is the solution to the mode equation: with unit boundary conditions: RHS

22. Dual description of QGP as a part of gauge/string duality. In the case of the energy-momentum tensor, the corresponding field is the 5D metric. 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

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

24. Conclusion Black Hole in AdS 5  static QGP in 4-dim Formula for correlates Static uniform plasma  BH AdS 2001-2011 formation of QGP  formation of BH in AdS Today: Tomorrow:2008-now

25. D=2 CFT at T=0 and BTZ black hole 2D CFT 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 (*)(*) Backup

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

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

28. 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. (* ) Backup

29. Relation between Bogoliubov-Tyablikov G R and Matsubara G E Green functions denotes Euclidean time ordering Matsubara Bogoliubov-Tyablikov BACKUP