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
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
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
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.
4.75NNs GeV
17.2NNs GeV
200NNs GeV
2.76NNs TeV
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
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
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.
(3) (2) (1)S WSU SU U
(gen.cov.)SMPSMGSMN LLL
ΛπG
R(g)πG
LNN 8
1
16
1SMG
Theory. Heavy Ions collisionsMacroscopic: 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
pp collisions vs heavy ions collisions
PbPb
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
Plot from: Alice Collaboration PRL, 2010
0.12~ NNsMModified Hologhrapic Model
Multiplicity: Landau’s/Hologhrapic formula vs experimental data
1/4~ NNsMLandau formula
1/3~ NNsMSimplest Hologhrapic Model
Plot from: ATLAS Collaboration 1108.6027
s1/2NN dependence of the charged particle dNch/ d | =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)
Multiplicity
Multiplicity in Landau model
Thermodynamic methods to investigating the process of high-energy collision.
energy p presure 0
T temperature
s entropy
0
1
3
Ts p
p
4
3Ts
d Tds
4
3
d ds
s
3/4 3/4
3/4 1/4 1/2 1/4
0
NN
s S V
E V S E V S E s
mV
E
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
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
“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
Holography and duality.
Duality: D=4 Gauge theoryN=4 YM with SU(Nc)
Holography: ‘t Hooft, 1993; Susskind, 1994
D=4
String theoryon AdS5xS5
Gravityin AdS5
D=5
Group symmetry
Strings
Maldacena 1997:
D=4Gauge Theory
bosonic part SO(2,4) x SU(4) <-> isometries of AdS5xS5 conformal symmetry and R-symmetry
D=4
D=3
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
= 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
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:4
5AdS RM M
1 1( , )... ( , ) , ( , )n n k kx x x M
O O 0
Me
O
0[ ( )]cSe
Renormalization0| ( )z C
Witten;Gubser, Klebanov,Polyakov,1998
IA, I.Volovich, PLB, 1998
O( , , ), [ ], [ ] 0cx z S S
0 0( ) ( , , | )c c c x z
AdS/CFT correspondence in Minkowski space.
45AdS MM M
1 1( , )... ( , ) , ( , )n n k kt x t x t x M
O O
( , , ), [ ], [ ] 0ct x z S S
0
Me
O
0[ ( )]cSe
2
2 2 2 22
Rds dt dx dz
z
incoming wave at z
( ) [ ( , ), (0,0)] , ( , )RG t t x t x M
O O
( ) 2 ( , ) |RzG k F k z
40 0 0[ ( )] ( ) ( , ) ( ) |zc zS d k k F k z k
AdS/CFT correspondence in Euclidean space. T=0
denotes Euclidean time ordering
Hzz 0
+ requirement of regularity at horizon
0
Me
O
0[ ( )]cSe ( , , ), [ ], [ ] 0g g cx z S S
g:
Ex
Ex
AdS/CFT correspondence. Minkowski. T=0
QFT at finite temperature (D=4) Classical gravity with black hole (D=5)
QFT with T
4
/
( ) ( ) [ ( ), (0)
/
R ikx
H T
G k i d ke t x
e Z
O O
O Otr
Bogoliubov-Tyablikov Green function
-- retarded temperature Green function in 4-dim MinkowskiRG
F -- kernel of the action for the (scalar) field in AdS5
Son, Starinets, 2002
LHS
AdS/CFT correspondence. T=0
Gravity: AdS5 with black hole
fk(z) is the solution to the mode equation:
with unit boundary conditions:
RHS
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
ˆT T
, , 0,...3T
ˆ " " MNIf then g O T
O
AdS/CFT: operators in the gauge theory correspond to fields in SUGRA
Solution:
Performing a change of coordinates
The standard AdSstatic black hole
Static uniform plasma
(4)2
, |MN MN boundary
ren
gG g g
z
Janik, 05
Dual description of QGP as a part of gauge/string duality
Conclusion
Black Hole in AdS5 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
D=2 CFT at T=0 and BTZ black hole
2D CFTEuclidean finite temperature correlates of the local operator with conformal dimension
)2
1,
2
1(),( hh
)sinh()sinh(
)(),(
40
wTwT
TCxGR
Conformal map from the infinite plane to the cylinder with circumference L=1/T
ixw
)(
)(')(
z
zz
(*)
Backup
D=2 CFT at T=0 and BTZ black hole
Gravity calculations
Backup
On shell
Boundaries:
1z
D=2 CFT at T=0 and BTZ black hole
(**)
Backup
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
Relation between Bogoliubov-Tyablikov GR
and Matsubara GE Green functions
,...1,0),,2(),2( nkTnGkiTnG ER
denotes Euclidean time ordering
TT
exexxx HHEE
/1/1
),0(ˆ)(ˆ);,(
OO
Matsubara
Bogoliubov-Tyablikov
BACKUP
