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

Lecture II: Holographic Description of Formation of Quark Gluon Plasma. Multiplicity

Lecture I: Holographic description of Quark Gluon Plasma (QGP in equilibrium)

Holography for thermal states

TQFT in MD-space-time Black holein AdSD+1-space-time =

Lecture I. : Hologhraphic description of Quark Gluon Plasma (QGP in equilibruum)

TQFT = QFT with temperature

AdS/CFT correspondence in Euclidean space. T=0

denotes Euclidean time ordering

Hzz 0

+ requirement of regularity at horizon

0M

ET e

O

0[ ( )]cSe

g:

( , , ), [ ], [ ] 0g g cx z S S

Ex

FROM Lecture 1

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 with BLACK HOLE

Son, Starinets, 2002

LHS

FROM Lecture 1

Hologhraphic thermalizationThermalization of QFT in Minkowski D-dim space-time

Black Hole formation in Anti de Sitter (D+1)-dim space-time

Lecture II.: Hologhraphic Description of Formation of Quark Gluon Plasma

Studies of BH formation in AdSD+1Time of thermalization in HIC

Multiplicity in HIC

HIC = heavy ions collisions

Formation of BH in AdS. Deformations of AdS metric leading to BH formation

• colliding gravitational shock waves

• drop of a shell of matter with vanishing rest mass

("null dust"), infalling shell geometry = Vaidya metric

• sudden perturbations of the metric near the boundary that propagate into the bulk

Gubser, Pufu, Yarom, Phys.Rev. , 2008 Alvarez-Gaume, C. Gomez, Vera, Tavanfar, Vazquez-Mozo, PRL, 2009IA, Bagrov, Guseva, Joukowskaya, 2009 JHEPKiritsis, Taliotis, 2011

Chesler, Yaffe, 2009

Danielsson, Keski-Vakkuri and Kruczenski…….

Deformations of MD metric by infalling shell

4-dimensional infalling shell geometry (Vaidya metric) Vaidya 1951

flat

Deformations of MD metric by infalling shell

4-dimensional Minkowski. Penrose diagram

Deformations of MD metric by infalling shell . Vaidya metric

=

Deformations of AdS metric by infalling shell

d+1-dimensional infalling shell geometry is described in Poincar'e coordinates by the Vaidya metric Danielsson, Keski-Vakkuri and Kruczenski

1)

Danielsson, Keski-Vakkuri and Kruczenski

Deformations of AdS metric infalling shell (Vaidya metric) )

Estimations of thermalization times )/(1~ cmTtherm

cfmtherm /3.0~

Hawking temperature (in 5) = temperature of GQP (in 4)

Balasubramanian at all 1103.26833 measures of thermalization:

two-point functions, Wilson loop v.e.v., entanglement entropy.

Geodesics and correlators

1 1

2 2

( , )

( , )

x M

x M

1 1 2 2( , ) ( , )x x O O

Geodesics and correlators

Dual description of ultrarelativistic nucleus by shock waves

An ultrarelativistic nucleus is a shock wave in 4d with the energy-momentum tensor

The metric of a shock wave in AdS corresponding to the ultrarelativistic nucleus in 4d is

22242

2

2

22 )(22 dzdxdxzxT

Ndxdx

zLds

C

Janik, Peschanksi ‘05

x

x

2 2 3

1~ ( )( )

T xL x

Woods-Saxon profile

Dual description of collision of 2 ultrarelativistic nucleus

)(~ xT

Two ultrarelativistic nucleus in D=4 correspond to the metric of two shock waves in AdS

2 2 22 4 2 4 2 2 2

2 2 2

2 22 ( ) ( )C C

Lds dx dx T x z dx T x z dx dx dzz N N

x

x

~ ( )T x

Question:

what happens in AdS after collision of two shock waves?

Dual description of collision of 2 ultrarelativistic nucleus

• 't Hooft and Amati, Ciafaloni and Veneziano 1987 Aichelburg-Sexl shock waves to describe particles, Shock Waves ------ > BH

• Colliding plane gravitation waves to describe particles Plane Gravitational Waves ----- > BH I.A., Viswanathan, I.Volovich, Nucl.Phys., 1995

• Boson stars (solitons) to describe particles M.Choptuik and F.Pretorius, PRL, 2010

4-dim 19PlM 10 GeV

BLACK HOLE FORMATION = Trapped Surface(TS)

• Theorem (Penrose): BH Formation = TS

Different profiles An arbitrary gravitational shock wave in AdS5

The chordal coordinate

• Point sourced shock waves

p

LzLzxq

4)( 22

)(q

X-X+

X

TS comprises two halves,which are matched along a “curve”

Trapped Surface for two shock waves in AdS

3 1,23( ) ( ) 0H L

AdS3

Trapped Surface (TS) for two shock waves in AdS

1,2 1,2

2 ( 2)1,2 (1,2)

1 2

0, , 0,

( ), ,

4,

D

the outer null normals have zeroconvergenc

X D X D

X X

e

no function in convergence

X D

X D

Eardley, Giddings; Kang, Nastase,….

• Theorem. TS for two shock waves = . solution to the following Dirichlet problem

D

D

Conjecture

Multiplicity in HIC in D=4 can be estimated by the area of TS in AdS5 formed in collisionof shock waves

Gubser, Pufu, Yarom

D=4 Multiplicity = Area of trapped surface in D=5

Lattice calculations

3 4

35

163

L E TG

4 11E T Hawking-Page relation

From a Woods-Saxon profile for the nuclear density

1: 4.3 4.3 5 ; : 4.4Au L fm Pb L fmGeV

Gubser, Pufu, Yarom, 0805.1551, Alvarez-Gaume, C. Gomez, Vera, Tavanfar, Vazquez-Mozo, 0811.3969IA, Bagrov, Guseva, 0905.1087

Multiplicity: Experimental data, Landau, AdS-estimation

1/4Landau NNS s

Phenomenological estimation: total multiplicity and the number of charged particle

chS NN

1/3trapped NNS s

0.14trapped NNS smodified

0.15data NNS s

0.14~ NNsMModified Hologhrapic Model

Multiplicity: Landau’s/Hologhrapic formula vs experimental data

1/4~ NNsMLandau’s formula

ATLAS Collaboration 1108.6027

Plots from: Alice Collaboration PRL, 2010

Different profiles different multiplicities

Cai, Ji, Soh, gr-qc/9801097IA, 0912.5481

Kiritsis, Taliotis, 1111.1931

• Dilaton shock waves

Multiplicity very closed to LHC data

Goal: try to find a profile to fit experimental data

2 2 2 2 2( )( ( , ) ( ) )ds b r dudv r x u du dr dx

0( )( )a

a

r rb rL

3( 1)

2(3 2)1/3

3 16

1

~

~

aa

a

aa

a

S s

S s

0.142 ~a S s

11 3

1/3

13( 1)

1

~

~

aa

aa

S s

S s

Cut-off

Gubser, Pufu, Yarom,II

• Plane shock waves

Wall-on-wall dual modelof heavy-ion collisions

• Simplify calculation• But strictly speaking not correct• One needs a regularization

I.A., A.Bagrov, E.Pozdeeva, JHEP, 2012

The Einstein equation Lin, Shuryak, 09Wu, Romatschke, Chesler, Yaffe, Kovchegov,….

Phase diagram from dual approach

Formation of trapped surfaces is only possible when Q<Qcr

Red for smeared matterBlue for point-like

I.A., A.Bagrov, Joukovskaya, 0909.1294I.A., A.Bagrov, E.Pozdeeva, 1201.6542

Conclusion

Black Hole formation in AdS5 formation of QGP of 4-dim QCD

Black Hole in AdS5 QGP in 4-dim 2001-2011

• Formula for multiplicity

2008-2012

Future directions: • multiplicity and quark potential for the same dual gravity model • New phase transitions related with parameters of gravity models