RHIC physics and AdS/CFT

Amos Yarom, Munich

together with: S. Gubser and S. Pufu

Overview

• The quark gluon plasma

Energy

gYM

1

~170 MeV

?

Overview

• The quark gluon plasma

• N=4 SYM plasma via AdS/CFT

AdS/CFT

J. Maldacena

Overview

• The quark gluon plasma

• N=4 SYM plasma via AdS/CFT

?

Overview

• The quark gluon plasma

• N=4 SYM plasma via AdS/CFT

• Energy loss of a moving quark

Overview

• The quark gluon plasma

• N=4 SYM plasma via AdS/CFT

• Energy loss of a moving quark

Overview

• The quark gluon plasma

• N=4 SYM plasma via AdS/CFT

• Energy loss of a moving quark

• Summary

The quark gluon plasma at RHICRELATIVISTIC

HEAVY

ION

COLLIDER

Jet quenching

197 ×

pT

RAA =Hadron yield in Au-Au

(rescaled) Hadron yield in p-pRAA (pT ) =

Hadron yield in Au-Au(pT )(rescaled) Hadron yield in p-p(pT )

Jet quenching(Phenix, 2005)

Friction coefficient for QCD plasma

_p= ¡ ´p+F

0:3f m=c< ´ < 8f m=c

N=4 SYM plasma via AdS/CFT

AdS/CFT

J. Maldacena

AdS5 CFT

Empty AdS5Vacuum

L4/’2 gYM2 N

L3/2 G5 N2

J. Maldacena hep-th/9711200

T>0

N=4 SYM plasma via AdS/CFT

AdS5 CFT

AdS5 BH Thermal state

L4/’2 gYM2 N

L3/2 G5N2

E. Witten hep-th/9802150

Horizon radius Temperature

Empty AdS5Vacuum

J. Maldacena hep-th/9711200

AdS Black holes

ds2 = L2=z2¡¡ g(z)dt2+dx2i +dz

2=g(z)¢

g(z) = 1¡µzz0

¶4

z0

z

0x1xi, t

AdS5 CFT

AdS5 BH Thermal state

L4/’2 gYM2 N

L3/2 G5N2

E. Witten hep-th/9802150

Horizon radius Temperature

z01/ T

AdS/CFT

J. Maldacena

Friction coefficient

AdS5 CFT

J. Maldacena hep-th/9803002

Massive parton

Endpoints of an open

string

z0

z

0

S =12¼®0

Z(G@X@X )1=2d2¾

±S±X

= 0X jb= (t;0;0;0;0)X jh =no outgoing

?

(Gubser 2006, Holzhey, Karch, Kovtun, Kozcaz, Yaffe, 2006, Teaney Cassalderrey-Solana, 2006)

Friction coefficient

AdS5 CFT

J. Maldacena hep-th/9803002

Massive parton

Endpoints of an open

string

z0

z

0

S =12¼®0

Z(G@X@X )1=2d2¾

±S±X

= 0X jb= (t;vt;0;0;0)X jh =no outgoing

?

X = (t;vt+»(z);0;0;z)

»(z) =vz04

µln1¡ z=z01+z=z0

+2arctanzz0

¶

(Gubser 2006, Holzhey, Karch, Kovtun, Kozcaz, Yaffe, 2006, Teaney Cassalderrey-Solana, 2006)

Friction coefficient

z0

z

0

X = (t;vt+»(z);0;0;z)

»(z) =vz04

µln1¡ z=z01+z=z0

+2arctanzz0

¶

F =¼pg2Y M NT

2p2m

F F

_p= ¡¼pg2Y M NT

2

2mp

´b » 0:5f m=c

´c » 1:2f m=c

(Gubser 2006, Holzhey, Karch, Kovtun, Kozcaz, Yaffe, 2006, Teaney Cassalderrey-Solana, 2006)

Friction coefficient

´b » 0:5f m=c

´c » 1:2f m=c

AdS/CFT

J. Maldacena

0:3f m=c< ´ < 8f m=c

(Gubser 2006, Holzhey, Karch, Kovtun, Kozcaz, Yaffe, 2006, Teaney Cassalderrey-Solana, 2006)

Measurables which have been compared

Friction coefficient

Energy density Shear viscosity Jet quenching parameter

0:3f m=c< ´ < 8f m=c

² » 0:8²SB

1< 4¼́ v=s < 2

5GeV2=f m< q? <15GeV2=f m

(Policastro, Son, Starinets, 2001)

(Liu, Rajagopal, Wiedemann, 2006)

(Gubser 2006, Holzhey, Karch, Kovtun, Kozcaz, Yaffe, 2006, Teaney Cassalderrey-Solana, 2006)

(Gubser, Klebanov, Peet, 1996)

Measuring jets

Measuring jets

Measuring di-jets

Measuring di-jets(STAR, 0701069)

Measuring di-jets(STAR, 0701069)

=

Measuring di-jets(STAR, 0701069)

»

Creation of sound waves(Casalderrey-Solana, Shuryak, Teaney, 2004, 2006)

Creation of sound waves(Casalderrey-Solana, Shuryak, Teaney, 2004, 2006)

Mach cones and di-jets(Casalderrey-Solana, Shuryak, Teaney, 2004, 2006)

»

Mach cones in N=4 SYM

AdS5 CFT

» hTrF 2i©jb

hTmn iGmn jb

z0

z

0

The energy momentum tensor

AdS5 CFT

» hTrF 2i©jbhTmn iGmn jb

hTmn i / Qmn

Gmn =G(0)mn +hmn

AdS black hole Metric fluctuationshmn = :::+Qmnz4+:::

S = SN G +SE H

SN G =12¼®0

Z(G@X@X )1=2d2¾

SE H =1

16¼G5

µR +

12L2

¶G1=2d5x

±S±X

= 0±S±G= 0

z0

z

0

Gmnz,k)

The energy momentum tensor

z0

z

0

S = SN G +SE H

SN G =12¼®0

Z(G@X@X )1=2d2¾

±S±X

= 0±S±G= 0D¹ º½¾h½¾= J ¹ º

Gmn =G(0)mn +hmn

SE H =1

16¼G5

µR +

12L2

¶G1=2d5x

X = (t;vt+»(z);0;0;z)

»(z) =vz04

µln1¡ z=z01+z=z0

+2arctanzz0

¶

The energy momentum tensor

D¹ º½¾h½¾= J ¹ º

hmn =

0

BBBB@

h00 h01 h02 h03 h04h10 h11 h12 h13 h14h20 h21 h22 h23 h24h30 h31 h32 h33 h34h40 h41 h42 h43 h44

1

CCCCA

hmn =

0

BBBB@

h00 h01 h02 h03 0h10 h11 h12 h13 0h20 h21 h22 h23 0h30 h31 h32 h33 00 0 0 0 0

1

CCCCA

hmn =

0

BBBB@

h00 h01 h02 0 0h10 h11 h12 0 0h20 h21 h22 0 00 0 0 h33 00 0 0 0 0

1

CCCCA

Gauge choiceCylindrical symmetry

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k21g(z)

¶¶©T = ze¡ ik1»(z)

©T =12v2

Ã

¡ h11+2µk1k?

¶2h22+

µkk?

¶2h33

!Tensor modesVector modes

µ©V1©V2

¶=

0

@12v

³h01 ¡ k1

k?h02´

12v2

³¡ h11+

³k1k?¡ k?

k1

´h12+h22

´

1

A

(x1;t) ! x1 ¡ vtZh½¾eikx

d3k(2¼)3

The energy momentum tensor

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k21g

¶¶©T = ze¡ ik1»(z)

¡@2z +K V@z +VV

¢µ©V1©V2

¶= ~SV

zge¡ ik1»

©T =12v2

Ã

¡ h11+2µk1k?

¶2h22+

µkk?

¶2h33

!

VV =k2

g

á g gv2

¡ k1k

¢2

¡ 1 v2¡ k1k

¢2

!

K V =

á 3z 00 ¡ 3z +

g0

g

!~SV =

µ11

¶

Tensor modes

Vector modesµ©V1©V2

¶=

0

@12v

³h01 ¡ k1

k?h02´

12v2

³¡ h11+

³k1k?¡ k?k¡

´h12+h22

´

1

A

+ first order constraint

The energy momentum tensor

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k2¡g

¶¶©T = ze¡ ik¡ »(z)

¡@2z +K V@z +VV

¢µ©V1©V2

¶= ~SV

zge¡ ik¡ »¡

@2z +K S@z +VS¢

0

BB@

©S1©S2©S3©S4

1

CCA =

~SSzge¡ ik¡ »

Tensor modes

Vector modes

+ first order constraint

Scalar modes

+ 3 first order constraints

D¹ º½¾h½¾= J ¹ º

Energy density for v=3/4

Over energy

Under energy

v=0.75 v=0.58

v=0.25

Small momentum approximations

E = ¡3iK 1v(1+v2)

2¼(K 2? +K21(1¡ 3v2))

+O(K 0)

D¹ º½¾h½¾= J ¹ º

h½¾=X

n

K nh(n)½¾ Zh½¾eiK X

d3K(2¼)3

(x1;t) ! x1 ¡ vt

1-3v2 > 0 (subsonic)

Small momentum approximations

E = ¡3iK 1v(1+v2)

2¼(K 2? +K21(1¡ 3v2))

+O(K 0)

1-3v2 > 0 (subsonic)

Re(K1)

Im(K1)

§iK ?p1¡ 3v2

v decreasesv increases

ZEeiK 1X 1 dK 1

2¼

X1 > 0

X1 < 0

Small momentum approximations

E = ¡3iK 1v(1+v2)

2¼(K 2? +K21(1¡ 3v2))

+O(K 0)

1-3v2 < 0 (supersonic)

Re(K1)

Im(K1)

ZEeiK 1X 1 dK 1

2¼

1-3v2 = 0

v increases

??

X1 > 0

X1 < 0X1

Small momentum approximations

E = ¡3iK 1v(1+v2)

2¼(K 2? +K21(1¡ 3v2))

+O(K 0)

1-3v2 < 0 (supersonic)

E(X ) =

(3v(1+v2 )4¼2

X 1(X 2

1+(1¡ 3v2)X 2

? )3=2 inside theMach cone.

0 outside theMach cone.

1-3v2 > 0 (subsonic)

E(X ) =3v(1+v2)8¼2

X 1(X 21 +(1¡ 3v2)X

2? )3=2

Small momentum approximations

E =

¡3K 21v

2(K 2? (2+v2) +2K 21(1+v

2)2¼(K 2? +K

21(1¡ 3v2))2

¡3iK 1v(1+v2)

2¼(K 2? +K21(1¡ 3v2))

+O(K 1)

E =¡3iK 1v(1+v2) +O(K 2)

2¼(K 2? +K21(1¡ 3v2) ¡ ivK 2K 1)

+O(K 1)

Small momentum approximations

E =¡3iK 1v(1+v2) +O(K 2)

2¼(K 2? +K21(1¡ 3v2) ¡ ivK 2K 1)

+O(K 1)

Re(K1)

Im(K1)

¡@2t +@

2x(c

2s +¡ s@t)

¢E = sources

cs2=1/3

s=1/3

Multi-scale analysis

Large distances – linear hydrodynamic

picture valid

Intermediate distances – nonlinear

hydrodynamics

Short momenta – Strong dissipative

effects

Energy density for v=3/4

0

v=0.75 v=0.58

v=0.25

Large momentum approximations

E =v2(1¡ v2)K 21 ¡ (2+v

2)(K 21(1¡ v2) +K 2? )

24pK 21(1¡ v2) +K

2?

+i¼vK 12v2(1¡ v2)K 21 +(5¡ 11v

2)(K 21(1¡ v2) +K 2? )

18(K 21(1¡ v2) +K2? )2

+O(K ¡ 3)

D¹ º½¾h½¾= T ¹ º

h½¾=X

n

K ¡ nh(n)½¾ Zh½¾eiK X

d3K(2¼)3

(x1;t) ! x1 ¡ vt

+O(X 0)

E(X ) =

¡ vX 1(5¡ 11v2)X 21 +(1¡ v

2)(5¡ 8v2)X 2?

72(1¡ v2)5=2³X 21

1¡ v2 +X2?

´5=2

X 21 +(1+v2)X 2?

12¼2p1¡ v2

³X 21

1¡ v2 +X2?

´3

Large momentum approximations

E =v2(1¡ v2)K 21 ¡ (2+v

2)(K 21(1¡ v2) +K 2? )

24pK 21(1¡ v2) +K

2?

+i¼vK 12v2(1¡ v2)K 21 +(5¡ 11v

2)(K 21(1¡ v2) +K 2? )

18(K 21(1¡ v2) +K2? )2

+O(K ¡ 3)

+O(X 0)

E(X ) =

¡ vX 1(5¡ 11v2)X 21 +(1¡ v

2)(5¡ 8v2)X 2?

72(1¡ v2)5=2³X 21

1¡ v2 +X2?

´5=2

X 21 +(1+v2)X 2?

12¼2p1¡ v2

³X 21

1¡ v2 +X2?

´3

Large momentum approximations

E(X ) = ¡ vX 1(5¡ 11v2)X 21 +(1¡ v

2)(5¡ 8v2)X 2?

72(1¡ v2)5=2³X 21

1¡ v2 +X2?

´5=2

Large momentum approximations

E(X ) = ¡ vX 1(5¡ 11v2)X 21 +(1¡ v

2)(5¡ 8v2)X 2?

72(1¡ v2)5=2³X 21

1¡ v2 +X2?

´5=2

Large momentum approximationsc2s =

515

Wakes

Mach cones, wakes and di-jets(Casalderrey-Solana, Shuryak, Teaney, 2004, 2006)

Mach cones, wakes and di-jets(Casalderrey-Solana, Shuryak, Teaney, 2004, 2006)

(STAR, 0701069)

The Poynting vector

z0

z

0

D¹ º½¾h½¾= J ¹ º

The Poynting vector

V=0.25

S1 S?

V=0.58

V=0.75

(Gubser, Pufu, AY, 2007)

Re(K1)

Im(K1)

Small momentum asymptotics

Sound Waves ?

S1 = ¡ iK 1(1+v2)

2¼(K 2 ¡ 3K 21v2)+ i

12¼K 1

+O(K 0)

K 21v2¼(K 2 ¡ 3K 21v2)2

+3K 41v

2(1+v2)2¼(K 2 ¡ 3K 21v2)2

+K 2

8¼K 21v¡

18¼v

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)+

4v+O(K )2¼(K 2 ¡ 4iK 1v)

(Gubser, Pufu, AY, 2007)

X1

Small momentum asymptotics

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)+

4v+O(K )2¼(K 2 ¡ 4iK 1v)

Z4v

2¼(K 2 ¡ 4iK 1v)eiK X

d3K(2¼)3

= ¡X 18¼2X 3

e¡ 2v(X 1+X )

X 1X 3e¡ 2v(X 1+X ) »

8<

:

X 1X 3 e¡ 4vX 1 jX j À 1; X 1 > 0

X 1X 3 e

¡ vX 2?X 1 jX j À 1; X 1 <0

(Gubser, Pufu, AY, 2007)

The poynting vector

V=0.25

S1 S?

V=0.58

V=0.75

(Gubser, Pufu, AY, 2007)

Energy analysis

S2 = ¡ iK 2(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)+

O(K )2¼(K 2 ¡ 4iK 1v)

S3 = ¡ iK 3(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)+

O(K )2¼(K 2 ¡ 4iK 1v)

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)+

4v+O(K )2¼(K 2 ¡ 4iK 1v)

² = ¡3iK 1v(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2+iK 2K 1v)

_² +@iSi = ¡@E@t

Zd3x

limK ! 0

(¡ iK 1v² + iK iSi ) = F 0drag

= F 0drag±(~X ¡ ~vT)

(Friess, Gubser, Michalogiorgakis, Pufu, 2006 Gubser, Pufu, AY, 2007)

_² +@iSi = ¡@E@t

Z

d3x

limK ! 0

(¡ iK 1v² + iK iSi ) = F 0drag

= F 0drag±(~X ¡ ~vT)

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, 2006 Gubser, Pufu, AY, 2007)

z0

z

0 F

?

Just been calculated

limK ! 0

(¡ iK 1v² + iK iSi ) = F 0draglimK ! 0

(¡ iK 1v² + iK iSi )¯¯wake+ lim

K ! 0(¡ iK 1v² + iK iSi )

¯¯sound

= F 0jwake+F 0jsound

F 0jwake : F 0jsound = ¡ 1 : 1+v2

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, 2006, Gubser, Pufu, AY, 2007)

_² +@iSi = ¡@E@t

Z

d3x = F 0drag±(~X ¡ ~vT)

F 0jwake : F 0jsound = ¡ 1 : 1+v2

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, 2006, Gubser, Pufu, AY, 2007)

=

Universality(Gubser, AY,2007)

SN G =12¼®0

Z(g@X@X )1=2q(ÁI )d2¾

SE H =1

16¼G5

Zg12Rd5x

SÁ =1

16¼G5

Zg12¡­ I J @ÁI@ÁJ +V(ÁI )d5x

¢

z0

z

0

ds2 =®(z)2¡¡ h(z)dt2+d~x2+dz2=h(z)

¢

®! L=z

h ! 1

h ! 0

Universality(Gubser, AY, 2007)

SN G =12¼®0

Z(g@X@X )1=2q(ÁI )d2¾

SE H =1

16¼G5

Zg12Rd5x

SÁ =1

16¼G5

Zg12¡­ I J @ÁI@ÁJ +V(ÁI )d5x

¢

z0

ds2 =®(z)2¡¡ h(z)dt2+d~x2+dz2=h(z)

¢

0

zD¹ º½¾h½¾+D¹ ºI Á

I = J ¹ ºF 0jwake : F 0jdrag = ¡ 1: v2

Summary

N=4 SYM plasma exhibits a Mach cone and a wake at large distances, where the hydrodynamic approximation is valid.

The laminar wake behind the quark is a universal feature of theories with string duals, and the ratio of energy carried by the wake to the drag force is 1:v2.

This wake is difficult to reconcile with current experimental data.