High energy collisions in AdS

Yoshitaka Hatta U. Tsukuba

Asian triangle heavy-ion conference 2008/10/13

Outline

Motivation Gluon saturation in QCD DIS and e+e- annihilation in sSYM Jets at strong coupling? Jet decay at finite temperature

　 Jet quenching at RHIC

Is the strong suppression entirelyof perturbative origin ?

Note: The data are also consistent with the log behavior

The soft Pomerontot

08.0s

s2ln

1.08

j

2m

2

GeVmG 22

Thermal hadron spectrum

Identified particle yields are well described by a thermal model

TMM

NN **

exp

The model works in e+e- annihilation, hadron collisions, and heavy-ion collisions

Becattini; Chliapnikov; Braun-Munzinger et al.

There are many phenomena at collider experiment which defy weak coupling approaches.

Study N=4 SYM as a toy model of QCD. (Interesting in its own right…) One can solve strong coupling problems using AdS/CFT. Think how it may (or may not?) be related to QCD later…

Possible applications to jet quenching at RHIC and LHC.

Lots of works on DIS. e+e- annihilation is a cross channel of DIS.

MotivationWhy N=4 SYM?

Why study jets ?

Regge limit of QCD

E

E

,...,,2 TpMtEs

One of the most challenging problems of QCD is the high energy limit.

Can we compute the total cross section ?What are the properties of the final states ?

Deep inelastic scattering

P X

e

2Q

sQQmm

QqP

Q

pX

x 2222

22

2

x

High energy = small- )1(x

Two independent kinematic variables

02 Qqq Photon virtuality

Bjorken-

Physical meaning : momentum fraction of the constituents (`partons’)

Gluons at HERA

The gluon distribution rises very fast at small-x

),( 22 QxF

)( 2

~ Qcx ),(~ 2Qxxgs

Small- resummationOrdinary perturbation theory

322 1 ssssT

At small- such that 　　　

sss xT 1ln12

2222 1ln1ln sss xx

xx ss 1ln1ln 2333

xs 1ln44

x

x ,1~1ln xs

sc

s

n

sn xx

cx

cn

11lnexp1ln

!1

The BKFL Pomeron

2ln4c

More precisely, solve the bootstrap equation

)()(1ln

xTKxTx s

Eigenvalue of : K

T = + T

Kg g

The ladder diagrams sum up to a Pomeron—like behavior

Gluon saturation

)()(1ln

xTKxTx s

)()()(1ln

2 xTxTKxTx s

Without interaction

With interaction (BK-JIMWLK)

Rapid growth of the gluon number tamed,leading to a Bose condensate of gluons,or the Color Glass Condensate. 　

`Phase diagram’ of QCD

2lnQ

s

xAxQs

9.4312 1)(

x

1ln Saturation

BFK

L

DGLAP

Recent progress on saturation

A proof of factorization for inclusive gluon production in AAGelis, Lappi & Venugopalan

Two gluon production and correlation in pA

production in pA

production in pA and AA

Evolution of glasma flux tubes

Saturation in Mueller-Navelet jets

Fukushima & Hidaka

' Fillion-Gourdean & Jeon

Fujii & Itakura; Iwazaki

Complete NLO BK equation

Running coupling effects for gluon production

Balitsky & Chirilli

/J Kharzeev, Levin & Tuchin

Iancu, Kugeratski & Triantafyllopoulos

Kovchegov & Weigert

Gluon correlation in impact parameter space

21ln

TTKTx s

BK equation

The mean field approximation OK for a large nucleus, but not OK for a small target (e.g., a proton).

,2TTT

)10(~/5.1 2 OTTT

Factorization violated due to the power-law correlation in impact parameter space from BFKL

YH & Mueller (2007)Avsar & YH (2008)

N=4 Super Yang-Mills

The ‘t Hooft coupling doesn’t run:

Global SU(4) R-symmetry choose a U(1) subgroup and gauge it.

0CYM Ng 2

N=4 SYM

QCD

Type IIB superstring Consistent superstring theory in D=10 Supergravity sector admits the black 3-bra

ne solution which is asymptotically

Our universe 5S5th dimension

55 SAdS

25

22

223

22

21

222

dR

zdzdxdxdxdtRds

(anomalous) dimension mass`t Hooft parameter curvature radius number of colors string coupling constant

The correspondence

Take the limits and N=4 SYM at strong coupling is dual to weak

coupling type IIB on Spectrums of the two theories match

CN CYM Ng 2

Maldacena (1997)

2'4 RCN1 sg

CFT string

55 SAdS

What one would expect at strong coupling…

Rapid fragmentation. Most interesting physics is at small-x.

String S-matrix dominated by J=2 singularity. 　　 Pomeron graviton in AdS.

There are no jets. Final states look spherical.

22 Pomj

2ln41 sBFKLj cf.

Polchinski & Strassler (2002)

Kotikov et al. (2005); Brower et al (2006)

Hofman & Maldacena (2008); YH, Iancu & Mueller (2008); YH & Matsuo (2008)

Shock wave picture

Lz

Characteristic size wavefunction localized at

L

‘Hadron’ closed string state in cutoff AdS

Weak coupling Strong coupling

Large nucleus (CGC) random color sources

non-abelian Weiszacker-Williams field (boosted color-Coulomb field)

gravitational shock wave(boosted Schwartzschild metric)

figure from Gubser, Pufu & Yarom (2008)

Dilaton localized at

DIS at strong coupling

R-charge current excites metric fluctuations in the bulk, which then scatters off a dilaton

z

1~z

Cut off the space at (mimic confinement)

022 Qq

Polchinski & Strassler (2002)

We are here

Photon localized at Qz 1~

1z

)0( z

String S-matrix

dilaton gauge bosonvertex op. vertex op.

Insert t-channel string statesdual to twist-2 operators

j

jj VV1

j

AdS version of the graviton Regge trajectory

Phase diagram at strong couplingxY 1ln

YH, Iancu & Mueller (2007)

DIS vs. e+e- annihilation

P

e 022 Qq

e

022 Qq

e

Bjorken variable Feynman variable

P

qPQx

2

2

2

2Q

qPx

Parton distribution function Fragmentation function

),( 2QxDS ),( 2QxDT

crossing

The reciprocity relation

),()(),(ln

2//

2/2 QjDjQjD

Q TSTSTS DGRAP equation

Dokshitzer, Marchesini & Salam (2006)

The two anomalous dimensions derive from a single function

Basso & Korchemsky (2007)Application to AdS/CFT

Assume this is valid at strong coupling and see where it leads to.

Nontrivial check up to three loops (!) in QCD Mitov, Moch & Vogt (2006)

Average multiplicity at strong coupling

)(221

2)( 0jjjjS

22

1)(2

0jjjjT

231)1(2 )()()( QQQn T

c.f. in perturbation theory, 22)(

QQn

crossing

c.f. heuristic argument QQn )( YH, Iancu & Mueller (2008)

YH & Matsuo (2008)

spacelike anomalous dimension timelike anomalous dimension

Jets at strong coupling?

The inclusive distribution is peaked at the kinematic lower limit

1QQEx 2

QxFQQxDT

22 ),(

Rapidly decaying function for Qx

21)( jjT in the supergravity limit

Branching is so fast. Nothing remains at large-x !All the particles have the minimal four momentumThere are no jets at strong coupling !

Qn ~ |~| p

Thermal hadron production from gauge/string duality 　　　　　　　　　 　 YH & Matsuo (2008)

Matrix element between a photon and particles.

)()( )1(1 QzHzA )exp(iQz~

Q

complex saddle point in the z-integral

Qn

Finite temperature AdS/CFT

　 AdS 　　 Schwartzschild AdS

Our universe

Hawking temperature = gauge theory temperature

Witten (1999)

25

22

40

422240

422 )1()1(

dRz

zzdzxddtzzRds

)1( 0 Tz

Event horizon

Solve the 5D Maxwell equation

in the background of Schwarzschild AdS_5

Evolution of jets in a N=4 plasma

0z

Tz 1Event horizon

)(),,( 3 zAezxtA iqxti

Time-dependent Schrödinger equation

z

Solutions available only piecewise.

A new characteristic scale

t=0

horizon

312 )( TQs Minkowskiboundary plasma saturation momentum

),(),,( 3 ztAezxtA iqxti

To study time-evolution, add a weak t-dependence and keep only the 1st t-derivative

YH, Iancu & Mueller (2008)

40

42

22

2

2281

21

zzQ

zzti

0z

(naive) Gauge theory interpretation

T1

s

s

QqQ

T11

L

disappear into the plasma

||~ 2

sQt

Lz

breakup into a “ pair”qq

Use the correspondence

||~ 2

sQt

The scale is the meson screening length

Relation to other works

Tv

qQ

TL zs

412 )1(1

Liu, Rajagopal & Wiedemann (2006)

WKP solution after the breakup features the trailing string solution

))((exp 3 ztvxiq z Herzog, et al, Gubser (2006)

312

sf Q

t

Time to reach the horizon (penetration length)

cf. damping time of a gluon 31tGubser, Gulotta & Pufu (2008)

cf. weak coupling result (BDMPS) 21t

Branching picture at strong coupling

Energy and virtuality of partons in the n-th generation

At strong coupling, branching is as fast as allowed by the uncertainty principle

nn 2 nn

QQ2

21n

nnn Qtt

Final state cannot be just a pair of partons.

)(2ns

n

Q

(vacuum)

(medium)

Trajectory of the parton pair Enveloping curve of the parton shower.

Conclusions Various aspects of high energy scattering at strong coupli

ng—including some details of the final state—are accessible from gauge/string duality techniques.

Going to phenomenology, it is important to think when AdS-based approaches may be a good starting point and when it is not. e.g., Mueller (2008)

If the initial hard scattering were described by a strongly coupled theory, there would be no jets to begin with.

pp or AA collisions not fully explored yet. Sin, Shuryak & Zahed (2005); Albacete, Kovchegov & Taliotis (2008)