Jets in N=4 SYM from AdS/CFT

Yoshitaka HattaU. Tsukuba

Y.H., E. Iancu, A. Mueller, arXiv:0803.2481 [hep-th] (JHEP)Y.H., T. Matsuo, arXiv:0804.4733 [hep-th]

Contents

Introduction e^+e^- annihilation and Jets in QCD Jet structure at strong coupling Jet evolution at finite temperature

Strongly interacting matter at RHIC

Observed jet quenching of high-pt hadrons is stronger than pQCD predictions.

dypd

dN

coll

dypd

dN

AA

T

pp

T

AA

NR

2

2

Strongly interacting matter at RHIC

Ideal hydro simulation works,suggesting short mean free path.

2cos)(21 2 Tpvd

dn

low viscosity, strong jet quenching

The Regge limit of QCD

tot 08.0ss2ln

Hadron-hadron total cross section grows with energy (‘soft Pomeron’)

c.f., pQCD prediction (BFKL, ‘hard Pomeron’)

ss 2ln4

or

?

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, or in the `unparticle’ sector at the LHC?

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

Motivation

Why N=4 SYM?

Why study jets ?

Strassler, arXiv:0801.0629 [hep-ph]

Deep inelastic scattering in QCD

P X

e

2Q

222

22

2 Qmm

Q

qP

Q

pX

x

x

Two independent kinematic variables

22 Qq Photon virtuality

Bjorken-

Momentum fraction of a parton

),( 2QxDS : Parton distribution function Count how many partons are there inside a proton.

2122

,)(),(ln

Qz

xDzP

z

dzQxD

Q SSxSDGLAP equation

DIS vs. e^+e^- annihilation

P

e 022 Qq

e

022 Qq

e

Bjorken variable Feynman variable

P

qP

Qx

2

2

2

2

Q

qPx

Parton distribution function Fragmentation function

),( 2QxDS),( 2QxDT

crossing

Jets in QCD

ee

Average angular distributionreflecting fermionic degrees of freedom (quarks)

2cos1

Observation of jets in `75 provides one of the most striking confirmations of QCD

Fragmentation function

ee

P Count how many hadrons are there inside a quark.

),( 2QxDT

Q

E

Q

qPx 222

Feynman-x

First moment

nQxDdx T ),( 21

0 average multiplicity

2),( 21

0 QxxDdx T energy conservation

Second moment

022 Qq

Evolution equation

The fragmentation functions satisfy a DGLAP-type equation

),()(),(ln

222

QjDjQjDQ TTT

),(),( 211

0

2 QxDxdxQjD Tj

TTake a Mellin transform

Timelike anomalous dimension

)1(22 ),1( TQQDn T (assume )0

2122

,)(),(ln

Qz

xDzP

z

dzQxD

Q TTxT

Anomalous dimension in QCDLowest order perturbation

Soft singularity

~x

11

)(

j

j sT

!!)1( T

ResummationAngle-ordering

)1(

8)1(

4

1)( 2 j

Njj s

T

Mueller, `81

Inclusive spectrum

largel-x small-x

x1ln

),( 2QxxDdx

dxT

roughly an inverse Gaussian peaked at

Q

x

1

N=4 Super Yang-Mills

SU(Nc) local gauge symmetry Conformal symmetry SO(4,2)

The ‘t Hooft coupling doesn’t run. Global SU(4) R-symmetry

choose a U(1) subgroup and gauge it.

0CYM Ng 2

Type IIB superstring

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

ne solution which is asymptotically

Our universe

5S

AdS `radius’ coordinate

55 SAdS

(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, `97

2'4 RCN1 sg

CFT string

55 SAdS

Dilaton localized at small

DIS at strong coupling

R-charge current excites metric fluctuations in the bulk, which then scatters off a dilaton (`glueball’)

r

r

Cut off the space at (mimic confinement)

022 Qq

Polchinski, Strassler, `02Y.H. Iancu, Mueller, `07

We are here

Photon localized at large Qr

0rr

)( r

e^+e^- annihilation at strong coupling

Hofman, MaldacenaY.H., Iancu, Mueller,Y.H. Matsuo

arXiv:0803.1467 [hep-th]arXiv:0803.2481 [hep-th]arXiv:0804.4733 [hep-th]

022 Qq

5D Maxwell equation

Dual to the 4D R-current

)(2

1)(

2

1)( 433465562211 DDDDxJ

A reciprocity relation

),()(),(ln

2//

2/2

QjDjQjDQ TSTSTS

DGRAP equation

Dokshitzer, Marchesini, Salam, ‘06

The two anomalous dimensions derive from a single function

Basso, Korchemsky, ‘07Application to AdS/CFT

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

Confirmed up to three loops (!) in QCD Mitov, Moch, Vogt, `06

Anomalous dimension in N=4 SYM

Gubser, Klebanov, Polyakov, `02

)(22

1

2)( 0jj

jjS 2j Kotikov, Lipatov, Onishchenko, Velizhanin `05

Brower, Polchinski, Strassler, Tan, `06

)(22 jj S

2~j

jj XXrV

Leading Regge trajectory Twist—two operators

lowest mass state for given j lowest dimension operator for given j

The `cusp’ anomalous dimension

Average multiplicity

)(22

1

2)( 0jj

jjS

22

1)(

2

0

jjjjT

231)1(2)( QQQn T

c.f. in perturbation theory, 22)(

QQn

crossing

c.f. heuristic argument QQn )( Y.H., Iancu, Mueller ‘08

Y.H., Matsuo ‘08

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

At strong coupling, branching is so fast and efficient. There are no partons at large-x !

Q

Energy correlation functionHofman, Maldacena `08

Energy distribution is spherical for anyCorrelations disappear as

QAll the particles have the minimal four momentum~ and are spherically emitted. There are no jets at strong coupling !

1)()3( OS

241 weak coupling

strong coupling

1

2

Evolution of jets in a N=4 plasmaY.H., Iancu, Mueller `08

Solve the Maxwell equation

in the background of Schwarzschild AdS_5

25

2244

02

222

4

40

2

22

)1()(1

dRdr

rrr

Rxddt

r

r

R

rds

TRr 20

)(),,( rAerxtA iqzti

To compute correlation functions :2222 TQq

r

0rr Event horizon

Time-dependent Schrödinger equation

2

,2 0

r

r

Solutions available only piecewise.Qualitative difference between

t=0

horizon

Tk

2

312 )( TQQ s sQQ

Minkowskiboundary

‘low energy’ and ‘high energy’

T

QK

2

sQ : plasma saturation momentum

),(),,( rtAerxtA iqzti

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

low-energy,

Early time diffusion

AV BV CV

solution with AV

This represents diffusion

up to time || 2Q

qt

312 )(qTQQ s

Gauge theory interpretation

IR/UV correspondence

TLr

r 02

L Q1

c.f., general argument from pQCD Farrar, Liu, Strikman, Frankfurt ‘88

is the formation time of a parton pair (a.k.a., the coherence time in the spacelike case)|| 2Q

qt

low-energy

Intermediate free streaming region

AV BV CV

solution with BV

constant (group-) velocity motion

tq

Q

312 )(qTQQ s

Gauge theory interpretation

IR/UV correspondence

Q1

21 zvq

Q is the transverse velocity

tvL T2

Lq

Q

T

1

TL Linear expansion of the pair

low-energy

Falling down the potential

AV BV CV

solution with CV

A classical particle with mass falling down the potential

k

312 )(qTQQ s

Gauge theory interpretation

Q1 q

Q

T

1T

1

disappear into the plasma

Tp ||

start to `feel’ the plasma

In-medium acceleration

The high energy case

AV BV CV

312 )(qTQQ s

A new characteristic time

22 Q

q

Q

qt

s

sQ1T

1

No difference between the timelike/spacelike cases

The scale

is the meson screening length

Energy loss, meson screening length, and all that

T

v

q

Q

TL z

412 )1(1

Liu, Rajagopal, Wiedemann, `06 Q1q

Q

T

1

WKP solution after the breakup features the trailing string solution

))((exp rtvziq z

00

0 arctanln2

1)(

r

r

rr

rr

Tr

Herzog, et al, ‘06, Gubser, ‘06

breakup

Energy loss, meson screening length,and all that

Rate of enery flow towards the horizon

Identical to the motion of our wavepacket

312

qQ

qt

sf

sQ1T

1 Time to reach the horizon

ftc.f., damping time of a gluon

31qGubser, Gulotta, Pufu, Rocha, 0803.1470 [hep-th]

Branching picture at strong coupling

Energy and virtuality of partons in n-th generation

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

nn

qq

2

nn

QQ

2

221 2Q

q

Q

qtt n

n

nnn

QQn )(

,1)( ttQ or ttL )(

Final state cannot be just a pair of partons

Medium-induced branching at finite-T

Mach cone?

)(221ns

n

n

nnn qQ

q

Q

qtt

22 )()()(

TttQdt

tdqs where

31

T

q

Q

q

s

time-dependent drag force

Conclusion

Various aspects of jets at strong coupling—including the details of the final state—are accessible from gauge/string duality techniques.

Photon evolution problem sheds new light on the physics of energy loss, etc. in a strongly coupled plasma.