DIS and e+e- annihilationat strong coupling

Yoshitaka Hatta

(U. Tsukuba)

arXiv:0710.2148 [hep-th] with E. Iancu and A. Mueller arXiv:0804.4733 [hep-th] with T. MatsuoarXiv:0807.0098 [hep-ph] with T. Matsuo

Based on:

Contents

Introduction DIS in QCD DIS in N=4 SYM e+e- annihilation and jets in QCD Jets in N=4 SYM ? Thermal (exponential) hadron spectrum

Motivations

Longstanding problems in the small- regime of QCD (Froissart’s theorem, soft vs. hard Pomerons, hadronization...) for which pQCD methods do not apply. How does the nature of scattering change as one goes from weak to strong coupling ? N=4 SYM

Possible applications to the phenomenology of `sQGP’ (finite-T ) at RHIC, or in the `unparticle’ sector at the LHC

BF xx

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, etc ?What are the properties of the final states ?

Experimental knowledge

Regge trajectory

2')0( mj

Quadratic relation between mass and spin of hadrons

Interpretable as a spinning stringwith tension

intercept slope

'1~ 0.5

Heuristic picture (life before QCD)

j

j

j

jj

j

jj tj

sg

tm

sg

')0(~

2

s

t

Suppose scattering is dominated by the t-channel exchange of particles on some Regge trajectory.

=

=

spin-j

Complex angular momentum

j

tjji

stj

s

j

edjtsT ')0(

')0(sin

1),(

j

1)0()0,(Im ss

sTtot

Analytically continue the partial waves to the complex J-plane and pick up the leading Regge pole

Regge limit in string theory (flat space)

The leading Regge trajectory (closed string) '

)2(22

j

m j

stot Amplitude dominated by the spin-2 graviton exchange

2'2sin

1),( 2

tj

s

j

edjgtsT

jji

s

2'22 1 ts sit

g

Deep inelastic scattering in QCD

P X

e

2Q

sQQmm

Q

qP

QB

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’)

DIS structure functions

Imaginary part of the forward Compton scattering amplitude

PJxJPexdi ememiqx |)0()(|Im 4

qP

QxFq

q

qPPq

q

qPPQxF

q

qq

),(),(

22

222

12

p p

Probe of the parton distribution in the target hadron.

),(~),(),(),( 22222 QxxgQxqxQxxqQxF s

The operator product expansion

)(

2

24 ~~

2

j

jjQj

jj

ememixq QcOcJJexd

,)( 1qDqO jj

The moment sum rule)(

2

22

221

0~),(

j

j QQxFxdx

Invert

)(

2

212

2 )(2

~),(j

j Qxjc

i

djQxF

continued to arbitrary j

Twist-2, spin-j operators

)(22 jj

FDF j 2)(

The BFKL Pomeron Balitsky, Fadin, Kraev, Lipatov, `78

,1s 11ln xs

T = T+

5.0 1sx 2ln4

)(

2

21

0

1

2~

j

j Qx

jji

dj

),( 22 QxF

s2ln41

but )(j

‘Bootstrap’

`Phase diagram’ of QCD

2lnQ

s

xxQs

9.42 1)(

x

1ln Saturation

BF

KL

DGLAP

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.

0CYMNg 2

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

The correspondence

N=4 SYM at strong coupling is dual to weak coupling type IIB string theory

on

Spectrums of the two theories match

Maldacena, `97

2'4 RCN1 sg

SYM string

55 SAdS

Our universe 5-th dim.

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)

2Q

Polchinski, Strassler, `02

We are here

Photon localized at large Qr

0rr

)( r

High energy 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

Anomalous dimension in N=4 SYM

Gubser, Klebanov, Polyakov, `02

)(22

1

2)( 0jj

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

Brower, Polchinski, Strassler, Tan, `06

)(22 jj

2~j

jj XXrV

`Pomeron vertex operators’ Twist—two operators

Diffusion in the 5th dimension, important when

Pomeron at strong couplingPhoton wavefunction

Dilaton wavefunction

Qr ~

~'r

220 j

~ln s

Pomeron = massive graviton in AdS with intercept

s

Q

N

s

c ln8

/lnexp

222

2

21

Graviton dominance — high energy behavior even worse than in QCD.

Multiple Pomeron exchange (higher genus) goes like

Keep Nc finite and study the onset of unitarity corrections. We shall be interested in the regime

Not suppressed when resummation

Beyond the single Pomeron exchange

212

1 sN

Tc

2~ cNs

but to comply with unitarity1T

ncn

s Nssg 22 ~)(

eNs c 2~

The real part strikes back

In DIS, typically 1~

'

Q

r

r

The saddle point in the j—integration

s

rrjsaddle 2

222

ln8

'ln22

0j

Resurrection of massless gravitons

j

2 4 60j

j

2 4 60j

small large 2Q 2Q

graviton propagator in AdS

Eikonal approximation

The real part much bigger than the imaginary part (unlike in QCD ! )Typical in theories with gravity

Large real part resummed by the eikonal approximation `t Hooft, `87 Amati, Ciafaloni, Veneziano `87Can be extended to AdS Cornalba, Costa, Penedones, Schiappa `06 Brower, Strassler, Tan `07

Valid when the impact parameter is large. This is the case in DIS at large !

Impact parameter

2Q

b

)()( biTebT

Diffractive dissociation

flat space 68)(

b

set

skdbT bik

AdS 2)(

r

srT

62

2

2

22 )(

b

s

b

lbT

bl ss

22

22 1

)(r

srT

rGl rr

s

Dominant source of inelasticity from cuts between pomerons.

Amati, et al, `87

Tidal force stretches a string,causing excitation.

Only mildly suppressed,comparable to the elastic phase

Giddings, `06

`Phase diagram’ of QCD

2lnQ

s

xxQs

9.42 1)(

x

1ln Saturation

BF

KL

DGLAP

Phase diagram at strong coupling

xY 1ln

Jets in QCD

ee

Average angular distributionreflecting fermionic degrees of freedom (quarks)

2cos1

Observation of jets in `75 marks one of the most striking confirmations of QCD and the parton picture

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

Timelike anomalous dimension in QCD

Lowest order perturbation

Soft singularity

~x

11

)(

j

j sT

!!)1( T

ResummationAngle-ordering

)1(

8)1(

4

1)( 2 j

Njj s

T

Basseto, Ciafaloni, Marchesini, ‘80 Mueller, `81

Inclusive spectrum

largel-x small-x

),( 2QxxDdx

dxT

Roughly an inverse Gaussian in peaked at

Q

x

1

‘Hump-backed’ shape consistent with pQCD with coherence.

Mueller,Dokshitzer, Fadin, Khoze

x1ln

e+e- annihilation in strongly coupled N=4 SYM

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

arXiv:0803.1467 [hep-th]arXiv:0803.2481 [hep-th]arXiv:0804.4733 [hep-th]arXiv:0807.0098 [hep-ph]

022 Qq

5D Maxwell equation

Want to compute at strong coupling )( jT

But there is no corresponding operator in gauge theory…

DIS vs. e+e- : crossing symmetry

P

e 022 Qq

e

022 Qq

e

Bjorken variable Feynman variable

P

qP

QBx

2

2

2

2

Q

qPFx

Parton distribution function Fragmentation function

),( 2QxD BS ),( 2QxD FT

crossing

Reciprocity respecting evolution

),()(),(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

is consistent with the Gribov-Lipatov reciprocity)( jf

Average multiplicity at strong coupling

)(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

1Q QEx 2

Qx

FQQxDT22 ),( Rapidly decaying

for Qx

21)(

jjT in the supergravity limit

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

Q

Energy correlation functionHofman, Maldacena `08

Energy distribution is spherical for anyCorrelations disappear as

1)()3( OS

241 weak coupling

strong coupling

1

2

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

Q

Thermal hadron spectrum

Identified particle yields are well described by a thermal model

T

MM

N

N **

exp

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

Becattini,Chliapnikov,Braun-Munzinger et al.

The origin of the exponential law is not understood, though there are many speculations (QGP? Hawking radiation?) around…

A statistical modelBjorken and Brodsky, ‘70

Cross section to produce exactly n ‘pions’

Assume

Then Strikingly similar to theAdS/CFT predictions !

Total cross section given by the one-loop result !

Qn

n

nc

tot Q

Ne

2

24

32

Computing the matrix element

5D scalars5D photon

AdS metric

string amplitude

)()( )1(1 QzHNzA c

QnSince is large, do the saddle point approximation in the z-integral

)exp(iQz~

2.0c

The saddle point has an imaginary part.

This leads to

Probably absent once the decay width is Included in the propagators. Einhorn, ‘77

5.0c

for ‘mesons’

for `partons’

2

1

nV *)( 2

cNO )1(O

*zz

Summary

DIS and e+e- accessible at strong coupling The amplitude is dominantly real, final states mo

stly diffractive (unlike in QCD…) There are no jets, multiplicity rises linearly with

energy (unlike in QCD…) The multiplicity ratio exhibits ‘‘thermal’’ behavior

(like in QCD?)