Relating e+e- annihilation to 　 high energy scattering at 　 weak and strong coupling

Yoshitaka Hatta(U. Tsukuba)

JHEP 11 (2008) 057; arXiv:0810.0889 [hep-ph]

Outline

Jets in QCD Jets and BFKL? e+e- annihilation in AdS/CFT Soft gluons away from jets

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 gives the average multiplicity

)1(221

0),(

TQnQxDdx T

022 Qq

Timelike anomalous dimension

Lowest order perturbation

Soft singularity

~x

11

)(

j

j sT

!!)1( T

Resummationangle-ordering

)1(

8)1(

4

1)( 2 j

Njj s

T

2

)1( sT

N

Inclusive spectrum

largel-x small-x

x1ln

),(~ 2QxxDdx

dxT

Structure of jets well understood in pQCD

DLA + QCD coherence

Away-from-jets region

Gluons emitted at large angle, insensitive to the collinear singularity

Resum only the soft logarithms

ns x1ln

Marchesini-Mueller equationDifferential probability for the soft gluon emission

kbpap

large Nc

kbpap

)1ln( xY

BFKL equation

large Nc

x

yz

x

yz

Differential probability for the dipole splitting22

22

)()(

)(

yzzx

yxzd

ddP s

x

BFKL dynamics in jets

The two equations become formally identical after the small angle approximation

The interjet soft gluon number grows like the BFKL Pomeron !

Question : Is this just a coincidence, or is there any deep relationship between the two processes ?

The AdS/CFT correspondenceMaldacena `98

N=4 SYM at strong coupling is dual to weakly coupled type IIB superstring theory on

CYMNg 2

55 SAdS

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

2'4 RCN1 sg

CFT string

Gauge theory correlators calculable from string theory

e+e- annihilation in AdS/CFTHofman & Maldacena, 0803.1467; YH, Iancu & Mueller, 0803.2481; YH & Matsuo, 0804.4733, 0807.0098; YH, 0810.0889.

)( x

Calorimeters here

Properties at strong coupling

Energy distribution spherical, there are no jets !

In a sense, the entire solid angle is like an interjet region…

Branching is so fast and efficient. The total multiplicity grows linearlywith the energy

231)1(2)( QQQn T YH, Iancu & MuellerYH & Matsuo

Hofman & Maldacena

The inclusive spectrum is ‘thermal’

YH & Matsuo

The Poincare coordinates

Introduce two Poincare coordinate systems

Poincare 1 :

Poincare 2 :

Our universe

5AdS as a hypersurface in 6D

Related via a conformal transformation on the boundary

Shock wave picture of e+e- annihilation

Treat the photon as a shock wave in Poincare 2,solve the 5D Einstein equation

Energy density on the boundary ),(1

lim)( 525

0

)4(

5

yygy

yTy

Want to compute the total energy flow

Ty

Use the stereographic map to find the distribution on a sphere

Shock wave picture of a high energy hadron

A color singlet state lives in the bulk. At high energy, it is a shock wave in Poincare 1.

Energy distribution on the boundary transverse plane

Gubser, Pufu & Yarom, `08

The stereographic map

x

High energy, Regge

e+e- annihilation

Exact relationship between the final state in e+e- annihilationand the high energy hadronic Wavefunction !

Revisit the weak coupling problem

The same stereographic map transforms BFKL into the Marchesini-Mueller equation

Interjet gluon angular distribution

Exact solution to the BFKL equation known. Due to conformal symmetry, it is a function only of the anharmonic ratio.

Angular distribution of soft gluonsac

Related to the BFKL anomalous dimension 21

The exact solution to the Marchesini-Mueller equation

Conclusions

Novel correspondence between the final state in e+e- annihilation and the small-x hadronic wavefunction.

At strong coupling, the correspondence is exact. At weak coupling, the correspondence is useful to stu

dy interjet observables. Energy correlation functions are also related.