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.