Jets at strong coupling

Yoshitaka Hatta

(U. Tsukuba)

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Contents

• Jets in QCD

• Jets in strongly coupled N=4 SYM

1.Fragmentation function

2.Relation to high energy scattering

Jets in QCD

In e+e- annihilation, the most stringent

tests of pQCD have been done.

High pt jets at the LHC could be an

important discovery channel of BSM

Jets in heavy-ion collisions

Observed jet quenching of high-pt hadrons seems to be stronger

than pQCD predictions.

Important probe of the ``sQGP”—strongly coupled quark gluon plasma

The stopping distance at strong coupling

Massless , or more generally, Q2 = 0

Q2T < ET2

Arnold, Vaman Chesler, Ho, Rajagopal

Very small width !

…like in classical electrodynamics.

Conflict with quantum mechanics?

YH, Iancu, Mueller, Triantafyllopoulos

Wavepacket analysis Synchrotron radiation

Jets in QCD:

The inclusive spectrum

ee

Q

Cross section to produce one hadron with energy plus anything else

Q

Ex 2

Feynman-x

),(1 2QxD

dx

dT

tot

Fragmentation function

E

E

Counts how many hadrons are there inside a quark.

DGLAP equation

),()(),(ln

22

2QjDjQjD

QTTT

Mellin transform in

Timelike anomalous dimension

)1(22),1( TQQDn T

21

2

2,)(),(

lnQ

z

xDzP

z

dzQxD

QTT

xT

First moment gives the average multiplicity

x),(),( 21

1

0

2 QxDxdxQjD Tj

T

Timelike anomalous dimension

Lowest order perturbation

Soft singularity

~ x

1

1~)(

jj sT

!!)1( T

Resummation angle-ordering

)1(

8)1(

4

1)( 2 j

Njj sT

2)1( sT

N Mueller (1981)

Nonsense !

Inclusive spectrum

largel-x small-x

),( 2QxxDdx

dxT

“hump-backed” distribution

peaked at

Q

x~

1

Double logs + QCD coherence. Structure of jets well understood in pQCD.

e+e- annihilation in N=4 SYM

at strong coupling

ds2 =dx¹dx¹ + dz

2

z2

Introduce a “photon” associated

with a U(1) subgroup of

the SU(4) R-symmetry.

AdS metric in the Poincare coordinates

calorimeters here

DIS vs. e+e- : crossing symmetry

P

e 022 Qq

e

022 Qq

e

Bjorken variable Feynman variable

P

qP

Q

Bx

2

2

2

2

Q

qP

Fx

Parton distribution function Fragmentation function

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

Gribov-Lipatov reciprocity

),()(),(ln

2

//

2

/2QjDjQjD

QTSTSTS

DGLAP equation

Dokshitzer, Marchesini, Salam

The two anomalous dimensions derive from a single function

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

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 )( YH, Iancu, Mueller

YH, Matsuo

Lipatov et al. Brower et al.

Jets at strong coupling?

The inclusive distribution is peaked at the kinematic lower limit

1 Q QEx 2

QxFQQxDT

22 ),( Decays faster than the power-law when

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 function

Hofman, Maldacena

Energy distribution is spherical for any Correlations 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

Away-from-jets region

Gluons emitted at large angle, insensitive to the collinear singularity

Resum only the soft logarithms

n

jet

sk

E

ln

There are two types of logarithms.

Sudakov logs. (emission from primary partons) Kidonakis, Oderda, Sterman

Non-global logs. (emission from secondary gluons) Dasgupta, Salam

k

Marchesini-Mueller equation (2003)

k bpap

Evolution of the gluon distribution between jets.

Y = lnEjetk

Probability of gluon emission

BFKL equation

Evolution of the gluons in the transverse plane

Y = ln Ek

Exact solution known

thanks to 2D conformal symmetry.

n » e4®s ln 2Y =µ1

x

¶4®s ln 2

~xk

Two Poincaré coordinates

Introduce two Poincaré coordinate systems

Poincaré 1 : Poincaré 2 :

5AdS as a hypersurface in 6D

Cornalba

①

②

ds2 =dx¹dx¹ + dz

2

z2

in Poincare coordinates 5AdS

Our universe

The sphere can be mapped onto the transverse plane of Poincare 2 via the stereographic projection

Shock wave picture of e+e- annihilation

Boundary energy from the holographic renormalization

Angular distribution of energy

Hofman, Maldacena

Treat the photon as a shock wave in Poincare 2

Y.H.

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

Exact map at strong coupling

Tx

High energy, Regge

e+e- annihilation

The two processes are mathematically identical.

The only difference is the choice of the coordinate systems in AdS !

Exact map at weak coupling

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

Exact solution to the Marchesini—Mueller equation

NLO evolution

The NLO BFKL in coordinate space has been calculated and shown

to be conformal in N=4 SYM. Balitsky & Chirilli

NLO Marchesini-Mueller equation in N=4 Avsar, YH, Matsuo.

Summary

• Heavy-ion collisions give us strong

motivations to study jets at strong coupling

• Jet structure very different between weak

coupling QCD and strong coupling N=4.

(Of course!)

• Still, certain features are universal.