Jets in N=4 SYM from AdS/CFT Yoshitaka Hatta U. Tsukuba Y.H., E. Iancu, A. Mueller, arXiv:0803.2481...
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Transcript of Jets in N=4 SYM from AdS/CFT Yoshitaka Hatta U. Tsukuba Y.H., E. Iancu, A. Mueller, arXiv:0803.2481...
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
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
2
nn
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