Transcript Jets in N=4 SYM

Jets in N=4 SYM from AdS/CFT
Yoshitaka Hatta
U. 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
dN AA
RAA 
N
d 2 pT dy
dN pp
coll d 2 p dy
T
Observed jet quenching of high-pt
hadrons is stronger than pQCD
predictions.
Strongly interacting matter at RHIC
dn
 1  2v2 ( pT ) cos 2  
d
Ideal hydro simulation works,
suggesting short mean free path.
low viscosity,
strong jet quenching
The Regge limit of QCD
Hadron-hadron total cross section
grows with energy (‘soft Pomeron’)
 tot  s 0.08
 ln 2 s
or
?
c.f., pQCD prediction
(BFKL, ‘hard Pomeron’)
 s 4 ln 2 s
Motivation
Why N=4 SYM?

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…
Why study jets ?

Possible applications to jet quenching at RHIC,
or in the `unparticle’ sector at the LHC? Strassler, arXiv:0801.0629 [hep-ph]

Lots of works on DIS. e^+e^- annihilation is a cross channel of DIS.
Deep inelastic scattering in QCD
Q2
e
Two independent kinematic variables

 Photon virtuality
X
P
 Bjorken-
x
q 2  Q2
Q2
Q2
x

2P  q mX2  m2p  Q2
Momentum fraction of a parton
DS ( x, Q2 )
: Parton distribution function
Count how many partons are there inside a proton.
DGLAP equation
1 dz

x 2
2
DS ( x, Q )   PS ( z ) DS  , Q 
x z
 ln Q 2
z

DIS vs. e^+e^- annihilation
P
e


q 2  Q2  0
q 2  Q2  0
P
e

e

crossing
Parton distribution function
DT ( x, Q2 )
DS ( x, Q2 )
Bjorken variable
Fragmentation function
Q2
x
2P  q
Feynman variable
x  2P  q
Q2
Jets in QCD
Observation of jets in `75 provides one of the
most striking confirmations of QCD
e

e
Average angular distribution 1 cos 
reflecting fermionic degrees of freedom (quarks)
2
Fragmentation function
P
Count how many hadrons are there inside a quark.
DT ( x, Q2 )
Feynman-x
q 2  Q2  0
e

e

x  2P  q  2E
Q2
Q
First moment
1

0
dx DT ( x, Q 2 )  n
average multiplicity
Second moment
1

0
dx xDT ( x, Q 2 )  2
energy conservation
Evolution equation
The fragmentation functions satisfy a DGLAP-type equation
1 dz

x 2
2
DT ( x, Q )   PT ( z ) DT  , Q 
2
x z
 ln Q
z

Take a Mellin transform
1
DT ( j, Q )   dx x j 1DT ( x, Q 2 )
2
0

2
2
D
(
j
,
Q
)


(
j
)
D
(
j
,
Q
)
T
T
T
2
 ln Q
Timelike anomalous dimension
n  DT (1, Q2 )  Q2 T (1)
(assume
 0
)
Anomalous dimension in QCD
Lowest order perturbation
1
~
x
Soft singularity
 T ( j) 
s
j 1
 T (1)   !!
Resummation
Angle-ordering

8 N s
1
2
 T ( j )   ( j  1) 
 ( j  1)
4


Mueller, `81
Inclusive spectrum
x d
 xDT ( x, Q 2 )
 dx
roughly an inverse Gaussian
peaked at
1
Q

x

 ln1 x
largel-x
small-x
N=4 Super Yang-Mills
SU(Nc) local gauge symmetry
 Conformal symmetry SO(4,2)   0
2
NC doesn’t run.
The ‘t Hooft coupling   gYM
 Global SU(4) R-symmetry
 choose a U(1) subgroup and gauge it.

Type IIB superstring


Consistent superstring theory in D=10
Supergravity sector admits the black 3brane solution which is asymptotically
AdS5  S
5
Our universe
S5
AdS `radius’
coordinate
The correspondence
Maldacena, `97



2
NC   and NC  
Take the limits   gYM
N=4 SYM at strong coupling is dual to weak
coupling type IIB on AdS5  S 5
Spectrums of the two theories match
CFT
(anomalous) dimension
`t Hooft parameter 
number of colors 1 N C
string
mass
curvature radius R 4  '2
string coupling constant g s
DIS at strong coupling
R-charge current excites metric fluctuations in the bulk,
which then scatters off a dilaton (`glueball’)
Polchinski, Strassler, `02
Y.H. Iancu, Mueller, `07
r
q 2  Q2  0
We are here
(r  )
Photon localized
at large r  Q

Dilaton localized
at small r  
Cut off the space at
r  r0
(mimic confinement)
e^+e^- annihilation at strong coupling
Hofman, Maldacena
Y.H., Iancu, Mueller,
Y.H. Matsuo
arXiv:0803.1467 [hep-th]
arXiv:0803.2481 [hep-th]
arXiv:0804.4733 [hep-th]
q 2  Q2  0
5D Maxwell equation
Dual to the 4D R-current
J  ( x) 
1
1
( 1  1  2  2 )  (6 D 5  5 D 6  4 D 3  3 D 4 )
2
2
A reciprocity relation
DGRAP equation

2
2
D
(
j
,
Q
)


(
j
)
D
(
j
,
Q
)
S /T
S /T
S /T
2
 ln Q
The two anomalous dimensions derive from a single function
Dokshitzer, Marchesini, Salam, ‘06
Confirmed up to three loops (!) in QCD
Application to AdS/CFT
Mitov, Moch, Vogt, `06
Basso, Korchemsky, ‘07
Assume this is valid at strong coupling and see where it leads to.
Anomalous dimension in N=4 SYM
Leading Regge trajectory

V j ~ r j  X  X 

Twist—two operators
j
2
lowest mass state for given j
lowest dimension operator for given j
  j  2  2 S ( j)
Gubser, Klebanov, Polyakov, `02
The `cusp’ anomalous dimension
 S ( j) 
j 1

2  ( j  j0 )
2 2
j2
Kotikov, Lipatov, Onishchenko, Velizhanin `05
Brower, Polchinski, Strassler, Tan, `06
Average multiplicity
j 1
 S ( j)  
2  ( j  j0 )
2 2
crossing
1
j2
 T ( j )    j  j0 
2
2 
n(Q)  Q2 T (1)  Q13 2
c.f. in perturbation theory,
c.f. heuristic argument
n(Q)  Q
n(Q)  Q




Y.H., Matsuo ‘08

2 2
Y.H., Iancu, Mueller ‘08
Jets at strong coupling?
 T ( j)  1 
j
2
in the supergravity limit

The inclusive distribution is peaked at the kinematic lower limit
 Qx 
DT ( x, Q 2 )  Q 2 F 




Rapidly decaying function
for x   Q
Q
Q
1
x  2E Q
At strong coupling, branching is so fast and efficient. There are no partons at large-x !
Energy correlation function
Hofman, Maldacena `08
Energy distribution is spherical for any 
Correlations disappear as   
 S (3)  O( )  1
  1 4 2
1
weak coupling
strong coupling
All the Q  particles have the minimal four momentum~  and
are spherically emitted. There are no jets at strong coupling !
2
Evolution of jets in a N=4 plasma
Y.H., Iancu, Mueller `08
Solve the Maxwell equation
in the background of Schwarzschild AdS_5
2
r 2  r04 
R2
2
2
2
2
ds  2 1  4 (dt  dx )  2
dr

R
d

5
R  r 
r (1  r04 r 4 )
2
r0  R2T
r 
Event horizon
To compute correlation functions :

A (t, x, r )  eit iqz A (r )
r  r0
 2  q 2  Q2  T 2
Time-dependent Schrödinger equation
To study time-evolution, add a weak t-dependence and keep only the 1st t-derivative

A (t, x, r )  eit iqz A (t, r )

2 r0
,
r
k

2T
K
Q
2T

Solutions available only piecewise.
Qualitative difference between
t=0
‘low energy’

Minkowski
boundary
2
horizon
Q  Qs  (T 2 )1 3
Qs
and
‘high energy’
Q  Qs
: plasma saturation momentum
low-energy, Q  Qs  (qT 2 )1 3
Early time diffusion
VA
solution with VA
This represents diffusion
up to time
t
q
| Q2 |
VB
VC
Gauge theory interpretation
t
q
| Q2 |
is the formation time of a parton pair
(a.k.a., the coherence time in the spacelike case)
IR/UV correspondence
2r0

 TL
r
c.f., general argument from pQCD
Farrar, Liu, Strikman, Frankfurt ‘88
L
1Q
2 13
low-energy Q  Qs  (qT )
Intermediate free streaming region
VA
solution with
VB
VB
constant (group-) velocity motion
Q
 t
q
VC
Gauge theory interpretation
Q
 1  vz2 is the transverse velocity
q
IR/UV correspondence
  TL
L  2vT t
1Q
L
Linear expansion of the pair
1 Q
T q
low-energy
Q  Qs  (qT 2 )1 3
Falling down the potential
VA
solution with
VB
VC
A classical particle with mass
falling down the potential
k
VC
Gauge theory interpretation
In-medium acceleration
1Q
1
T
1 Q
T q
start to `feel’ the plasma
| p  | T
disappear into the plasma
The high energy case Q  Qs  (qT )
2 13
No difference between the
timelike/spacelike cases
VA
VB
A new characteristic time
t
q
q

Qs2 Q 2
1 Qs
1
T
VC
Energy loss, meson screening length,
and all that
The scale L 
1 Q (1  v )

T q
T
2 14
z
breakup
is the meson screening length
Liu, Rajagopal, Wiedemann, `06
WKP solution after the breakup
features the trailing string solution
  expiq( z  vz t   (r ))
 (r ) 
 1  r  r0
r
 ln
 arctan 
2T  r  r0
r0 
Herzog, et al, ‘06, Gubser, ‘06
1Q
1 Q
T q
Energy loss, meson screening length,
and all that
Rate of enery flow towards the horizon
Identical to the motion of our wavepacket
1 Qs
tf
1
T
Time to reach the horizon
tf 
c.f., damping time of a gluon
q
13

q
Qs2
 q1 3
Gubser, Gulotta, Pufu, Rocha, 0803.1470 [hep-th]
Branching picture at strong coupling
Final state cannot be just
a pair of partons
Energy and virtuality
of partons in n-th generation
q
qn  n
2
Q
Qn  n
2
At strong coupling, branching is as fast
as allowed by the uncertainty principle
tn  tn1 
qn
n q

2
Qn2
Q2
Q(t )  1 t ,
n(Q)  Q
or
L(t )  t
Medium-induced branching at finite-T
qn
qn
tn  tn1  2  2
Qn
Qs (qn )
Mach cone?
dq (t )

 Qs2 (t )   (t )T 2
dt
13
where
q q

 
Qs  T 
time-dependent drag force
Conclusion

Various aspects of jets at strong coupling—
including the details of the final state—are
accessible from gauge/string duality
techniques.

Photon evolution problem sheds new light
on the physics of energy loss, etc. in a
strongly coupled plasma.