jets at weak and strong coupling - keio universityimportant probe of the ``sqgp”—strongly...
TRANSCRIPT
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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
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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
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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
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The stopping distance at strong coupling
Massless , or more generally, Q2 = 0
Q2T < ET2
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Arnold, Vaman Chesler, Ho, Rajagopal
Very small width !
…like in classical electrodynamics.
Conflict with quantum mechanics?
YH, Iancu, Mueller, Triantafyllopoulos
Wavepacket analysis Synchrotron radiation
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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.
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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
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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 !
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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.
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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
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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
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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
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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.
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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
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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
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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
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Marchesini-Mueller equation (2003)
k bpap
Evolution of the gluon distribution between jets.
Y = lnEjetk
Probability of gluon emission
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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
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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
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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.
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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
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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 !
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Exact map at weak coupling
The same stereographic map transforms BFKL into the Marchesini-Mueller equation
Exact solution to the Marchesini—Mueller equation
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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.
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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.