Recent results on
saturation and CGC
Kazunori Itakura
Theory Group, KEK
JAPAN
Stained glass by M. Chagall (1964) at United Nations in New York
Outline
• Introduction/Motivation
Basic questions, Important experimental results
• High energy limit of QCD
Color Glass Condensate
• The Balitsky-Kovchegov Equation
A fresh look at the equation from the statistical physics
(The Logistic equation and the FKPP equation)
• Recent Progress in Phenomenology
Deuteron-Au collisions at RHIC, predictions for LHC
• Recent Progress in Theory
Beyond the BK equation
• Summary
Introduction/Motivation
Basic questions/problems which we want to answer/understand:
What is the “high energy limit” of QCD ?
If it indeed exists, …
- Is it different from the ordinary picture of
hadrons ?
- Is it already seen in experiments ?
- What is the evidence for it ?
- Can we treat it in weak-coupling techniques ?
as<<1, in scatt. with high Q2, or at high temperature/density
What is the information of nucleons relevant for
high energy scattering ?
instead of static information such as mass, radius at rest, etc
Important Experimental Results
Deep inelastic scattering (DIS) at HERA
 Steep rise of F2 (and gluon density) at small x
g*
1/Q
1/xP+
Q2 = qT2 : transverse resolution
x =p+/P+ : longitudinal mom. fraction
High density gluons appear at small x =“high energy scatt.”
Important Experimental Results
Hadronic cross section at high energy (total cross sec. for pp)
 ab  Z ab  B ln 2 ( s / s0 )  
Including cosmic ray data
of AKENO and Fly’s eye
S1/2
10
102
103
104GeV
Most recent PDG  consistent with ln2 s.
[COMPETE Collab.]
-- saturating unitarity (Froissart) bound
--
The coefficient B is universal (B=0.308mb) for pp, p p, p p, etc…..
High energy limit of QCD
Keys: many gluons, unitarity, universality
A universal form of matter at high energy
 Color Glass Condensate (CGC) !!
Gluons
have
“color”
created from “frozen” random
color source, that evolves slowly
compared to natural time scale
High density !
occupation number
~ 1/as at saturation
higher energy
Dilute gas
CGC: high density gluons
The Balitsky-Kovchegov equation
A basic equation for the CGC
[Balitsky, Kovchegov, Braun]
_

d 2z
( x  y) 2
 NY ( x, y)  NY ( x, z)  NY ( z, y)  NY ( x, z) NY ( z, y)
NY ( x, y)  a s 
2
2
Y
2p ( x  z) ( y  z )
An “evolution” equation, describing
the change of the dipole scattering amplitude NY(x,y) ~ gluon number
under the change of scattering energy √s (Y~ ln s : rapidity)
Derived from QCD by using
resummation w.r.t. (as ln s)n &
strong gluonic field in the target
A nonlinear differential equation, solved
numerically with/without impact parameter
in coordinate/momentum space
[Braun,Golec-Biernat,Motyka,Stasto,Marquet,Soyez]
analytically in some separate kinematical regimes
[Levin,Tuchin,Iancu,KI,McLerran,Mueller,Triantafyllopoulos,Kozlov]
Global energy dependence
Population dynamics
N(t) : (normalized) polulation density
When N<< 1
[Malthus 1798]
d
N (t )  N (t )
dt
t
 N (t )  N 0 e
population explosion
When N~1 [Verhulst 1838]
The Logistic equation
Gluon dynamics
NY: gluon density
The BFKL eq. [’75~]
Multiple gluon emission
NY  exp( PY ),  P  4a S ln 2
unitarity violation
The BK eq. [’99~]
Gluon recombination
d
N (t )   ( N (t )  N (t ) 2 )
nonlinear
dt
Exponential growth is tamed by the
nonlinear term  saturation !
Initial condition dependence disappears
at late time  universal !
stable
unstable
Time
(energy)
Reaction-diffusion dynamics
Munier & Peschanski (2003~)
With a reasonable approximation*, the BK equation in
momentum space is rewritten as the FKPP equation
(Fisher, Kolmogorov, Petrovsky, Piscounov)
where t ~ Y, x ~ ln k2 and u(t, x) ~ NY(k).
Well-understood in non-equilibrium statistical physics including directed percolation,
pattern formation, spreading of epidemics…
FKPP = “logistic” + “diffusion”
: “reaction” part, transition
from unstable to stable states
Diffusion : expansion of stable region
u=1: stable
Logistic

t
t’ > t
Traveling wave solution
u=0:unstable
*take the 2nd order expansion of the BFKL kernel around its saddle point
Saturation scale & Geometric scaling
Fact 1: For a “traveling wave” solution, one can define
the position of a “wave front” x(t) = v(t)t .
 x(t) ~ ln Qs2(Y) Saturation scale !
1/QS(Y) : transverse size of gluons when the
transverse plane of a target is filled by gluons.
“Boundary” btw dilute and saturated regimes
Precise form of QS(Y) determined
saturated
R
dilute
NLO BFKL : QS2 ( x)  (1 / x)   e Y ,
Fact 2: At late time, the shape of a traveling wave is
preserved, and the solution is only a function of x – vt.
 x - v(t)t ~ ln k2/Qs2(Y) Geometric scaling !!
Observed in HERA DIS at small x [Stasto,Golec-Biernat,Kwiecinski]
QS(Y) from the data consistent with theoretical results.
Geometric scaling approximately holds even outside of CGC!!
 “Scaling window”
[Iancu,KI,McLerran]
“Phase diagram”
1/x
in log scale
CGC
LQCD2
QS2(x) ~ 1/x: grows as x  0
as(QS2) << 1 weak coupling
Extended
scaling
regime
Non-perturbative (Regge)
Higher energies 
of a proton as seen in DIS
QS4(x)/LQCD2
Parton gas
Q2
in log scale
BFKL,
BK
DGLAP
Fine transverse resolution 
Recent progress in
phenomenology
Phenomenological applications
DIS at HERA
* [Golec-Biernat,Wusthoff, Bartels, Kowalski,Teaney] @[Iancu,KI,Munier]
• A cornerstone providing the most precise information about CGC
•
Golec-Biernat & Wusthoff model and its improvements* work relatively well.
• The CGC fit@ (based on the BK eq) works very well for F2 at small x.
QS2 (x) = (1 GeV)2 (x0/x) , x0=0.26 x 10-4, =0.25
Au-Au at RHIC

• CGC  initial condition for heavy ion collision; “seeds” of QGP
•
•
Most of produced particles have pt < 1 GeV ~ QS(RHIC)  bulk quantities
Multiplicities (centrality and rapidity dependences) [Kharzeev,Levin]
Hydro-dynamical calculation with CGC initial condition [Hirano,Nara]
Numerical studies [Krasnitz,Nara,Venugopalan,Lappi]
Deuteron-Au at RHIC
next slides
p-Pb and Pb-Pb at LHC (predictions)
High Energy Cosmic Rays [Jalilian-Marian,Dumitru,Drescher,Strikman]
Ideal situation for CGC with x ~ 10-9 or smaller
Effects of saturation examined for pA and n-p scattering
Deuteron-Au at RHIC
Going forward in p(d)-A collision corresponds
to probing nuclear wavefunction at smaller x
Nuclear modification factor (Brahms)
If RdAu=1, d-Au is just
1 dN d  Au / d 2 pt dh
RdAu 
a summation of pp
N coll dN p  p / d 2 pt dh
(up to iso-spin effect)
h(h-+h+)/2
d
q, g
x1 
pt h
e
s
x2 
pt h
e
s
g
Au
Cronin enhancement
at h=0,
suppression at h=3.2
Lots of studies in the CGC framework (see a review by Kovchegov & Jalilian-Marian)
• Qualitative behaviors consistent with predictions of CGC.
• Cronin peak  multiple Glauber-Mueller scattering (McL.-V. model)
• High pt suppression  due to mismatch between “evolution speeds”
of proton & nucleus. Nucleus grows only slowly due to saturation.
• Quantitative results also available
[Albacete,Armesto,Kovner,Salgado,Wiedemann,Gelis,Jalilian-Marian,Kharzeev,Kovchegov,Tuchin,
Accardi,Gyulassy,Levin,McLerran,Iancu,KI,Triantafyllopoulos,Venugopalan]
Deuteron-Au at RHIC
• Running coupling effects evaluated
[Iancu,KI,Triantafyllopoulos]
• DGLAP improvements on the projectile side (deuteron) necessary
Averaged x2 in 21 kinematics is ~ 10-3 for RHIC y~3 (private communication)
[Dumitru,Hayashigaki,Jalilian-Marian]  see talk by Jalilian-Marian
• Various observables show “suppression” due to saturation.
EM probes: dileptons, photons[Jalilian-Marian,Baier,Mueller,Shiff,Gay-Ducati,Betemps]
qqbar (meson) production [Blaizot,Fujii,Gelis,Venugopalan,Kharzeev,Tuchin]
 see talks by Fujii & Lappi
Jet azimuthal correlations disappear due to “mono-jet” production.
[Kharzeev,Levin,McLerran,Baier,Kovner,Nardi,Wiedemann]
Other approaches….
• Standard nuclear shadowing (NLO Leading Twist) with 22 process
[Vogt,Guzey,Strikman,Vogelsang]
• Parton recombination  identified particle dependence [Hwa,Yang,Fries]
• Re-scattering effects with factorization formalism of pQCD [Qiu,Vitev]
 Need more detailed investigation to be convinced …
Phase diagram with numbers
From the CGC fit
Qs2(x)~(10-4/x)0.25
proton
x in log
x A1/3 ~ 6
nucleus (A~200)
x2 in log
Extended
CGC Scaling
~BFKL
LHC
10-4
10-4 y=4
Parton gas
y=2
HERA
10-2
10-2
y=0
y=0
RHIC
100
103
Q2 in log
A1/3 x100
103
k2 in log
CGC at LHC
LHC
√sNN = 14 TeV for pp, 5.5 TeV for PbPb
For the same pt, Qs2(LHC) is increased by a factor of 3 than Qs2 (RHIC).
Qs2(LHC) ~ 3 -- 10 GeV2
mid forward
Number of gluons in the saturation regime increases.
Predictions
 Effects of saturation will be more visible!!
Multiplicities in PbPb and p-Pb
[Kharzeev,Levin,Nardi]
RpA (red dashed line)
[Kharzeev,Kovchegov,Tuchin]
Recent progress in
theory
Beyond the BK equation
Beyond the BK equation
The complete picture of high energy scattering in QCD will contain
Pomeron : 2 gluon exchange, C-even state
Odderon : 3 gluon exchange, C-odd state
Reggeon : quark-antiquark exchange,…..
and interaction among them
C-odd
C-even
projectile
Beyond BK
BK
The BK equation -- multiple exchange of P, and P-merging PPP
 Need to go beyond the BK equation !!
PPP
Fan diagram
Pomeron appears as a “collective” state of the “JIMWLK” Hamiltonian
which governs the small-x evolution. The same is true for the other
loop
exchanges. JIMWLK Hamiltonian correctly
PPPdescribes Odderon
P
P
O
In order to correctly describe the interaction among them, one needs to
modify JIMWLK Hamiltonian so that it contains “P-splitting” PPP .
This allows one to have Pomeron loops.
target
Beyond the BK equation
Small-x physics beyond the Kovchegov equation, Mueller and Shoshi, Nucl.Phys. B692 (2004) 175-208
Universal behavior of QCD amplitudes at high energy from general tools of statistical physics,
Iancu, Mueller, and Munier, Phys. Lett. B606 (2005) 342-350
A Langevin equation for high energy evolution with pomeron loops,
Iancu and Triantafyllopoulos, Nucl.Phys. A756 (2005) 419-467
Extension of the JIMWLK Equation in the Low Gluon Density Region
Mueller, Shoshi and Wong, Nucl.Phys. B715 (2005) 440-460
Non-linear QCD evolution with improved triple-pomeron vertices
Iancu and Triantafyllopoulos, Phys.Lett. B610 (2005) 253-261
In pursuit of Pomeron loops: the JIMWLK equation and the Wess-Zumino term
Kovner and Lublinsky, Phys.Rev. D71 (2005) 085004
From target to projectile and back again: selfduality of high energy evolution
Kovner and Lublinsky, Phys.Rev.Lett. 94 (2005) 181603
Duality and Pomeron effective theory for QCD at high energy and large Nc
Blaizot, Iancu, Itakura, Triantafyllopoulos, Phys.Lett. B615 (2005) 221-230
High energy amplitude in the dipole approach with Pomeron loops: asymptotic solution
Levin, hep-ph/0502243
Effective Hamiltonian for QCD evolution at high energy
Hatta, Iancu, McLerran, Stasto, Triantafyllopoulos, hep-ph/0504182, see also hep-ph/0505235
The high energy asymptotics of scattering processes in QCD
Enberg, Golec-Biernat, Munier, hep-ph/0505101
On the Projectile-Target Duality of the Color Glass Condensate in the Dipole Picture
Marquet, Mueller, Shoshi, Wong, hep-ph/0505229
Fluctuations effects in high-energy evolution of QCD, Soyez, hep-ph/0504129.
Perturbative Odderon in the Dipole Model, Kovchegov, Szymanowski, Wallon, Phys. Lett. B586 (2004) 267
Odderon in the Color Glass Condensate, Hatta, Iancu, Itakura, McLerran, hep-ph/0501171
A classical Odderon in QCD at high energies, Jeon and Venugopalan, Phys. Rev. D71 (2005) 125003
Summary
High enegy limit of QCD is the Color Glass Condensate
- high density gluonic matter which shows
 saturation of gluon distribution (non-linearity),
 unitarization of scattering amplitude,
 universal (insensitive to initial conditions)
 provides natural interpretation of geometric scaling
All of these are confirmed by the close analogy with
the FKPP equation for “reaction-diffusion dynamics”.
CGC can be compared with experiments
 small x data in DIS at HERA
 suppression of RpA in deuteron-Au at forward rapidity
Theoretical framework under re-construction:
new direction: BEYOND the BK equation
We are now approaching the complete description of high energy
scattering in QCD.
Thanks to
My collaborators (chronological)
Larry McLerran, Edmond Iancu, Elena Ferreiro,
Yuri Kovchegov, Derek Teaney, Stephen Munier,
Dionysis Triantafyllopoulos, Yoshitaka Hatta,
Jean-Paul Blaizot
My colleagues (possible future collaborators, alphabetical)
Adrian Dumitru, Rikard Enberg, Hiro Fujii,
Francois Gelis, Arata Hayashigaki, Tetsu Hirano,
Jamal Jalilian-Marian, Dmitri Kharzeev, Cyrille Marquet,
Al Mueller, Yasushi Nara, Robi Peschanski,
Gregory Soyez, Kirill Tuchin, Raju Venugopalan,
and all the people who are interested in CGG !!
Backup slides
Pomeron Loops
Necessary ingredient for the complete description of the
high energy limit of QCD
The BK equation describes
multiple exchange of BFKL Pomerons and “fan” diagrams (merging)
projectile
target
BUT, not the opposite “Pomeron splitting” diagrams
 asymmetric under the exchange btw projectile and target
Need to supply “Pomeron splitting”
to obtain a Lorentz inv. description !
 a new concept : duality btw proj. & target
 related to “fluctuation” (BK is the mean
field approximation)
Modification to BK (and JIMWLK) done:
 stochastic FKPP equation
Odderon
The BK eq. is for the hard Pomeron = two reggeized gluon exchange
even under the charge conjugation.
Perturbative QCD  “hard” Odderon
3 reggeized gluon exchange in C-odd
state, obeys the BKP equation
[Bartels, Kwiecinski-Praszalowicz]
Recent progress
C-even
C-odd
New description of Odderon in CGC
[Kovchegov,Szymanovsky,Wallon,Hatta,Iancu,KI,McLerran,Jeon,Venugopalan]
• Can define relevant C-odd operators for dipole-CGC & 3quark-CGC scatt.
• Reproduce the BKP equation in the linear regime
• In the dipole-CGC scattering, nonlinear effects kills the Odderon.
A big step towards the description of n-reggeized gluon exchange !!
Geometric scaling
Observed in HERA DIS at small x and moderate Q2
[Stasto,Kwiecinski,Golec-Biernat]
The saturation scale from the data is
consistent with the theoretical results
CGC
g*p
total
cross
section
Extended
Scaling
regime
QS2 ( x)  (1 / x)   e Y
Geometric scaling approximately
exists even outside of CGC!!
 “Scaling window”
Geometric scaling with fluctuation
Inclusion of Pomeron loops
 Stochastic FKPP equation [Iancu, Mueller, Munier]
Geometric scaling is strongly violated by the “fluctuation”
Numerical analysis
by R.Enberg et al.
 Geometric scaling is
still valid for not so small x
More about deuteron-Au @ RHIC
pt spectrum in the CGC
Kharzeev-Kovchegov-Tuchin
Jalilian-Marian
quark+gluon production
Valence quark
distribution
+ KKT param.
+ FF(LO,KKP)
+nonpert.Cronin
quark production
LO GRV98
for deuteron
+IIM param.
(the CGC fit)
+FF(LOKKP)
+K factor
Dumitru-HayashigakiJalilian-Marian
Quark + gluon production
DGLAP for deuteron
+ FF(LO KPP)
+ LO CTEQ5 with K factor
+ KKT param.
x- and DGLAP evolution