Saturation Physics
Yuri KovchegovThe Ohio State University
Outline Breakdown of DGLAP evolution at small-
x and low Q2. What we expect to find at small-x in
nuclei: strong gluon fields, nonlinear dynamics, near-black cross sections.
What we have seen so far. How EIC can solve many of the existing
puzzles. Open theory questions.
Standard Approach: DGLAP Equation
xG (x 0.05)
xq (x 0.05)
Gluons and Quarks at Low-xGluons and Quarks at Low-xDistribution functions xq(x,Q2) and xG(x,Q2) rise steeply at low Bjorken x.
Gluons onlyGluons and Quarks
Is all this well-described by the standard DGLAP evolution?
Negative gluon distribution!
NLO global fitting based on leading twist DGLAP evolution leads to negative gluon distribution
MRST PDF’s have the same features
Does it mean that we have no gluons at x < 10-3 and Q=1 GeV? No!
Why does DGLAP fail? Indeed we know that at low Q2 the higher twist effects scaling as ~1/Q2 become important.
These higher twist corrections are enhanced at small-x:
For large nuclei there is also enhancement by the atomic number A:
xQ
1~
2
2
x
A
Q
3/1
2
2
~
What We Expect to See at Small-x in Nuclei
Imagine an UR nucleusor hadron with valencequarks and sea gluonsand quarks.
Nuclear/Hadronic Wave Function Nuclear/Hadronic Wave Function
Boost to the rest frame:
NBjBjcoh mxpxk
l1
~1
~1
~
for small enough xBj we get with R the nuclear radius. (e.g. for x=10-3 get lcoh=100 fm)
Rxm
lBjN
coh 2
1
Color Charge Density Color Charge Density
Small-x gluon “sees” the whole nucleus coherently
in the longitudinal direction! It “sees” many color charges which
form a net effective color charge Q = g (# charges)1/2, such
that Q2 = g2 #charges (random walk). Define color charge
density
such that for a large nucleus (A>>1)
Nuclear small-x wave function is perturbative!!!
3/1222
2 ~~charges#
AS
Ag
S
g
S
Q
1)(~ 223/122 SQCDQCD A
McLerranVenugopalan’93-’94
McLerran-Venugopalan ModelMcLerran-Venugopalan Model
As we have seen, the waveAs we have seen, the wave
function of a single nucleus hasfunction of a single nucleus has
many small-x quarks and gluonsmany small-x quarks and gluons
in it. in it.
In the transverse plane the nucleusIn the transverse plane the nucleus
is densely packed with gluons andis densely packed with gluons and
quarks.quarks.
Large occupation number Large occupation number Classical Field Classical Field
McLerran-Venugopalan ModelMcLerran-Venugopalan Model Leading gluon field is Leading gluon field is classical! classical! To find the classical gluon field To find the classical gluon field
AAμμ of the nucleus one has to solve the non-linear analogue of Maxwell of the nucleus one has to solve the non-linear analogue of Maxwell
equations – the equations – the Yang-Mills equationsYang-Mills equations, with the nucleus as, with the nucleus as
a source of color charge: a source of color charge:
JFD
Yu. K. ’96Yu. K. ’96
J. Jalilian-Marian et al, ‘96J. Jalilian-Marian et al, ‘96
Classical Gluon Field of a Nucleus Classical Gluon Field of a Nucleus Using the obtained classical
gluon field one can construct
corresponding gluon distribution
function 2( , ) ~ ( ) ( )A x k A k A k
Note a change in concept: instead of writing an evolution equation a la DGLAP, we can simply write down a closed expression for the distribution of gluons. The calculation is non-perturbative (classical).
Gluon field is A~1/g, which is what one would expect for a classical field: gluon fields are strong!
Classical Gluon DistributionClassical Gluon Distribution
A good object to plot is
the gluon distribution
multiplied by the phase
space kT:
Most gluons in the nuclear wave function have transversemomentum of the order of kT ~ QS and We have a small coupling description of the whole wave function in the classical approximation.
3/12 ~ AQS
ATk
BFKL EquationBFKL Equation
The BFKL equation for the number of partons N reads:
),(),()/1ln(
22 QxNKQxNx BFKLS
Balitsky, Fadin, Kuraev, Lipatov ‘78
The powers of the parameter The powers of the parameter ln s ln s withoutwithout multiple rescatterings are multiple rescatterings are
resummed by the BFKL equation. Start with N particles in the proton’sresummed by the BFKL equation. Start with N particles in the proton’s
wave function. As we increase the energy a new particle can be emitted bywave function. As we increase the energy a new particle can be emitted by
either one of the N particles. The number of newly emitted particles iseither one of the N particles. The number of newly emitted particles is
proportional to N. proportional to N.
As energy increases BFKL evolution produces more partons, As energy increases BFKL evolution produces more partons, roughly of the same size. The partons overlap each other creating roughly of the same size. The partons overlap each other creating areas of very high density.areas of very high density.
Number density of partons, along with corresponding cross Number density of partons, along with corresponding cross sections grows as a power of energysections grows as a power of energy
But can parton densities rise forever? Can gluon fields be infinitely But can parton densities rise forever? Can gluon fields be infinitely strong? Can the cross sections rise forever?strong? Can the cross sections rise forever?
No! There exists a black disk limit for cross sections, which we No! There exists a black disk limit for cross sections, which we know from Quantum Mechanics: for a scattering on a disk of know from Quantum Mechanics: for a scattering on a disk of radius R the total cross section is bounded byradius R the total cross section is bounded by
BFKL Equation as a High Density MachineBFKL Equation as a High Density Machine
sN ~22 Rtotal
Nonlinear EquationNonlinear Equation
2222
)],([),()/1ln(
),(kxNkxNK
x
kxNsBFKLs
I. Balitsky ’96 (effective lagrangian)Yu. K. ’99 (large NC QCD)
At very high energy parton recombination becomes important. Partons not only split into more partons, but also recombine. Recombination reduces the number of partons in the wave function.
Number of parton pairs ~ 2N
Nonlinear Equation: SaturationNonlinear Equation: Saturation
Gluon recombination tries to reduce the number of gluons in the wave function. At very high energy recombination begins to compensate gluon splitting. Gluon density reaches a limit and does not grow anymore. So do total DIS cross sections. Unitarity is restored!
Black DiskBlack Disk
LimitLimit
Nonlinear Evolution at WorkNonlinear Evolution at Work
First partons are producedFirst partons are produced
overlapping each other, all of themoverlapping each other, all of them
about the same size.about the same size.
When some critical density isWhen some critical density is
reached no more partons of given reached no more partons of given
size can fit in the wave function.size can fit in the wave function.
The proton starts producing smaller The proton starts producing smaller
partons to fit them in.partons to fit them in.
Color Color Glass Glass CondensateCondensate
ProtonProton
Game Plan of High Energy QCDGame Plan of High Energy QCDSaturation physicsSaturation physics allows us allows us
to study regions of high to study regions of high
parton density in the parton density in the small small
coupling regimecoupling regime, where , where
calculations are still calculations are still
under control!under control!
Transition to saturation region isTransition to saturation region is
characterized by the characterized by the saturation scalesaturation scale
(or pT2)
The Little That We (Probably) Know Already
How to Calculate Observables
Start by finding the classical field of the McLerran-Venugopalan model.
Continue by including the quantum corrections of the BK/JIMWLK evolution equations.
Dipole Models in DISDipole Models in DISThe DIS process in the rest frame of the target is shown below.It factorizes into
))/1ln(,(),( *2*Bj
qqBj
Atot xYxNQx
QCD dynamics is all in N.
HERA DIS Results
Most of HERA DIS data iswell-described by dipole models based on CGC/saturation physics. This is particularly true inthe low-x low-Q region, where DGLAP-based pdf’sfail.
from Gotsman, Levin, Lublinsky, Maor ‘02
To find the gluon productionTo find the gluon production
cross section in pA onecross section in pA one
has to solve the samehas to solve the same
classical Yang-Millsclassical Yang-Mills
equationsequations
for two sources – proton and for two sources – proton and
nucleus. nucleus.
Gluon Production in Proton-Nucleus Gluon Production in Proton-Nucleus Collisions (pA): Classical FieldCollisions (pA): Classical Field
JFD
Yu. K., A.H. Mueller in ‘98Yu. K., A.H. Mueller in ‘98
Gluon Production in pA: Classical FieldGluon Production in pA: Classical Field
kdd
EA
kdd
ER pp
pA
pA
3
3
To understand how the gluon To understand how the gluon
production in pA is different fromproduction in pA is different from
independent superposition ofindependent superposition of
AA proton-proton (pp) collisions proton-proton (pp) collisions
one constructs the quantityone constructs the quantity
which is = 1 for independent which is = 1 for independent
superposition of sub-collisions. superposition of sub-collisions.
EnhancementEnhancement
(Cronin Effect)(Cronin Effect)
The quantity RThe quantity RpA pA plotted for theplotted for the
classical solution.classical solution.
Nucleus pushes gluons to higher transverse momentum!Nucleus pushes gluons to higher transverse momentum!
Gluon Production in pA: BK EvolutionGluon Production in pA: BK Evolution
Including quantum Including quantum
corrections to gluon corrections to gluon
production cross sectionproduction cross section
in pA using BK/JIMWLK in pA using BK/JIMWLK
evolution equations evolution equations
introduces introduces
suppression in Rsuppression in RpA pA withwith
increasing energy!increasing energy!
RpA
k / QS
EnergyEnergy
IncreasesIncreases
The plot is from D. Kharzeev, Yu. K., K. Tuchin ’03The plot is from D. Kharzeev, Yu. K., K. Tuchin ’03
(see also Kharzeev, Levin, McLerran, ’02 – original prediction,(see also Kharzeev, Levin, McLerran, ’02 – original prediction,
Albacete, Armesto, Kovner, Salgado, Wiedemann, ’03)Albacete, Armesto, Kovner, Salgado, Wiedemann, ’03)
RdAu at different rapidities
Most recent data from BRAHMS Collaboration nucl-ex/0403005
RdAu
RCP – centralto peripheralratio
Our prediction of suppression was confirmed!(indeed quarks have to be included too to describe the data)
What We May Know Already Saturation/CGC effects appear to
manifest themselves at x~10-3 and pT up to 3.5 GeV for gold nuclei at RHIC via breakdown of naïve factorization.
Saturation-based dipole models are hugely successful in describing HERA data, especially in the low-x low-Q region where DGLAP-based pdf’s fail.
eRHIC is almost certainly going to probe deep into the saturation region.
EM probes would be more convincing: no fragmentation effects there.
How EIC can solve many of the puzzles
EIC EIC/eRHIC would produce dedicated data
on nuclear structure functions and would allow one to answer many questions in small-x physics.
DIS on a nucleus with atomic number A would allow to test the physics equivalent to that of DIS on the proton at
xproton =xnucleus /A.
This is a much lower effective x!
3/12 ~
x
AQS
eA Landscape and a new Electron Ion ColliderThe x, Q2 plane looks well mapped out – doesn’t it?
Except for ℓ+A (A)
many of those with small A and very low statistics
Electron Ion Collider (EIC):
Ee = 10 GeV (20 GeV)
EA = 100 GeV
seN = 63 GeV (90 GeV)
High LeAu ~ 6·1030 cm-2 s-1
Terra incognita: small-x, Q Qs
high-x, large Q2
What Happens to F2 at Small-x?
?
EIC
EIC/eRHIC would allow us to map out the high energy behavior of QCD!
Open Theoretical Questions
Pomeron Loops
Important deep inside the saturation region for nuclei:
Resummation is still an open problem!
Here Be Ploops?
Higher Order Corrections To have the predictions of BK/JIMWLK
under control one needs to understand higher order corrections.
Recently there has been some progress on running coupling corrections.
NLO is still an open question.
2222
)],([),()/1ln(
),(kxNkxNK
x
kxNsBFKLs
(???)S
Marriage with DGLAP
Can we make BK/JIMWLK equations match smoothly onto DGLAP to have the right physics at very large Q2 ? Still an open problem.
Conclusions: Big Questions
What is the nature of glue at high density? How do strong fields appear in hadronic or
nuclear wave functions at high energies? What are the appropriate degrees of
freedom? How do they respond to external probes or
scattering? Is this response universal (ep,pp,eA,pA,AA)?An Electron Ion Collider (EIC) can provide definitive
answers to these questions.
Backup Slides
Discontinuity in x-dependence
The problem with theDGLAP fit can also beseen in x-dependence.
Define
q
G
xQxxq
xQxxG
xQxF
sea
~),(
~),(
~),(
2
2
22
and plot as a function of Q.
Gluons appear to have a problem at low Q in this DGLAP fit!
McLerran-Venugopalan ModelMcLerran-Venugopalan Model The density of partons in the nucleus (number of partons perThe density of partons in the nucleus (number of partons per
unit transverse area) is given by theunit transverse area) is given by the scale scale 2 2 ~ A~ ARR22 This scale is large, This scale is large, >> >> ΛΛQCDQCD , so that the strong coupling , so that the strong coupling
constant is small, constant is small, ααSS ( () << 1) << 1. .
Leading gluon field is Leading gluon field is classical! classical! To find the classical gluon field To find the classical gluon field
AAμμ of the nucleus one has to solve the of the nucleus one has to solve the Yang-Mills equationsYang-Mills equations, ,
with the nucleus as a source with the nucleus as a source
of color charge: of color charge:
JFD
Yu. K. ’96Yu. K. ’96
J. Jalilian-Marian et al, ‘96J. Jalilian-Marian et al, ‘96
DIS in the Classical ApproximationDIS in the Classical Approximation
The DIS process in the rest frame of the target is shown below.It factorizes into
))/1ln(,(),( *2*Bj
qqBj
Atot xYxNQx
with rapidity Y=ln(1/x)
DIS in the Classical ApproximationDIS in the Classical Approximation
The dipole-nucleus amplitude can be plotted:
x
QxYxN S 1
ln4
exp1),(22
1/QS
Colortransparency
Black disklimit,
22tot R
DIS in the Classical ApproximationDIS in the Classical Approximation
The dipole-nucleus forward scattering amplitude is
x
QxYxN S 1
ln4
exp1),(22
A.H. Mueller, ‘90
For DIS, the size of the dipole such that the aboveamplitude resums powers of
These are the promised A-enhanced higher twists.
3/12
2
2
2
~ AQQ
QS
Qx
1~
Quantum EvolutionQuantum Evolution
As energy increases
the higher Fock states
including gluons on top
of the quark-antiquark
pair become important.
They generate a
cascade of gluons.
These extra gluons bring in powers of S ln s, such thatwhen S << 1 and ln s >>1 this parameter is S ln s ~ 1.
Conclusions There is a lot of interesting
nonlinear phenomena that theorists expect at small-x.
We are in the position to test many of the predictions experimentally, mapping out the new and unexplored region of QCD.
BFKL EquationBFKL EquationIn the conventional Feynman diagram picture the BFKL equation can be represented by a ladder graph shown here. Each rung ofthe ladder brings in a power of ln s.
The resulting dipole amplitudegrows as a power of energy
sN ~violating Froissart unitarity bound
sconsttot2ln
How can we fix the problem?Let’s first resum the cascadeof gluons shown before.
Color Glass Picture of Heavy Ion Color Glass Picture of Heavy Ion CollisionsCollisions
T. Ludlam and L. McLerran, Physics Today, Oct. ‘03T. Ludlam and L. McLerran, Physics Today, Oct. ‘03
For more info on the roleof Color Glass in heavy ion collisions see thetalks by Dima Kharzeevand Brian Cole
Resumming Gluonic CascadeResumming Gluonic Cascade
In the large-NIn the large-NC C limit oflimit of
QCD the gluon correctionsQCD the gluon corrections
become color dipoles. become color dipoles.
Gluon cascade becomes Gluon cascade becomes
a dipole cascade.a dipole cascade.
A. H. Mueller, ’93-’94A. H. Mueller, ’93-’94
We need to resumdipole cascade, with each finalstate dipoleinteracting withthe target. Yu. K. ‘99
NonlinearNonlinear EvolutionEvolution EquationEquation
)],,(),,(),,(),,(),,([
2
),,(
1220101220
212
202
201
22
210
YxxNYxxNYxxNYxxNYxxN
xx
xxd
N
Y
YxxN CS
Defining rapidity Y=ln s we can resum the dipole cascade
I. Balitsky, ’96, HE effective lagrangianYu. K., ’99, large NC QCD
Linear part is BFKL, quadratic term brings in damping
x
QxYxxN S 1
ln4
exp1)0,,(22
0110 initial condition
Energy Dependence of the Saturation ScaleEnergy Dependence of the Saturation Scale
Single BFKL ladder gives scatteringamplitude of the order
sk
N ~
Nonlinear saturation effects becomeimportant when N ~ N2 N ~ 1. Thishappens at sQk ST ~
Saturation scale grows with energy!
Typical partons in the wave function have kT ~ QS, so that theircharacteristic size is of the order r ~ 1/kT ~ 1/QS. Typical parton size decreases with energy!
Going Beyond Large NGoing Beyond Large NCC: JIMWLK: JIMWLK
To do calculations beyond the large-NTo do calculations beyond the large-NCC limit on has to use a functional limit on has to use a functional
integro-differential equation written by integro-differential equation written by Iancu, Jalilian-Marian, Kovner, Iancu, Jalilian-Marian, Kovner,
Leonidov, McLerran and Weigert (JIMWLK):Leonidov, McLerran and Weigert (JIMWLK):
21[ ( , )] [ ( )]
2 ( ) ( ) ( )S
ZZ u v Z u
Y u v u
where the functional Z can then be used for obtaining wave function-averaged observables (like Wilson loops for DIS):
[ ] [ ]
[ ]
D Z OO
D Z
A lot of progress on solvingJIMWLK on the lattice has recently been achieved byK. Rummukainen and H. Weigert
Di-lepton Production
from J. Jalilian-Marian, hep-ph/0402014
One should get suppression at forward rapidities at RHIC.
Here we plot Rp(d)A
integrated over kT, for both pA and dA,for y=2.2 as a functionof invariant mass M.
Direct Photon ProductionAgain, CGC prediction for direct photon RdA is suppression.
Jamal Jalilian-Marian, ‘04
EIC vs RHIC II vs LHC
RHIC II is likely to produce good data on EM probes (prompt photons, di-leptons) in the forward region, providing an independent check of the origin of the observed forward suppression of hadrons.
Di-lepton ProductionThe suppression at forward rapidities at RHIC can also be viewed as a function of kT:
from Baier, Mueller, Schiff, hep-ph/0403201
RpA as a function of kT
for M=2GeV fory=1.5 (short-dashed)and y=3 (dashed),as well as for M=4GeVand y=3 (lower solid line).
Pomeron Loops
An example of the “fan” diagramincluded in BK/JIMWLK.
A diagram which is not included: a pomeron loop (ploop).
Rest frame of the hadron/nucleus Rest frame of the hadron/nucleus
Longitudinal coherence length (wavelength) of a gluon/ quark-antiquark pair is
such that for small enough xBj we get with R the nuclear radius. (e.g. for x=10-3 get lcoh=100 fm)
NBjBjcoh mxpxk
l1
~1
~1
~
Rxm
lBjN
coh 2
1
Conclusions: Big Questions
Can we understand the strong interactions at high energy?
Where do saturation effects come in? Is there a universality at the high energy
limit (ep vs eA vs pp vs pA vs AA)? What drives saturation? Which degrees
of freedom? Strong gluonic fields?
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