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Saturation Physics Yuri Kovchegov The Ohio State University

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Saturation Physics. Yuri Kovchegov The Ohio State University. Outline. Breakdown of DGLAP evolution at small-x and low Q 2 . 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. - PowerPoint PPT Presentation

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Page 1: Saturation Physics

Saturation Physics

Yuri KovchegovThe Ohio State University

Page 2: Saturation Physics

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.

Page 3: Saturation Physics

Standard Approach: DGLAP Equation

Page 4: Saturation Physics

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?

Page 5: Saturation Physics

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!

Page 6: Saturation Physics

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

~

Page 7: Saturation Physics

What We Expect to See at Small-x in Nuclei

Page 8: Saturation Physics

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

Page 9: Saturation Physics

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

Page 10: Saturation Physics

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

Page 11: Saturation Physics

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

Page 12: Saturation Physics

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!

Page 13: Saturation Physics

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

Page 14: Saturation Physics

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.

Page 15: Saturation Physics

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

Page 16: Saturation Physics

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

Page 17: Saturation Physics

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

Page 18: Saturation Physics

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

Page 19: Saturation Physics

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)

Page 20: Saturation Physics

The Little That We (Probably) Know Already

Page 21: Saturation Physics

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.

Page 22: Saturation Physics

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.

Page 23: Saturation Physics

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

Page 24: Saturation Physics

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

Page 25: Saturation Physics

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!

Page 26: Saturation Physics

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)

Page 27: Saturation Physics

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)

Page 28: Saturation Physics

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.

Page 29: Saturation Physics

How EIC can solve many of the puzzles

Page 30: Saturation Physics

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

Page 31: Saturation Physics

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

Page 32: Saturation Physics

What Happens to F2 at Small-x?

?

Page 33: Saturation Physics

EIC

EIC/eRHIC would allow us to map out the high energy behavior of QCD!

Page 34: Saturation Physics

Open Theoretical Questions

Page 35: Saturation Physics

Pomeron Loops

Important deep inside the saturation region for nuclei:

Resummation is still an open problem!

Here Be Ploops?

Page 36: Saturation Physics

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

Page 37: Saturation Physics

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.

Page 38: Saturation Physics

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.

Page 39: Saturation Physics

Backup Slides

Page 40: Saturation Physics

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!

Page 41: Saturation Physics

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

Page 42: Saturation Physics

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)

Page 43: Saturation Physics

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

Page 44: Saturation Physics

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~

Page 45: Saturation Physics

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.

Page 46: Saturation Physics

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.

Page 47: Saturation Physics

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.

Page 48: Saturation Physics

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

Page 49: Saturation Physics

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

Page 50: Saturation Physics

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

Page 51: Saturation Physics

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!

Page 52: Saturation Physics

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

Page 53: Saturation Physics

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.

Page 54: Saturation Physics

Direct Photon ProductionAgain, CGC prediction for direct photon RdA is suppression.

Jamal Jalilian-Marian, ‘04

Page 55: Saturation Physics

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.

Page 56: Saturation Physics

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).

Page 57: Saturation Physics

Pomeron Loops

An example of the “fan” diagramincluded in BK/JIMWLK.

A diagram which is not included: a pomeron loop (ploop).

Page 58: Saturation Physics

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

Page 59: Saturation Physics

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?