color glass condensate and initial state effects in heavy

Color Glass Condensateand

Initial State Effects in

Heavy Ion Collisions

Javier L. AlbaceteIPhT CEA/Saclay

International School on Quark Gluon Plasma and Heavy Ion Collisions Torino, 7-12 March 2011

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

Heavy Ion Collisions: the quest for the Quark Gluon Plasma

x− x+General introduction

Introduction to QCD

Parton model

Gluon saturation

Color Glass Condensate

Phenomenology of saturation

CERN

François Gelis – 2007 Lecture I / IV – Advanced School on QGP, IIT, Mumbai, July 2007 - p. 3

Stages of a nucleus-nucleus collision

z

t

strong fields classical EOMs

gluons & quarks out of eq. kinetic theory

gluons & quarks in eq.hydrodynamics

hadrons in eq.

freeze out

! τ → +∞! Chemical freeze-out :density too small to have inelastic interactions

! Kinetic freeze-out :no more elastic interactions

Quark GluonPlasma

RHIC: Au (A=197) Cu (A=63), D(A=2). Collision energy 200 GeV per nucleon

LHC: Pb (A=207). Collision energy 2.75 TeV per nucleon

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

Heavy Ion Collisions: the quest for the Quark Gluon Plasma

x− x+General introduction

Introduction to QCD

Parton model

Gluon saturation



CERN


Stages of a nucleus-nucleus collision

z

t

strong fields classical EOMs

gluons & quarks out of eq. kinetic theory

gluons & quarks in eq.hydrodynamics

hadrons in eq.

freeze out

! τ → +∞! Chemical freeze-out :density too small to have inelastic interactions

! Kinetic freeze-out :no more elastic interactions

Quark GluonPlasma

RHIC: Au (A=197) Cu (A=63), D(A=2). Collision energy 200 GeV per nucleon

LHC: Pb (A=207). Collision energy 2.75 TeV per nucleon

THESE LECTURES˚

What shall we call “initial state efffects” (in these lectures)?

• At high energies nuclei are Lorentz contracted (colliding “pancakes”), ... so the gluon and quark fields of different nucleons will strongly overlap...

corresponding to the simple addition of its constituent nucleons is commonly refered

to as the EMC effect [17, 18].

Whether there is enhancement or suppression of the nuclear structure functions with

respect to those of the nucleon depends on the kinematical region of interest. The

general Bjorken-x dependence of such modification is as follows:

RNA

1

x

• RAN > 1 for x→ 1.

• RAN < 1 for 0.3 ! x ! 0.8.

• RAN > 1 for 0.1 ! x ! 0.25.

• RAN < 1 for x ! 0.05

At high energies, small-x, nuclear structure functions are suppressed with respect to

those in a nucleon. This phenomenon is known as nuclear shadowing, and its physical

interpretation depends strongly on the choice of the reference frame. In a frame in

which the nucleus is fast moving, the infinite momentum frame, the constituent nucle-

ons necessarily overlap due to Lorentz contraction and partons associated to different

nucleons can interact with each other, as shown in Figure 2, which can result in gluon

recombination.

A A

γ∗ γ∗

Ν Ν. . .

~ 1/(m x) >>RN Alc

v

R/ γR

IMF r. f. at rest

Figure 2: Picture of nuclear shadowing in the infinite momentum frame (left), and in

a reference system at rest (right).

12

γRHIC ∼ 100

γLHC ∼ 3000

... “anything” that is “different” with respect to proton-proton collisions (the baseline). Hints

Modification of nuclear wave functions: Shadowing, saturation...

What shall we call “initial state efffects” (in these lectures)?... “anything” that is “different” with respect to proton-proton collisions (the baseline). Hints

General introduction

Introduction to QCD

!QCD reminder

!Confinement

!How to test QCD?

!Factorization

Parton model

Gluon saturation



CERN


Multi-parton interactions?

" Collinear or kt-factorization : only one parton in eachprojectile take part in the process of interest

• In proton-proton collisions “only one parton per hadron” participates in the scattering process.

• At high energies nuclei are Lorentz contracted (colliding “pancakes”), ... so the gluon and quark fields of different nucleons will strongly overlap...






RNA

1

x

• RAN > 1 for x→ 1.

• RAN < 1 for 0.3 ! x ! 0.8.

• RAN > 1 for 0.1 ! x ! 0.25.

• RAN < 1 for x ! 0.05







recombination.

A A

γ∗ γ∗

Ν Ν. . .

~ 1/(m x) >>RN Alc

v

R/ γR

IMF r. f. at rest



12

γRHIC ∼ 100

γLHC ∼ 3000


What shall we call “initial state efffects” (in these lectures)?

At high energies nuclei are Lorentz contracted (colliding “pancakes”), ... so the gluon and quark fields of different nucleons will strongly overlap...






RNA

1

x

• RAN > 1 for x→ 1.

• RAN < 1 for 0.3 ! x ! 0.8.

• RAN > 1 for 0.1 ! x ! 0.25.

• RAN < 1 for x ! 0.05







recombination.

A A

γ∗ γ∗

Ν Ν. . .

~ 1/(m x) >>RN Alc

v

R/ γR

IMF r. f. at rest



12

γRHIC ∼ 100

γLHC ∼ 3000

... “anything” that is “different” with respect to proton-proton collisions (the baseline). Hints


• Is that a good approximation when colliding 2 nuclei??

François Gelis

CGC

Why small-x gluons matter


Factorization

Stages of AA collisions

Leading Order

Leading Logs

Glasma fields

Initial color fields

Link to the Lund model

Rapidity correlations

Matching to hydro

Glasma stress tensor

Glasma instabilities

Summary

6

Implications for a QCD approach

• Main difficulty: How to treat collisions involving a large

number of partons?

• Dense regime : multiparton processes become crucial

! new techniques are required

! multi-parton distributions are needed

Multi-parton interactionsBreak down of factorization theorems

✓ Lecture 1: Introduction, general ideas• The baseline: e+p and p+p collisions• pQCD evolution equations• Saturation: non-linear evolution equations

✓ Lecture II: The Color Glass Condensate effective• Classical methods. McLerran Venugopalan • Quantum evolution: BK and B-JIMWLK equations• Particle Production in dense environements

✓ Lecture III: Phenomenology, from RHIC to the LHC• e+p collisions• d+Au, A+A collisions• Challenges and open issues

Lectures Plan

Warning: Few (or none :)) complete equations (emphasis is on the physics idea)

Comments/questions: contact me at [email protected]

Check the bibliography!

mailto:[email protected]

mailto:[email protected]

INTRODUCTIONStrong Interactions ⇒ Quantum Chromodynamics


Introduction to QCD

!QCD reminder

!Confinement

!How to test QCD?

!Factorization

Parton model

Gluon saturation



CERN


Quarks and gluons

" Electromagnetic interaction : Quantum electrodynamics# Matter : electron , interaction carrier : photon

# Interaction :

∼ e (electric charge of the electron)

" Strong interaction : Quantum chromo-dynamics# Matter : quarks , interaction carriers : gluons

# Interactions :

a

i

j

∼ g (ta)ija

b

c

∼ g (T a)bc

# i, j : colors of the quarks (3 possible values)

# a, b, c : colors of the gluons (8 possible values)

# (ta)ij : 3 × 3 matrix , (T a)bc : 8 × 8 matrix


Introduction to QCD

!QCD reminder

!Confinement

!How to test QCD?

!Factorization

Parton model

Gluon saturation



CERN


Quarks and gluons


# Interaction :



# Interactions :

a

i

j

∼ g (ta)ija

b

c

∼ g (T a)bc





Introduction to QCD

!QCD reminder

!Confinement

!How to test QCD?

!Factorization

Parton model

Gluon saturation



CERN


Quarks and gluons


# Interaction :



# Interactions :

a

i

j

∼ g (ta)ija

b

c

∼ g (T a)bc




Dµ = ∂µ − i g Aµ

Fµν = ∂µAν − ∂νAµ − i g [Aµ, Aν ]

• QCD: Non-abelian gauge theory for quark (matter) and gluons (interaction carriers)• Gauge group: SU(Nc=3)

quarks

gluons Aµ,a →{

µ = 1, . . . 4 Lorentz indexa = 1 . . . N2

c − 1 = 8 Color index

qα, af →

α = 1, . . . 4 Lorentz indexa = 1 . . . Nc = 3 Color indexf = u, d, s, c, b, t Flavor index

LQCD = −12trFµνFµν − ψ̄ (i /D −m) ψ

Deeply inelastic electron-proton scattering.

kk′

q = (k− k′)

P

electron

Proton x =Q2

2 P · q≈ Q2

W 2

Q2 = −(k − k′)2

Lorentz invariants: Infinite momentum frame

Pµ = (P, 0, 0, P )

qµ = (q0, q⊥, 0)

✓ Parton distribution functions (pdf’s): Probability of finding a parton within the hadron carrying a fraction of longitudinal momentum x when observed at a resolution scale Q2

• Photon as a microscope: The virtual photon resolves the electromagnetic constituents of the proton over times

within a transverse area

∆τDIS ∼ 1fmxP

Q" P

m

1ΛQCD

∆ r⊥ ∼1Q" 1

ΛQCD

• The proton sub-structure varies with the resolution scales: At small-x the proton structure is dominated by gluons:

H1 and ZEUS

HER

A In

clus

ive

Wor

king

Gro

upA

ugus

t 201

0

x = 0.00005, i=21x = 0.00008, i=20

x = 0.00013, i=19x = 0.00020, i=18

x = 0.00032, i=17x = 0.0005, i=16

x = 0.0008, i=15x = 0.0013, i=14

x = 0.0020, i=13x = 0.0032, i=12

x = 0.005, i=11x = 0.008, i=10

x = 0.013, i=9x = 0.02, i=8

x = 0.032, i=7x = 0.05, i=6

x = 0.08, i=5x = 0.13, i=4

x = 0.18, i=3

x = 0.25, i=2

x = 0.40, i=1

x = 0.65, i=0

Q2/ GeV2

!r,

NC

(x,Q

2 ) x 2

i

+

HERA I+II NC e+p (prel.)Fixed TargetHERAPDF1.5

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10 6

10 7

1 10 10 2 10 3 10 4 10 5

gluons/20seaquarks

valencequarks

✓Results from HERA (collider)

Remember: High energy ~ small-x x =Q2

2 P · q≈ Q2

W 2

Can such strong growth of the gluon distribution continue for ever??

• Nuclear structure functions do not correspond to a mere superposition of nucleons:

✓Results from e+A scattering

SHADOWING: At small-x, the nuclear structure functions are supressed w.r.t that of a free nucleon: There are less partons at small-x

3.1 Introduction

x

A

2FR

0.1 0.3 0.8

1

shadowing

antishadowing

EMC

Fermi

motion

Figure 3.1: Illustration of the generic behavior of the nuclear ratioRAF2

= FA2 /F d

2as a function of x for a given fixed Q2 [143].

while others are based on DGLAP evolution of nuclear ratios of parton densitiesfAi (x,Q

2). es verdad en este contexto?

Paralleling the determination of proton PDFs, several global QCD analysesof nPDFs have been made within the last decade [162, 163, 164, 165, 166] basedon DGLAP evolution: nuclear ratios are parametrized at some value Q2

o ∼ 1÷ 2GeV2 which is assumed to be large enough for perturbative DGLAP evolutionto be applied. These initial parametrizations for every parton density have tocover the full x range 0 < x < 1. The nuclear size appears as an additionalvariable. Then these initial conditions are evolved through DGLAP equationstowards larger values of Q2 where there are experimental data. Comparing thedata and the calculation the initial parameters are adjusted.

Up until recently, these analyses were based solely on fixed-target nuclear DISand DY data. Compared to the data constraining proton PDFs, these are oflower precision and lie in a much more limited range of Q2 and x. Constraints onnuclear gluon distribution functions are particularly poor, since they cannot beobtained from the absolute values of DIS structure functions, but only from theirlogarithmic Q2-evolution, for which a wideQ2-range is mandatory. To improve onthis deficiency, recent global nPDF analysis [162, 163] have included for the firsttime data from inclusive high-pT hadron production in hadron-nucleus scatteringmeasured at RHIC [167, 168, 169].

However, in contrast to the theoretical basis for global analyses of protonPDFs, the separability of nuclear effects into process-independent nPDFs andprocess-dependent but A-independent hard processes is not established withinthe framework of collinear factorized QCD. In particular, some of the charac-teristic nuclear dependencies in hadron-nucleus collisions, such as the Cronineffect [49], may have a dynamical origin that cannot, or can only partly, be ab-

41

1A

FA2

FN2

F2A != AF2N

• Quantum fluctuations “bigger” and longer lived than the probe can be resolved during interaction

Motivations from RHIC Partons at strong coupling Phenomenology Conclusions Backup

Quenched QCD

... or in quenched QCD (no quark loops), where C (µ2) = 0 !

1/T

1/Q

RG

Measure the quark energy density in quenched lattice QCD...compare the result with the weak coupling expectation (SB)

If the difference is less than 30% =⇒ weak coupling

A reduction by a large factor ! 5 =⇒ strong coupling

Rencontres QGP–France 2009, Etretat, 15–18 Septembre AdS/CFT and Heavy Ion Collisions

1/Q0

Motivations from RHIC Partons at strong coupling Phenomenology Conclusions Backup

Quenched QCD

... or in quenched QCD (no quark loops), where C (µ2) = 0 !

1/T

1/Q

RG

Measure the quark energy density in quenched lattice QCD...compare the result with the weak coupling expectation (SB)

If the difference is less than 30% =⇒ weak coupling

A reduction by a large factor ! 5 =⇒ strong coupling

Rencontres QGP–France 2009, Etretat, 15–18 Septembre AdS/CFT and Heavy Ion Collisions

1/Q

Q>>Q0

RenormalizationGroup

• Renormalization Group QCD equations: while the ultimate origin of pdf’s is non-perturbative, their change with the resolution scales can be described by perturbative techniques

∆τDIS ∼ xP/Q Decreasing-x increases the chances of catching short-lived fluctuations

• Probability for a quark (or gluon) p to emit a small-x gluon (Light-cone perturbation theory):

φ(x,k⊥) =dNg

dxd2k⊥=

αs CF(A)

π

1x

1k2⊥

k⊥

p

Gluon distribution of a single quark:

k‖ = xp

xG(x,Q2) = x∫ Q2

d2k⊥φ(x,k⊥) ∼ αs CF

πln

(Q2/Λ2

QCD

)

soft collineardivergences:

• Multiple emissions generate large logarithmic corrections.

φ(x,k⊥) =dNg

dxd2k⊥=

αs CF(A)

π

1x

1k2⊥

In the limit of strong transverse momentum ordering

k⊥n" ! · · ·! k⊥2 ! k⊥1

(αs)n

∫ Q2

Q0

d2k⊥n

k2⊥n

. . .

∫ k⊥3

Q0

d2k⊥2

k2⊥2

∫ k⊥2

Q0

d2k⊥1

k2⊥1

∝ 1n!

(αs ln(Q2/Q2

0))n

it is possible to resum leading logarithmic contributions

and recast them in the form of an evolution equation for the pdf’s

∂ xG(x, Q2)∂ ln(Q2/Q2

0)=

Cαs

π

∫ 1

xdzPgg(z) xG

(x

z, Q2

)

p

(xn,k⊥n)

(x1,k⊥1)

(x2,k⊥2)

• DGLAP evolution towards large Q2

✓ DGLAP evolution towards large Q2

In the limit of strong transverse momentum ordering

(z1, k⊥1)

(z2, k⊥2)

(zn, k⊥n)k⊥n" ! · · ·! k⊥2 ! k⊥1

(αs)n

∫ Q2

Q0

d2k⊥n

k2⊥n

. . .

∫ k⊥3

Q0

d2k⊥2

k2⊥2

∫ k⊥2

Q0

d2k⊥1

k2⊥1

∝ 1n!

(αs ln(Q2/Q2

0))n


and recast them in the form of an evolution equation for the pdf’s

Pqq0(z)

q(y)

q(x)

g(y-x)

Pgq0(z)

q(y)

g(x)

q(y-x)

Pqg0(z)

g(y)

q(x)

q(y-x)

Pgg0(z)

g(y)

g(x)

g(y-x)

The splitting functions are calculated perturbatively as a power series in αs. Their

explicit expressions can be found in [1, 4].

The complete DGLAP evolution equations can be written in a compact matrix way

that explicitly shows how different components are mixed through evolution:

∂

∂ ln Q2

Σ(x, Q2)

g(x, Q2)

=αs(Q2)

2π

Pqq 2NfPqg

Pgq Pgg

⊗

Σ(y, Q2)

g(y, Q2)

, (18)

where Σ(x, Q2) = q(x, Q2) + q̄(x, Q2).

3.2 BFKL

At asymptotically large energies, it is believed that the theoretically correct descrip-

tion of the structure function of DIS processes is given by the Balitsky-Fadin-Kuraev-

Lipatov (BFKL) equation [20, 21]. It provides the evolution of hadron structure with

increasing center of mass energy of the virtual photon-hadron system, W 2, for a fixed

value of the photon virtuality, Q2. The high-energy limit in which it is formally derived

is defined by the conditions

W 2 →∞ , Q2 fixed,

x $ Q2

W 2 → 0 , Y = ln(1/x)→∞,(19)

where Y is the rapidity variable. In this limit of very small values of Bjorken-x, the

relevant degrees of freedom are gluons, and gluon radiation is the leading mechanism

for evolution. Contrary to DGLAP, the leading contribution for BFKL evolution comes

from diagrams in which the longitudinal momenta of the successively radiated gluons

are strongly ordered, so that each new gluon takes a very small fraction of the energy

of the propagating gluon,

x1 % x2 · · ·% xn. (20)

The transverse momenta of the gluons in the radiative cascade are no longer ordered,

contrary to the case for DGLAP evolution. Rather, they describe a random walk in kt-

space, which leads to a diffusion of the initial distribution to larger and smaller values

of kt.

16





∂

∂ ln Q2

Σ(x, Q2)

g(x, Q2)

=αs(Q2)

2π

Pqq 2NfPqg

Pgq Pgg

⊗

Σ(y, Q2)

g(y, Q2)

, (18)

where Σ(x, Q2) = q(x, Q2) + q̄(x, Q2).

3.2 BFKL







W 2 →∞ , Q2 fixed,

x $ Q2

W 2 → 0 , Y = ln(1/x)→∞,(19)







x1 % x2 · · ·% xn. (20)




of kt.

16

✓ High energy limit. BFKL evolution: Q2 fixed,

In the limit of strong longitudinal momentum ordering


BFKL evolution is non-local in transverse momentum:





∂

∂ ln Q2

Σ(x, Q2)

g(x, Q2)

=αs(Q2)

2π

Pqq 2NfPqg

Pgq Pgg

⊗

Σ(y, Q2)

g(y, Q2)

, (18)

where Σ(x, Q2) = q(x, Q2) + q̄(x, Q2).

3.2 BFKL







W 2 →∞ , Q2 fixed,

x $ Q2

W 2 → 0 , Y = ln(1/x)→∞,(19)







x1 % x2 · · ·% xn. (20)




of kt.

16

Under these conditions BFKL resums leading terms αs ln 1/x to all orders.

At leading logarithmic accuracy, the BFKL equation reads [28]:

∂φ(k, Y )

∂ ln(1/x)=

αsNc

π

∫d2q

(k − q)2

[φ(q, Y )

q2− φ(k, Y )

q2 + (k − q)2

]. (21)

The evolved object is the unintegrated gluon distribution function, φ(x, k). It gives

the probability of finding a gluon in the parent hadron with fraction of longitudinal

momentum x and a transverse momentum k. The unintegrated gluon density can be

related to the usual integrated one xG(x, Q2) by

xG(x, Q2) "∫ Q2

d2kφ(x, k). (22)

The sign " in the above equation indicates that there is not a strict equality between

integrated and unintegrated distributions, as neither are observables. Indeed, a precise

definition of them requires the use of light-front quantization [29].

3.3 Small-x solutions

In order to extract the small-x behaviour of the parton distributions from the DGLAP

equations, one has to consider the case where both logarithms, lnQ2 and ln 1/x, are

large. This approximation, the double logarithm approximation of DGLAP (DLA), is

valid in the kinematical region where both longitudinal and transverse momenta are

strongly ordered:

kt1 # kt2 · · ·# ktn # Q ! s, (23)

x1 $ x2 · · ·$ xn. (24)

The DLA solution for the gluon distribution for running coupling is [30,31]

xGDLA(x, Q2) ∼ exp

{(48

11− 23Nf

lnln Q2/Λ2

ln Q20/Λ

2ln 1/x

)1/2}

, (25)

showing a fast increase with decreasing x.

17

φ(x, kt) =dNg

d2k⊥ dYxG(x, Q2) =

∫ Q2

d2k⊥φ(x, kt)unintegrated gluon distributions:

p

(xn,k⊥n)

(x1,k⊥1)

(x2,k⊥2)

(αs)n

∫ 1

xn−1

dxn

xn. . .

∫ 1

x3

dx2

x2

∫ 1

x2

dx1

x1∼ 1

n!(αs ln (1/x))n

x= Q2

W20









∂

∂ ln Q2

Σ(x, Q2)

g(x, Q2)

=αs(Q2)

2π

Pqq 2NfPqg

Pgq Pgg

⊗

Σ(y, Q2)

g(y, Q2)

, (18)

where Σ(x, Q2) = q(x, Q2) + q̄(x, Q2).

3.2 BFKL







W 2 →∞ , Q2 fixed,

x $ Q2

W 2 → 0 , Y = ln(1/x)→∞,(19)







x1 % x2 · · ·% xn. (20)




of kt.

16



∂φ(k, Y )

∂ ln(1/x)=

αsNc

π

∫d2q

(k − q)2

[φ(q, Y )

q2− φ(k, Y )

q2 + (k − q)2

]. (21)





xG(x, Q2) "∫ Q2

d2kφ(x, k). (22)









strongly ordered:

kt1 # kt2 · · ·# ktn # Q ! s, (23)

x1 $ x2 · · ·$ xn. (24)



{(48

11− 23Nf

lnln Q2/Λ2

ln Q20/Λ

2ln 1/x

)1/2}

, (25)


17

p

(xn,k⊥n)

(x1,k⊥1)

(x2,k⊥2)

(αs)n

∫ 1

xn−1

dxn

xn. . .

∫ 1

x3

dx2

x2

∫ 1

x2

dx1

x1∼ 1

n!(αs ln (1/x))n

x= Q2

W20

Note: BFKL evolution does not require momentum ordering momentum: Gluons in the cascade are “of the same size”:

k⊥n ∼ · · · ∼ k⊥2 ∼ k⊥1 ∼ Q

In turn, the life times of successive gluons is shorter and shorter

∆ti ∼xip

k2i⊥

So each new gluon sees all the previous ones as frozen -> Emission is coherent!!









∂

∂ ln Q2

Σ(x, Q2)

g(x, Q2)

=αs(Q2)

2π

Pqq 2NfPqg

Pgq Pgg

⊗

Σ(y, Q2)

g(y, Q2)

, (18)

where Σ(x, Q2) = q(x, Q2) + q̄(x, Q2).

3.2 BFKL







W 2 →∞ , Q2 fixed,

x $ Q2

W 2 → 0 , Y = ln(1/x)→∞,(19)







x1 % x2 · · ·% xn. (20)




of kt.

16



∂φ(k, Y )

∂ ln(1/x)=

αsNc

π

∫d2q

(k − q)2

[φ(q, Y )

q2− φ(k, Y )

q2 + (k − q)2

]. (21)





xG(x, Q2) "∫ Q2

d2kφ(x, k). (22)









strongly ordered:

kt1 # kt2 · · ·# ktn # Q ! s, (23)

x1 $ x2 · · ·$ xn. (24)



{(48

11− 23Nf

lnln Q2/Λ2

ln Q20/Λ

2ln 1/x

)1/2}

, (25)


17

p

(xn,k⊥n)

(x1,k⊥1)

(x2,k⊥2)

(αs)n

∫ 1

xn−1

dxn

xn. . .

∫ 1

x3

dx2

x2

∫ 1

x2

dx1

x1∼ 1

n!(αs ln (1/x))n

x= Q2

W20

NON!PERTURBATIVE

DGLAP

BFKL

ln!QCD

Y=ln

ln

(1/x)

Q

✓ Linear vs non-linear evolution and saturation

•Both BFKL and DGLAP are linear evolution equations: The newly emitted partons act as sources of new partons, thus yielding an endless growth of the gluon distribution at small-x

In the case of BFKL with fixed coupling, for which the equation is derived, the gluon

distribution behaves in an even more singular way

xGBFKL(x, Q2) ∼ x−4Nc ln 2αsπ ∼ x−0.5, for αs ∼ 0.2. (26)

3.4 The small-x problem

As discussed in the previous section, both linear evolution schemes, DGLAP and BFKL,

predict a sharp rise of gluon densities at high energies (small-x). This singular behaviour

for the distribution of soft gluons poses many theoretical problems. The most important

one is that it leads to the violation of unitarity, an essential property of quantum field

theories.

The energy dependence of total hadron-hadron cross sections is constrained by the

Froissart bound [32], which establishes that they cannot grow faster than a logarithm

squared of the center of mass energy of the collision, s,

σhh(s) !1

m2π

ln2 s, (27)

where mπ is the pion mass.

This bound is obtained from unitarity, analiticity properties of the scattering amplitude

and from the short range nature of hadronic interactions. Despite the fact that the

Froissart bound is not directly applicable to DIS processes, the small-x behaviour of

the solutions of linear evolution equations indicate a stronger energy dependence than

allowed by this bound, thus violating unitarity.

The strong growth of gluon distributions is rooted in the linearity of evolution equa-

tions. In them, only radiative processes that increase the number of partons in the

hadron wave function are taken into account, and they implicitly assume that the vir-

tual photon interacts with a single parton in the hadron. Such a picture is only valid if

the average distance between partons is larger than the resolution power of the probe

or, in other words, if the hadron is a dilute system and parton-parton interactions can

be safely neglected. However, due to the radiative processes that drive the evolution

to higher energies, more and more partons add to the hadron wave function and a

18

e.g:

dNg

dY∼ K ⊗Ng Ng = N0 exp [K Y ]


•Both BFKL and DGLAP are linear evolution equations: The newly emitted partons act as sources of new partons, thus yielding an endless growth of the gluon distribution at small-x

Malthus equation for population growth. K ~ birth rateAn essay on the principle of population as it affects the future improvement of society. Reverend T R Malthus 1798

dNg

dY∼ K ⊗Ng

dNg

dY∼ K ⊗Ng − µN2

g

"The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race...”

µ~ mortality rate


• Both BFKL and DGLAP are linear evolution equations: The newly emitted partons act as sources of new partons, thus yielding an endless growth of the gluon distribution at small-x

• At some point, gluon densities become large and gluon recombination processes become also possible: They tame the strong growth of gluon densities towards small-x: Saturation

dNg

dY∼ K ⊗Ng − µN2

grecombination ~ gluon death


• Saturation scale:

• # gluons per unit area:

• gluon-gluon x-section:

ρ ∼ xG(x,Q2)πR2

p

σgg→g ∼ αs(Q2)Q2

• gluons will recombine if ρ · σgg→g ! 1 i.e for Q2 ! Q2s(x)

Q2s(x) ∼ αs xG(x,Q2

s)π R2

p

∼ x−0.3

Saturation criterium:





ρ ∼ xG(x,Q2)πR2

p


• gluons will recombine if ρ · σgg→g ! 1 i.e for Q2 ! Q2s(x)


s)π R2

p

∼ x−0.3

Saturation criterium:Qs ∼

1Rs

Rs

Rs

Rs

decreasing-xQ2 fixed





ρ ∼ xG(x,Q2)πR2

p


• gluons will recombine if ρ · σgg→g ! 1 i.e for


s)π R2

p

∼ x−0.3


Nuclear enhancement: large # of gluons, even at moderate energies

xGA(x, Q2) ∼ A xGp(x, Q2)

RA ∼ A1/3 Rp

Qs ∼1

Rs

Rs

Q2sA ∼ A1/3 Q2

sp

Rs

Rs

decreasing-x

Weak coupling methods can be used to describe the saturation domain if

Qs ! ΛQCD ⇒ αs(Qs)# 1

SHADOWING!





ρ ∼ xG(x,Q2)πR2

p


• gluons will recombine if ρ · σgg→g ! 1 i.e for


s)π R2

p

∼ x−0.3


Nuclear enhancement: large # of gluons, even at moderate energies

Qs ∼1

Rs

Rs

Rs

Rs

decreasing-x


Introduction to QCD

Parton model

Gluon saturation


!Degrees of freedom

!Deep Inelastic Scattering

!Energy dependence

!MV model


CERN


Initial condition - MV model

" The JIMWLK equation must be completed by an initialcondition, given at some moderate x0

" As with DGLAP, the problem of finding the initial condition isin general non-perturbative

" The McLerran-Venugopalan model is often used as an initialcondition at moderate x0 for a large nucleus :

z

# partons distributed randomly

# many partons in a small tube

# no correlations at different !x⊥

" The MV model assumes that the density of color charges

ρ("x⊥) has a Gaussian distribution :

Wx0 [ρ] = exp

»−

Zd2!x⊥

ρa(!x⊥)ρa(!x⊥)2µ2(!x⊥)

–

b=impact parameter

Q2sA(x, b) ∼ αs xG(x, Q2) TA(b)

✓ Roadmap of hadron structure

NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q

“BFKL”∂φ(x,kt)∂ ln(x0/x)

≈ K ⊗ φ(x,kt)

∂φ(x,kt)∂ ln(x0/x)

≈ K ⊗ φ(x,kt)− φ(x,kt)2“BK-JIMWLK”

• In the saturation domain, non-linear corrections to the evolution equations are needed to account for the saturation phenomenon:

radiation recombination


NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q


≈ K ⊗ φ(x,kt)





lnQ2

s

k2⊥

With decreasing x, gluons are produced mostly atlarge momenta k⊥ > Qs(x), and thus cannot be“seen” by a probe (γ∗) with Q2 < Qs(x)

=⇒ unitarization of DIS!

kY!

Y!

Y2

Y2

k2

! QCD

1

"s

ng

Qs(

sQ ( )"

Y> 1

f(Y,k)

• Gluon saturation is a natural solution to the small–xproblems of the linear evolution equations.

• How to compute in this non–linear regime ?

Large occupation numbers ←→ Strong (A ∼ 1/g)classical fields. (cf. Introduction)

Classical effective theory for the small–x gluons

φ(x,k⊥)

Q2s

k2⊥


NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q


≈ K ⊗ φ(x,kt)





lnQ2

s

k2⊥



kY!

Y!

Y2

Y2

k2

! QCD

1

"s

ng

Qs(

sQ ( )"

Y> 1

f(Y,k)





φ(x,k⊥)

Q2s

k2⊥

SPS


NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q


≈ K ⊗ φ(x,kt)





lnQ2

s

k2⊥



kY!

Y!

Y2

Y2

k2

! QCD

1

"s

ng

Qs(

sQ ( )"

Y> 1

f(Y,k)





φ(x,k⊥)

Q2s

k2⊥

SPSRHIC


NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q


≈ K ⊗ φ(x,kt)





lnQ2

s

k2⊥



kY!

Y!

Y2

Y2

k2

! QCD

1

"s

ng

Qs(

sQ ( )"

Y> 1

f(Y,k)





φ(x,k⊥)

Q2s

k2⊥

SPSRHIC

LHC


NO

N−P

ERTU

RBA

TIV

E

CGC: JIMWLK−BK

BFKL

DGLAP

lnΛQCD

Y=ln (1/x)

ln

Q(x)S

High density

Low density

extendedscaling

Q


≈ K ⊗ φ(x,kt)





Saturation ~ large color fields.

φ(x, k⊥) ∼ dNg

dY d2k⊥∼ 〈AA〉 ∝ 1

αs

for k⊥ ! Qs(x)

S! ∼

1!

∫d4x FµνFµν " 1

Classical scenario

These two ideas are at the heart of the Color Glass Condensate... tomorrow!!

• Factorization theorems: separation of long and short distance physics in scattering processes:

A

B

C

D

i

j

k

h

✓Hard processes in proton-proton collisions

Assumptions: • Only one parton per hadron participates in the production process (no multiple scatterings)• Partons enter the collision on-shell and collinearly with the hadron i.e they have zero transverse momentum (this assumption is modified in kt-factorization theorems)

pdf’s fragmentation functions

cross section at partonic level

dσAB→CD ∝ fi/A fj/B ⊗ dσij→hk ⊗DC/h DD/k +O(1/Q4)

“higher twists”(multiple scatterings)

• Factorization theorems: separation of long and short distance physics in scattering processes:

A

B

C

D

i

j

k

h

✓Hard processes (Q2>>) in proton-proton collisions

Initial state radiation absorbed in the pdf’s Multiple scatterings become important when the momentum transfer is of the order of the saturation scale

pdf’s fragmentation functions

cross section at partonic level

dσAB→CD ∝ fi/A fj/B ⊗ dσij→hk ⊗DC/h DD/k +O(1/Q4)

“higher twists”(multiple scatterings)

color glass condensate and initial state effects in heavy

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