New perspective of QCDNew perspective of QCD at high energy at high energy

- introduction to Color Glass Condensate -

Kazunori ItakuraService de Physique Theorique

CEA/Saclay, France

Plan of the lecturesPlan of the lectures1. Introduction and overview2. Kinematics and Evolution equations Bjorken x and virtuality Q2 in DIS, DGLAP and BFKL equations

3. Growth of the gluon distribution and unitarity violation Solution to the BFKL equation, High energy behaviour of cross section

4. Color Glass Condensate McLerran-Venugopalan model, Quantum evolution and JIMWLK equation, the Balitsky-Kovchegov equation

5. The Balitsky-Kovchegov equation Solutions (analytical and numerical), saturation scale, geometric scaling, abs

ence of infrared diffusion

6. Recent progress in phenomenology HERA DIS, RHIC (Au-Au, d-Au)

7. Recent progress in theory Traveling wave, Structure of JIMWLK equation, Evolution equation for n-point functions, attempts beyond the BK equation (fluctuation)

I

II

III

IV

Lecture ILecture I

1. Introduction and overview1. Introduction and overview

IntroductionIntroductionQuestions/problems which we would like to answer/understand:

What is the high energy limit of QCD? Can we see it in experiments?

What is the high energy limit of “scattering” involving hadrons? How can we treat it from first principles (QCD)? Is the energy in the present experiments enough large to see it?

Is there an unexplored regime which is still approachable in weak coupling techniques?

In fact, most of the “success stories” in QCD is due to identifying new perturbative regimes. What’s next? Can we define a perturbative regime in relation to the high energy limit of scattering?

What is the dynamical information of the nucleons? Static properties such as nucleon’s constituents, mass, radius, charg

e etc. do not help to describe the actual scattering. We need information about, say, higher Fock components, which becomes relevant at high energy. This is the dynamical aspects of the nucleon!

References:References: review and lecture notes-- Iancu & Venugopalan, hep-ph/0303204, published in “QGP3” “The Color Glass Condensate and High Energy Scattering in QCD”-- Iancu, Leonidov & McLerran, hep-ph/020227 Cargese lectures “The Colour Glass Condensate: An introduction” (actually this is not an introductio

n)

-- Al Mueller, hep-ph/0111244 Cargese lectures “Parton Saturation - An Overview”

For japanese reading (only introductory discussion),-- Itakura, JPS membership journal (February 2004) “Color Glass Condensate – a universal picture of hadrons and nuclei”

Textbooks-- Devenish & Cooper-Sarkar, Oxford University Press, April 2004 “Deep Inelastic Scattering”

For basic understanding before CGC-- Forshaw & Ross, Cambridge University Press, 1997 “Quantum Chromodynamics and the Pomeron”-- Ellis, Stirling, & Webber, Cambridge University Press, “QCD and Collider Physics”

Overview (I) Phase diagramOverview (I) Phase diagram

ancient < 1970 old ~ 1980 new > 2002

Reg

ge t

heor

y

1/x

Q2

saturation

parton gas

Extended scaling regimecolor glass

condensate

-Better understanding of the saturation line

-new phase = extended scaling regime

-proton scatt.

Gribov,Levin,Ryskinalready considered gluon saturation

Overview (II) UniversalityOverview (II) Universality

As the scattering energy is increased, hadrons and nuclei eventually become the color glass condensate, irrespective of the details of the system.

This is due to the multiple production of gluons, while the species properties are carried by the valence quarks.

Universal fixed point of the evolution.

Indeed, in the saturation regime, the gluon distributions of a nucleon and a nucleus are expressed in the same functional form. The only difference is the magnitude of the saturation scale.

High energy limit of QCD is the Color Glass Condensate.

Overview (III) Overview (III) Important Experimental ResultsImportant Experimental Results

1993: Beginning of excitement in hard small x physics (not the soft Pomeron physic

s)

Steep rise of F2 at small x

evidence for BFKL ?

LO, NLO(98),

resumed NLO (99)

Overview (III) Overview (III) Important Experimental ResultsImportant Experimental Results

Hadronic cross section at high energy (pp, total)ln s, ln2 s (Froissart bound), or s(Pomeron) ??

Most recent PDG consistent with ln2 s.

include cosmic ray pp data of AKENO & Fly’s eye

Seems to saturate the Froissart bound….

How can the usual soft Pomeron description be modified so that the unitarity bound is satisfied??

The same question should be asked to the hard Pomeron, too.

S1/2 10 102 103 104 GeV

Overview (III)Overview (III)Important Experimental ResultsImportant Experimental Results

2001: Discovery of geometric scaling in DIS (ep)

2002: eA

Need saturation to understand this phenomena

total * p cross section

Overview (III) Overview (III) Important Experimental ResultsImportant Experimental Results

2004: high pt suppression at forward rapidity in dAu at RHIC

Another evidence of CGC??

“Shattered Glass” by D. AppellScientific American, April 2004(Nikkei Science, June 2004, p8)

2. Kinematics and evolution2. Kinematics and evolution

equationsequations

Light Cone variablesmomentum

q= (q0, q1, q2, q3) q+ = (q0 + q3)/ longitudinal momentum

q- = (q0 – q3)/ LC energy

qT=(q1,q2) transverse momentum

√2

√2

Infinite momentum frame = a frame in which the target proton is moving very fast (in z direction). p=(p0,p1,p2,p3)~(p+M2/2p, 0, 0, p)~(p, 0, 0, p).

p+ = (p0 + p3)/ ~ p very large

p- = (p0 - p3 )/ ~ 0

pT=(p1,p2) = 0

√2

√2

√2

DIS Kinematics (I)DIS Kinematics (I)Two basic kinematical variables in deep inelastic scattering

electron(k) + proton(p) electron(k’) + X

　　　　　　　　　　　　　　　　　　　 Lorentz invariant quantiti

es

q

p

Q2 = - q2 > 0 : virtuality of photon

x = Q2/2p q : Bjorken variable.

DIS Kinematics (II)DIS Kinematics (II)

q

p

1) Infinite momentum frame p=(p+m2/2p,0,0,p)~(p,0,0,p), where p is large. Take q= (q0,q1,q2,0),(Breit frame) so that q0~pq/p 0 as p infinity . Thus, one finds Q2 ~ qT 2

2) On-shell condition for the struck quark (pq+ q)2=2p q (- x)+2 Mp

2=0 Ignoring the proton mass x = : x is the fraction of momentum carried by the parton to the total nucleon momentum. 0 < x < 1 . In IMF, this becomes just the longitudinal momentum fraction. p+= p, p-~ 0.

Q2 = - q2 : transverse resolution, transverse size of measured partons

x = Q2/2p q : fraction of longitudinal momentum of a parton

Physical meaning of Q2 and x

pq=p

q

.

√2

.

DIS Kinematics (III)DIS Kinematics (III)

• Q2 = qT2 : transverse resolution

• x =p+/P+ : longitudinal momentum fraction

transverse

longitudinal

Structure functions of a protonStructure functions of a protonF1 and F2 structure functions (neglecting the proton mass)

from Lorentz decomposition of hadronic tensor W

The Bjorken limit: Q2, =p q infinity with x fixed x=Q2/2 Fi (x,Q2) Fi (x) Bjorken Scaling-- A proton is made of point-like objects (otherwise Q2 dependent)-- naïve parton model: a proton is an incoherent collection of partons whose distributi

on is given by the probability q(x)dxwith xbeing a fraction of longitudinal momentum carried by a parton.

F2(x)= 2 x F1(x)= q eq2 x q(x) q(x) quark distribution function

-- there is a weak violation of Bjorken scaling log Q2 dependence (QCD effect!)

y=q p/k p. .

.

Structure functions and DGLAP equationStructure functions and DGLAP equationOne gluon emission gives the logarithmic Q2 dependence

Change of the resolution scale Q2

evolution equation for the parton distribution functions

DGLAP equation (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi)

“Splitting function” Pij (x):

a probability of finding a parton

of type “i” in a parton of type “j”.

The equation for the change of

“transverse resolution”

DGLAP equation DGLAP equation at small xat small xSplitting functions at leading order O(S

0)

At small x, only Pgq and Pgg are relevant.

　　　　　　　　　　　　　　　　 Gluon dominant at small x!

The double log approximation (DLA) of

DGLAP is easily solved.

-- increase of gluon distribution at small x

BFKL evolutionBFKL evolutionEvolution with respect to x or rapidity y = ln 1/x

Resum all the contributions (S ln 1/x)n (n>0) in gluon distribution xg(x,Q2) even if they are not accompanied by powers of ln Q2.

A0 ~ S ln 1/x ~ (S ln 1/x)n

x1 >> x2 >>….>> xn >> x strong ordering in x

gluon number ~ Cn (1/n!) (S ln 1/x)n ~ exp{ S ln 1/x }

n

BFKL equationBFKL equation• Linear equation for the unintegrated gluon distribution f(x,k)

recursive equation

• Relation to the gluon distribution

Summary for lecture I Summary for lecture I

• DIS process is described by two independent kinematical variables x and Q2

• High energy = small x

• Change of energy multiple gluon production BFKL equation at high energy [(S ln 1/x) is not small]