Download New perspective of QCD at high energy

New perspective of QCD
at high energy
- introduction to Color Glass Condensate Kazunori Itakura
Service de Physique Theorique
CEA/Saclay, France
Plan of the lectures
1. Introduction and overview
2. Kinematics and Evolution equations
Bjorken x and virtuality
Q2
I
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
II
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,
absence of infrared diffusion
III
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)
IV
Lecture I
1. Introduction and overview
Introduction
Questions/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,
charge 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:
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 introduction)
-- 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 diagram
Extended
scaling
regime
saturation
1/x
Regge theory
color glass
condensate
parton gas
ancient < 1970
old
~ 1980
new
> 2002
Gribov,Levin,Ryskin
already considered
gluon saturation
-Better understanding of
the saturation line
-new phase = extended
scaling regime
g-proton scatt.
Q2
Overview (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)
Important Experimental Results
1993: Beginning of excitement in hard small x physics
(not the soft Pomeron
physics)
Steep rise of F2 at small x
evidence for BFKL ?
 LO, NLO(98),
resumed NLO (99)
Overview (III)
Important Experimental Results
Hadronic cross section at high energy (pp, total)
ln s, ln2 s (Froissart bound), or sl (l=0.08) (Pomeron) ??
Most recent PDG  consistent with ln2 s.
include cosmic ray pp data of AKENO & Fly’s eye
S1/2
10
102
103
104 GeV
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.
Overview (III)
Important Experimental Results
2001: Discovery of geometric scaling in DIS (ep)
2002: eA
total g* p
cross
section
Need saturation to understand this phenomena
Overview (III)
Important Experimental Results
2004: high pt suppression at forward rapidity in dAu at RHIC
Another evidence of CGC??
“Shattered Glass” by D. Appell
Scientific American, April 2004
(Nikkei Science, June 2004, p8)
2. Kinematics and evolution
equations
Light Cone variables
momentum
qm = (q0, q1, q2, q3)  q+ = (q0 + q3)/√2
q- = (q0 – q3)/√ 2
qT=(q1,q2)
longitudinal momentum
LC energy
transverse momentum
Infinite momentum frame = a frame in which the target proton is moving
very fast (in z direction). pm=(p0,p1,p2,p3)~(p+M2/2p, 0, 0, p)~(p, 0, 0, p).
p+ = (p0 + p3)/√2
~ √2 p
p- = (p0 - p3 )/√2
~
0
pT=(p1,p2)
=
0
very large
DIS Kinematics (I)
Two basic kinematical variables in deep inelastic scattering
electron(k) + proton(p)  electron(k’) + X
Lorentz invariant quantities
Q2 = - q2 > 0 : virtuality of photon
qm
pm
x = Q2/2p .q
: Bjorken variable
DIS Kinematics (II)
Physical meaning of Q2 and x
Q2 = - q2 : transverse resolution, transverse size of measured partons
x = Q2/2p .q : fraction of longitudinal momentum of a parton
1) Infinite momentum frame pm=(p+m2/2p,0,0,p)~(p,0,0,p), where p is large.
Take qm = (q0,q1,q2,0),(Breit frame) so that q0~pmqm /p  0 as p infinity .
Thus, one finds Q2 ~ qT 2
2) On-shell condition for the struck quark (pqm + qm)2=2p . q (x - x)+x2 Mp2=0
Ignoring the proton mass  x = 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+= √2p, p-~ 0.
qm
qm
pm
pqm=x pm
DIS Kinematics (III)
• Q2 = qT2 : transverse resolution
• x =p+/P+ : longitudinal momentum fraction
transverse
longitudinal
Structure functions of a proton
F1 and F2 structure functions
(neglecting the proton mass)
y=q .p/k . p
from Lorentz decomposition of hadronic tensor Wab
The Bjorken limit: Q2, n =p . q  infinity with x fixed x=Q2/2n
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
distribution is given by the probability q(x)dx with x being a fraction of longitudinal
momentum carried by a parton.
F2(x)= 2 x F1(x)= S q eq2 x q(x)
q(x) quark distribution function
-- there is a weak violation of Bjorken scaling  log Q2 dependence (QCD effect!)
Structure functions and DGLAP equation
One 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 at small x
Splitting functions at leading order
O(aS0)
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 evolution
Evolution with respect to x or rapidity y = ln 1/x
Resum all the contributions (aS ln 1/x)n (n>0) in gluon distribution xg(x,Q2)
even if they are not accompanied by powers of ln Q2.
A0
 gluon number ~
~ aS ln 1/x
~ (aS ln 1/x)n
x1 >> x2 >>….>> xn >> x
strong ordering in x
Sn Cn (1/n!) (aS ln 1/x)n ~ exp{ w aS ln 1/x }
BFKL equation
• Linear equation for the unintegrated gluon distribution f(x,k)
recursive equation
•
Relation to the gluon distribution
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 [(aS ln 1/x) is not small]