Transcript Lecture II

Lecture II
3. Growth of the gluon distribution
and unitarity violation
Solution to the BFKL equation
• Coordinate space representation t = ln 1/x
• Mellin transform + saddle point approximation

• Asymptotic solution at high energy
dominant energy dep.
t}
is given by exp{ w aS
Can BFKL explain the rise of F2 ?
Can BFKL explain the rise of F2 ?
Actually, exponent is too large w = 4ln2 = 2.8
 Resummation tried
 Marginally consistent
with the data
(but power behavior always has
a problem cf: soft Pomeron)
exponent
 NLO analysis necessary!
 But! The NLO correction is too large and the exponent becomes
NEGATIVE!
High energy behavior of the hadronic
cross sections – Froissart bound
Intuitive derivation of the Froissart bound ( by Heisenberg)
Total energy
Saturation is implicit
BFKL solution violates the unitarity bound.
4. Color Glass Condensate
Saturation & Quantum Evolution - overview
BFKL eq.
Low energy
dilute
[Balitsky, Fadin,Kraev,Lipatov ‘78]
N : scattering amp. ~ gluon number
t : rapidity t = ln 1/x ~ ln s
exponential growth of gluon number
 violation of unitarity
Balitsky-Kovchegov eq.
High energy
[Balitsky ‘96,
Kovchegov ’99]
Gluon recombination
dense,
 nonlinearity  saturation,
saturated,
unitarization,
random
universality
Population growth
-- ignoring transverse dynamics -T.R.Malthus (1798) N : polulation density
Growth rate is proportional to the population at that time.
 Solution
P.F.Verhulst (1838)
population explosion
Growth constant k decreases as N increases.
(due to lack of food, limit of area, etc)
Logistic equation
linear regime
exp growth
1. Exp-growth is tamed by nonlinear term
 saturation !! (balanced)
2. Initial condition dependence disappears
at late time dN/dt =0  universal !
3. In QCD, N2 is from the gluon
recombination ggg.
non-linear
saturation
universal
 Time (energy)
McLerran-Venugopalan model
(Primitive) Effective theory of saturated gluons with high
occupation number (sometimes called classical saturation model)
Separation of degrees of freedom in a fast moving hadron
Large x partons slowly moving in transverse plane  random source,
 Gaussian weight function
Small x partons
LC gauge
(A+=0)
classical gluon field induced by the source
Effective at fixed x,
no energy dependence in m
Result is the same as independent
multiple interactions (Glauber).
Color Glass Condensate
Color : gluons have “color” in QCD.
Glass : the small x gluons are created by slowly moving
valence-like partons (with large x) which are distributed
randomly over the transverse plane
 almost frozen over the natural time scale of scattering
This is very similar to the spin glass, where the spins are distributed
randomly, and moves very slowly.
Condensate: It’s a dense matter of gluons. Coherent state
with high occupancy (~1/as at saturation). Can be better
described as a field rather than as a point particle.
CGC as quantum evolution of MV
Include quantum evolution wrt t = ln 1/x into MV model
- Higher energy  new distribution Wt[r]
- Renormalization group equation is a linear functional differential
equation for Wt[r], but nonlinear wrt r.
- Reproduces the Balitsky equation
- Can be formulated for a(x) (gauge field)
through the Yang-Mills eq.
[Dn , Fnm] = dm+r (xT)
 - T2 a (xT) = r (xT)
D
(r is a covariant gauge source)
 JIMWLK equation
JIMWLK equation
Evolution equation for Wt [a], wrt rapidity t = ln 1/x
Wilson line in the adjoint representation gluon propagator
Evolution equation for an operator O
JIMWLK eq. as Fokker-Planck eq.
The probability density P(x,t) to find a stochastic particle at point x
at time t obeys the Fokker-Planck equation
D is the diffusion coefficient, and Fi(x) is the external force.
When Fi(x) =0, the equation is just a diffusion equation and its
solution is given by the Gaussian:
JIMWLK eq. has a similar expression, but in a functional form
Gaussian (MV model) is a solution when the second term is absent.
DIS at small x : dipole formalism
_
Life time of qq fluctuation is very long >> proton size
This is a bare dipole (onium).
1/ Mp x
 1/(Eqq-Eg*)
 Dipole factorization
DIS at small x : dipole formalism
N: Scattering amplitude
S-matrix in DIS at small x
Dipole-CGC scattering in eikonal approximation
scattering of a dipole in one gauge configuration
stay at the same transverse positions
Quark propagation in a background gauge field
average over the random
gauge field should be taken
in the weak field limit,
this gives gluon distribution
~ (a(x)-a(y))2
The Balitsky equation
Take O=tr(Vx+Vy) as the operator
Vx+ is in the fundamental representation
The Balitsky equation
-- Originally derived by Balitsky (shock wave approximation in QCD) ’96
-- Two point function is coupled to 4 point function (product of 2pt fnc)
Evolution of 4 pt fnc includes 6 pt fnc.
-- In general, CGC generates infinite series of evolution equations.
The Balitsky equation is the first lowest equation of this hierarchy.
The Balitsky-Kovchegov equation (I)
The Balitsky equation
The Balitsky-Kovchegov equation
A closed equation for <tr(V+V)>
First derived by Kovchegov (99) by the independent multiple interaction
Balitsky eq.

Balitsky-Kovchegov eq.
<tr(V+V) tr(V+V)>
<tr(V+V)> <tr(V+V)> (large Nc? Large A)
Nt(x,y) = 1 - (Nc-1)<tr(Vx+Vy)>t is the scattering amplitude
The Balitsky-Kovchegov equation (II)
Evolution eq. for the onium (color dipole) scattering amplitude
- evolution under the change of scattering energy s (not Q2)
resummation of (as ln s)n  necessary at high energy
- nonlinear differential equation
resummation of strong gluonic field of the target
- in the weak field limit
reproduces the BFKL equation (linear)
scattering amplitude becomes proportional to unintegrated gluon density of target
t = ln 1/x ~ ln s is the rapidity
Saturation scale
1/QS(x) : transverse size of gluons when the transverse
plane of a hadron/nucleus is filled by gluons
R
- Boundary between CGC and non-saturated regimes
- Energy and nuclear A dependences
LO BFKL
NLO BFKL
[Gribov,Levin,Ryskin 83, Mueller
99 ,Iancu,Itakura,McLerran’02]
A dependence is modified in running coupling case
[Triantafyllopoulos, ’03]
[Al
Mueller ’03]
- Similarity between HERA (x~10-4, A=1) and RHIC (x~10-2, A=200)
QS(HERA) ~ QS(RHIC)
Geometric scaling
DIS cross section s(x,Q) depends only on Qs(x)/Q at small x
[Stasto,Golec-Biernat,Kwiecinski,’01]
Natural interpretation in CGC
Qs(x)/Q=(1/Q)/(1/Qs) : number of overlapping
1/Q: gluon size
times
Qs(x)/Q=1
=
Once transverse area is filled with gluons, the only
relevant variable is “number of covering times”.
 Geometric scaling!!
Saturation scale from the data
consistent with theoretical results
Geometric scaling persists even outside of CGC!!
 “Scaling window”
[Iancu,Itakura,McLerran,’02]
Scaling window = BFKL window
Summary for lecture II
• BFKL gives increasing gluon density at high energy, which however
contradicts with the unitarity bound.
• CGC is an effective theory of QCD at high energy
– describes evolution of the system under the change of energy
-- very nonlinear (due to )
-- derives a nonlinear evolution equation for 2 point function which
corresponds to the unintegrated gluon distribution in the weak field limit
• Geometric scaling can be naturally understood within CGC framework.