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 ggg.
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.