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Chaos in the Color Glass Condensate

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Chaos in the Color Glass Condensate

Kirill Tuchin

P

- Interaction time int~1/qz=1/Q
- Life-time of a parton part~k+/mt2.
- Since kz=xpz, part~Q/mt2.
- Thus, part>> int : photon is a “microscope” of resolution ~1/Q

q

e

- Proton’s radius R~ln1/2(s)

- Density of gluons:

- Number of gluons
- resolved by a photon:

~1:

High parton

density

Qs2

(Gribov,Levin,Ryskin,82)

- Life-time of a dipole is

- Total cross section

is a forward scattering amplitude

- Quasi-classical regime:

(McLerran,Venugopalan,94)

- High energy linear evolution regime

(Fadin,Kuraev,Lipatov,Balitsky,75,78)

- Evolution equation:

- Fourier image of the forward amplitude

- The BFKL equation:

where

- Evolution in a Color Glass Condensate:

(Balitski,Kovchegov,96,00)

- Equivalently:

(Kovchegov,01)

(Kharzeev, K.T., 05)

- At small x emission of a gluon into a wave function of
a high energy hadron happens when sln(1/x)~1

- Let’s impose the boundary condition by putting a system in
- a box of size L~

- We can think of evolution as a discrete process of gluon
- emission when parameter n=sln(1/x)changes by unity.

- Evolution equation can be written as

- Diffusion approximation:

- Let’s keep only the first term

- Rescale

- Discrete equation:

- For fixed kT this is the ‘logistic map’.
- It is used to describe

- population growth in the ecological systems.
- It’s properties are very different from those
of the continuous equation. (von Neumann,47)

- It’s properties are very different from those

n

n=1

n=4

n=8

n=11

kT

continuous

- Stable fixed point:

discrete

- Unstable fixed point:

- is a bifurcation point: fixed point condition admits two
- new solutions (period doubling).

- Unstable fixed points

- Stable fixed points:

n

n=4

n=7

n=1

n=10

kT

continuous

discrete

n

n=1

n=5

n=7

n=9

kT

n

n=1

n=5

n=8

n=11

kT

FFeigenbaum’s number)

- In pertubative QCD:
- min=1+4ln2=3.77>F

Canadian Lynx population (Hudson Bay Company’s archives)

Note large

fluctuations

Fixed points

High energy evolution

starts here.

- Diffraction cross section is the statistical dispersion in the
- absorption probabilities of different eigenstates.

diff=<2>-<>2

- Large fluctuations in the scattering amplitude imply large
- target independent diffractive cross sections at highest
- energies.

- Optimistic/pessimistic point of view:
there are an interesting non-linear effects in the

Color Glass Condensate beyond the continuum limit.

- Pessimistic/optimistic point of view:
appearance of chaos in the high energy evolution

signals breakdown of a perturbation theory in vacuum.