Chaos in the Color Glass Condensate

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# Chaos in the Color Glass Condensate - PowerPoint PPT Presentation

Chaos in the Color Glass Condensate. Kirill Tuchin. DIS in the Breit frame. P. Interaction time  int ~1/q z =1/Q Life-time of a parton  part ~k + /m t 2 . Since k z =xp z ,  part ~Q/m t 2 . Thus,  part &gt;&gt;  int : photon is a “microscope” of resolution ~1/Q. q. . e.

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### Chaos in the Color Glass Condensate

Kirill Tuchin

DIS in the Breit frame

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

How many gluons are resolved
• Density of gluons:
• Number of gluons
• resolved by a photon:

~1:

High parton

density

Qs2

(Gribov,Levin,Ryskin,82)

Target rest frame



• Life-time of a dipole is
• Total cross section

is a forward scattering amplitude

• Quasi-classical regime:

(McLerran,Venugopalan,94)

Linear evolution
• High energy linear evolution regime

• Evolution equation:
Operator form of BFKL
• Fourier image of the forward amplitude
• The BFKL equation:

where

Evolution in a dense system
• Evolution in a Color Glass Condensate:



(Balitski,Kovchegov,96,00)

• Equivalently:

(Kovchegov,01)

Discretization of BK equation

(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
• 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)

n

n=1

n=4

n=8

n=11

kT

1<<3

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

Period doubling

continuous

discrete

n

n=1

n=5

n=8

n=11

kT

Onset of chaos

FFeigenbaum’s number)

• In pertubative QCD:
• min=1+4ln2=3.77>F

Chaos in ecology

Canadian Lynx population (Hudson Bay Company’s archives)

Bifurcation diagram

Note large

fluctuations

Fixed points

High energy evolution

starts here.

Implication to diffraction
• 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.
Summary
• 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.