Chaos in the color glass condensate
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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 >>  int : photon is a “microscope” of resolution ~1/Q. q. . e.

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Chaos in the Color Glass Condensate

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Chaos in the color glass condensate

Chaos in the Color Glass Condensate

Kirill Tuchin


Dis in the breit frame

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

How many gluons are resolved

  • 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)


Target rest frame

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

Linear evolution

  • High energy linear evolution regime

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

  • Evolution equation:


Operator form of bfkl

Operator form of BFKL

  • Fourier image of the forward amplitude

  • The BFKL equation:

where


Evolution in a dense system

Evolution in a dense system

  • Evolution in a Color Glass Condensate:



(Balitski,Kovchegov,96,00)

  • Equivalently:

(Kovchegov,01)


Discretization of bk equation

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

  • 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)


Chaos in the color glass condensate

n

n=1

n=4

n=8

n=11

kT

1<<3

continuous

  • Stable fixed point:

discrete

  • Unstable fixed point:


Chaos in the color glass condensate



  • is a bifurcation point: fixed point condition admits two

  • new solutions (period doubling).

  • Unstable fixed points

  • Stable fixed points:


Period doubling

n

n=4

n=7

n=1

n=10

kT

Period doubling

continuous

discrete


Chaos in the color glass condensate

n

n=1

n=5

n=7

n=9

kT




Onset of chaos

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

Chaos in ecology

Canadian Lynx population (Hudson Bay Company’s archives)


Bifurcation diagram

Bifurcation diagram

Note large

fluctuations

Fixed points

High energy evolution

starts here.


Implication to diffraction

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

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.


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