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

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

  • 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



  • 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

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

  • 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=7

n=9

kT




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.


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