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Chaos in the Color Glass Condensate

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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  1. Chaos in the Color Glass Condensate Kirill Tuchin

  2. 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

  3. 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)

  4. Target rest frame  • Life-time of a dipole is • Total cross section is a forward scattering amplitude • Quasi-classical regime: (McLerran,Venugopalan,94)

  5. Linear evolution • High energy linear evolution regime (Fadin,Kuraev,Lipatov,Balitsky,75,78) • Evolution equation:

  6. Operator form of BFKL • Fourier image of the forward amplitude • The BFKL equation: where

  7. Evolution in a dense system • Evolution in a Color Glass Condensate:  (Balitski,Kovchegov,96,00) • Equivalently: (Kovchegov,01)

  8. 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

  9. 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)

  10. n n=1 n=4 n=8 n=11 kT 1<<3 continuous • Stable fixed point: discrete • Unstable fixed point: 

  11.  • is a bifurcation point: fixed point condition admits two • new solutions (period doubling). • Unstable fixed points • Stable fixed points:

  12. n n=4 n=7 n=1 n=10 kT Period doubling continuous discrete 

  13. n n=1 n=5 n=7 n=9 kT  

  14. 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 

  15. Chaos in ecology Canadian Lynx population (Hudson Bay Company’s archives)

  16. Bifurcation diagram Note large fluctuations Fixed points  High energy evolution starts here.

  17. 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.

  18. 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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