1. Thermalization and Plasma Instabilities Question: What is the (local) thermalization time for QGPs in heavy ion collisions? A simpler question: What is it for arbitrarily high energy collisions, where  s ¿ 1? A much simpler question: How does that time depend on  s ? In the saturation picture,

2. Aside: plasma physics is complicated Image of solar coronal filament from NASA’s TRACE satellite

3. Why are QGP papers typically less complicated than Often study equilibrium QGP. In local equilibrium, magnetic confinement » 1/(g 2 T) expect no long-range color magnetic fields (HD instead of MHD). Study physics at scales where everything is weakly interacting. ?

4. Bottom-Up Thermalization (Baier, Mueller, Schiff, Son) Starting point t » 1/Q s p » Q s ´ “hard” gluons in this talk 1/Q s System expands density decreases more perturbative f(x,p) » 1/  s initially non-perturbative

5. Bottom-Up Thermalization (Baier, Mueller, Schiff, Son) Later, if interactions ignored t ¿ 1/Q s  z » v z t p » Q s ´ “hard” gluons in this talk

6. Bottom-Up Thermalization (Baier, Mueller, Schiff, Son) Later, if interactions ignored t ¿ 1/Q s p » Q s pp pzpz QsQs

7. Original bottom-up scenario: 2-body collisions (with some LPM thrown in)  Stage I: 1 ¿ Q s t ¿  -5/2 small angle scattering: soft Brem: pp pzpz “hard” gluons ´ p » Q s “soft” gluons ´  p ¿ Q s QsQs

8.  Stage II:  -5/2 ¿ Q s t ¿  -13/5 n soft ¿ n hard E hard À E soft soft collisions  soft gluons thermalize hard particles lose energy by Brem + cascade  Local thermalization complete: Q s t »  -13/5

9. Plasma instabilities Major proponent that they are important for the QGP: Stan Mrówczyński (`88, `93, `94, `97, `00, `03) Application to bottom-up thermalization: Peter Arnold, Jonathan Lenaghan, Guy Moore (`03) But also Heinz (`84) Pokrovsky and Selikhov (`88, `90) Pavlenko (’92) Mrówczyński and Thoma (`00) Randrup and Mrówczyński (’03) Romatschke and Strickland (`03) Birse, Kao, Nayak (`03)

10. Anisotropic HTL self-energy Calculate HTL self energy  ( ,k) or  negative eigenvalues of  ij (0,k)  instabilities at small k  exponentially growing soft gauge fields Anisotropic f(p) [Note: I always assume f is parity symmetric, f(p) = f(–p).] orlinearized kinetic theory

11. Scalar Analogy Consider :Scalar theory at finite temperature. Integrate out hard particles. massless  2  = (k 2 + m 0 2 +  )  ´ (k 2 + m eff 2 )  k 2 +  < 0   = § i   exponentially growing solutions with  » T 2 Effective potential : Linearized eq. : V eff (  ) = m eff 2   +  4 =   2 +  4 If  < 0 (some multi-scalar models; Rochelle salts),   =0 unstable exponential growth of k < (-  ) 1/2 modes growth stops once  is non-perturbative

12. A picture of the Weibel (or filamentation) instability z x [adapted from Mrówczyński ’97]

13. A picture of the Weibel (or filamentation) instability z x [adapted from Mrówczyński ’97] J makes B grow

14. A contradictory picture z x

15. z x J makes B shrink

16. 1 st picture revisited2nd picture “trapped” trajectories focused  instability “untrapped” trajectories  stability For isotropic f(p), these effects cancel   ij (0,k)=0.

17. p>p> pzpz k unstable k stable k>k> kzkz Note for later: Typical unstable modes have k pointing very close to the z direction

18. When does growth stop? Answer: When fields become non-perturbatively large. D »  – igA  A »  / g  Possibility 1 (QED and QCD): effects on hard particles non-perturbative A » p / g  Possibility 2 (QCD only): self-interaction of soft modes non-perturbative A » k / g  These are parametrically different scales: k ¿ p

19. Conjecture: Growing instabilities “abelianize” in QCD.  A » p / g (same as QED)  The complicated stuff that happens next is closely related to mainstream plasma physics of (collisionless) relativistic QED plasmas.

20. Suggestive arguments for abelianization [Arnold & Lenaghan, in preparation] Start with (anisotropic) HTL effective action [adapted from Mrówczyński, Rebhan, Strickland ’04] Find effective potential V eff by looking at S eff for time-independent configurations in A 0 = 0 gauge. Problem: too complicated!

21. pzpz kzkz Inspired by ignore k T and consider A = A(z) depending on z only.  A miracle [Iancu `98]: is linear in A.  is quadratic in A.

22. pzpz kzkz Inspired by ignore k T and consider A = A(z) depending on z only.  A miracle [Iancu `98]: is linear in A.  is quadratic in A.

23. pzpz kzkz Inspired by ignore k T and consider A = A(z) depending on z only.  A miracle [Iancu `98]: is linear in A.  is quadratic in A.

24. Look at k z ! 0 limit: System runs away in Abelian directions e.g.orabove.

25. Conjecture : Weibel instabilities “abelianize” as SU 2 gauge theory  U 1 plasma physics SU 3 gauge theory  U 1 £ U 1 plasma physics

26. What happens next? Does it stabilize as something that looks like this, requiring individual collisions to equilibrate further?

27. What happens next? Does it stabilize as something that looks like this, requiring individual collisions to equilibrate further?

28. What happens next? Does it stabilize as something that looks like this, requiring individual collisions to equilibrate further?

29. What happens next? Current filaments attract 

30. What happens next? Current filaments attract  magnetic tear instability

31. non-relativistic 2D+3D simulations by Califano et al. [from Phys. Rev. Lett. 86 (2001) 5293]

32. But not all collisionless problems isotropize... Simulations by Honda et al. of uniform relativistic beam of electrons injected into a plasma: [from Phys. Plasmas 7 (2000) 1302] known as a Bennett self-pinch equilibrium (1934)

33. Original question : How does the thermalization time depend on  s ? A lower bound : Near equilibrium, thermalization time After expansion time t, initial saturation energy density is diluted to Settingand combing the above, This is indeed smaller than the original bottom-up’s  -13/5. And it might be the right answer for instability-driven thermalization.

34. A wish list  Full simulations of QCD kinetic theory to check the “abelianization” conjecture.  Simulations of QED and/or QCD kinetic theory to investigate isotropization of ultra-relativistic systems that start from pzpz What is parametric time scale for isotropization in each case?