Parametric instabilities of color magnetic field

1. Parametric instabilities of color magnetic field Shoichiro Tsutsui (Kyoto) In collaboration with Hideaki Iida (Kyoto), TeijiKunihiro (Kyoto), Akira Ohnishi (YITP) lunch seminar @ BNL

2. Outlines • Introduction • Early thermalization problem • Instabilities triggered by color magnetic field • Formalism • Floquettheory • Application to CYM • Numerical results • Summary lunch seminar @ BNL

3. Early thermalization problem hydrodynamics is applicable Heinz(2005) QGP How can we understand the considerably short time ? lunch seminar @ BNL ？ time

4. Instabilities in HIC the earliest stage of HIC… (under strong EM fields) • non-Abelian Weibelinstability • Nielsen-Olesen instability • parametric resonance instability has been expected to be the triggersleading to early thermalization turbulence chaos isotropization Berges, Boguslavski, Schlichting,Venugopalan (2014) Epelbaum,Gelis(2014) Kunihiro, Muller, Ohnishi, Schafer (2009) Kunihiroet. al. (2010) Arnold, Lenaghan, Moore, Yaffe (2005) Micha, Tkachev (2004) Iida et. al. (2013)

5. Instabilities triggered by color magnetic field color magnetic field “non-abelian” configuration gauge fields are also homogeneous “abelian” configuration homogeneous Nielsen-Oleseninstability Fujii, Itakura(2008) Iwazaki(2009) Tanji, Itakura (2012) parametric resonance lunch seminar @ BNL Berges, Scheffler, Schlichting, Sexty(2012)

6. Previous work Berges, Scheffler, Schlichting, Sexty(2012) time evolution of gauge field time evolution under the “non-abelian” configuration performed by classical statistical simulation unstable behavior is observed lunch seminar @ BNL

7. Growth rate Berges, Scheffler, Schlichting, Sexty(2012) growth rate is given by time average NO-like instability parametric resonance sub-dominant instability band is also discussed lunch seminar @ BNL

8. What is the origin ? magnetic field Landau quantization ? however… system is spatially homogeneous “non-abelian” configuration does not form Landau orbit ! growth rate is still unknown lunch seminar @ BNL

9. Our work • SU(2) pure Yang-Mills • temporal gauge • non-expanding geometry • under homogenous and time dependent magnetic field • linear analysis same as Berges, Scheffler, Schlichting, Sexty but enable to determine instability bands precisely We show that instability under non-abelian conf. is completely different from Nielsen-Olesen instability lunch seminar @ BNL

10. Outlines • Introduction • Early thermalization problem • Instabilities triggered by color magnetic field • Formalism • Floquettheory • Application to CYM • Numerical results • Summary lunch seminar @ BNL

11. ODE with periodic function differential eq. with periodic coefficient periodic function n- independent solutions solution matrix lunch seminar @ BNL

12. Monodromy matrix follow same equation periodic function solution matrix monodromy matrix is given by lunch seminar @ BNL

13. Floquet’stheorem Floquet’s theorem solution matrix is given by eigenvalue of M stability of the solution unstable (exponential growth) (anti-)periodic or polynomial growth stable lunch seminar @ BNL

14. Algorithm set initial conditions EOM solve the EOM for a period get the monodromy matrix fixed parameter calculate eigenvalues (characteristic exponent) unstable mode lunch seminar @ BNL

15. Merit of Floquet analysis Floquet analysis Numerical calc. linear non-linear non-linearity low high (solve EOM for only one period) cost how to find instability bands check fitting Floquet analysis is economical criterion of instability is clear lunch seminar @ BNL

16. Example: Lame eq. the simplest case (n=2) unstable mode Lameeq. instability bands lunch seminar @ BNL

17. Outlines • Introduction • Early thermalization problem • Instabilities triggered by color magnetic field • Formalism • Floquettheory • Application to CYM • Numerical results • Summary lunch seminar @ BNL

18. Time evolution of background Classical Yang-Mills eq. (temporal gauge) “non-abelian” configuration analytic solution is available solution is periodic period lunch seminar @ BNL

19. Time evolution of fluctuations Classical Yang-Mills eq. without loss of generality EOM of fluctuation (multi component Hill’s eq. ) 2nd order ODE with periodic function lunch seminar @ BNL

20. Outlines • Introduction • Early thermalization problem • Instabilities triggered by color magnetic field • Formalism • Floquettheory • Application to CYM • Numerical results • Summary lunch seminar @ BNL

21. Complete band structure characteristic exponent map view lunch seminar @ BNL

22. Complete band structure NO-likeband agreewith Berges et.al. (2012) lunch seminar @ BNL

23. Complete band structure lunch seminar @ BNL broad instability region in pT direction

24. Complete band structure broad instability region in pT direction many bands around NO-like band at pT=0 plane should be regarded as a part of parametric resonance bands parametric resonancebands Lowest Landau mode (n=0) is only unstable in NO instability NO-likeband quantized transverse mom lunch seminar @ BNL

25. Summary • We investigate parametric resonance in Yang-Mills theory • Floquet theory enable us to perform in a systematic way • Instability considered here is completely different from Nielsen-Olesen instability (there exists broad instability region in pT direction) • Can we see variousunstable modes in full numerical simulation? (work in progress) outlook lunch seminar @ BNL

26. back up lunch seminar @ BNL

27. CYM under color magnetic field EOM of fluctuation (multi component Hill’s eq. ) without loss of generality lunch seminar @ BNL

28. CYM under color magnetic field lunch seminar @ BNL

29. notation color (a=1,2,3) Lorentz(i=x,y,z) lunch seminar @ BNL