Calculation of Dielectric Function of a Quark-gluon Plasma

1. Calculation of Dielectric Function of a Quark-gluon Plasma Jiang Bing-feng Li Jia-rong IOPP CCNU 2008.4

2. Outline • Motivation • Gluon self-energy and dielectric function of QGP calculated with the effective perturbative theory---the HTL resummation technique • Numerical curves of the dielectric function and discussion on the results • Summary

3. Motivation • The medium effects of hot and/or dense nuclear matter---hot topics in Heavy ion collision. • The dielectric function is very important. The field in medium is different from that in vacu-um and the dielectric function reflects that differences. • The dielectric function in the HTL approxi-mation is incomplete

4. The formula for dielectric function ---longitudinal part of gluon self energy

5. In general, it is not possible to derive an analytic formula of the gluon self-energy, unless in the case of the HTL approximation. The gluon self-energy in the HTL approximation:

6. A brief Introduction to HTL resummation technique • Hard momentum~T, soft momentum~gT • Soft inner line---effective propagator Hard inner line---bare propagator • All legs of a vertex are soft---effective vertex, otherwise---bare vertex

7. Effective propagator • Effective propagator is constructed with Dyson-schwinger equation

8. Calculation with HTL resummation technique • Loop integrals seperated by two parts ---hard momentum and soft momentum Braatten,Pisarski Phys.Rev.Lett 64.1138 Thoma hep-ph/0010164 our purpose: calculate the dielectric function excited by the hard gluon

9. Hard contribution External line and inner lines both are hard momentum, bare propagator and bare vertex are enough. Hard contribution ~ the HTL gluon self energy at high temperature.

10. Soft contribution = +

11. Effective propagator at finite temperature in the real time formula in Coulomb gauge

12. Gluon self-energy calculated with HTL resummation technique = + → →

13. Hard part

14. Soft part

15. Parameters: g=0.1 T=500MeV p=350MeV

16. Numerical curves and discuss on the dielectric function of QGP • Real part of the dielectric function Left picture: the HTL resummation result Right picture: the HTL result

17. The time-like region • The singularity • The space-like region • Magnitude

18. The imaginary part of the dielectric function • Solid curve: the HTL resummation result dashed curve: the HTL result

19. The positions of the inflexion and the maximum of the imaginary part are exactly the same as that of the maximum and the minimum of the real part • Nonzero imaginary dielectric function related to some energy exchange phenomenon • Exchange phenomenon in space-like region---Landau damping • Nontrivial behavior of real dielectric function in space-like region related to Landau damping

20. In the HTL approximation, dispersion relation is time-like, Landau damping is absent. • In the case of the HTL resummation, although adopting time-like dispersion relation, we obtain the resonance absorbing structure of the imaginary dielectric function, which comes from the nonlinear Landau damping.

21. Because of non-Abelian and nonlinear interactions of eigenwaves, the time-like eigenwaves may produce space-like pulsations the space-like pulsations interacting with plasma particles may cause nonlinear Landau damping. Xiao-fei Zhang and Jia-rong LiPhys.Rev.C52.964 Xiao-ping Zheng and Jia-rong Li Phys.Lett.B409.45

22. Rusummation : gluon self-polarization contains multi-waves processes which reflects non-Abelian and nonlinear interaction of QGP,which causes the Landau damping lost in the HTL approximation

23. Summary • In space-like region, the dielectric function of QGP have two extrema,which reflects Landau damping mechanism. • HTL approximation lost some imformation in the space-like region.

24. Thanks!