Calculation of Dielectric Function of a Quark-gluon

Plasma

Jiang Bing-feng Li Jia-rong

IOPP CCNU

2008.4

Outline

• Motivation• Gluon self-energy and dielectric function of Q

GP calculated with the effective perturbative theory---the HTL resummation technique

• Numerical curves of the dielectric function and discussion on the results

• Summary

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

• The formula for dielectric function

---longitudinal part of gluon self energy

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:

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

Effective propagator

• Effective propagator is constructed with Dyson-schwinger equation

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

• 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.

• Soft contribution

= +

• Effective propagator at finite temperature in the

real time formula in Coulomb gauge

• Gluon self-energy calculated with HTL resummation technique

= +

→

→

• Hard part

• Soft part

• Parameters:

g=0.1

T=500MeV

p=350MeV

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

• The time-like region

• The singularity

• The space-like region

• Magnitude

• Solid curve: the HTL resummation result

dashed curve: the HTL result

The imaginary part of the dielectric function

• 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

• 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.

• 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 Li Phys.Rev.C52.964

Xiao-ping Zheng and Jia-rong Li Phys.Lett.B409.45

• 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

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.

Thanks!