1. Gluon Propagator and Static Potential for a heavy Quark-antiquark Pair in an Anisotropic Plasma Yun Guo Helmholtz Research School Graduate Days 19 July 2007

2. Outlines ：  Introduction & Hard-Thermal-Loop Gluon Self-Energy Diagrammatic Approach Semi-Classical Transport Theory  Gluon Propagator in an Anisotropic Plasma Tensor Decomposition Self-Energy Structure Functions Gluon Propagator in Covariant Gauge  Static Potential for a Quark-Antiquark Pair Static Potential: in an Isotropic Plasma Static Potential: in an Anisotropic Plasma Results  Summary & Outlook

3. Introduction At the early stage of ultrarelativistic heavy ion collisions at RHIC or LHC, the generated parton system has an anisotropic distribution. The parton momentum distribution is strongly elongated along the beam direction. Why anisotropy ? With an anisotropic distribution, new physical results come out as compared to the isotropic case. Eg, the unstable mode of an anisotropic plasma ( Weibel instabilities). See: P. Romatschke and M. Strickland, “Collective modes of an anisotropic quark gluon plasma,” Phys. Rev. D 68, 036004 (2003)

4. Hard-Thermal-Loop Gluon Self-Energy Diagrammatic Approach: Feynman graphs for gluon self-energy in the one-loop approximation : Gluon self-energy :. In hard thermal loop (HTL) approximation, the leading contribution has a - behaviour. gluon self-energy in Euclidean space Hard momentum Soft momentum

5. Hard-Thermal-Loop Gluon Self-Energy Semi-classical transport theory: Within this approach, partons are described by their phase-space density (distribution function) and their time evolution is given by collisionless transport equations (Vlasov-type transport equations). Fluctuating part of the parton densities Gluon field strength tensor The distribution functions are assumed to be the combination of the colorless part and the fluctuating part Linearize the transport equations colorless part of the parton densities

6. Hard-Thermal-Loop Gluon Self-Energy By solving the transport equations, the induced current can be expressed as In this expression, we have neglected terms of subleading order in and performed a Fourier transform to momentum space. The distribution function is completely arbitrary This result is identical to the one get by the diagrammatic approach if we use an isotropic distribution function symmetric transverse

7. Gluon Propagator in an Anisotropic Plasma In an anisotropic system, the gluon propagator depends on : the anisotropy direction and the heat bath direction, as well as the four-momentum. Anisotropy direction: Heat bath direction: From isotropy to anisotropy the anisotropic distribution function is obtained from an arbitrary isotropic distribution function by the rescaling of only one direction in momentum space

8. Gluon Propagator in an Anisotropic Plasma The four structure functions can be determined by the following contractions: tensor basis for an anisotropic system

9. Gluon Propagator in an Anisotropic Plasma The inverse propagator (in covariant gauge) can be expressed as The anisotropic gluon propagator obtained by inverting the above tensor is For, the structure functions and are 0, the coefficient of and vanish, we get the isotropic propagator. is the gauge fixing parameter

10. Consider the heavy quark-antiquark pair (heavy quarkonium systems), or in the nonrelativistic limit, we can determine the potential for the heavy quarkonium by the following expression Static Potential for a Quark-Antiquark pair the unlike charges of the heavy quarkonium give the overall minus sign. in the nonrelativistic limit, the spatial current of the quark or antiquark vanishes, and the main contributions come from the zero component of the gluon propagator. in the nonrelativistic limit, the zero component of the gluon four momentum can be set to zero approximately.

11. Static Potential for a Quark-Antiquark pair The isotropic potential for a heavy quark-antiquark pair: Taking, the isotropic potential can be expressed as the following The isotropic potential depends only on the modulus of r. We get the general Debye-screened potential after completing the contour integral Also see: M. Laine, O. Philipsen, P. Romatschke, and M. Tassler, J. High Energy Phys. 03 (2007) 054

12. Static Potential for a Quark-Antiquark pair The anisotropic potential for a heavy quark-antiquark pair: Assumptions:  because of the complication of the four structure functions, we consider is a small number so that we can expand the four structure functions to the linear order of.  unlike the isotropic potential, the anisotropic potential depends not only on the modulus of r, but also on the angle between r and q. For simplicity, we consider the following two cases.

13. Static Potential for a Quark-Antiquark pair For the first case, an analytic result can be obtained after completing the integral

14. Static Potential for a Quark-Antiquark pair Preliminary results:

15. Static Potential for a Quark-Antiquark pair Preliminary results:

16. Static Potential for a Quark-Antiquark pair Preliminary results:

17. Summary & outlook  By introducing the tensor basis for an anisotropic system, we derived gluon self energy and gluon propagator in covariant gauge.  Using this anisotropic gluon propagator, we can determine the potential for a heavy quark pair.  For an anisotropic plasma there is an angular dependence of the potential. For small r, the effect of the anisotropy becomes very weak and we can use the isotropic potential approximately.  Results show stronger binding along beam direction than transversally.  It is worthwhile to consider an extremely anisotropic distribution.  It is expected there will be a large difference between the anisotropic potential and isotropic potential. Angular dependence should also be a feature for the extreme anisotropy but for small r, the isotropic approximation probably can not be used any more.