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
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
Introduction
Why anisotropy ?
 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.
 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)
Hard-Thermal-Loop Gluon Self-Energy
Gluon self-energy :
 Diagrammatic Approach:
Feynman graphs for gluon self-energy in the one-loop approximation :
Hard momentum
Soft momentum
. In hard thermal loop (HTL) approximation,
the leading contribution has a T 2 - behaviour.
gluon self-energy in Euclidean space
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).
Linearize the
transport equations
Fluctuating part of the
parton densities
The distribution functions are assumed to be
the combination of the colorless part and the
fluctuating part
Gluon field strength
tensor
colorless part of the
parton densities
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 g 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
Gluon Propagator in an Anisotropic Plasma
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
In an anisotropic system , the gluon propagator depends on : the anisotropy
direction and the heat bath direction, as well as the four-momentum p .
Anisotropy direction:
Heat bath direction:
Gluon Propagator in an Anisotropic Plasma
tensor basis for an anisotropic system
The four structure functions can
be determined by the following
contractions:
Gluon Propagator in an Anisotropic Plasma
The inverse propagator (in covariant gauge) can be expressed as
is the gauge fixing parameter
The anisotropic gluon propagator obtained by inverting the above tensor is
For
, the structure functions and  are 0, the coefficient of C  and D 
vanish, we get the isotropic propagator.
Static Potential for a Quark-Antiquark pair
Consider the heavy quark-antiquark pair (heavy quarkonium systems), cc or bb in
the nonrelativistic limit, we can determine the potential for the heavy quarkonium by
the following expression
 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.
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
We get the general Debye-screened potential after completing the contour integral
The isotropic potential depends only on the modulus of r .
Also see: M. Laine, O. Philipsen, P. Romatschke, and M. Tassler,
J. High Energy Phys. 03 (2007) 054
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.


Static Potential for a Quark-Antiquark pair


For the first case, an analytic result can be obtained after completing the integral
Static Potential for a Quark-Antiquark pair
Preliminary results:
Static Potential for a Quark-Antiquark pair
Preliminary results:
Static Potential for a Quark-Antiquark pair
Preliminary results:
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.