Gluon Propagator and Static Potential for a heavy Quarkantiquark Pair in an Anisotropic Plasma

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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
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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

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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)
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Hard-Thermal-Loop Gluon Self-Energy
Gluon self-energy

• Diagrammatic Approach
• Feynman graphs for gluon self-energy in the
  one-loop approximation

. In hard thermal loop (HTL) approximation, the
leading contribution has a - behaviour.
Hard momentum
Soft momentum
gluon self-energy in Euclidean space
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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).
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
Fluctuating part of the parton densities
Gluon field strength tensor
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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
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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
.
Anisotropy direction
Heat bath direction
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Gluon Propagator in an Anisotropic Plasma
tensor basis for an anisotropic system
The four structure functions can be determined by
the following contractions
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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 and vanish,
we get the isotropic propagator.
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Static Potential for a Quark-Antiquark pair
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

• 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.

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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
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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.

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Static Potential for a Quark-Antiquark pair

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