Title: Heavy Quark Potential in an Anisotropic Viscous Plasma
1Heavy Quark Potential in an Anisotropic (Viscous)
Plasma
Yun Guo
Helmholtz Research School, Johann Wolfgang Goethe
Universität
Co-Authors Adrian Dumitru
Michael Strickland
Institut für Theoretische Physik, Johann Wolfgang
Goethe Universität
Palaver 14 Jan 2008
Reference arXiv 0711. 4277 hep-ph
2Outlines
- Introduction Motivation
- 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 anisotropic plasma
- Static Potential in some limit cases
- General Results and Comparisons
- Summary outlook
3 Introduction Motivation
In classic theory, the potential between two
unlike charges can be determined through the
Poisson equation
-
Solving the equation with the boundary condition
that the potential vanishes at infinity produce
the Coulomb potential
-
Solving the equation with the boundary condition
that the potential vanishes at infinity produce
the Debye Screened potential
with
4 Introduction Motivation
In quantum theory (QCD or QED), at leading order,
considering one gauge boson exchange, the same
potential can be determined from the Fourier
transform of the static photo (gluon) propagator
Vacuum gauge boson propagator
Coulomb potential
Hard Thermal Loop (HTL) gauge boson propagator
Debye Screened potential
5 Introduction Motivation
PT
PT
Why anisotropy ?
Pz
Pz
- 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
stretched along the beam direction.
- With the 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, Phys. Rev.
D 68, 036004 (2003)
6Hard-Thermal-Loop Gluon Self-Energy
Gluon self-energy
- Diagrammatic Approach
- Feynman graphs for gluon self-energy
in the one-loop approximation
Hard momentum
. In hard thermal loop (HTL) approximation, the
leading contribution has a T2 -behaviour.
Soft momentum
- Gluon Self-energy in Euclidean Space
7Hard-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
Gluon field strength tensor
Fluctuating part of the parton densities
colorless part of the parton densities
8Hard-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
9Gluon Propagator in an Anisotropic Plasma
From isotropy to anisotropy
fiso is the general Fermi-Dirac or
Bose-Einstein distribution function and the
parameter ? determines the degree of 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 anisotropic direction and the
heat bath direction, as well as the four-momentum
p .
Anisotropic direction
Heat bath direction
10Gluon Propagator in an Anisotropic Plasma
The gluon self-energy tensor can be decomposed
with 4 tensor bases
tensor bases for an anisotropic system
Since the self-energy tensor is symmetric and
transverse, not all of its components are
independent. We can therefore restrict our
considerations to the spatial part
11Gluon Propagator in an Anisotropic Plasma
The four structure functions (the coefficient of
the tensor basis) can be determined by the
following contractions
The inverse propagator (in covariant gauge) can
be expressed as
Gauge fixing term with gauge parameter l
Free part
Upon inversion, the propagator is written as
By definition
12Gluon Propagator in an Anisotropic Plasma
The anisotropic gluon propagator
with
For x 0, the structure function g and d are 0,
the coefficient of Cmn and Dmn vanish, we get
the isotropic propagator.
13Static Potential for a Quark-Antiquark pair
Consider the heavy quark-antiquark pair (heavy
quarkonium systems) in the nonrelativistic
limit, at leading order, we can determine the
potential for the heavy quarkonium from the
Fourier transform of the static gluon propagator
- the unlike charges of the heavy quarkonium
gives 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.
14Static Potential for a Quark-Antiquark pair
The four mass scales in the above expression are
With
The above expression apply when n(0,0,1) points
along the z-axis, in general case, pz and p? get
replaced by pn and p-n(pn), respectively.
15Static Potential for a Quark-Antiquark pair
Some limit cases
I. isotropic case where x 0
Taking x 0, the isotropic potential can be
expressed as the following
We get the general Debye-screen potential after
completing the contour integral
The isotropic potential only depends on the
modulus of r .
Also see M. Laine, O. Philipsen, P. Romatschke,
and M. Tassler, J. High Energy Phys. 03 (2007) 054
16Static Potential for a Quark-Antiquark pair
II. the limit r ?0 for arbitrary x
The phase factor of the Fourier transform is
essentially constant up to momenta of order p
1/r
the mass scales are bounded as p ? 8 they can
be neglected
the potential coincides with the vacuum Coulomb
potential
In this limit, there is no medium effect no mater
it is isotropic or anisotropic
17Static Potential for a Quark-Antiquark pair
III. extreme anisotropy
The same potential emerges for extreme anisotropy
since all mass scales approach to 0 as ? ? 8
the potential coincides with the vacuum Coulomb
potential
due to the fact that at ? 8 the phase space
density f(p) has support only in a
two-dimensional plane orthogonal to the direction
n of anisotropy. As a consequence, the density of
the medium vanishes in this limit.
18Static Potential for a Quark-Antiquark pair
IV. nonzero but small anisotropy
the analytic result of the potential in small x
approximation can be expressed as
this expression does not apply for rmD much
larger than 1, which is a shortcoming of the
direct Taylor expansion of the potential in
powers of ?.
unlike the isotropic potential, the anisotropic
potential depends not only on the modulus of r ,
but also on the angle between r and p. To
simplify the angular dependence, we consider the
following two cases.
19Static Potential for a Quark-Antiquark pair
- for r parallel to the direction of anisotropy
- for r perpendicular to the direction of
anisotropy
with
for rmD ? 1, the coefficient of ? is positive, (
) 0.27 for rmD 1, and thus a slightly
deeper potential than in an isotropic plasma
emerges at distance scales r 1/mD.
for rmD ? 1, the coefficient of ? is positive
again, ( ) 0.115 for rmD 1, but smaller
than the case where r is parallel to n.
quark-antiquark pair aligned along the direction
of momentum anisotropy and separated by a
distance r 1/mD is expected to attract more
strongly than a pair aligned in the transverse
plane.
For general ? and r , the integral in the
potential expression has to be performed
numerically.
20Static Potential for a Quark-Antiquark pair
Numerical results for general x I
Heavy-quark potential at leading order as a
function of distance for r parallel to the
direction of anisotropy.
Left the potential divided by the Debye mass and
by the coupling Right potential relative to that
in vacuum.
21Static Potential for a Quark-Antiquark pair
Numerical results for general x II
angular dependence of the potential r parallel
to the direction of anisotropy vs. r
perpendicular to the direction of anisotropy for
different x
22Static Potential for a Quark-Antiquark pair
Comparison with lattice results
From the lattice results, the potential can be
modeled as
when
with
See A. Mocsy and P. Petreczky, Phys. Rev. Lett.
99, 211602 (2007)
For low T
For large T (T 2 TC )
- In this regime around rmed , quarkonium states
are either unaffected by the medium
- Coulomb contribution dominates
- for states with a root-mean square radius larger
than rmed , it is insufficient to consider only
the (screened) Coulomb-part of the potential
which arises from one-gluon exchange
- our result is directly relevant for quarkonium
states with wavefunctions which are sensitive to
the length scale l rmed
23Summary 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
from the Fourier transform of the static gluon
propagator
- In general screening effect is reduced in an
anisotropic system, the potential is deeper and
closer to the vacuum potential than for an
isotropic medium. (partly caused by the lower
density of the anisotropic plasma.)
- Angular dependence appears in an anisotropic
system, the potential is closer to that in
vacuum, if the quark pair is aligned along the
direction of anisotropy.
24Summary outlook
- Detailed numerical solutions of the Schrödinger
equation in our anisotropic potential is
important which can be used to determine the
binding energy of the heavy quark pair.
- It is also worthwhile to consider the imaginary
part of the potential in an anisotropic system
which gives the damping rate.
25Thank You for Your Attention