Heavy quark potential in an anisotropic plasma

We determine the real part of the heavy-quark potential in the nonrelativistic limit, at leading order, from the Fourier transform of the static gluon propagator

Some limiting cases: When , we recover the isotropic Debye Screening potential

When for an arbitrary , the potential coincides with the vacuum Coulomb potential

When is infinite large for the extreme anisotropy, the same potential emerges

Numerical results

Conclusions

Our results are applicable when the momentum of the exchanged gluon is on the order of theDebye mass or higher, i.e. for distances on the order of Debye length or less. For realistic values of the coupling, , the Debye length is approximately equal to the typical length . 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. We may therefore expect that quarkonium states whose wave-functions are sensitive to the regime are bound more strongly in an anisotropic medium, in particular if the quark-antiquark pair is aligned along n. According to the potential model, at high temperature, our result is directly relevant for quarkonium states with wave-functions which are sensitive to the length scale . For lower temperature, quarkonium states whose typical momentum component in the wave-functions is much large than are either unaffected by the medium.

Bibliography

[1] A. Jakovac, P. Petreczky, K. Petrov and A. Velytsky, Phys. Rev. D 75, 014506 (2007) [arXiv:hep-lat/0611017]; G. Aarts, C. Allton, M. B. Oktay, M. Peardon and J. I. Skullerud, Phys. Rev. D 76, 094513 (2007) [arXiv:0705.2198 [hep-lat]].[2] A. Mocsy and P. Petreczky, Phys. Rev. D 73, 074007 (2006) [arXiv:hep-ph/0512156].[3] A. Mocsy and P. Petreczky, Phys. Rev. Lett. 99, 211602 (2007) [arXiv:0705.2559 [hep-ph]].[4] A. Dumitru, Y. Guo, and M. Strickland, [arXiv:0711.4722 [hep-ph]].[5] S. Mrowczynski and M. H. Thoma, Phys. Rev. D 62, 036011 (2000) [arXiv:hep-ph/0001164].[6] P. Romatschke and M. Strickland, Phys. Rev. D 68, 036004 (2003) [arXiv:hep-ph/0304092].

Fig. 2. Comparison of and

Adrian Dumitru †, Yun Guo ‡ and Michael Strickland †

† Institut für Theoretische Physik, J.W. Goethe-Universität, D-60438 Frankfurt am Main, Germany

‡ Helmholtz Research School, J.W. Goethe-Universität, D-60438 Frankfurt am Main, Germany

The heavy-quark potential in an anisotropic plasma

Hard-thermal-loop self-energy in an anisotropic plasma

The retarded gauge-field self-energy in the hard-loop approximation is given by [5]

Here, is a light-like vector describing the propagation of a plasma particle in space-time. The self-energy is symmetric and transverse.

Decomposition of the self-energy

In a suitable tensor basis the components of the self-energy can be determined explicitly. For anisotropic systems, we introduce the four-tensor basis appropriate for use in general covariant gauges. Specifically,

with

Here, is the heat-bath vector, which in the local rest frame is given by . The direction of anisotropy in momentum space is determined by the vector , where is a three-dimensional unit vector. The self-energy can now be written as

Anisotropic distribution

In order to determine the four structure functions explicitly we need to specify the phase-spacedistribution function. We employ the following ansatz:Thus, is obtained from an isotropic distribution by removing particles with a largemomentum component along . In the expression of the anisotropic distribution, we use the parameter to determine the degree of the anisotropy.

Calculation of the four structure functions

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 of self-energy tensor

with

and employ the following contractions:

Then, the four structure functions and can be determined analytically [6].

Gluon propagator in an anisotropic plasma

In covariant gauge, the inverse propagator, by definition, equals to with the gauge parameter . Upon inversion, we can determine the anisotropic propagator which has a following form

with

Introduction

Information on quarkonium spectral functions at high temperature has started to emerge from lattice-QCD simulations [1]. The information from lattice simulations has motivated a number of attempts to understand the lattice measurements within non-relativistic potential models including finite temperature effects such as screening [2]. According to the potential model [3], when the distance between quark pair is smaller than some typical length, the potential can be expressed as when

At sufficiently high temperature, roughly speaking, when , the perturbative Coulomb contribution dominates over the linear confining potential. At low temperature, states with a root-mean square radius larger than the typical length do experience medium modifications. For such states, one should then sum the medium-dependent contributions due to one-gluon exchange (Coulomb-part) and due to the string (linear-part). Using the Hard Thermal Loop (HTL) approximation, we derive the gluon propagator in an anisotropic system. The heavy-quark potential (perturbative Coulomb part) can be determined from the Fourier transform of the static gluon propagator. This work is a first attempt to consider the effects due to a local anisotropy of the plasma in momentum space on the heavy-quark potential [4].

Abstract

We determine the hard-loop resummed propagator in an anisotropic QCD plasma in general covariant gauges and define a potential between heavy quarks from the Fourier transform of its static limit. We find that there is stronger attraction on distance scales on the order of the inverse Debye mass for quark pairs aligned along the direction of anisotropy than for transverse alignment.

Fig. 1. Heavy-quark potential at leading order as a function of distance ( ) for r parallel to the direction n of anisotropy. The anisotropy parameter of the plasma is denoted by . Left: the potential divided by the Debye mass and by the coupling, Right: potential relative to that in vacuum.