Heavy Quark Potential in an Anisotropic (Viscous) Plasma

1. Heavy 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]

2. Outlines： • 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 + - • in vacuum: Solving the equation with the boundary condition that the potential vanishes at infinity produce the Coulomb potential + - • in plasma (isotropic): 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 • in vacuum: Vacuum gauge boson propagator : Coulomb potential : • in plasma (isotropic): 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)

6. Hard-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

7. 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 Gluon field strength tensor Fluctuating part of the parton densities colorless part of the parton densities

8. 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

9. Gluon Propagator in an Anisotropic Plasma From isotropy to anisotropy fiso is the general Fermi-Dirac or Bose-Einstein distribution function andthe parameter ξ 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:

10. Gluon 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

11. Gluon 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

12. Gluon 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.

13. Static 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.

14. Static 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 p·n and p-n(p·n), respectively.

15. Static 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

16. Static 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| → ∞ 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

17. Static Potential for a Quark-Antiquark pair III. extreme anisotropy The same potential emerges for extreme anisotropy since allmass scales approach to0 asξ → ∞ the potential coincides with the vacuum Coulomb potential due to the fact that atξ = ∞ 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.

18. Static 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 rmDmuch larger than 1, which is a shortcoming of the direct Taylor expansion of the potential in powers of ξ. 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.

19. Static 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 ξ 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 ξ 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 ξ and r , the integral in the potential expression has to be performed numerically.

20. Static 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.

21. Static 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

22. Static 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

23. 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 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.

24. Summary & 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.

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