Analyzing HTL Approximation Losses in Dispersion Relations

1. What does HTL lose? Hui Liu Jinan University, China

2. HTL approximation • Hard thermal loop (HTL) • Widely used approximation to self-energy • Advantages • Simple and convenient for further calculations • Gauge invariant • Restrictions • Requires weak coupling • Equivalent to high temperature approximation

3. Lattice result, hep-lat/0011006v1 Why not HTL • Some facts in RHIC experiment STAR collaboration, PRL 89 (2002) 132301 • Strongly coupled? Does HTL still work? • What information does HTL approximation lose compared to the complete loop?

4. Dispersion Relations • Dispersion relation • Fundamental property of a many-body system • Energy-momentum relation determined by the pole of effective propagator. ΠL: longitudinal component of self-energy ΠT: transverse component Equation of dispersion

5. Comparison • Self-energy of toy model QED • HTL • Complete one loop (C1L)

6. HTL C1L • Dispersion relations • Solve the equation of dispersion and find out the relation between ω and q q=iqi Above the plasma frequency, q is real and qi=0. While below it, q is complex. We plot ω-qi relation in the left area. The main difference between the two curves is the appearance of a threshold frequency on the C1L curve.

7. Dynamical screening • Dynamical screening regime • Below the plasma frequency, the modes are complex, which can be signaled by the screening of external charges . q=qr+iqi Above , is consis- tent zero. While below it has values, which indicates an expectant change in physics pro-perties.

8. Oscillatory potential • Static limit • HTL: Purely imaginary modes • C1L: Complex modes • Static screening potential —Debye screening —Oscillatory screening ~~~~~~~~~~~~~~ where

9. solid liquid gas Radial Distribution Function • RDF • Probability of finding two particles at a distance r • Density-density autocorrelation function • Typical RDFs of different states of matter

10. RDF of a liquid? • Which potential can result in a liquid? • RDF and the potential of “mean force” • Short range order • Indicate a liquid state? Molecular dynamical simulation Gelman, Shuryak, and Zahed, PRC 74, 044908 (06)

11. Conclusion • Physics concealed in the C1L dispersion relation might not be revealed by the HTL • Comparing the dispersion relations we found the existence of a threshold frequency in the dynamical screening regime of the C1L • Below the threshold frequency the modes contain both real and imaginary parts • In the static limit, the complex mode leads to an oscillatory screening potential, which is contrast to the Debye-like potential in the HTL case • The oscillatory potential could result in a liquid-like RDF, which might indicate the liquid QGP

12. RDF in hot QGP • Gluon polarization • RDF of QGP • Short range order. Very similar to the typical shape of liquid. Footprint of liquid QGP!? • Enhanced oscillations at lower temperatures

13. Non-zero frequencies • Frequency-dependent screening • For HTL, the potential is always Debye-like in the whole range of frequencies below the plasma frequency. • For C1L, the potential can be either Debye-like or oscillatory. • Above the threshold frequency the screening potential is Debye-like • below that frequency, the potential is oscillating.

14. Screening of a moving particle? • Current-current correlation • Static case – density correlation • Static screening potential • Non-static case • Frequency dependent screening potential