What does HTL lose?What does HTL lose?What does HTL lose?What does HTL lose?

Hui Liu Hui Liu Jinan University, ChinaJinan University, China

Hui Liu, SQM2008 Beijing, October 6-10 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

Hui Liu, SQM2008 Beijing, October 6-10 3

Why not HTL• Some facts in RHIC experiment

Lattice result, hep-lat/0011006v1

• Strongly coupled? Does HTL still work?― What information does HTL approximation

lose compared to the complete loop?

ST

AR

collab

oratio

n, P

RL

89 (2002

) 132301

Hui Liu, SQM2008 Beijing, October 6-10 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

Hui Liu, SQM2008 Beijing, October 6-10 5

Comparison

• Self-energy of toy model QED

– HTL

– Complete one loop (C1L)

Q

P

Hui Liu, SQM2008 Beijing, October 6-10 6

• Dispersion relations– Solve the equation of dispersion and find out t

he relation between ω and q

The main difference between the two curves is the appearance of a threshold frequency on the C1L curve.

HTL

C1L

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.

q=iqi

Hui Liu, SQM2008 Beijing, October 6-10 7

Dynamical screening

• Dynamical screening regime– Below the plasma frequency, the modes are

complex, which can be signaled by the screening of external charges .

cω

Above , is consis- tent zero. While below it has values, which indicates an expectant change in physics pro-perties.

q=qr+iqi

Hui Liu, SQM2008 Beijing, October 6-10 8

Oscillatory potential

• Static limit– HTL: Purely imaginary modes

– C1L: Complex modes

• Static screening potential

where

— Debye screening— Oscillatory screening

~~~~~~~~~~~~~~

Hui Liu, SQM2008 Beijing, October 6-10 9

Radial Distribution Function

• RDF– Probability of finding two particles at a distance r– Density-density autocorrelation function

– Typical RDFs of different states of matter

solid liqui

d gas

Hui Liu, SQM2008 Beijing, October 6-10 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?

Gelman, Shuryak, and Zahed, PRC 74, 044908 (06)

Molecular dynamical simulation

Hui Liu, SQM2008 Beijing, October 6-10 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

Hui Liu, SQM2008 Beijing, October 6-10 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

0.5GeVT0.40.3

Hui Liu, SQM2008 Beijing, October 6-10 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.

Hui Liu, SQM2008 Beijing, October 6-10 14

Screening of a moving particle?

• Current-current correlation

• Static case – density correlation– Static screening potential

• Non-static case– Frequency dependent screening potential