what does htl lose?

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