Thermodynamics of Quasi-

Particles

Fernanda Steffens

UPM

Collaboration with F. G. Gardim

Hadronic Matter New State, dominated by degrees of freedom

of quarks and gluons

Lattice QCD: Phase transition at Tc. Stephan-Boltzmann limit at very large T

Perturbative QCD: up to order gs6 ln(1/gs) – Kajantie et al. PRD67:105008, 2003

Series is weakly convergent

Valid only for T ~ 105Tc

Resum: Hard Thermal Loops effective action

Andersen,Strickland, Annals Phys. 317: 281, 2005

2-loop derivable approximation

Blaizot, Iancu, Rebhan, Phys. Rev. D63:065003, 2001

Region close to Tc: quasi-particles?

Quasi-Particles: modified dispersion relations

Quark and gluon masses dependent on the

temperature T and/or the chemical potential

Goal: To calculate thermodynamics functions that reproduce the data

from lattice QCD and the results from perturbative QCD at large

T and/or

Thermodynamics in a grand canonical ensemble

If the mass is independent of

T e de , then the grand potential

Partition Function

= - T lnZ V; T)

However, in general:

Not zero if H depends

on T and on

The extra terms lead to an inconsistency in the

thermodynamics relations

Generalization

Extra term forces

a consistent formulation

With

What is the meaning of B?

Quantum interpretation

Density Operator

The internal energy:

Zero point energy

For T=0, we subtract the zero point energy

For finite T (and ), the dispersion relation depends on T

So does the zero point energy

It can not be subtracted

is the energy of the system in the absence of quasi-particles

The lowest energy of the system

The thermodynamics functions of the system are then

From all possible solutions, which ones are physically relevant?

= 0 Entropy unchanged

Originally developed for =0

Solution of the type Gorenstein – Yang

Extension to finite : Peshier, Cashing, etc

GY1 Solution

Set = , Entropy unchanged

Internal energy unchanged

Simpler

Smaller number of constants

Other solutions of the kind Gorenstein – Yang? Yes

GY2 Solution

This solution allows us to write explicit expressions for the thermodynamics

functions

Reduced entropy: s’(T, ) – s’(T,0)

HTL mass was used

Number density Pressure

Comparison to lattice QCD

Unpublished

What about perturbative QCD at T >> Tc ? (HTL mass)

GY1 Solution

GY2 Solution

QCD

Both solutions fail!!

FG,FMS, NP A825: 222, 2009

Is there a solution that reproduces both, lattice QCD and

perturbative QCD?

YES

Solution with = 0

Doing the integrals...

And similar for the entropy density, energy density and number density...

Lattice data:

FG,FMS, NP A825: 222, 2009

Factor of 1/2! Disagreement:

Hard Thermal Loop (HTL) masses were used

Redefinition of the mass:

And agreement is found with both pQCD and Lattice QCD...

Main points:

• General formulation of thermodynamics consistency for a system whose

masses depend on both T and

• Multiple ways to obtain consistency

• First explicit calculation of the thermodynamics functions

• Good agreement with lattice QCD with a smaller number of free

parameters

• Possible agreement with perturbative QCD and lattice QCD for finite T

and for a particular solution

• The usual quasi-particle approach (Gorenstein-Yang) does not reproduce

perturbative QCD and lattice QCD at finite chemical potential

• Single framework to study a large portion of the T plane