Thermodynamics of Quasi-Particles

Fernanda Steffens

Mackenzie – São Paulo

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 convergentValid 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

What is the thermodynamics of quasi-particles?

Originally: Gorenstein and Yang – PRD 52 (1995) 5206Follow up: Peshier, Cassing, Kampfer, Blaizot, Rebhan, Weise, Bluhm, etc

Peshier et al. PRD 54 (1996) 2399

Goal: To calculate thermodynamics functions that reproduce the data from lattice QCD and the results from perturbative QCD at large T and/or

What about finite chemical potential? Peshier et al., PRC 61 (2000) 045203 Thaler, Schneider, Weise, PRC 69 (2004) 035210 Bluhm et al., PRC 76 (2007) 034901

Thermodynamics in a grand canonical ensemble

If the mass is independent of T and , 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 thethermodynamics relations

Generalization

Extra term forcesa 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 unchangedOriginally developed for =0

Solution of the type Gorenstein – Yang

Extension to finite : Peshier, Cashing, etc

GY1 Solution

Set = , Entropy unchangedInternal energy unchanged

SimplerSmaller number of constants

Other solutions of the kind Gorenstein – Yang? Yes

GY2 Solution

This solution allows us to write explicit expressions for the thermodynamicsfunctions

Reduced entropy: s’(T,) – s’(T,0)

HTL mass was used

Number density Pressure

Comparison to lattice QCD

Unpublished

HTL = Hard Thermal Loop – loops dominated by k~T

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 andperturbative 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

HTL mass in NLO was used, and

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

Feliz aniversário, Tony!

E obrigada pela sua amizade e por todo o resto!!!!