Thermodynamics of Quasi-Particles

1. Thermodynamics of Quasi-Particles Fernanda Steffens Mackenzie – São Paulo Collaboration with F. G. Gardim

2. Hadronic Matter New State, dominated by degrees of freedom of quarks and gluons

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

4. 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 F derivable approximation Blaizot, Iancu, Rebhan, Phys. Rev. D63:065003, 2001 Region close to Tc: quasi-particles?

5. Quasi-Particles: modified dispersion relations Quark and gluon masses dependent on the temperature T and/or the chemical potential m What is the thermodynamics of quasi-particles? Originally: Gorenstein and Yang – PRD 52 (1995) 5206 Follow up: Peshier, Cassing, Kampfer, Blaizot, Rebhan, Weise, Bluhm, etc

6. Peshier et al. PRD 54 (1996) 2399

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

8. Thermodynamics in a grand canonical ensemble F = - T lnZ(m; V; T) Partition Function If the mass is independent of T and m, then Ff:the grand potential

9. However, in general: Not zero if H depends on T and on m The extra terms lead to an inconsistency in the thermodynamics relations

10. Generalization Extra term forces a consistent formulation With

11. What is the meaning of B? Quantum interpretation Density Operator The internal energy: Zero point energy

12. For T=0, we subtract the zero point energy For finite T (and m), 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

13. The thermodynamics functions of the system are then From all possible solutions, which ones are physically relevant?

14. Solution of the type Gorenstein – Yang Entropy unchanged Originally developed for m=0 g = 0 Extension to finite m: Peshier, Cashing, etc GY1 Solution

15. Other solutions of the kind Gorenstein – Yang? Yes Entropy unchanged Internal energy unchanged Set a = l, h = g = 0 GY2 Solution Simpler Smaller number of constants

16. This solution allows us to write explicit expressions for the thermodynamics functions Reduced entropy: s’(T,m) – s’(T,0)

17. Comparison to lattice QCD Number density Pressure HTL mass was used HTL = Hard Thermal Loop – loops dominated by k~T Unpublished

18. What about perturbative QCD at T >> Tc ? (HTL mass) GY1 Solution GY2 Solution QCD Both solutions fail!! FG,FMS, NP A825: 222, 2009

19. Is there a solution that reproduces both, lattice QCD and perturbative QCD? YES

20. Solution with a = 0

21. Doing the integrals... And similar for the entropy density, energy density and number density...

22. HTL mass in NLO was used, and Lattice data: FG,FMS, NP A825: 222, 2009

23. What about perturbative QCD? Disagreement: Factor of 1/2!

24. Hard Thermal Loop (HTL) masses were used Redefinition of the mass: And agreement is found with both pQCD and Lattice QCD...

25. Main points: • General formulation of thermodynamics consistency for a system whose • masses depend on both T and m • 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 m 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 Tm plane

26. Feliz aniversário, Tony! E obrigada pela sua amizade e por todo o resto!!!!