1. XXXI Bienal de la RSEF, Granada, España, 10-14 septiembre 2007. Angel Gómez Nicola Universidad Complutense Madrid COEFICIENTES DE TRANSPORTE EN UN GAS DE PIONES

2. Hadronic matter at high Temperature and Density Heavy Ion Collisions:  (fm/c) T (MeV) 5-10 120-200  300-400 50 ¿QGP? HADRON GAS ( (  K  PERTURBATIVE QCD and quarks and gluons EFFECTIVE LOW-ENERGY MODELS Light hadrons

3. Hadronic matter at high Temperature and Density Heavy Ion Collisions: QCD phase diagram

4. RELEVANT D.O.F AT LOW ENERGIES E<<    1 GeV “NGB” are the lightest pseudoscalars ( m q  0  m BG  0) N f = 2:  ’s (3) N f = 3:  ’s (3), K’s (4),  QCD with N f massless quarks is chiral invariant: Chiral Symmetry of QCD: Spontaneously Broken at T=0 (no chiral degeneracy of SU(N f ) multiplets): (ISOSPIN FOR N f =2)

5. T similarly to a ferromagnet with Chiral Symmetry is restored at T Chiral Symmetry is restored at T c  150-200 MeV The order parameter is the quark condensate: free energy density

6. Chiral Perturbation Theory: Relevant for low and moderate temperatures below Chiral SSB Weinberg’s chiral power counting: NLSM Most general derivative and mass expansion of NGB mesons compatible with the SSB pattern of QCD model-independent low-energy predictions.

7. Transport coefficients of a pion gas in ChPT: D.Fernández-Fraile,AGN: PRD73:045025,2006, NPA785:166-169,2007, EPJA31:848,2007 TC measure the response of the system to restore equilibrium after the action of an external force coupled to a conserved current. Diagramatically, nonperturbative corrections needed to account for the inverse collisional width dependence (“pinching poles”): as expected from Kinetic Theory  Im Linear Response (Kubo formula) in terms of retarded correlators: In equilibrium at temperature T

8. In the dilute gas regime: Data up to 1 GeV.  (770), f 0 (600) Exactly Unitary.

9. In the dilute gas regime: Data up to 1 GeV.  (770), f 0 (600) Exactly Unitary.

10. Transport coefficients: revising chiral counting Identify diagrams with larger powers of Y: Ladders Bubbles “Double lines” with equal momentum carry  ≠0 propagators and carry a “nonperturbative” factor Y for chiral loops:

11. Potentially large Detailed analysis of the effective ladder vertex Loop spectral function ~ T dependence factorizes PERTURBATIVE IN CHPT

12. Unitarization is important ! Ladders still formally ChPT perturbative. Unitarity gives larger T-dependence.

13.    0 and resonances in  scattering important near the origin ! W.Liu, R.Rapp nucl-th/0604031

15. Sound attenuation length (related to elliptic flow): Results for shear and bulk viscosities

16. Sound attenuation length (related to elliptic flow): Results for shear and bulk viscosities

18. The leading order gives rigourously the T<<m  behaviour, consistently with nonrelativistic Kinetic Theory. Transport coefficients in ChPT require a modification of the standard chiral power counting to account for O(1/   ) collision effects. Unitarity is crucial to ensure the correct high-T behaviour. As T  T c ladder-type diagrams may have to be sumed, although results suggest strong cancellations. Kaons, chemical potentials … At higher temperatures, the LO agrees reasonably with KT analysis and phenomenological/theoretical results.

20. Analytically continued to s   yields correct  (770) and f 0 (600) poles. Exactly Unitary. t IAM  t (2) + t (4) +... matches the low-energy ChPT predictions. Reproduces scattering data up to  s  1 GeV, much beyond ChPT. Can be extended to coupled channels in SU(3) and to finite temperature. J.A.Oller, E.Oset,J.R.Peláez, 1998&1999 F.Guerrero,J.A.Oller, 1999 AGN,J.R.Peláez 2002 A.Dobado, F.J.LLanes-Estrada, AGN,J.R.Peláez 2002 T.N.Truong, 1988&1991 A.Dobado, M.J.Herrero,T.N.Truong, 1990 A.Dobado,J.R.Peláez, 1993&1997 The Inverse Amplitude Method: