XXXI Bienal de la RSEF, Granada, España, septiembre Angel Gómez Nicola Universidad Complutense Madrid COEFICIENTES DE TRANSPORTE EN UN GAS DE PIONES

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

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

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)

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

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.

Transport coefficients of a pion gas in ChPT: D.Fernández-Fraile,AGN: PRD73:045025,2006, NPA785: ,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

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

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

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:

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

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

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

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

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

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.

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: