1. Daniel Gómez Dumm IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas Universidad de La Plata, Argentina Issues on nonlocal chiral quark models

2. Plan of the talk Motivation Description of two-flavor nonlocal chiral quark models Phase regions in the T –  plane Phase diagram under neutrality conditions Extension to three flavors Description of deconfinement transition Meson masses at finite temperature Summary

3. Motivation The understanding of the behaviour of strongly interacting matter at finite temperature and/or density is a subject of fundamental interest Cosmology (early Universe) Astrophysics (neutron stars) RHIC physics Several important applications : QCD Phase Diagram

4. Essential problem: one has to deal with strong interactions in nonperturbative regimes. Full theoretical analysis from first principles not developed yet Lattice QCD techniques – Difficult to implement for nonzero chemical potentials Effective quark models – Systematic inclusion of quark couplings satisfying QCD symmetry properties Nambu  Jona-Lasinio (NJL) model: most popular theory of this type. Local scalar and pseudoscalar four-fermion couplings + regularization prescription (ultraviolet cutoff) NJL (Euclidean) action Two main approaches: Nambu, Jona-Lasinio, PR (61)

5. Several advantages over the standard NJL model: No need to introduce sharp momentum cut-offs Relatively low dependence on model parameters Consistent treatment of anomalies Successful description of low energy meson properties A step towards a more realistic modeling of QCD: Extension to NJL  like theories that include nonlocal quark interactions In fact, the occurrence of nonlocal quark couplings is natural in the context of many approaches to low-energy quark dynamics, such as the instanton liquid model and the Schwinger-Dyson resummation techniques. Also in lattice QCD. Bowler, Birse, NPA (95); Ripka (97)

6. Nonlocal chiral quark model Euclidean action: Theoretical formalism (two active flavors, isospin symmetry) m c : u, d current quark mass; G S : free model parameter (based on OGE interactions) g (z) : nonlocal, well behaved covariant form factor j a (x) : nonlocal quark-quark current

7.  Hubbard-Stratonovich transformation: standard bosonization of the fermion theory. Introduction of bosonic fields  and  i  Mean field approximation (MFA) : expansion of boson fields in powers of meson fluctuations Further steps:  Minimization of S E at the mean field level gap equation where Momentum-dependent effective mass Lattice (Furui et al., 2006) Gaussian fit Lorentzian fit Lattice QCD : (also presence of a wave function form factor)

8. Beyond the MFA : low energy meson phenomenology where Quark condensate :, Pion mass from Pion decay constant from – nontrivial gauge transformation due to nonlocality – General, DGD, Scoccola, PLB(01); DGD, Scoccola, PRD(02); DGD, Grunfeld, Scoccola, PRD (06)

9. Numerics Model inputs : Fit of m c, G S and  so as to get the empirical values of m  and f   Parameters : G S, m c  Form factor & scale Gaussian n- Lorentzian G S  2 = 18.78 m c = 5.1 MeV  = 827 MeV Gaussian, covariant form factor : Consistency with ChPT results in the chiral limit : GT relation GOR relation    coupling

10. Phase transitions in the T –  plane Extension to finite T and  : partition function obtained through the standard replacement, with Possibility of quark-quark condensates: Inclusion of diquark interactions where New effective coupling Bosonization : quark-quark bosonic excitations additional bosonic fields  a – in principle, possible nonzero mean field values   2,  5,  7 – Breakdown of color symmetry – Arbitrary election of the orientation of  in SU(3) C space (residual SU(2) C symmetry) Obtained trhough Fierz transformation – then H / G S = 0.75 (quark-quark current)

11. Assumption : G S, H, independent of T and  Grand canonical thermodynamical potential (in the MFA) given by S  : Inverse of the propagator, 48 x 48 matrix in Dirac, flavor, color and Nambu-Gorkov spaces ( 4 x 2 x 3 x 2 ) Low energy physics not affected by the new parameter H parameter fit unchanged “Reasonable” values for the ratio H / G S in the range between 0.5 and 1.0 (Fierz : H / G S = 0.75) Large , low T : nonzero mean field value  – two-flavor superconducting phase (2SC) paired quarks uuuuuu dddddd unpaired Pairing between quarks of different colors and flavors : Alford, Rajagopal, Wilczek, NPB (99)

12. 1 st order transition 2 nd order transition crossover EP End point 3P Triple point ( H / G S = 0.75 ) T = 100 MeV : CSB – NQM crossover Behavior of MF values of qq and qq collective excitations ( T  0, 70 MeV, 100 MeV) T = 0 : 1 st order CSB – 2SC phase transition T = 70 MeV : 2 nd order NQM – 2SC phase transition Typical phase diagram : Duhau, Grunfeld, Scoccola, PRD (04)

13. Low  : chiral restoration shows up as a smooth crossover Peaks in the chiral susceptibility – T c (  = 0) ~ 120 - 140 MeV – somewhat low … T c ~ 160 - 200 MeV from lattice QCD – DGD, Grunfeld, Scoccola, PRD (06) Increasing  : End Point Figure: ( , T ) EP = (200 MeV, 80 MeV) -- Strongly model-dependent

14. Application to the description of compact star cores Color charge neutrality Compact star interior : quark matter + electrons Electrons included as a free fermion gas, Electric charge neutrality where Need to introduce a different chemical potential for each fermion flavor and color (no gluons in this effective chiral quark model) (  i for i = 1, … 7 trivially vanishing ) with,

15. Beta equilibrium (no neutrino trapping assumed) Quark – electron equilibrium through the reaction Residual color symmetry : not all chemical potentials are independent from each other (only two independent chemical potentials needed,  e and  8 ) Procedure: find values of , ,  e and  8 that satisfy the gap equations for  and  together with the color and electric charge neutrality conditions Numerical results: phase diagrams for neutral quark matter DGD, Blaschke, Grunfeld, Scoccola, PRD(06) Mixed phase : global electric charge neutrality( 2SC – NQM coexisting at a common pressure )

16. Hybrid compact star models: are they compatible with observations? Modern compact star observations : stringent constraints on the equation of state for strongly interacting matter at high densities Two-phase description of hadronic matter Nuclear matter EoS Quark matter EoS NJL model : relatively low compact star masses – more stiff EoS needed Nuclear matter : Dirac-Brückner-Hartree-Fock model Quark matter : nonlocal chiral quark model + vector-vector coupling Our approach (nonlocal quark-quark currents) Mass vs. radius relations obtained from Tolman- Oppenheimer-Volkoff equations of general-relativistic hydrodynamic stability for self-gravitating matter Compact stars with quark matter cores not ruled out by observations Blaschke, DGD, Grunfeld, Klähn, Scoccola, PRC(07); EPJA(07)

17. Extension to three flavors : SU(3) f symmetry Nonlocal scalar quark-antiquark coupling + six-fermion ‘t Hooft interaction where Currents given by a = 0, 1,... 8  Momentum-dependent effective quark masses  q ( p ), q = u, d, s  u, d and s quark-antiquark condensates  Bosonic fields , K,  0,  8, a 0, ,  0,  8 Phenomenology (MFA + large N C ) :

18. Model parameters G, H’, , m u, m s (  gaussian form factor choice ) Numerical results : meson masses, decay constants and mixing angles Scarpettini, DGD, Scoccola, PRD (04); Contrera, DGD, Scoccola, PRD (10) Our inputs: m u, m , m K, m  ’, f  Adequate overall description of meson phenomenology  8 –  0 sector : two mixing angles Four decay constants Current algebra: NLO ChPT + 1/Nc :

19. Quarks coupled to a background color field SU(3) C gauge fields Traced Polyakov loop Taken as order parameter of deconfinement transition, related with Z(3) center symmetry of color SU(3): Gauge field potential Group theory constraints satisfied – a ( T ), b ( T ) fitted from lattice QCD results Fukushima, PLB(04); Megias, Ruiz Arriola, Salcedo, PRD(06); Roessner, Ratti, Weise, PRD(07) Extension to finite T  description of deconfinement transition Polyakov gauge :  diagonal, From QCD symmetry properties, assuming that fields are real-valued, one has Confinement, deconfinement T = 0 :, color field decouples

20. Chiral restoration & deconfinement :  u and  as functions of the temperature ( 3 flavors )  SU(2) chiral transition temperature increased up to 200 MeV (in agreement with lattice QCD results)  Deconfinement transition (smoother)  Both chiral and deconfinement transition occurring at approximately same temperature Main qualitative features : Contrera, DGD, Scoccola, PLB (08) MFA : Grand canonical thermodynamical potential given by (coupling to fermions) finite T : sum over Matsubara frequencies MF values from

21. Beyond mean field: quadratic fluctuations at finite T (pseudoscalar sector) Here, while are bosonic Matsubara frequencies Functions G given by loop integrals, e.g. where with Meson masses and decay constants given by

22. Finite T : meson masses and mixing angles “Ideal” mixing  Masses dominated by thermal energy at large T  Mass degeneracy of scalar – pseudoscalar partners  Matching at T = T c for nonstrange mesons  Regions where G M (  p 2,0) show no real zeros  “Ideal” mixing at large T Main qualitative features : Contrera, DGD, Scoccola, PRD (10)

23. We have studied quark models that include effective covariant nonlocal quark-antiquark and quark-quark interactions, within the mean field approximation. These models can be viewed as an improvement of the NJL model towards a more realistic description of QCD Summary  GT and GOR chiral relations satisfied. Pion decay to two photons properly described.  Good description of low energy scalar and pseudoscalar meson phenomenology.  Extension to finite T and , with the inclusion of quark-quark interactions – SU(2) case  = 0 : chiral transition (crossover) at relatively low T c (120 – 130 MeV)  Phase diagram for finite T and  : CSB, NQM and 2SC phases.  Neutral matter + beta equilibrium : need of color chemical potential  8. Hybrid star model compatble with compact star observations.  Coupling with the Polyakov loop increases T c up to 200 MeV. Chiral restoration and deconfinement occurr in the same temperature range.  Behavior of meson masses with temperature: scalar and pseudoscalar chiral partners become degenerate right after the chiral restoration. Ideal mixing at large T.

24. Extension of the phase diagram to higher  – Inclusion of strangeness (CFL phases) Many possibilites of quark pairing Extension of Polyakov loop model for finite chemical potential Form factors from Lattice QCD – effective mass & wave function form factors Description of vector meson sector... To be done Final look of the full phase diagram ? NJL

25. VS. Vamos pra revanche!!