Daniel Gómez Dumm IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas

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

2. Plan of the talk • Motivation • Description of two-flavor nonlocal chiral quark models • Phase regions in the T – m 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 Two main approaches: • 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 Nambu, Jona-Lasinio, PR (61)

5. A step towards a more realistic modeling of QCD: Extension to NJL-like theories that include nonlocal quark interactions Bowler, Birse, NPA (95); Ripka (97) 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. 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

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

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

8. Quark condensate : Beyond the MFA : low energy meson phenomenology where , 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. GS L2 = 18.78 mc = 5.1 MeV L = 827 MeV • GT relation • GOR relation • p0gg coupling Consistency with ChPT results in the chiral limit : Numerics Model inputs : • Parameters : GS , mc • Form factor & scale Gaussian n-Lorentzian Fit of mc , GS and L so as to get the empirical values of mp and fp Gaussian, covariant form factor :

10. Bosonization : quark-quark bosonic excitations additional bosonic fields D a Phase transitions in the T – m plane Extension to finite T and m : partition function obtained through the standard replacement , with Possibility of quark-quark condensates: Inclusion of diquark interactions New effective coupling where (quark-quark current) Obtained trhough Fierz transformation – then H / GS = 0.75 – in principle, possible nonzero mean field values D2 , D5 , D7 – Breakdown of color symmetry – Arbitrary election of the orientation of D in SU(3)C space (residual SU(2)C symmetry)

11. Large m, low T : nonzero mean field value D – two-flavor superconducting phase (2SC) Grand canonical thermodynamical potential (in the MFA) given by S -1 : Inverse of the propagator, 48 x 48 matrix in Dirac, flavor, color and Nambu-Gorkov spaces ( 4 x 2 x 3 x 2 ) Assumption : GS , H, independent of T and m Low energy physics not affected by the new parameter H parameter fit unchanged “Reasonable” values for the ratio H / GSin the range between 0.5 and 1.0 (Fierz : H / GS= 0.75) paired quarks unpaired Pairing between quarks of different colors and flavors : uuu ddd Alford, Rajagopal, Wilczek , NPB (99)

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

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

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

15. Beta equilibrium Quark – electron equilibrium through the reaction (no neutrino trapping assumed) Residual color symmetry : not all chemical potentials are independent from each other (only two independent chemical potentials needed, me and m8) Procedure: find values of D , s , me and m8 that satisfy the gap equations for D and s together with the color and electric charge neutrality conditions Numerical results: phase diagrams for neutral quark matter Mixed phase : global electric charge neutrality ( 2SC – NQM coexisting at a common pressure ) DGD, Blaschke, Grunfeld, Scoccola, PRD(06)

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 Nuclear matter EoS Two-phase description of hadronic matter Quark matter EoS Mass vs. radius relations obtained from Tolman-Oppenheimer-Volkoff equations of general-relativistic hydrodynamic stability for self-gravitating matter NJL model : relatively low compact star masses – more stiff EoS needed Nuclear matter : Dirac-Brückner-Hartree-Fock model Our approach Quark matter : nonlocal chiral quark model + vector-vector coupling (nonlocal quark-quark currents) 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 a = 0, 1, ... 8 Currents given by Phenomenology (MFA + large NC) : • Momentum-dependent effective quark masses Sq(p) , q = u , d , s • u, dand s quark-antiquark condensates • Bosonic fields p , K , h0 , h8 , a0 , k , s0 , s8

18. Numerical results : meson masses, decay constants and mixing angles Model parameters G , H’ , L , mu , ms ( + gaussian form factor choice ) Our inputs: mu , mp , mK , mh’ , fp h8 – h0 sector : two mixing angles Four decay constants NLO ChPT + 1/Nc : Current algebra: Adequate overall description of meson phenomenology Scarpettini, DGD, Scoccola, PRD (04); Contrera, DGD, Scoccola, PRD (10)

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

20. Chiral restoration & deconfinement : su and F as functions of the temperature ( 3 flavors ) MFA : Grand canonical thermodynamical potential given by (coupling to fermions) finite T : sum over Matsubara frequencies MF values from Main qualitative features : • 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 Contrera, DGD, Scoccola, PLB (08)

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 Main qualitative features : • Masses dominated by thermal energy at large T • Mass degeneracy of scalar – pseudoscalar partners • Matching at T = Tc for nonstrange mesons • Regions where GM(-p2,0) show no real zeros • “Ideal” mixing at large T “Ideal” mixing Contrera, DGD, Scoccola, PRD (10)

23. Summary 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 • 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 m, with the inclusion of quark-quark interactions – SU(2) case • m = 0 : chiral transition (crossover) at relatively low Tc(120 – 130 MeV) • Phase diagram for finite T and m : CSB, NQM and 2SC phases. • Neutral matter + beta equilibrium : need of color chemical potential m8. Hybrid star model • compatble with compact star observations. • Coupling with the Polyakov loop increases Tc 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. To be done • Extension of the phase diagram to higher m – 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 • . . . NJL Final look of the full phase diagram ?

25. VS. Vamos pra revanche!!