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

1. QCD phase diagram in 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 models • Phase regions in the T – m plane • Phase diagram under neutrality conditions • Extension to three flavors • Description of deconfinement transition • Summary & outlook

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: • Consistent treatment of anomalies • No need to introduce sharp momentum cut-offs • Small next-to-leading order corrections Blaschke et al., PRC (96) • Successful description of meson properties at T=  = 0 • Plant, Birse, NPA (98); Scarpettini, DGD, Scoccola, PRD (04)

6. Nonlocal chiral quark models Theoretical formalism Euclidean action: (two active flavors, isospin symmetry) mc : u, d current quark mass; GS : free model parameter js(x) : nonlocal quark-quark current M I M II Two alternative ways of introducing the nonlocality : Model I (inspired on the ILM) Model II (based on OGE interactions) r(x), g(x) : nonlocal, well behaved covariant form factors,

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 ( M I ) where , with ( M II ) Momentum-dependent effective mass S(p) Instanton liquid model (MFA) : Lattice QCD : Schaefer, Shuryak, RMP (98)

8. Quark condensate : Beyond the MFA : low energy meson phenomenology where , Pion mass from M I ¹ M II Pion decay constant from – nontrivial gauge transformation due to nonlocality – • GT relation • GOR relation • p0gg coupling Consistency with ChPT results in the chiral limit : General, DGD, Scoccola, PLB(01); DGD, Scoccola, PRD(02); DGD, Grunfeld, Scoccola, PRD (06)

9. Numerics Gaussian n-Lorentzian Model inputs : • Form factor & scale • Parameters : GS , mc Fit of mc , GS and L so as to get the empirical values of mp and fp Covariant vs. “instantaneous” models M I →GS L2 = 15.41 mc = 5.1 MeV L = 971 MeV M II → GS L2 = 18.78 mc = 5.1 MeV L = 827 MeV DGD, Grunfeld, Scoccola, PRD (06)

10. Phase transitions in the T – m plane Extension to finite T and m : partition function obtained through the standard replacement , with Inclusion of diquark interactions : new effective coupling M I where M II , a = 2, 5, 7 (Pauli principle) Assumption : GS , H, independent of T and m

11. Bosonization : quark-quark bosonic excitations additional bosonic fields D a Artificial duplication of the number of d.o.f. : Nambu – Gorkov spinors Diquark currents written as – in principle, possible nonzero mean field values D2 , D5 , D8 – Breakdown of color symmetry – Arbitrary election of the orientation of D in SU(3)C space (residual SU(2)C symmetry) 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 )

12. Peaks in the chiral susceptibility – Tc(m = 0) ~ 120 - 140 MeV – somewhat low … Tc ~ 160 - 200 MeV from lattice QCD – Low energy physics not affected by the new parameter H parameter fits unchanged “Reasonable” values for the ratio H / GSin the range between 0.5 and 1.0 (Fierz : H / GS= 0.75) M II M I Typical phase diagrams ( H / GS = 0.75 ) 1st order transition 2nd order transition crossover EP End point 3P Triple point Low m : chiral restoration shows up as a smooth crossover DGD, Grunfeld, Scoccola, PRD (06)

13. M I : EP = (80 MeV, 208 MeV) M II : EP = (235 MeV, 33 MeV) Increasing m : end point & first order chiral phase transition Large m, low T : nonzero mean field value D – two-flavor superconducting phase (2SC) paired quarks T = 0 : 1st order CSB – 2SC phase transition uuu unpaired M I ddd Behavior of MF values of qq and qq collective excitations with increasing chemical potential (T fixed) T = 100 MeV : 2nd order NQM – 2SC phase transition T = 100 MeV : CSB – NQM crossover Duhau, Grunfeld, Scoccola, PRD (04)

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 effective chiral quark models) 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 where From b-equilibrium , 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

16. Numerical results: phase diagram for neutral quark matter As in the color symmetric case, parameters obtained from low energy physics remain unchanged. Considered ratio H / GSin the range between 0.5 and 1.0 M I M II DGD, Blaschke, Grunfeld, Scoccola, PRD(06)

17. Behavior of MF values D and s and chemical potentials me and m8 as functions of the baryonic chemical potential for fixed values of T ( M I , H / GS = 0.75 ) Mixed phase : global electric charge neutrality coexisting 2SC – NQM phases at a common pressure mc (T = 0) in the 250 – 300 MeV range (larger for M II)

18. “Gapless” superconducting phases : NJL (local) model : quasiparticle dispersion relations blue quarks (unpaired) (degenerate) diquark condensates Ei = (12 quark degrees of freedom) In the region of small diquark gaps, two gapless modes Shovkovy, Huang, PLB (03) Nonlocal chiral quark models: complicated dispersion relations, same qualitative behavior – Look at the imaginary part of the poles of the (Euclidean) quark propagator • From our analysis: • Tiny regions of gapless phase close to the 2nd order 2SC – NQM transition • Size only significant for low values of H / GS • Never extends to zero T , thus not relevant for compact star physics

19. 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)

20. New bosonic fields – additional nonvanishing MF value for the isospin zero channel, w Numerical analysis : parameter set GS L2 = 23.7 , mc= 6.5 MeV , L = 678 MeV leading to a quark condensate ( T = m = 0 ) Compatibility with observations for low values of GV , H / GS close to 0.75 g = GV/ GS h = H / GS Symmetric matter : allowed region from elliptic flow data Neutral matter : constraints from compact star observations Compact stars with quark matter cores not ruled out by observations ! Blaschke, DGD, Grunfeld, Klähn, Scoccola, PRC(07); EPJA(07)

21. 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 ( M I – analogous for M II ) Phenomenology (MFA + large NC) : • Momentum-dependent effective quark masses Sq(p2) , q = u , d , s • u, dand s quark-antiquark condensates • Bosonic fields p , K , h0 , h8

22. Numerical results : values for pseudoscalar meson masses and decay constants ( M I ) (Gaussian) Model parameters G , H’ , L , mu , ms ( + choice of the form factor ) • r Adequate overall description of meson phenomenology • r • r • r • r Qualitatively similar results for Model II, and different form factors Scarpettini, DGD, Scoccola, PRD (04)

23. Finite T : chiral restoration ( m = 0 ) • Qualitative features of SU(2) chiral symmetry restoration not significantly changed by flavor mixing • Effective strange quark mass of about 650 MeV • SU(3) chiral restoration not well defined Tc (m = 0) ~ 115 MeV (too low compared with lattice results) Flavor mixing : shoulder in cs at the SU(2) chiral restoration temperature Contrera, DGD, Scoccola, arXiv:hep-ph (07)

24. Description of confinement : coupling with the Polyakov loop Quarks moving in a background color field SU(3)C gauge fields Traced Polyakov loop ( taken as order parameter of deconfinement transition ) Polyakov gauge : f diagonal , MFA : Grand canonical thermodynamical potential given by (coupling to fermions) finite T : sum over Matsubara frequencies where 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)

25. From QCD symmetry properties, assuming that f fields are real-valued, one has As usual, mean field value f 3 obtained from minimizing the thermodynamical potential Numerical results for su and F as functions of the temperature ( 3 flavors – Model I – Gaussian ) Main qualitative features : • Transition temperature increased up to 200 MeV (as suggested by lattice QCD results) • Deconfinement transition • Both chiral and deconfinement transition occurring at approximately same temperature Contrera, DGD, Scoccola, arXiv:hep-ph (07)

26. Summary & outlook 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 • Noncolality can be introduced in different ways. We have considered two possibilities, inspired in ILM and OGE-like interactions. Main qualitative results are similar in both cases • Chiral relations GT and GOR are satisfied. Pion decay to two photons is properly described. • A reasonably good description of low energy meson phenomenology is obtained, even with the inclusion of strangeness and flavor mixing • In general, results not strongly dependent on form factor shapes. Instantaneous form factors lead to too low values of quark-antiquark condensates. • Extension to finite T and m, with the inclusion of quark-quark interactions – SU(2) case • m = 0 : chiral transition (crossover) at relatively low critical T (120 – 130 MeV) • QCD phase diagram for finite T and m showing various phases : NQM phase, hadronic (CSB) phase and – for low T and intermediate m – 2SC phase (quark pairing) • Neutral matter + beta equilibrium : need of color chemical potential m8. Mixed and gapless phases. Compatibility with compact star observations and ellyptic flow constraints • Coupling with the Polyakov loop increases Tc(m = 0) up to 200 MeV. Chiral restoration and deconfinement occurring in the same temperature range.

27. To be done • Extension of the phase diagram to higher m – Inclusion of strangeness (CFL phases) • Many possibilites of quark pairing ! • Polyakov loop + neutrality : need of color chemical potential m8 • Inclusion of Polyakov loop for finite chemical potential • Form factors from Lattice QCD – effective mass & wave function form factors • Neutrino trapping effects in compact stars • Description of vector meson sector • . . . NJL Final look of the full phase diagram ?