1. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Daniel Gómez Dumm IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas Universidad de La Plata, Argentina QCD phase diagram in nonlocal chiral quark models

2. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Plan of the talk Motivation Description of two-flavor nonlocal models Phase regions in the T –  plane Phase diagram under neutrality conditions Extension to three flavors Description of deconfinement transition Summary & outlook

3. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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) 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. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Nonlocal chiral quark models Euclidean action: Theoretical formalism (two active flavors, isospin symmetry) m c : u, d current quark mass; G S : free model parameter j s (x) : nonlocal quark-quark current Two alternative ways of introducing the nonlocality : Model I (inspired on the ILM) Model II (based on OGE interactions) M IM II r(x), g(x) : nonlocal, well behaved covariant form factors,

7. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil  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, with ( M I ) ( M II ) Momentum-dependent effective mass  ( p ) Instanton liquid model (MFA) : Lattice (Furui et al., 2006) Gaussian fit Lorentzian fit Lattice QCD : Schaefer, Shuryak, RMP (98)

8. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Beyond the MFA : low energy meson phenomenology where Quark condensate :, M I  M II Pion mass from Pion decay constant from – nontrivial gauge transformation due to nonlocality – Consistency with ChPT results in the chiral limit : GT relation GOR relation    coupling General, DGD, Scoccola, PLB(01); DGD, Scoccola, PRD(02); DGD, Grunfeld, Scoccola, PRD (06)

9. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Numerics Model inputs : M I → G S  2 = 15.41 m c = 5.1 MeV  = 971 MeV M II → G S  2 = 18.78 m c = 5.1 MeV  = 827 MeV Fit of m c, G S and  so as to get the empirical values of m  and f   Form factor & scale  Parameters : G S, m c Gaussian n- Lorentzian Covariant vs. “instantaneous” models DGD, Grunfeld, Scoccola, PRD (06)

10. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Phase transitions in the T –  plane Extension to finite T and  : partition function obtained through the standard replacement, with Inclusion of diquark interactions : new effective coupling where M I M II a = 2, 5, 7 (Pauli principle), Assumption : G S, H, independent of T and 

11. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Bosonization : quark-quark bosonic excitations additional bosonic fields  a Artificial duplication of the number of d.o.f. : Nambu – Gorkov spinors 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 Diquark currents written as – in principle, possible nonzero mean field values   2,  5,  8 – Breakdown of color symmetry – Arbitrary election of the orientation of  in SU(3) C space (residual SU(2) C symmetry) ( 4 x 2 x 3 x 2 )

12. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Low energy physics not affected by the new parameter H parameter fits unchanged “Reasonable” values for the ratio H / G S in the range between 0.5 and 1.0 (Fierz : H / G S = 0.75) Typical phase diagrams ( H / G S = 0.75 ) M I M II 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 – 1 st order transition 2 nd order transition crossover EP End point 3P Triple point DGD, Grunfeld, Scoccola, PRD (06)

13. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Increasing  : end point & first order chiral phase transition Large , low T : nonzero mean field value  – two-flavor superconducting phase (2SC) M I : EP = (80 MeV, 208 MeV) M II : EP = (235 MeV, 33 MeV) paired quarks uuuuuu dddddd unpaired T = 0 : 1 st order CSB – 2SC phase transition M I T = 100 MeV : 2 nd order NQM – 2SC phase transition T = 100 MeV : CSB – NQM crossover Behavior of MF values of qq and qq collective excitations with increasing chemical potential (T fixed) Duhau, Grunfeld, Scoccola, PRD (04)

14. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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 effective chiral quark models) (  i for i = 1, … 7 trivially vanishing ) with,

15. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Beta equilibrium (no neutrino trapping assumed) Quark – electron equilibrium through the reaction Residual color symmetry : not all chemical potentials are independent from each other where From  -equilibrium, 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

16. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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 / G S in the range between 0.5 and 1.0 DGD, Blaschke, Grunfeld, Scoccola, PRD(06) M I M II

17. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Behavior of MF values  and  and chemical potentials  e and   as functions of the baryonic chemical potential for fixed values of T ( M I, H / G S = 0.75 ) Mixed phase : global electric charge neutrality coexisting 2SC – NQM phases at a common pressure  c ( T = 0) in the 250 – 300 MeV range (larger for M II)

18. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil “Gapless” superconducting phases : NJL (local) model : quasiparticle dispersion relations (12 quark degrees of freedom) E i = In the region of small diquark gaps,two gapless modes Nonlocal chiral quark models: complicated dispersion relations, same qualitative behavior From our analysis:  Tiny regions of gapless phase close to the 2 nd order 2SC – NQM transition  Size only significant for low values of H / G S  Never extends to zero T, thus not relevant for compact star physics blue quarks (unpaired) (degenerate) diquark condensates – Look at the imaginary part of the poles of the (Euclidean) quark propagator Shovkovy, Huang, PLB (03)

19. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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

20. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Symmetric matter : allowed region from elliptic flow data Neutral matter : constraints from compact star observations New bosonic fields – additional nonvanishing MF value for the isospin zero channel,  G S  2 = 23.7, m c = 6.5 MeV,  = 678 MeV Numerical analysis : parameter set leading to a quark condensate ( T =  = 0 ) Compatibility with observations for low values of G V, H / G S close to 0.75 Compact stars with quark matter cores not ruled out by observations ! g = G V / G S h = H / G S Blaschke, DGD, Grunfeld, Klähn, Scoccola, PRC(07); EPJA(07)

21. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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 2 ), q = u, d, s  u, d and s quark-antiquark condensates  Bosonic fields , K,  ,   Phenomenology (MFA + large N C ) : ( M I – analogous for M II )

22. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Model parameters G, H’, , m u, m s (  choice of the form factor ) Numerical results : values for pseudoscalar meson masses and decay constants ( M I ) Adequate overall description of meson phenomenology r r r r r Qualitatively similar results for Model II, and different form factors (Gaussian) Scarpettini, DGD, Scoccola, PRD (04)

23. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Finite T : chiral restoration (  = 0 ) T c (  = 0) ~ 115 MeV (too low compared with lattice results)  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 Flavor mixing : shoulder in  s at the SU(2) chiral restoration temperature Contrera, DGD, Scoccola, arXiv:hep-ph (07)

24. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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 :  diagonal, MFA : Grand canonical thermodynamical potential given by where Group theory constraints satisfied – a ( T ), b ( T ) fitted from lattice QCD results (coupling to fermions) finite T : sum over Matsubara frequencies Fukushima, PLB(04); Megias, Ruiz Arriola, Salcedo, PRD(06); Roessner, Ratti, Weise, PRD(07)

25. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil From QCD symmetry properties, assuming that  fields are real-valued, one has As usual, mean field value  3 obtained from minimizing the thermodynamical potential Numerical results for  u and  as functions of the temperature ( 3 flavors – Model I – Gaussian )  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 Main qualitative features : Contrera, DGD, Scoccola, arXiv:hep-ph (07)

26. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil 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 & outlook  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 , with the inclusion of quark-quark interactions – SU(2) case   = 0 : chiral transition (crossover) at relatively low critical T (120 – 130 MeV)  QCD phase diagram for finite T and  showing various phases : NQM phase, hadronic (CSB) phase and – for low T and intermediate  – 2SC phase (quark pairing)  Neutral matter + beta equilibrium : need of color chemical potential  8. Mixed and gapless phases. Compatibility with compact star observations and ellyptic flow constraints  Coupling with the Polyakov loop increases T c (  = 0) up to 200 MeV. Chiral restoration and deconfinement occurring in the same temperature range.

27. II Latin American Workshop on High Energy Phenomenology December 3-7, 2007, São Miguel das Missões, RS, Brazil Extension of the phase diagram to higher  – Inclusion of strangeness (CFL phases) Many possibilites of quark pairing ! Polyakov loop + neutrality : need of color chemical potential  8 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... To be done Final look of the full phase diagram ? NJL