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

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 ExactasUniversidad de La Plata, Argentina

QCD phase diagram

in nonlocal chiral quark models


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


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 :

QCDPhase Diagram


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

NambuJona-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)


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 NJLlike 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)


Nonlocal chiral quark models

Euclidean action:

Theoretical formalism

(two active flavors, isospin symmetry)

mc : u, d current quark mass; GS : free model parameter

js(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 I M II

r(x), g(x) : nonlocal, well behaved covariant form factors,


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 SE 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 fitLattice QCD :

Schaefer, Shuryak, RMP (98)


Beyond the MFA : low energy meson phenomenology

where

Quark condensate :

,

M I M IIPion 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)


Numerics

Model inputs :

M I → GS 2 = 15.41 mc = 5.1 MeV = 971 MeV

M II → GS 2 = 18.78 mc = 5.1 MeV = 827 MeV

Fit of mc , GS and so as to get the empirical values of m and f

MeV2503/1 qq

Form factor & scale

Parameters : GS , mc

Gaussian

n-Lorentzian

Covariantvs.

“instantaneous”models

DGD, Grunfeld, Scoccola, PRD (06)


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

whereM I

M II

a = 2, 5, 7 (Pauli principle),

Assumption : GS , H, independent of T and


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 )


Low energy physics not affected by the new parameter H parameter fits unchanged

“Reasonable” values for the ratio H / GS in the range between 0.5 and 1.0

(Fierz : H / GS = 0.75)

Typical phase diagrams( H / GS = 0.75 )

M I M II

Low : chiral restoration shows up as a smooth crossover

Peaks in the chiral susceptibility – Tc( = 0) ~ 120 - 140 MeV–

somewhat low … Tc ~ 160 - 200 MeV from lattice QCD –

1st order transition

2nd order transition

crossover

EP End point

3P Triple point

DGD, Grunfeld, Scoccola, PRD (06)


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

u u u

d d dunpaired

T = 0 : 1st order CSB – 2SC phase transition

M I

T = 100 MeV : 2nd orderNQM – 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)


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

,


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


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 / GS in the range between 0.5 and 1.0

DGD, Blaschke, Grunfeld, Scoccola, PRD(06)

M I

M II


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 / GS = 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)


“Gapless” superconducting phases :

NJL (local) model : quasiparticle dispersion relations

(12 quark degrees of freedom)

Ei =

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 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

blue quarks (unpaired)

(degenerate) diquarkcondensates

– Look at the imaginary part of the poles of the (Euclidean) quark propagator

Shovkovy, Huang, PLB (03)


Hybrid compact star models: are they compatible with observations?

Modern compact star observations : stringent constraints on the equation of state forstrongly interacting matter at high densities

Two-phase descriptionof 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 couplingOur approach

(nonlocal quark-quark currents)

Mass vs. radius relations obtained from Tolman-Oppenheimer-Volkoff equations of general-relativistic hydrodynamic stability for self-gravitating matter


Symmetric matter : allowedregion from elliptic flow data

Neutral matter : constraintsfrom compact star observations

New bosonic fields – additional nonvanishing MF value for the isospin zero channel,

GS 2 = 23.7 , mc = 6.5 MeV , = 678 MeV1/3 230 MeVqq

Numerical analysis : parameter set

leading to a quark condensate ( T = = 0 )

Compatibility with observations for low values of GV , H / GS close to 0.75

Compact stars with quark matter cores not ruled out by observations !

g = GV / GS

h = H / GS

Blaschke, DGD, Grunfeld, Klähn, Scoccola, PRC(07); EPJA(07)


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(p2) , q = u , d , s

u, d and s quark-antiquark condensates

Bosonic fields , K , ,

Phenomenology (MFA + large NC) :

( M I – analogous for M II )


Model parameters

G , H’ , , mu , ms

( choice of the form factor )

Numerical results : values forpseudoscalar meson massesand decay constants ( M I )

Adequate overall descriptionof meson phenomenology

r

r

r

r

r

Qualitatively similar results for Model II, and different form factors

(Gaussian)

Scarpettini, DGD, Scoccola, PRD (04)


Finite T : chiral restoration ( = 0 )

Tc ( = 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 theSU(2) chiral restoration temperature

Contrera, DGD, Scoccola, arXiv:hep-ph (07)


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)


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)


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 Tc( = 0) up to 200 MeV. Chiral restoration and deconfinement occurring in the same temperature range.


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

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