Daniel Gómez Dumm

IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias ExactasUniversidad de La Plata, Argentina

Issues on nonlocal

chiral quark models

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

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:

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 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 model

Euclidean action:

Theoretical formalism

(two active flavors, isospin symmetry)

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

(based on OGE interactions)

g(z) : nonlocal, well behaved covariant form factor

ja(x) : nonlocal quark-quark current

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 Momentum-dependent effective mass

Lattice (Furui et al., 2006)

Gaussian fit

Lorentzian fit

Lattice QCD :

(also presence of a wave function form factor)

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)

Numerics

Model inputs :

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

Parameters : GS , mc

Form factor & scale

Gaussian

n-Lorentzian

GS 2 = 18.78 mc = 5.1 MeV = 827 MeVMeV2503/1 qq

Gaussian, covariant form factor :

Consistency with ChPT results in the chiral limit :

GT relation

GOR relation

coupling

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 / GS = 0.75 (quark-quark current)

Assumption : GS , 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 / GS in the range between 0.5 and 1.0

(Fierz : H / GS = 0.75)

Large , low T : nonzero mean field value – two-flavor superconducting phase (2SC)

paired quarks

u u u

d d d

unpaired

Pairing between quarks ofdifferent colors and flavors :

Alford, Rajagopal, Wilczek , NPB (99)

1st order transition

2nd order transition

crossover

EP End point

3P Triple point

( H / GS = 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 : 1st order CSB – 2SC phase transition

T = 70 MeV : 2nd order NQM – 2SC phase transition

Typical phase diagram :

Duhau, Grunfeld, Scoccola, PRD (04)

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 –

DGD, Grunfeld, Scoccola, PRD (06)

Increasing : End Point Figure: ( , T )EP = (200 MeV, 80 MeV) -- Strongly model-dependent

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

,

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 )

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

Compact stars with quark matter cores not ruled out by observations

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(p) , q = u , d , s

u, d and s quark-antiquark condensates

Bosonic fields , K , 0 , 8 , a0 , , 0 , 8

Phenomenology (MFA + large NC) :

Model parameters

G , H’ , , mu , ms

( 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: mu , m , mK , m’ , f

Adequate overall description of meson phenomenology

8 – 0 sector : two mixing angles

Four decay constants

Current algebra:

NLO ChPT + 1/Nc :

Quarks coupled to a background color fieldSU(3)C gauge fields

Traced Polyakov loop

Taken as order parameter of deconfinement transition, related with Z(3) center symmetryof 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

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

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

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 = Tc for nonstrange mesons

Regions where GM(p2,0) show no real zeros

“Ideal” mixing at large T

Main qualitative features :

Contrera, DGD, Scoccola, PRD (10)

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 Tc (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 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.

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

VS.

Vamos pra revanche!!