Daniel Gómez Dumm IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas
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Transcript of Daniel Gómez Dumm IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias Exactas
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Daniel Gómez Dumm
IFLP (CONICET) – Dpto. de Física, Fac. de Ciencias ExactasUniversidad de La Plata, Argentina
Issues on nonlocal
chiral quark models
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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
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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
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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)
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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)
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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
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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)
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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)
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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
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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)
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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)
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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)
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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
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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
,
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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 )
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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)
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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) :
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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 :
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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
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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
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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
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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)
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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.
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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
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VS.
Vamos pra revanche!!