PNJL model: phase diagram and nonlocal extensions

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PNJL model phase diagram and non-local extensions
N. N. Scoccola Tandar Lab -CNEA Buenos Aires
In collaboration with G. Contrera (CNEA) D.
Gomez Dumm (Univ.LaPlata) M. Orsaria (CNEA)
PLAN OF THE TALK

• Introduction
• Non-local chiral quark models
• Two light flavor non-local models with Polyakov
  loop
• Extension to 21 flavors
• Outlook Conclusions

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Introduction
The understanding of the behavior of strongly
interacting matter at finite T and/or density is
of fundamental interest and has important
applications in cosmology, in the astrophysics of
neutron stars and in the physics of RHIC. Full
treatment of QCD at finite densities and
temperatures is a problem of high complexity for
which rigorous approaches are not yet available.
One possible approach is to develop effective
field-theoretical models which are built as close
as possible to the symmetry requirements of QCD
but still offer the possibility of the treatment
of the simplified interactions in a systematic
way.
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A type of model that has recently received
attention is the Polyakov-Nambu-Jona-Lasinio
model Fukushima (03), Megias,Ruiz Arriola,
Salcedo (06), Ratti, Thaler, Weise (06),

NJL model (CHIRAL DYNAMICS)
Synthesis of
Polyakov loop
dynamics (CONFINEMENT)

• NJL model is the most simple and widely used
  model with chiral quark interactions. Local
  scalar and pseudoscalar four-fermion couplings
  regularization prescription (ultraviolet cutoff)

NJL (Euclidean) action
Nambu, Jona-Lasinio, PR (61)

Z(3) symmetry

• Polyakov loop
• Polyakov, PLB (78)

Effective potential
deconfinement Z(3) symmetry spontaneously broken
confinement Z(3) symmetry not broken
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Non-local quark models

• Non-local quark models represent a step towards a
  more realistic modeling of the QCD interactions.
  Nonlocal quark couplings are natural in the
  context of many approaches to low-energy quark
  dynamics i.e. instanton liquid model,
  Schwinger-Dyson resummation techniques, etc. Also
  in lattice QCD.
• Several advantages over the standard local NJL
  model
• Consistent treatment of anomalies
• No need to introduce sharp momentum cut-offs
• Small next-to-leading order corrections
• Successful description of meson properties at
  T ? 0 Plant, Birse, NPA (98)
• Scarpettini, Gomez Dumm, NNS, PRD (04),
  Noguera, NNS, PRD(08)

Euclidean action for two flavors
where
g(x) nonlocal, well behaved covariant form
factors,
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Model parameters and form factor are chosen so as
to obtain a good description of the vacuum and
its mesonic excitations. They are determined as
follows

• Standard bosonization of the fermion theory is
  performed boson fields
• s and pi are introduced.

• Mean field approximation (MFA) expansion
• of boson fields in powers of meson
  fluctuations

• Minimization of SE at the mean field level
  leads to gap equation

NJL
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• Beyond the MFA low energy meson phenomenology

Pion mass from
Pion decay constant from

• GT relation
• GOR relation
• p0gg coupling

Consistency with ChPT results in the chiral limit

Plant, Birse, NPA (98), Scarpettini, Gomez Dumm,
NNS, PRD (04), Gomez Dumm, Grunfeld, NNS, PRD (06)
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Extension to finite temperature T and chemical
potential ? is obtained by using Matsubara
formalism
, with
General, Gomez Dumm, NNS, PLB (01), Gomez Dumm,
NNS, PRD (02), Duhau, Grunfeld, NNS PRD (04)
Tc120-140 MeV Rather Low ! Typical NJL model
Tc 175 MeV Typical lattice QCD Tc 160
200 MeV
Inclusion of diquark interactions
where
with
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Two light flavors nonlocal models with Polyakov
loop
To account for chiral restoration and quark
deconfinement we will work with a non-local model
coupled to the Polyakov. Specifically for the
quark sector we will use
Usual scalar-pseudoscalar non-local
current-current term
Accounts for wave function renormalization (WFR)
of quark propapagator
We bosonize this action introducing, as usual,
scalar and pseudoscalar boson fields. Due to the
presence of the WFR term a second scalar field
?2(x) has to be introduced. Next we perform the
MFA approximation
In this approximation the quark propagator is
where
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Using the Matsubara formalism thermodynamical
potential in the presence of the Polyakov loop
(PL) reads
with
since in the Polyakov gauge the MFA value
of the PL can be expressed as
with
For the PL effective potential we take
a(T) , b(T) fitted to lattice QCD results.
Ratti, Thaler, Weise (06)
In our calculations we take to have
real ?MFA. In addition we choose for simplicity
exponential forms for functions g(r) and f(r).
Model parameters are adjusted so that at T?0 we
reproduce empirical f? and m? and
Suggested by QCD Sum Rules Suggested by lattice
QCD calculations
This leads to good phenomenology for mesons (i.e.
?-? scat. parameters, etc) Noguera, NNS (08)
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From ?MFA we derive gap equations
At vanishing chemical potential we obtain
Both chiral restoration and deconfinement
transitions are smooth crossovers.
In the absence of quarkPL coupling chiral
restoration is crossover Tch?130 MeV while
deconfinement transition is 1st order with T? ?
270 MeV
The peaks of the chiral susceptibility ?ch and
the PL susceptibility ?? occur at approximately
the same T ? 210 MeV
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Phase diagram
(TEP, ?EP) ?(170, 200) MeV
?c(T0)?312 MeV
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Extension to 21 flavors
Contrera, Gomez Dumm, NNS, Phys.Lett.B 661(08)113
In the quark sector we consider
where
We use exponential form factors and adjust
parameters to reproduce T0 meson properties,
i.e. m?138 MeV, mK495 MeV, m?, 958 MeV,
f?92.4 MeV. Predictions m? 520 MeV (547 MeV
exp), fK/f? 1.29 (1.22 exp), anomalous
photodecays of ?0,, ?, ? well described, etc.
Scarpettini, Gomez Dumm, NNS, PRD (04). Coupling
to Polyakov loop implemented as in the two flavor
case.
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?u and ?s peaked at roughly the same T while ?s
at somewhat higher T Tcu? Tc? ? 198 MeV in good
agreement with Lattice QCD value from
RBC-Bielefeld group
M. Cheng et al., Phys. Rev. D74 (2006) 054507
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Meson masses and weak decay constants
Roughly 70 MeV above Tc all the pseudoscalar
masses tend to coincide
As expected f? goes to zero very fast above Tc.
fK remains non-negligible up to higher T
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Summary Conclusions

• PNJL-type models provide a simultaneous dynamical
  description of the DECONFINEMENT and CHIRAL
  cross-over transitions.
• Non-local extension of the PNJL represents a step
  towards a more realistic modeling of the QCD
  interactions. In this framework the coupling of
  the non-local quark effective action to the
  Polyakov loop tends to increase the otherwise too
  low value of Tc(?0).
• The main effect of the WFR seems to be in the
  prediction for the location of the EP.
• The extension to 21 flavors leads to a Tc(?0)
  which is in good agreement with the lattice QCD
  RBC-Bielefeld value.
• To be done extension of 21 flavor calculations
  to finite ?, inclusion of diquark channels,
  determination of EOS under compact stars
  conditions, etc.

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Comparison of momentum dependence of M(p) and
Z(p) with lattice results from Parappilly et al,
Phys. Rev.D73(06)054504