G. Contrera a,b ,M. Orsaria a ,N. N. Scoccola a,c,d
1
2 2 2 2 2 , , 2 1 2 2 ,, 3 3 , (,) 2 ln , (2) 2 2 c c n np np p c c rgbn S S np MF A M dp T T G G TZ T ù U In eq.(1), represent the MFA values of the sigma fields, G S , P the corresponding coupling constants (see Ref.[3] for details), and is the MFA Polyakov loop. The latter is, in general defined as In order to keep MFA real valued, we set 8 =0 in the present calculations. The parametrizations of the non-local regulators we consider are Nonlocal PNJL model with wavefunction renormalization at finite temperature and chemical potential We study the chiral phase transition at finite temperature and chemical potential considering a non-local Polyakov chiral quark model with wave-function renormalization (WFR) in the propagator. In particular we analyze the role played by the Polyakov loop and compare results obtained with two different parametrizations and without the WFR. G. Contrera a,b ,M. Orsaria a ,N. N. Scoccola a,c,d a Physics Department, Centro Atómico Constituyentes, Buenos Aires, Argentina., [email protected] b Universidad Nacional de San Martín, Buenos Aires, Argentina, [email protected] C CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina., [email protected] d Universidad Favaloro, Solis 453, 1078 Buenos Aires, Argentina. REFERENCES [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Phys. Rev. 124, 246 (1961). [2] P. N. Meisinger, M. C. Ogilvie, Phys. Lett. B 379 (1996) 163; K. Fukushima, Phys. Lett. B 591 (2004) 277; K. Fukushima, Phys. Lett. B 591(2004) 277; E. Megias, E. Ruiz Arriola, L. L. Salcedo, Phys. Rev. D 74 (2006) 065005; C. Ratti, M. A. Thaler, W. Weise, Phys. Rev. D 73 (2006) 014019; S. Roessner, C. Ratti, W. Weise, Phys. Rev. D 75 (2007) 0340007. [3] S. Noguera, N. N. Scoccola, Phys.Rev. D78 (2008) 114002. Depending on the value of temperature one is expected to find two different types of transitions: i) a crossover at high values of T. In this region the position of the chiral transition is determined by the peak of the chiral susceptibility defined by ii) a first order transition at low values of T. As usual, in this case the transition point is determined by the value of (T,) at which the two minima of the thermodynamical potential have the same value. where f (p) NONLOCAL g (p) REGULATORS and The presence of the PL implies the existence of three coupled gap equations, RESULTS 2 2 2 , 2 1 c np c n T i p 3 3 8 8 1 () exp ( )/ =diag( , , ) 3 c r g b x Tr i x T For the effective potential we consider (Ratti et al in Ref.[2]) a(T) , b(T) fitted to lattice QCD results. To calculate the numerical results we use a set of parameters that allow for a good description of the pion and sigma vacuum properties [3]. At very low temperatures and densities the two principal features in the QCD theory are chiral symmetry breaking and confinement. But when the temperature and/or chemical potential increase a chiral symmetry restoration is expected: the interaction among color charges is screened rather than confined because the short range interactions are weak due to the asymptotic freedom. To study the strongly interacting quark-gluon matter at low energies and try to interpret lattice QCD results it is necessary resort to effective models due to the non-perturbative behavior of the QCD. One of the most popular is the Nambu-Jona Lasinio (NJL) model [1], in which the quark fields interact via local four point vertices, with the quark condensate and pions as Goldstone bosons emerging from a chiral invariant if such interaction is strong enough. The problem with NJL model is that the confinement is absent. In the context of NJL-like models the Polyakov loop (PL) has been considered recently [2] as order parameter to describe the confinement- deconfinement phase transition. We study here a non local extension of the NJL, which includes wave function renormalization (WFR) and the coupling to the PL, at finite temperature and chemical potential. Two parametrizations are used: a) an exponential form, b) a parametrization based on a fit to the mass and renormalization functions obtained in lattice calculations was considered. Parametrization b) has been considered in a previous work [3] to study the properties of pion and sigma meson at T==0. Starting with the Euclidean action for the nonlocal chiral model in the case of two light flavors [3], and including the coupling to the PL, we obtain that the mean field thermodynamical potential MFA is Fig2: Phase diagram showing the chiral transition curves in the T- plane for parametrization a) with (black line) and without (red line) PL. Fig3: Phase diagram showing the chiral transition curves in the T- plane for parametrization a) with (black line) and without (red line) WRF, in comparison with parametrization b) (blue line). z q m z z q m m z z z z z z z m p f m p f p f p f p f p p f ) ( ) ( ) ( 1 1 ) ( ) ( 1 1 ) ( ) ( g 2 / 5 2 1 2 1 2 / 3 2 0 2 )] ( 1 [ ) ( ; ] ) ( 1 [ ) ( p p f p p f z m MeV m z 309 , 3 . 0 Fig1: Parametrization a) in the presence of the PL. The upper panel displays the behavior of mean field value 1 and the traced PL, and the bottom panel shows the behavior of the susceptibilities ch and .. Exponential parametrization Fig4: Phase diagram showing the chiral transition curves in the T- plane in the model with PL and WRF, using the parametrization b) (blue line). This figure includes the spinodal curves. Lattice adjusted parametrization (1) 1,2 with a) b) P
description
G. Contrera a,b ,M. Orsaria a ,N. N. Scoccola a,c,d. a Physics Department, Centro Atómico Constituyentes, Buenos Aires, Argentina., morsaria @tandar.cnea.gov.ar b Universidad Nacional de San Martín, Buenos Aires, Argentina , [email protected] - PowerPoint PPT Presentation
Transcript of G. Contrera a,b ,M. Orsaria a ,N. N. Scoccola a,c,d
2 2 2 22, , 21
2 2, ,
3
3,
( , ) 2 ln ,(2 ) 2 2
c cnn p n p p
cc r g b n S Sn p
MFA
Md pT T
G GT ZT
ùU
In eq.(1), represent the MFA values of the sigma fields, GS, P the corresponding coupling constants (see Ref.[3] for details), and is the MFA Polyakov loop. The latter is, in general defined as
In order to keep MFA real valued, we set 8=0 in the present calculations.
The parametrizations of the non-local regulators we consider are
Nonlocal PNJL model with wavefunction renormalization at finite temperature and chemical potential
We study the chiral phase transition at finite temperature and chemical potential considering a non-local Polyakov chiral quark model with wave-function renormalization (WFR) in the propagator. In particular we analyze the role played by the Polyakov loop and compare results obtained with two different parametrizations and without the WFR.
G. Contreraa,b,M. Orsariaa,N. N. Scoccolaa,c,d
aPhysics Department, Centro Atómico Constituyentes, Buenos Aires, Argentina., [email protected] b Universidad Nacional de San Martín, Buenos Aires, Argentina, [email protected]
C CONICET, Rivadavia 1917, 1033 Buenos Aires, Argentina., [email protected] Universidad Favaloro, Solis 453, 1078 Buenos Aires, Argentina.
REFERENCES
[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Phys. Rev. 124, 246 (1961).[2] P. N. Meisinger, M. C. Ogilvie, Phys. Lett. B 379 (1996) 163; K. Fukushima, Phys. Lett. B 591 (2004) 277; K. Fukushima, Phys. Lett. B 591(2004) 277; E. Megias, E. Ruiz Arriola, L. L. Salcedo, Phys. Rev. D 74 (2006) 065005; C. Ratti, M. A. Thaler, W. Weise, Phys. Rev. D 73 (2006) 014019; S. Roessner, C. Ratti, W. Weise, Phys. Rev. D 75 (2007) 0340007.[3] S. Noguera, N. N. Scoccola, Phys.Rev. D78 (2008) 114002.
Depending on the value of temperature one is expected to find two different types of transitions: i) a crossover at high values of T. In this region the position of the chiral transition is determined by the peak of the chiral susceptibility defined by
ii) a first order transition at low values of T. As usual, in this case the transition point is determined by the value of (T,) at which the two minima of the thermodynamical potential have the same value.
where f (p) NONLOCAL
g (p) REGULATORS
and
The presence of the PL implies the existence of three coupled gap equations,
RESULTS
22 2, 2 1c
n p cn T i p
3 3 8 8
1 ( ) exp ( ) / = diag( , , )
3 c r g bx Tr i x T
For the effective potential we consider (Ratti et al in Ref.[2])
a(T) , b(T) fitted to lattice QCD results.
To calculate the numerical results we use a set of parameters that allow for a good description of the pion and sigma vacuum properties [3].
At very low temperatures and densities the two principal features in the QCD theory are chiral symmetry breaking and confinement. But when the temperature and/or chemical potential increase a chiral symmetry restoration is expected: the interaction among color charges is screened rather than confined because the short range interactions are weak due to the asymptotic freedom. To study the strongly interacting quark-gluon matter at low energies and try to interpret lattice QCD results it is necessary resort to effective models due to the non-perturbative behavior of the QCD. One of the most popular is the Nambu-Jona Lasinio (NJL) model [1], in which the quark fields interact via local four point vertices, with the quark condensate and pions as Goldstone bosons emerging from a chiral invariant if such interaction is strong enough. The problem with NJL model is that the confinement is absent. In the context of NJL-like models the Polyakov loop (PL) has been considered recently [2] as order parameter to describe the confinement-deconfinement phase transition. We study here a non local extension of the NJL, which includes wave function renormalization (WFR) and the coupling to the PL, at finite temperature and chemical potential. Two parametrizations are used: a) an exponential form, b) a parametrization based on a fit to the mass and renormalization functions obtained in lattice calculations was considered. Parametrization b) has been considered in a previous work [3] to study the properties of pion and sigma meson at T==0.Starting with the Euclidean action for the nonlocal chiral model in the case of two light flavors [3], and including the coupling to the PL, we obtain that the mean field thermodynamical potential MFA is
Fig2: Phase diagram showing the chiral transition curves in the T- plane for parametrization a) with (black line) and without (red line) PL.
Fig3: Phase diagram showing the chiral transition curves in the T- plane for parametrization a) with (black line) and without (red line) WRF, in comparison with parametrization b) (blue line).
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)(11
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2 )](1[)( ; ])(1[)( ppfppf zm
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Fig1: Parametrization a) in the presence of the PL. The upper panel displays the behavior of mean field value 1
and the traced PL, and the bottom panel shows the behavior of the susceptibilities ch and ..
Exponential parametrization
Fig4: Phase diagram showing the chiral transition curves in the T- plane in the model with PL and WRF, using the parametrization b) (blue line). This figure includes the spinodal curves.
Lattice adjusted
parametrization
(1)
1,2
with
a)
b)
P
