Download Tulio Eduardo Restrepo Medina RADIATIVELY INDUCED VECTOR REPULSION FOR LIGHT PDF

TitleTulio Eduardo Restrepo Medina RADIATIVELY INDUCED VECTOR REPULSION FOR LIGHT
LanguageEnglish
File Size2.6 MB
Total Pages82
Table of Contents
                            Introduction
	The QCD phase diagram
	The sign problem of QCD at finite densities
	Scope and outline of this dissertation
The Nambu–Jona-Lasinio Model
	The model and its symmetries
		Chiral Symmetry
	The effective potential
		The interpolated model
	Results at finite temperature and zero density
		Taylor expansion coefficients
	Repulsive vector interaction in the NJL model
The Polyakov–Nambu–Jona-Lasinio Model
	Center symmetry group Z(3)
	The PNJL free energy beyond the large-Nc limit
	Numerical results at finite temperature and zero chemical potential
		Evaluation of the second cumulant
	Results for hot and dense quark matter
The entangled Polyakov–Nambu–Jona-Lasinio Model
	Taylor expansion coefficients
Conclusions
Notation
	Relativistic notation
	Dirac Matrices
Matsubara Sums
Color-trace over two loop contributions
Bibliography
                        
Document Text Contents
Page 1

UNIVERSIDADE FEDERAL DE SANTA CATARINA - UFSC
CENTRO DE CIÊNCIAS FÍSICAS E MATEMÁTICAS - CFM

DEPARTAMENTO DE FÍSICA
CURSO DE PÓS-GRADUAÇÃO EM FÍSICA

Tulio Eduardo Restrepo Medina

RADIATIVELY INDUCED VECTOR REPULSION FOR LIGHT
QUARKS

Florianópolis
2014

Page 41

2.4. Repulsive vector interaction in the NJL model

PSB
T 4

=
8π2

45

[
1 +

21

32
Nf

(
1 +

120

7
µ̂

2
+

240

7
µ̂

4

)]
. (2.64)

Then, using Eq. (2.61) we �nd the Stefan�Boltzmann limit for c2

c
SB
2 = 1. (2.65)

At this stage it seems puzzling that the OPT, which supposedly should produce
results more accurate than the LN approximation, seems to perform rather poorly
at high temperatures. Moreover, none of the LQCD evaluations for this coe�cient
reproduce the maximum for c2 (T ) observed in the OPT results. Nevertheless, a result
similar to ours has been obtained by Steinheimer and Schramm [59] who used the
LN approximation to investigate the NJL model in the presence of a repulsive vector
channel. In the next section we review this version of the NJL model and the LN
result for the free energy in order to shed light into the OPT results.

2.4. Repulsive vector interaction in the NJL model

In the Introduction we have emphasized the importance of a vector interaction in the
studies of compressed quark matter. Within the NJL model such a term can be of
the form −GV

(
ψ̄γµψ

)2
with GV > 0 describing repulsion which is the case here and

GV < 0 describing attraction. Then the standard NJL lagrangian density becomes

LV = ψ̄
(
i/∂ − m̂0

)
ψ +GS

[(
ψ̄ψ
)2

+
(
ψ̄iγ5τψ

)2]−GV (ψ̄γµψ)2 , (2.66)
and the e�ective potential in the LN approximation reads [2]

FLN =
σ2

4GS
− 4GVN2fN

2
c I

2
3 (µ̃, T )− 2NfNcI1 (µ̃, T ) , (2.67)

where I1(µ̃, T ) and I3(µ̃, T ) can be readily obtained from Eqs. (2.38) and (2.40) upon
replacing µ→ µ̃ = µ− 2GV ρ, with ρ = 2NcNfI3 , and m0 + η → m0 +σ. Fukushima
[20] has shown that the combined e�ect of µ̃ and −4GVN2cN2f I

2
3 in the above equation

is to produce a net e�ect similar to +4GVN
2
cN

2
f I

2
3 . This interesting result allows us to

better understand the type of 1/Nc contributions radiatively generated by the OPT.
An inspection of Eq. (2.37) reveals that this approximation generates a +4GSNcNfI

2
3

term which is similar to the +4GVN
2
cN

2
f I

2
3 term appearing in the LN result.

Let us pursue this investigation by analyzing the chiral transition at high densities
obtained with the OPT (at GV = 0) and the LN approximation (at GV = 0 and
GV 6= 0).
Note that, in order to obtain thermodynamical results with the LN approximation,

at GV 6= 0, one needs to solve the following set of equations

∂F
∂σ

= 0, and
∂F
∂µ̃

= 0. (2.68)

Fig. 2.8 shows the predictions for the chiral transition at low temperatures and high

31

Page 42

Clipboard


2. The Nambu�Jona-Lasinio Model

æ

à

ò

310 320 330 340 350 360
0

10

20

30

40

50

Μ @MeVD

T
@M

e
V
D

Figure 2.8.: Phase diagram in the T�µ plane for the NJL model showing the �rst
order transition lines and the critical end points. The LN with GV = 0
is denoted by the circle, the LNGV with GV = GS/(NfNc) is denoted
by the triangle and the OPT is denoted by the square.

chemical potential values where the transition is of the �rst kind. This transition line
starts at T = 0 and terminates at a second order transition point which de�nes a
critical end point (when m0 6= 0) since at supercritical temperatures a crossover takes
place as already discussed (note that in the chiral limit case, the �rst order transition
line terminates at a tricritical point and at supercritical temperatures one has second
order phase transitions). This low-T/high-µ portion of the QCD phase diagram is
very important for astrophysical applications and the fact that the transition is of
the �rst kind has non-negligible consequences concerning the structure of compact
stellar objects. For example, within this kind of phase transition one may have two
substances, with distinct densities, coexisting at the same P , T and µ. In the case of
strongly interacting matter these two substances may represent hadronic and quark
matter which could lead to the formation a hybrid star instead of a pure neutron
star. Our results show that both, the OPT and the LN approximation with GV 6= 0
(LNGV ) reduce the �rst order transition in relation to the LN case with GV = 0.
Also, for a given temperature, the coexistence chemical potential value at which the
transition occurs is shifted to higher values within the OPT and the LNGV indicating
that these two di�erent model approximations produce a similar type of physics as
one could infer by comparing their free energies.

We have already emphasized the important role played by a repulsive vector channel
in the description of compressed quark matter. As we have just shown the OPT seems
to reproduce the same e�ects without the need to explicitly introduce such a term (and
one more parameter!) at the tree level. In the OPT case the radiative corrections,

32

Page 81

Bibliography

[58] Najmul Haque, Munshi G. Mustafa, and Michael Strickland. Two-loop hard
thermal loop pressure at �nite temperature and chemical potential. Phys. Rev.
D, 87:105007, May 2013.

[59] J. Steinheimer and S. Schramm. The problem of repulsive quark interactions �
lattice versus mean �eld models. Phys. Lett. B, 696(3):257 � 261, 2011.

[60] Murray Gell-Mann. Symmetries of baryons and mesons. Phys. Rev., 125:1067�
1084, Feb 1962.

[61] H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi, and
C. Ratti. Mesonic correlation functions at �nite temperature and density in
the nambu�jona-lasinio model with a polyakov loop. Phys. Rev. D, 75:065004,
Mar 2007.

[62] C. Ratti, S. Röÿner, M.A. Thaler, and W. Weise. Thermodynamics of the pnjl
model. Eur.Phys.J. C, 49(1):213�217, 2007.

[63] Kenji Fukushima. Chiral e�ective model with the polyakov loop. Phys. Lett. B,
591(3�4):277 � 284, 2004.

[64] Claudia Ratti, Michael A. Thaler, and Wolfram Weise. Phases of qcd: Lattice
thermodynamics and a �eld theoretical model. Phys. Rev. D, 73:014019, Jan
2006.

[65] J Greensite. The con�nement problem in lattice gauge theory.
Prog.Part.Nucl.Phys, 51(1):1 � 83, 2003.

[66] S. Ejiri, Y. Maezawa, N. Ukita, S. Aoki, T. Hatsuda, N. Ishii, K. Kanaya, and
T. Umeda. Equation of state and heavy-quark free energy at �nite temperature
and density in two �avor lattice qcd with wilson quark action. Phys. Rev. D,
82:014508, Jul 2010.

[67] Yuji Sakai, Kouji Kashiwa, Hiroaki Kouno, Masayuki Matsuzaki, and Masanobu
Yahiro. Determination of qcd phase diagram from the imaginary chemical po-
tential region. Phys. Rev. D, 79:096001, May 2009.

[68] Jan Steinheimer and Stefan Schramm. Do lattice data constrain the vector
interaction strength of qcd? Phys. Lett. B, 736(0):241 � 245, 2014.

[69] Yuji Sakai, Takahiro Sasaki, Hiroaki Kouno, and Masanobu Yahiro. Entangle-
ment between decon�nement transition and chiral symmetry restoration. Phys.
Rev. D, 82:076003, Oct 2010.

[70] Junpei Sugano, Junichi Takahashi, Masahiro Ishii, Hiroaki Kouno, and
Masanobu Yahiro. Determination of the strength of the vector-type four-
quark interaction in the entanglement polyakov-loop extended nambu�jona-
lasinio model. Phys. Rev. D, 90:037901, Aug 2014.

71

Page 82

Bibliography

[71] Frithjof Karsch and Edwin Laermann. Susceptibilities, the speci�c heat, and a
cumulant in two-�avor qcd. Phys. Rev. D, 50:6954�6962, Dec 1994.

72

Similer Documents