##### 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

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