1. Tomohiro Inagaki Hiroshima University
SI2009, Fujiyoshida, JAPAN
Dense star mass/radius in the NJL model with dimensional vs. cut-off regularization
Tomohiro Inagaki
Hiroshima University
Dense states of hadrons, i. e. neutron, proton, pi meson and K meson have an important role to understand the structure of neutron stars. All the hadrons are constructed by quarks and gluons. Recently many researchers ague that quark and gluon matter has rich phase structure at high density.
Especially, it is considered that quark and gluon matter may show color superconducting phase in cold and dense objects.
Today I review a story of color superconductivity which may realize inside a compact star and discuss its influence on the dense star.
T. Inagaki, D. Kimura and A. Kvinikhidze, Phys. Rev. D77 (2008) ,
T. Fujihara, D. Kimura, T. Inagaki and A. Kvinikhidze, Phys. Rev. D79 (2009)

2. Today’s Menu Symmetry in QCD New Phase of QCD Color Superconductivity
Analysis in Effective Model
Radius of Dense Stars
Conclusion
This is today’s contents.
First I briefly introduce some characteristic and symmetry in QCD.
Next I show rich phase structure of QCD at finite temperature and chemical potential.
Then I simply review the color superconductivity.
To evaluate the ground state of such color superconducting phase I have to use an effective model of QCD. I show how to obtain the EoS here.
Then I show some numerical results about the radius of dense stars in our simple model.
Finally I will give some concluding remarks.

3. Quark Matter p meson Hadron Proton Neutron
The target of my talk is the quark matter. As is known, all the matter around us are constructed by hadrons and leptons. There are many kinds of hadrons. For example, proton, neutron, pion and so on. Such hadrons are made from quarks and gluons.

4. Quark Matter Hadron Quark Matter Confinement Deconfinement u u d u u u
anti-red u u d u d d u d d
The proton is made by two up and one down quarks. The neutron is made by one up and two down quarks. Each three quarks in proton and neutron have different colors, red, green and blue. Then the protons and neutrons become color singlet. The neutral pi meson is constructed by up-quark and anti-particle of up quark, u u bar, or d d bar. It is also color singlet. Usually all the color charges are confined inside the hadrons. We can not observe any color charges.
However, it is expected that such a confinement may disappear and we may observe a deconfinement phase in an extreme situation, high temperature, high density and so on. In deconfinement phase quarks do not bound into a color singlet pair any longer.

5. Definition of QCD QCD Lagrangian SU(3) gauge symmetry
Lorentz invariance
Locality
Renormalizability
The first principle to describe the physics of quarks and gluons is Quantum chromodynamics (QCD). Imposing the local SU(3) gauge symmetry between quark colors, Lorentz invariance, Locality and Renormalizability, we obtain an unique QCD Lagrangian.
It is a start point to analyze the quark matter.

6. Asymptotic Freedom Running coupling g(m) Strong coupling g(m)
One of the characteristic features in QCD dynamics is “asymptotic freedom”. The coupling constant “g” of the QCD interaction grows up at low energy scale and all the color charges are confined inside hadrons. Then the hadronic phase is realized around us.
On the other hand the coupling constant goes down at high energy scale. So we expect that a deconfinement phase can realize at high temperature, high density and so on. m
(Energy scale)

7. Symmetry in QCD
Before we consider the property of the quark matter, we see the symmetry in QCD. It gives us some key words to understand the property of the quark matter.

8. Symmetry in QCD Lagrangian
(Apparent) Symmetry at massless limit
(Poincare invariance, P, C and T)
At the quark massless limit QCD Lagrangian has these symmetries.
SU(3) is a local gauge symmetry.
Nf describes the number of massless quark flavors.
If we consider that up and down quarks are massless, Nf is two.
If we consider that up, down and strange quarks are massless, Nf is three.
So SU(N)xSU(N) is a global flavor symmetry.
It shows baryon number symmetry, axial U(1) symmetry, scale invariance.
Of course, QCD Lagrangian has Poincare invariance.
The theory is also invariant under parity, charge conjugate and time reversal transformations without a special term which is called the theta term.
These symmetries are apparent symmetries in the Lagrangian, I mean that most of these symmetries are broken in the real world.

9. Symmetry Breaking (Explicit)
Non-zero quark mass
Quantization
Dimensional transmutation
Chiral anomaly
All the quark flavors are massive. The mass depends on its flavor.
So the global flavor symmetry is broken by quark mass.
The scale invariance of the theory is violated by the dimensional transmutation.
The QCD scale can be introduced from the scale where the coupling constant blows up.
The chiral U(1) symmetry is broken by quantum corrections.
These are explicit breaking of symmetry.

10. Symmetry Breaking (Dynamical)
Symmetry of the ground state
Chiral condensation at low T and low m
Di-quark condensation at low T and high m
I will discuss it later.
The QCD ground state is not invariant under some symmetry transformations. Then some symmetries are broken dynamically.
Under the chiral symmetry we can transform left quarks and right quarks separately. At low temperature and low density the QCD ground state violates this chiral symmetry. We only have the symmetry to transform both the left and the right quarks in the same way.
At low temperature and high density the QCD ground state violates local SU(3) symmetry. I will discuss it later.

11. Symmetry in QCD (Actual)
Physical Observable
Confinement
SU(3) gauge symmetry is not directly a property of any physical observable.
Actual symmetry in QCD at T=m=0
Because of confinement, SU(3) gauge symmetry is not directly a property of any physical observable at low T and low m.
So the actual symmetry in QCD is ….,
if we can neglect the mass of up and down quarks.
At the massless limit of up and down quarks, QCD phenomena have to be invariant under this symmetry transformation.
It is the property of QCD at low T and low m.

12. New Phase of QCD
Under some extreme conditions such a property may be changed and new phase of QCD realizes.

13. New Phase of QCD
Are symmetries restored or lost in extreme conditions?
High Density
High Temperature
Strong Magnetic Field
Strong Curvature
The question is,
Are symmetries restored or lost in extreme conditions?
What happens in high ….
In this talk I focus on the cold and dense object. I do not ague the contribution of the magnetic and gravitational field.

14. High Density Overlap of Hadron wave functions Asymptotic freedom
g small at high r u d u u d u u u u d
At high density overlap of the Hadron wave function seems to destroy the usual quark confinement. Some color singlet states overlap each other and many particle state realizes.
Furthermore the interaction between quarks becomes weaker.
Then Hadronic confinement phase disappears and varieties of new phases of quarks and gluons realizes at high density. d

15. Phase Diagram Heavy Ion Collision, Early Universe Compact Stars
Thomas Schafer, hep-ph/
Here we show a conjectured phase diagram of three flavors QCD from hep-ph/
The vertical axis show the temperature and the horizontal axis is the chemical potential (Fermi momentum for massless quarks). It becomes larger and larger for higher density.
Each phases are classified by the symmetry of the QCD ground state. I will discuss later.
The quark gluon plasma at high temperature and low density is observed in the high energy Au+Au collisions at RHIC. It also contributes some critical phenomena at early universe.
Today’s target is located here, cold and dense state. You can see that many phases are found in this area.
Compact Stars

16. Compact Star Compact star Condition Neutron star interiors Quark stars
High Density
Low Temperature
Weak Curvature
Magnetic field?
Quark Star
Hadron Phase
Quark
Matter
Such a cold and dense object may realize at neutron star interiors.
There is a possibility that most of the star is in deconfinement phase and constructed by quark matter (not a hadronic state).
It is expected that the theory of the cold and dense quark matter may predict a new type of stars and the theory is tested by observing compact stars.
To evaluate the properties of quark matter inside we can neglect the contribution from the temperature and curvature. Today I do not ague the magnetic field evolution.

17. Final Stage of Star Hot neutron star Quark matter radius & mass ??
Cool down
The deconfinement quark matter also contributes to the cool down process of a hot neutron star after a super nova. Neutrino emission rate and the mean free path of neutrino depend on the QCD ground state inside the stars.
Today I suppose the quark star is constructed and evaluate the possible radius and mass range for quark stars.
neutrino
H. Grigorian, astro-ph/

18. Color Superconductivity
New phase at high density
Now I discuss the di-quark condensation at low temperature and high density.
At high density Cooper pair of quarks who have Fermi momentum break the SU(3) gauge symmetry and induce a color superconductivity.

19. Superconductivity Cooper pair of electrons photon electron
Fermi momentum
electron
It is a schematic image of an origin of the usual superconductivity for the electromagnetic field. A weak attractive interaction between electrons inside solid state destabilizes the filled Fermi sea at low temperature.
It induces the Cooper instability and a single state is macroscopically occupied by electron-electron pairs.
As is explained by the BCS theory, the composite operator constructed by two electron pair develops a non-vanishing expectation value under the BCS ground state. Then the U(1) electro magnetic symmetry is broken.
It is the origin of the usual type I superconductivity at low temperature.
Fermi momentum
photon

20. Color Superconductivity
Cooper pair of “u, d, s” quarks
u quark
gluon
Fermi momentum
d quark
We also illustrate a schematic image of an origin of the color superconductivity. In this case not a weak attractive interaction between quarks destabilizes the filled Fermi sea.
It induces a kind of a Cooper type instability and a single state is macroscopically occupied by di-quark pairs.
Similar to the usual super conductor, the composite operator constructed by di-quark pair develops a non-vanishing expectation value under the QCD ground state at high density. The color SU(3) gauge symmetry is broken down.
Fermi momentum
gluon

21. History (1) One gluon exchange (1977- )
attractive force in color 3 channel
unstable Fermi surface (Cooper instability)
B. Barrois, Nucl. Phys. B129 (1977) 390; PhD thesis (1979).
D. Bailin and A. Love, Phys. Rep. 107 (1984) 325. R R R G - G G G R
The possibility of the color superconductivity was pointed out at 1977.
B. Barrois found that the attractive force exists in anti-triplet channel, i. e. such a asymmetric channel and it can induce the Cooper instability.
But at that time only the one gluon exchange interaction is considered.
Then the mass gap for this Cooper instability is too small to realize inside stars.
It was considered that the Cooper instability should be washed out below the temperature of compact stars. So there is no possibility to observe such a state in our Universe.

22. History (2) Contribution from instanton (1998- )
attractive force in color 3 channel
M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422 (1998) 247,
R. Rapp, T. Schafer, E. V. Shuryak and M. Velkovsky, P. R. L. 81 (1998) 53. L R L R
The situation was changed at 1998.
M. Alford, K. Rajagopal, F. Wilczek introduce the pairing force induced by instantons. It is also attractive in color anti-triplet channel.
They found that the mass gap for the color Cooper instability is not so small.
The critical temperature to wash out the color Cooper instability is QCD scale, about 100 MeV scale.
It is enough high to realize the color superconducting phase inside the neutron star.

23. Color Superconducting Phase
Quark Cooper pair
Fermi momentum
Color charged
Deconfinement
Hadron phase
Color singlet
Confinement u u
anti-red u d d
red d
I emphases the differences of the hadronic phase and the color superconducting phase again.
In color superconducting phase only quarks who have Fermi momentum contribute the quark Cooper pair. Other quarks are frozen in the Fermi sea and do not contribute to the QCD dynamics.
An attractive interaction appears in a color asymmetric state. The Cooper pair is color charged. Such a color charged pair can be only possible in a deconfinement phase.
On the other hand only a color singlet pair constructs a bound state. All the quarks construct bound states in the Hadron phase.

24. Color Superconductivity
Quark Cooper pair
2 flavors
Color Superconductivity d d d d u d u d u d d u d d u d
If we consider two massless flavors, up and down, the di-quark condensation breaks the color SU(3) symmetry to the SU(2) symmetry.
There are 8 generators in SU(3) group. SU(2) group has only 3 generators.
3 of generators survive.
It means that 5 gluons become massive through the Higgs mechanism.
Color Meissner effect takes place. 5 kinds of color magnetic fields are screened out of the superconductor. u d

25. Quark Cooper pair 3 flavors Color-flavor locking Color flavor lockingd d u d d u u d u u u d s s
If we consider the neutrality of the star the number density of the down type quarks have to be twice of the up type quarks. If we consider only down quarks Fermi surface for down type quarks are much higher than that for up type quarks. If the strange quarks are light enough, it is more stable to consider the state which has almost equal density for up, down and strange quarks.
In this case the color SU(3)x flavor SU(3) is broken down to the color and flavor mixed SU(3) symmetry. In this state all the gluons become massive.
This QCD ground state is called as the color-flavor locking phase. s s

26. Analysis in Effective Model
4-fermion interaction model
Next I explain how to analyze the properties of the quark matter at high density.

27. QCD in Dense Matter
Asymptotic freedom meets the Fermi surface. Is the interaction between quarks with suppressed?
Cooper instability
Highly degeneracy
Non-perturbative effects even for a very weak coupling. (c.f. BCS theory)
To analyze the QCD ground state one of the most important points is the validity of perturbative expansion in terms of the coupling constant.
The density considered here is enough for the asymptotic freedom to meet the Fermi surface.
Because of the asymptotic freedom the interaction between quarks is suppressed.
However, highly degeneracy of the states make the non-perturbative effects more important even for a very weak coupling.
It is similar to the BCS theory of usual super conductor.
So it is impossible to analyze the QCD ground state perturbatively.

28. Phenomenological Approach
We can not evaluate the ground state from the first principle, i. e. QCD. Lattice QCD can not be applied at high density too.
Phenomenological approach
Effective model
SD equation
It means that we can not evaluate the ground state from the first principle, i. e. QCD. Lattice simulation is also difficult to apply for high density situation.
We should use some phenomenological approach.
Here we use a phenomenological effective theory of QCD.

29. Effective Model of QCD NJL model
Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1960) 345; 124 (1961) 246.
We often use the NJL model to evaluate the pion phenomena at low energy scale.
The model is useful in Hadron phase.
It is not useful to consider the quark matter at high density.

30. Di-quark condensation
From Fermi statistics
Spin singlet (S=0)
S wave (L=0)
Iso-scalar (I=0)
Color 3
Flavor asymmetric
From Fermi statistics di-quark pair should have these properties.
Spin singlet,
S wave,
Iso scalar,
and Color anti-triplet.
So we consider the effective model which has a attractive force for this channel.

31. Effective Model of QCD Extended NJL model Nambu-Jona-Lasinio
D. Ebert, L. Kaschluhn and G. Kastelewicz, Phys. Lett. B264 (1991) 420
M. Buballa, Phys. Rept. 407 (2005) 205
Nambu-Jona-Lasinio
(NJL) type interaction
3 interaction
An anti-triplet channel interaction between quarks is included in the theory by some authors.
So we use the extended NJL model here.

32. Reguralization NJL type interactions: dim 6
In QFT we regularize the theory to avoid divergence in quantum corrections. The dim 6 operators are not renormalizable. The theory depends on regularization method.
Cut-off regularization
Dimensional regularization
The interaction in NJL model has mass dimensions 6.
In QFT we regularize the theory to avoid divergence in quantum corrections.
The dim 6 operators are not renormalizable.
The theory depends on regularization method.
I should noted that the fundamental theory QCD is renormalizable and the result is independent on the regularization method.
Here we compare two types of regularizations, cut-off regularization and dimensional regularization.
Since we consider the phenomenological effective theory, such a regularization dependence should sweep away phenomenologically.

33. Cut-off regularization
Introduce ultraviolet cut-off scale
Problem: for large .
In cut-off regularization we cut the higher momentum scale at the cut-off scale L. The physical meaning of the cut-off is clear. The scale is usually fixed to reproduce the pion physics at T=m=0.
But the cut-off scale may overtake the Fermi momentum for a large chemical potential.

34. Phase Diagram Thomas Schafer, hep-ph/0304281
Such area seems not valid for applying the naive cut-off regularization.
So we use the other regularization.

35. Dimensional regularization
Analytically continue the spacetime dimensions to a non-integer value.
It is useful for large .
Problem: physical meaning of D?
Here we apply the dimensional regularization to the extended NJL model.
It is a kind of an analytic regularization.
We regularize the divergent integrals to analytically continue the spacetime dimensions to a non-integer value.
It seems to be more useful at high density but the physical meaning of the non-integer dimension is not clear. It is determined to reproduce the pion phenomena at T=m=0.

36. Parameters in the Model
Coupling constant: Gs, Gd
Regularization parameter or D.
Gs and or D mp, fp and Tc
Gd = (3/4) Gs assumption
T. Hatsuda and T. Kunihiro, Phys. Rept. 247 (1994) 221;
T. Inagaki, D. Kimura and A. Kvinikhidze, Phys. Rev. D77 (2008) ,
T. Fujihara, D. Kimura, T. Inagaki and A. Kvinikhidze,
Phys. Rev. D79 (2009)
What are the parameters in the extended NJL model.
Two coupling constants and the parameter to regularize the divergent integral.
We fix two of the parameters to reproduce the pion mass and the pion decay constant.
This is assumption. We fix the other coupling for 3 channel interaction in this way. It is consistent with the perturbative QCD, but it is not necessary.

37. Radius of Dense Stars
Now we prepare to evaluate the property of dense stars.

38. Here we will consider two massless quarks
Flow Chart
Effective Potential
Stress Tensor
Here we will consider two massless quarks
Gap D and s
r(D, s) and P(D,s)
Radius and Mass
TOV equation
First we evaluate the effective potential and find the value of the mass gap in the ground state.
We also calculate the energy-momentum tensor. It gives us the equation of state, relationship between the energy density and pressure.
Then we solve the TOV equation to find the relationship between the mass and radius of dense stars.
Here we only consider the simplest case.
Two flavors of massless quarks exist.

39. Ground State
First we show the numerical result for the ground state in the extended NJL model.

40. Effective Potential Analysis
The ground state is found by observing the minimum of the effective potential.
V(D, s)
D, s
It is given by observing the minimum of the effective potential.
Ground state

41. Gap D and s Diquark condensation Chiral condensation
We introduce the mass gap D and s.
D shows the di-quark condensation.
Non-vanishing D means that the color super conducting phase is realized.
s shows the chiral condensation.
Finite s means that the system is in the Hadron phase.

42. Phase Diagram Thomas Schafer, hep-ph/0304281
We numerically evaluate these mass gap along this line.
For low m we find the chiral condensation and the di-quark condensation is found for a large chemical potential.

43. Numerical Result It is our numerical result.
In the cut-off regularization all the mass gaps disappear about m=800(MeV).
Because the chemical potential overtakes the cut-off scale.
We see the lower mass gaps in the dimensional regularization.
The critical chemical potential in which the mass gap s disappears seems to be too small in this simplest case.

44. Equation of State (EoS)
Next I show the equation of state in the extended NJL model.

45. Energy Density and Pressure
Stress tensor
Under the ground state
It is found by evaluating the energy-momentum tensor under the QCD ground state.

46. Relationship between m and r
This is numerical result about the energy density.
In the cut-off regularization the energy density becomes constant about m=800 (MeV).
In dimensional regularization there is no such an upper limit.
The energy density in dimensional regularization is much smaller than that in the cut-off regularization.

47. Relationship between m and P
This is the pressure.
The pressure also stop increasing at m=800MeV in the cut-off regularization.
The pressure in this range has similar scale in both the regularizations.

48. Relationship between r and P
Therefore the harder state is realized in the dimensional regularization.

49. Tolman-Ophenhimer-Volkov
TOV equation
Tolman-Ophenhimer-Volkov
Finally we solve the Tolman-Ophenhimer-Volkov equation.

50. Relationship between M and R
TOV equation
J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55 (1939) 374.
pressure
gravity
These are the equations to show the balance of the pressure and the gravitational force. Solving these equations we obtain the mass and radius of stars.

51. Numerical Result
In this figure I plot the mass of star as a function of the energy density at the center of the star.
Heavier star can be realize in the dimensional reguralization.
In this range the total mass increases as the center density increases.
So the state is stable against the gravitational collapse.

52. Numerical Result
This is the relationship between the mass and radius of the dense stars.
In the case of dimensional regularization larger star can be realized in the simplest model.
Neutron star is located here.

53. Summary and Future Problems
Conclusion
Summary and Future Problems
Now we come to the conclusion.

54. Not a small regularization dependences
Summary
Complex phase in QCD at high density
Color superconductivity
2SC (2 light flavors: u and d)
CFL (3 light flavors: u, d and s)
Effective model: extended NJL model
Effective potential ground state
Stress tensor EoS
TOV equation R-M
In this talk I briefly reviewed the complex phases in QCD at high density.
And we focused on the color superconductivity.
If two flavors of quarks are light compared with the chemical potential the SU(3) color symmetry is broken down SU(2) color symmetry.
It is called the 2 flavors color superconductivity.
For higher chemical potential the strange quark also contribute the QCD ground state.
Then the SU(3) color symmetry and the SU(3) flavor symmetry is broken down to the color and flavor mixed SU(3) symmetry.
Then I demonstrated the flow to evaluate the ground state by the effective potential, the equation of state by the energy-momentum tensor and the relationship between the radius and the mass of the dense stars by using the extended NJL model.
I show that there is not a small regularization dependences.
Not a small regularization dependences

55. Structure of Quark Star
Cut-off?
Quark Matter
Quark
Matter
We can also analyze the structure of dense stars.
In our simplest model the star considered here has almost this structure.
Most of stars are color superconducting state.
In cut-off regularization case we see a little Hadron phase.
But to distinguish these two cases we should include the neutron, pion and Kon in the model.
Dimensional?

56. What I did not talk today.
Neutrality
Contribution from electron
Contribution from strange quarks
K.Iida, G.Baym, PRD63 (2001)
I.Shovkovy, M.Huang, PLB564 (2003) 205
Mesons
K condensation
p condensation
Hadron Phase
Today we do not care the neutrality condition and nucleon and mesons.
It is also important to evaluate the property of compact stars.
If we adopt the neutrality conditions, it is predicted that the contribution from the massive strange quark increases the energy density, then decreases the pressure of the quark matter.

57. What I did not talk today.
Magnetic field evolution.
Cooling mechanism of neutron stars.
Other interesting phases in QCD
LOFF pairing
H. Muther and A. Sedrakian, Phys. Rev. Lett. 88 (2002)
Color glass condensation
Other extreme conditions
It is also interesting that the magnetic field evolution inside the stars,
Cooling mechanism of neutron stars.
There are a lot of papers to consider such a problem.
Other phases in dense QCD are also interesting.