Kyung Kiu Kim(Kyung-Hee University, CQUeST) With Youngman Kim(APCTP), Ik-jae Sin(APCTP) and Yumi Ko(APCTP)

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1 09-18 April 2012 Third Year of APCTP-WCU Focus program From dense matter to compact stars in QCD and in hqcd,apctp, Pohang, Korea Kyung Kiu Kim(Kyung-Hee University, CQUeST) With Youngman Kim(APCTP), Ik-jae Sin(APCTP) and Yumi Ko(APCTP)

2 Motivation and introduction Short note for AdS/CFT Nuclei from holography Compact stars from holography Summary and discussion

3 QCD is mysterious region in physics. Strong coupling behavior, density and temperature make field theory calculation very difficult. We haven t found complete method to control these situations yet in field theory framework. AdS/CFT gave us some possibilities for dealing with the situations(density and temperature). At least, for some super conformal field theories, we can overcome these difficulties completely. Although it is not clear whether this way can produce the quantitative explanation for real QCD or not, we may extend the AdS/CFT due to a possibility of qualitative explanations. There have been a lot of attempts to mimic real QCD and many successful works.

4 Most of the works are devoted to inhomogeneous structures. For example, uniform density, uniform chemical potential, flat space time, uniform condensation, uniform energy and so on. In our works, we try to realize inhomogeneous structure in models of holographic approach. There are two interesting topics related to inhomogeneous structures, the nuclei and compact stars. Understanding formation of nuclei is quite interesting in this business. We have constructed configurations which is similar to nuclei by giving boundary condition on radial direction. Using the result, we try to understand the density behavior of nuclei.

5 In addition, the compact star(neutron star, white dwarfs, quark stars, strange stars, ) is another interesting research topic, because they can tell us the equation of state for some region in the phase space of QCD. There have been many interesting observations from neutron stars. Since these objects are self bound objects with nuclear force and gravity, we have two obstacles, the density and the gravity. First one is already mentioned. It is well known that we can deal with this problem by introducing bulk gauge field. For the second one, we have to introduce the gravity degree of freedom in the boundary of AdS space. So the construction of gravity degrees of freedom at the boundary of AdS space is very important to realize compact star configuration in holographic QCD. Assuming RS model(uv cutoff) and other model(ir cutoff) within AdS/CFT, we can find configurations to mimic compact stars.

6 Type IIB string theory

7 Low energy effective field theory. -- Type IIB super gravity theory

8 black 3-brane solution Actually this black 3 brane solution is same physical object with D3 brane on end point of open strings. (Polchinski)

9 Let s consider N D3-brane. The D-brane has two description, D-brane in string theory and Black brane in supergravity. D-brane in stringy picture

10 D3 brane action becomes Non-abelian DBI action. The perturbation theory is welll defined, when

11 Let us consider low energy limit. The higher derivative terms can be ignored. The SUGRA fields are decoupled from D-brane. Then the action governing D3 brane turns out to be U(N) Super Yang- Mill theory with 4 super-charges.

12 Super gravity point of view Near horizon limit(r -> 0) -Low energy string limit by Red shift factor The SUGRA is valid when two following quantities are large.

13 There are two limit for same object N D3 brane This is the AdS/CFT correspondence. The strong coupling limit of SYM could be defined by SUGRA or string theory in AdS5. 5d classical gravity ~ 4d boundary QFT => Holography

14 This super conformal YM is very strange theory which is very far from real YM theory QCD. In order to make similar theory to real QCD, we need to deform the conjecture. There are two approaches.

15 Or we may think phenomenological model

16 An example ; EKSS(J. Erlich, E. Katz, Dam T. Son and M. A. Stephanov, 05 ), model

17 In order to describe chiral symmetry, the authors consider minimal bulk fields which describe the symmetry. Field contents Background AdS5 with cut off z_m (for confinement) Bulk matter action composed of the field contents

18 IR boundary condition at z=z_m For scalar fields For gauge fields (fluctuation) -simplest choice ; Neumann BC

19 Parameter matching Gauge coupling -from vector current current correlation ftn -from holographic calculation(g_5) -from QCD calculation(n_c) Hadrons - Normalizable modes of this background(ads + scalar field). - These are not sensitive to IR boundary condition

20 Input About 10% accuracy!

21 Based on JHEP 10(2010) 039, K.Kim, Y. Kim. And Y. Ko The self-bound objects in QCD Nuclei strange quark matter Kaon hyper- nuclei(could be produced from Heavy ion colliders.) To describe the objects in holographic model For simplicity, we consider an over-simplified inhomogeneous object: a structure-less and spherically symmetric object in EKSS. Thus we introduce configurations, whose number density and chiral condensation depend on radial coordinate r.

22 EKSS model Vector U(1) and real scalar with higher dimensional correction Equations of motion

23 For non-trivial structure of radial direction As a first approximation EoM are simple -separable!

24 It s easy to solve these equations. Equations for the gauge field Equations for the scalar field General solution

25 Putting IR boundary condition in this model. We considered In the Chiral limit (could be another approxmation) Gauge field with regularity at the origin

26 From the AdS/CFT dictionary Density Boundary condition for r direction

27 Take a choice for density and lowest mode. Select only n=1 Chemical potential and density function are completely determined

28 Scalar field Boundary condition In the chiral limit, condensation is BC for condensation

29 Consider lowest possible mode. We have to take two modes for this configuration. Condensation

30 A configuration We made decreasing density with increasing condensation

31 Real charge density can be obtained by experiments Relation between Nucleon density and charge distribution (A nucleons and Z protons) The experimental result of charge distribution is almost same with a function for A > 20

32 Radius of a nucleus

33 Comparison with our solutions Our results with EKSS cut off(~ 320MeV)

34 Changing cut-off

35 This z_m is a model parameter, we found that one has to put different model parameter from original EKSS work to describe inhomogeneous nucleon density. In more refined model, which consider more degrees of freedom, this cut off scale can be changed.(ex. Baryons in AdS/QCD,Deog Ki Hong, Takeo Inami, Ho-Ung Yee, Phys.Lett.B646:165,2007 )

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38 For inhomogeneous configuration with nonzero C_1, we regard C_1 as very small value and consider this problem perturbatively

39 For compact stars, AdS/CFT or gauge/gravity correspondence is not enough. We need (gauge+ gravity)/gravity correspondence to consider an selfbound object with QCD interaction and gravity interaction. Randall-Sundrum model was considered for the gravity degrees of freedom in our brane to explain hierachy. The gravity strength is represented by warping factor. The first work investigating gravity wave functions is given by Gherghetta et al, 05. In the model, they considered a mass term for IR brane and obtained composite graviton. Another work is given by Kiritsis and Nitti 06. They considered massless 4d gravitons in the Asymptotically AdS5 geometries. (IR cutoff)

40 In Randal Sundrum model can be embedded in AdS5. So we may consider the model in terms of AdS/CFT. In the model, dynamical gravity degrees of freedom can be introduced with the strength given by warping factor. In our case, the only requirement is that range of z is larger than small value. This small value corresponds to UV cutoff and gives small Newton constant. (e < z <.) No gravity in EKSS type

41 Another suggestion : IR cut-off(kiritsis and Nitti) If we introduce the gravity fluctuation in the AdS space with Poincare coordinates. Solving Einstein equation, this gravity wave function is not a normalizable function., where y is from 0 to infinity.

42 One idea for normalizable wave function is introducing IR cutoff and suitable boundary condition for gravity wave function. This makes the existence of graviton possible on the boundary. But this is not a popular way to introduce gravity degrees freedom. The UV cutoff is more popular and accepted way than this way.

43 Following these ways, one can construct existence of gravity d.o.f on the boundary of AdS space (z= small value) by introducing cut-offs and boundary conditions on gravity wave function. Now I review how neutron star was studied so far in elementary level to compare our approach. For studying Neutron star, one has to consider the 4 dimensional Einstein gravity with simplest perfect fluid stars. where the energy momentum tensor is perfect fluid energy momentum tensor.

44 The metric is given by Where Then Einstein equation gives

45 If we consider hydrostatic equation, one can obtain TOV equation. Thus we have to solve following three equations The first one can be integrated. We have three unknown with two equations except for first integration equation,.

46 Usual study requires one more equation, the equation of state. If we know the equation of state, one can solve the three equations. As a result, we can see full functions of mass, pressure and energy density. From these, we can read the radius and the mass of a compact star. This computation can be related to observation. The EOS is from the microscopic nature of the nuclear matter, the radius and the mass are from the macroscopic observation. Thus the compact stars are good object to understand microscopic physics from observation. In this elementary study, we have to assume or obtain EOS from QCD. But we don t know yet.

47 If we believe possibility for existence of gravity degrees of freedom, we may consider holographic model in a curved spacetime. This is a deformation from the gauge/gravity correspondence. In order to obtain first intuition, we had better take a simplest toy model. So we are going to consider the Einstein-Hilbert action with negative cosmological constant in 5 dimension. The general behavior was studied by Skenderis et al(2000).

48 The summary of their work is as follows. If we take the Fefferman-Graham coordinates, The general behavior of metric is The expectation value of energy-momentum tensor is

49 Again, the bulk metric is The first term is the scheme dependent term, when we consider holographic renormalization. For simplicity, we assume that this term vanishes(polynomial Ansatz). The second term is interpreted as the metric of boundary system. Usually, this is taken as flat metric. In our case we will put nontrivial non-flat metric into this boundary metric.

50 Since we will take star object into account, we assume that our metric has spherical symmetry. Then our ansatz for bulk metric is From this ansatz, we may solve the bulk equation.

51 Now we take a star metric as the boundary metric. Then we can solve the bulk Einstein equation order by order in z. Up to zeroth order equation, g^(2) is given in terms of g^(0).

52 Up to second order, the metric must satisfy following constraint equation. In other words, If the boundary metric satisfies this equation, then the full bulk metric could be a solution of the bulk Einstein equation up to second order.

53 We take one more assumption. If the boundary metric is a perfect fluid star, we have to consider following constraints. These come from the boundary Einstein equation and TOV equation. Then previous complicate constraint becomes simple.

54 If we solve the following equations with initial condition P(0),rho(0) and m(0)=0, then we can obtain pressure and energy density as functions of the radial coordinate.

55 Indeed, these equations can be solved, the solution is well known uniform density solution. This model is a simplest toy model, so we expect that a model with matter can give more realistic equation of state.

56 Now we move to realistic configuration. First one we can think is giving a simplest matter to the system. So we may add a neutral scalar bulk field as follows. Eom for matter fields

57 The ansatz(polynomial type) for the matter field is as follows. The metric ansatz is same with the previous case. We can solve the Einstein equation and the equations of motion for matter field near boundary(z=small value) of the AdS5 order by order in z.

58 For the matter fields, we can obtain the result Again a complicate equation for metric

59 And for the scalar field If we give perfect fluid metric condition and TOV equation to these solution

60 The two equations become much simpler

61 Thus the equations we have to solve are

62 By regularity condition of the star metric, the relevant parameter for this solution are P(0), rho(0), phi_1(0) If we give three parameters, we can obtain holographic star solutions.

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66 By holography, we have obtained equation of state depending on some parameters. The back reaction was included in our construction. And we can make a situation whose surface energy density does not vanish. In this case, we can interpret the scalar field parameter as a parameter related to surface energy density. So the scalar field has clear physical meaning.

67 Inhomogeneous structure in AdS/CFT and extended gauge/gravity correspondence is very interesting topic. In this work, we considered nuclei and compact stars in holographic way. First, we have considered the nuclei in EKSS model. We considered a boundary condition on radial direction and we compare the resulting density to the experimental data. We found that the model parameter z_m should be changed to describe density profile. This boundary condition is similar to BC of Infinite well potential. This is just a starting point, we expect more similar type structure to real density profile could be possible. We may take more degrees of freedom into account in the model. Then we will see more complex structure.

68 In order to consider compact stars, we have deformed boundary metric which is not flat. To embed the compact star geometry, we have taken spherical symmetry on the boundary metric and polynomial ansatz. Since the bulk Einstein equation gives one constraint than the boundary Einstein equation, our system gives one more differential equation. By taking perfect fluid star as the boundary metric, the constraint became very simple equation we can solve. The simplest solution is well-known uniform density star. We introduced a scalar field as a matter field. It turns out that this scalar field has a meaning related to surface energy density. We can obtain EOS for each parameter point(center values of Pressure and density) We can consider more structure by more matter fields. As a future direction, we may study full bulk geometry by solving partial differential equation with boundary metric, then we can understand more on this holographic stars. Holographic BH is another promising topic.