The interaction of glueball and heavy-light flavoured meson in holographic QCD

We construct the D4/D8 brane configuration in the Witten-Sakai-Sugimoto model by introducing a pair of heavy flavour brane with a heavy-light open string. The multiplets created by the heavy-light string acquire mass due to the finite separation of the heavy and light flavour branes thus they could be identified as the heavy-light meson fields in this model. On the other hand the glueball field is identified as the gravitational fluctuations carried by the close string in the bulk, so this model is able to describe the interaction of glueball and heavy-light meson through the open-close string interaction in gauge-gravity duality. We explicitly derive the effective action for the various glueballs and heavy-light mesons then numerically evaluate the associated coupling constants. Afterwards the decay widths of various glueballs to the lowest heavy-light meson, which is identified as $D^{0}$ meson, are calculated by using our effective action. This work extends the previous investigations of glueball in holographic QCD and it is also a further prediction of glueball-meson interaction.


Introduction
In nuclear physics, Quantum Chromodynamics (QCD) as the underlying theory of the strong interaction allows to from bound states of pure gauge bosons due to its nonabelian nature. These bound states are named as "glueball"s [1,2,3] which are usually denoted by a range of quantum numbers J P C where J, P, C represents total spin, parity and charge conjugation respectively. For instance, the lowest glueball takes the quantum numbers of the vacuum as J P C = 0 ++ . In Yang-Mills theory, glueball is believed as the only possible composite particle states and its spectrum has been detailedly studied by using lattice method [4,5,6]. While the evident existence of the glueball is expected, their experimental identification in the hadron spectra is challenging. This difficulty is mostly due to the mixing of glueball with qq (quark and anti-quark) states which have the same quantum numbers of the glueball in the presence of quarks. Nonetheless the mass of the lightest scalar glueball is predicted to be around 1600-1800MeV by the simulations of lattice QCD [4,7]. However the coupling and decay products/widths of glueball have not been explicitly provided by the current status of lattice QCD since lattice theory including dynamical quarks expressed with real-time quantities would become extremely complexed.
Additional to the lattice gauge theory, an alternatively different approach to investigate strongly coupled quantum field theory has been proposed in [8,9] which is the famous anti-de Sitter/conformal field theory (AdS/CFT) correspondence, or more generally, the gauge-gravity (string) duality. In the AdS/CFT correspondence, the correlation functions of gauge invariant operators with the limitation of large N c (colour number) and large 't Hooft coupling λ are mapped to the perturbations of the geometrical backgrounds in classical supergravity. So the correlation functions of strongly coupled quantum field theory could be calculated by the weakly coupled gravity theory in which the perturbation method is valid [10]. Significantly, a top-down construction for QCD based on N c D4-branes compactified on a circle in type-IIA string theory was proposed by Witten in 1998 [11] in which both supersymmetry and conformal symmetry are broken below a Kaluza-Klein mass scale M KK , so that the dual field theory is four-dimensional pure Yang-Mills theory in the large-N c limit. By embedding N f pairs of coincident D8-and anti D8-branes (D8/D8-brane) as flavours into this D4-brane configuration, quarks in the fundamental representation of flavour and colour have been introduced into this model by Sakai and Sugimoto in 2004 [12,13]. Hence this model, named as the Witten-Sakai-Sugimoto (WSS) model, becomes remarkably successful since it almost contains all necessary ingredients of QCD and it is able to reproduce various fundamental features of low-energy nuclear physics with few parameters [14,15,16,17,18,19,20,21].
Particularly the glueball field in this model is identified as the gravitational perturbations in the bulk supergravity while meson and baryon states are created by the open strings on the flavoured D8/D8-brane and "baryonic D4-branes" 2 . So this model naturally includes the interaction of glueball and hadron which is nothing but the open-close string interaction in the viewpoint of string theory. Specifically the mass spectrum of glueball could be obtained by evaluating the eigen equation of the bulk graviton and the glueball-hadron interaction would arise in the Dirac-Born-Infield (DBI) action of the D-brane once the gravitational perturbation is involved [23,24,25,26,27].
In this paper, we will focus on the glueball-meson interaction by including the heavy flavour in order to extend the investigation in [25,26] with this model. Since the fundamental quarks in this model are created by the open string with one end attached to the D4-brane and the other end to the D8/D8-brane which has zero length, it implies the fundamental quarks are massless. Thus the meson states consisted of these massless quarks are usually identified as the light flavour mesons, such as π, ρ meson. To involve the heavy flavour, we are going to employ the mechanism proposed in [28,29], that is to introduce another pair of flavoured D8/D8-brane (as heavy flavour branes) separated from the other coincident D8/D8-brane (as light flavour branes) with a open string (as heavy-light string) stretched between them as illustrated in Figure  2. Accordingly the multiplets created by the heavy-light string acquire mass due to the finite separation of the heavy and light flavour branes. So the model is able to include the heavy-light flavoured mesons with massive quarks in this approach and the mechanism is recognized as the "Higgs mechanism" in string theory [30,31]. Then the action for the interaction of glueball and heavy-light flavoured mesons could be derived by combining the gravitational perturbations in the bulk with the heavy-light meson fields on the flavour branes. We finally evaluate the decay rates of glueball into two lowest scalar heavy-light mesons (i.e. the D 0 meson) by using our effective action and compare our results with the previous works in [25,26].
The organization of this manuscript is as follows. In section 2, we review the relation of 11d M-theory and type IIA string theory by their corresponding supergravity. We also give the dynamics of the glueball from these supergravity systems in holography. In section 3, we briefly review the setup of the flavours in the WSS model and discuss how to involve the heavy flavour and heavy-light mesons. In section 4, we detailedly derive the effective action for the interaction of various glueball and heavy-light meson in holography. In section 5, we identify the lowest heavy-light meson in the model as the lowest D-meson through some experimental data and we use our effective action to compute the decay width of the glueball into scalar D 0 meson. The final section is our conclusion and summary about this work. At the end of this manuscript, we collect some essential discussion about the supergravity, the dynamics of D-brane, the supersymmetry and the glueball-light meson interaction in Appendix A, B, C respectively which are also useful to this work.

The geometry in supergravity
The WSS model is based on the AdS 7 /CFT 6 correspondence obtained by coincident N c M5branes in 11-dimensional (11d) M-theory and the dual field theory on the M5-brane is a 6dimensional (0, 2) superconformal field theory. The effective supergravity description provides a 11d geometrical near-horizon solution which takes the topology of AdS 7 × S 4 . Let us denote the extended directions of the M5-branes by X µ , X 4 , X 11 with indices µ, ν = 0, 1, 2, 3, so the 11d non-extremal black brane solution is given as [30] 3 , where Ω 4 , L denotes the volume element on S 4 and the curvature radius of the AdS 7 respectively. The indices i, j run from 1 to 3 and r represents the radius coordinate of the transverse directions.
Note that this background also contains a non-vanished Romand-Romand strength F 4 with N c units of flux on S 4 . As the 11d M-theory compactified on a cycle is equivalent to 10d type-IIA string theory, one can obtain non-conformal D4-branes by the dimensional reduction which preserves circle with Afterwards, the dual field theory becomes a 5d super Yang-Mills theory on the D4-branes.
In order to obtain a pure Yang-Mills or QCD-like theory, a further compactification with a double Wick-rotation was proposed by Witten based on the above holographic duality. Accordingly the metric (2.1) with the double Wick-rotation X 0 → −iX 4 , X 4 → −iX 0 corresponds to a bubble configuration of D4-brane which is given as [11,12], so that the dual field theory exhibits confinements in this geometry. The supersymmetry of the D4-branes is broken down by compactifying X 4 on another circle, hence the fermionic gluinos become massive at tree level by imposing the antiperiodic boundary conditions and the adjoint scalars also acquire masses through its loop corrections. Therefore the gauge bosons are the only degrees of freedom in the limit of large Kaluza-Klein (KK) mass scale. Employing the relations of the dimensional reduction, the 10d metric could be written as [25,26,32], where M = 0, 1...9 and Φ = 3 2 ln r L is the 10d dilaton field. Note that according to the above discussion we have A m = 0 and a periodic condition for X 4 , The relation of r KK and M KK can be determined by eliminating the conical singularity at r = r KK . Using alternative radial coordinates U ∈ [U KK , +∞) and Z ∈ [−∞, +∞) defined as the 10d metric in (2.4) can be explicitly written as, where 4 denotes a unit volume element on S 4 and F 4 is the field strength of the Romand-Romand form C 3 . Note that the metric in (2.7) corresponds to D4 bubble configuration which is obtained by a double Wick-rotation from the black brane solution. And the 11th direction X 11 becomes vanished in the large N c limit. In terms of QCD variables we additionally have the following relations, where λ, g YM , g s , l s respectively represents the 't Hooft coupling constant, Yang-Mills coupling constant, string coupling constant and the length of string.

Gravitational fluctuations as the glueball field in holography
Since glueball is the gauge invariant composite state in the Yang-Mills theory, they holographically corresponds to the gravitational perturbations of the near-horizon geometry in the supergravity description [23,24,25,26,27]. Such metric fluctuations are sourced by the operators in the dual field theory i.e. the 5d compactified Yang-Mills theory. Hence let us introduce the gravitational perturbations to the background (2.3) by rewriting the 11d metric as M N refers to the metric in (2.3). And it would be convenient to integrate over the S 4 in the 11d supergravity action since S 4 is not very necessary in the following discussion. So we can obtain the kinetic action for glueball by imposing the metric (2.3) into the 11d supergravity action and it leads to, where S G denotes the kinetic term of the glueball and C E,D,T is a constant determined by the various gravitational fluctuations. Note that the pre-factor in the above action has to be normalized as 1 4 Ω 4 C E,D,T = 1 in which C E,D,T is completely fixed. We will use this action to discuss the dynamics of the glueball with various gravitational perturbations in the the rest of the manuscript.

A. The exotic scalar glueball
The lowest exotic scalar glueball has quantum number J CP = 0 ++ which corresponds to the exotic polarizations of the bulk graviton. The 11d components of δG M N are given as [26], where the eigenvalue equation for function H E (r) is, Note that the above components in (2.10) involve asymptotics in the bulk as δG 44 = −4δG 11 = −4δG 22 = −4δG 33 = −4δG 11,11 for r → ∞. Imposing the metric (2.3) with fluctuations (2.10) and the eigenvalue equation for function H E (r) into action (2.9), we obtain the kinetic term of the exotic scalar glueball which is,

B. The dilatonic and tensor glueball
The scalar glueball of 0 ++ mode corresponds to the fluctuations of the metric as [26], (2.14) We refer to the upon mode as "dilatonic" since δG 11,11 reduces to the 10d dilaton. The tensor glueball with J CP = 2 ++ corresponds to the metric fluctuations which includes a transverse traceless polarization. A choice of the non-vanishing components of the graviton polarizations could be, . T µν is a constant symmetric tensor normalized as T µν T µν = 1 and satisfies the traceless condition η µν T µν = 0. The eigenvalue equation for functions H D,T (r) are given as, Plugging metric (2.3) with fluctuations (2.14) and (2.15) into action (2.9) respectively leads to the dynamics of the dilatonic scalar and tensor glueball, where the corresponding normalization constant is, 3 The flavours and mesons

The light flavour and meson
There are also flavours in this model which are introduced by embedding a stack of N f probe D8-and anti-D8-branes (D8/D8-brane) at the antipodal position of the background geometry. The configurations of the various branes are illustrated in Table 1. Table 1: The D-brane configurations: "-" denotes the world volume directions of the D-branes.
The N f flavoured fermions arise from the lowest modes of the 4 − 8 strings 4 and 4 −8 strings which belong to the fundamental representation of the colour group U (N c ) and flavour group U (N f ). In order to define the chirality of these fundamental fermions, a natural choice is to perform the GSO (Gliozzi-Scherk-Olive) projection [12,33]. Since the GSO projection for 4 − 8 strings is opposite to the projection chosen for the 4−8 strings, these fundamental fermions could be interpreted as quarks with opposite chiralities in QCD. And the gauge symmetry introduced by D8/D8-branes can be therefore denoted as U (N f ) L × U (N f ) R , namely the chiral symmetry. Note that the 4 − 8/8 strings has zero length in this D4/D8 configuration, the fundamental fermions are therefore massless which correspond accordingly to the light flavoured quarks in QCD. In this sense it motivates us to name these flavour branes as "light flavour branes". As shown in Table 1, the quarks and anti-quarks live on the separate position of X 4 . However the global chiral symmetry U (N f ) L × U (N f ) R is broken spontaneously since the geometry of the D4 bubble configuration forces the D8/D8-brane to be connected at the bottom of the bulk as illustrated in Figure1. Thus the induced metric on the flavour branes could be straightforward obtained by putting D8/D8-branes onshell i.e. imposing the condition dx 4 dU = 0, then we have, where the indices M, N run over the flavour brane. So the action of the joined light flavour branes describes the dynamics of qq mesons, which contains the light quarks only, through the flavoured gauge fields on the worldvolume of the D8/D8-branes. The explicit form of the action is, Here g and Φ is the induced metric and dilaton given in (3.1) (2.7) respectively. F refers to the field strength of the flavoured gauge field A M on the D8-branes which is defined as Note that in this model the non-zero components of the gauge field A M are only {A µ , A Z } which are functions dependent on the coordinates {x µ , Z}.
5 For N f D8-branes, all the actions in (3.3) should include a pre-factor N f . Nonetheless we do not pick it up because a pre-factor does not affect any results. 6 In this notation, the gauge field AM is anti-Hermitian which means Since the flavour branes are probes, we could obtain the following 5d quadratic action after imposing the solution (3.1) (2.7) to (3.3) and integrating over S 4 which is, The light flavour meson fields are identified as the expansion modes of A M by a complete sets {ψ n } in the model. Explicitly A M is assumed to be expanded as 7 , so the lowest mode of A µ and A Z could be interpreted as the lightest vector ρ meson and scalar π meson. Therefore if we identify B (1) the effective 4d action of π and ρ meson denoted by S eff π−ρ can be obtained from (3.4) whose form is 8 , The Chern-Simons (CS) term in (3.3) corresponds to the Wess-Zumino-Witten (WZW) term in chiral QCD. However we will not discuss its explicit form in this manuscript because it is independent on the metric thus does not include the interaction of meson and glueball.
creates additional multiplets which could be approximated near the worldvolume of the light flavour branes by local vector fields. Due to the finite separation of the heavy and light flavour branes, the HL-string has a non-zero length so that those multiplets acquire mass through the non-zero vacuum expectation value (VEV) of the HL-string. Hence the multiplets could be interpreted as the heavy-flavoured mesons with massive quarks. We note that this mechanism to acquire the massive fields is exactly the "Higgs mechanism" in string theory [30].
In order to include HL-multiple states in the actions (3.3), we need to replace the gauge fields on the light flavour branes by its matrix-valued formula according to string theory, Thus the field strength of A M becomes a matrix-valued 2-form as, . (3.10) And we have assumed that the non-zero components of Φ M are {Φ µ , Φ Z } which depends only on {x µ , Z} in order to be compatible with A M . Plugging (3.9) (3.10) into the quadratic action (3.4), we obtain the action as, where "Tr" refers to the "flavour trace" and complete DBI action with non-Abelian excitation in the Appendix B. Then let us denote the only one transverse "coordinate" as Ψ for D8-branes and its T-dualitized action is given as, By using the solution (3.15), the action (3.14) could be rewritten as, We note here the total action of the light flavour branes involving the HL fields is collected as, where S CS takes the same formula as (3.3) but replacing F → F. Along the discussion in the sector of light flavour , we further assume that the HL-multiplet Φ M could be expanded as 9 , with the normalization conditions then we can obtain the kinetic term of the massive HL-fields from (3.12) (3.16) which is, where

The interactions of glueball and heavy-light flavoured mesons
Since the gravitational perturbation signals the glueball states in holography, let us include the gravitational fluctuations in this section in order to obtain the effective Lagrangian of glueball and mesons. The glueball field could be involved by replacing the metric by G → G + δG as we discussed in Section 1, the interaction of glueball and various mesons will therefore arise once we add the gravitational fluctuations to the terms depended on the metric in the action. Accordingly it only concerns the Yang-Mills action (3.10) and the mass term (3.16) since the Chern-Simons action in (3.3) is independent on the metric. Thus let us first rewrite the Yang-Mills action (3.10) as, where S L , S HL refers to the kinetic terms of light and HL-mesons given respectively in (3.8) (3.20) and The coupling terms of light mesons and glueball, HL-mesons and glueball, HL-and light meson, light and HL-meson with glueball have been respectively denoted by using the index "L-G" "HL-G" "HL-L" "HL-L-G" in the action (4.1). Note that only the linear terms of the glueball field are valid since the gravitational fluctuations are perturbations in the supergravity description. Then the 10d metric involving the 11d fluctuations is given as, ,n Note that I µνρσ satisfies the relation I µνρσ = −I µνσρ = −I νµρσ , TrI µνρσ = TrI ρσµν . And we also have, (4.14) For the terms in S 3 , respectively they are, and all the coefficients are given as, Additionally we have another coupling from the mass terms (3.16) which is, where v V and the coefficients are, Eφmφn s (5) mn = κˆdZ The total action of the exotic scalar glueball and various mesons could be obtained by collecting all the terms in (4.1) and (4.19) with the kinetic term of the glueball in (2.12).

Involving the dilatonic scalar glueball
The effective action of the dilatonic glueball and mesons could be directly obtained by plugging (4.3) (4.4) with the dilatonic polarizations (2.14) into action (4.2). We find that the interaction terms S HL−L i in (4.2) remains since they are independent on the metric fluctuations. So the interaction terms of dilatonic glueball and mesons can be collected as takes the same formulas as in the case of the exotic scalar glueball while the coefficients have to be replaced by And the corresponding coefficients are given as, The coupling term S HL−G D m for the dilatonic glueball from the mass terms (3.16) also takes the same formula as in (4.19) while the coefficients need to be replaced as s So the total action for the interaction of dilatonic scalar glueball and meson could also be collected from (4.5) -(4.17) and (4.19) while the coefficients are replaced by (4.22) (4.23).

Involving the tensor glueball
The interaction terms of tensor glueball and mesons can be collected by inserting (4.3) (4.4) with the polarizations (2.15) into (3.11) (3.16). As before, let us rewrite the interaction terms involving the tensor glueball in the action as, where their explicit formulas are collected as, and The coupling from the mass term (3.16) is given as, where the corresponding coefficients in above actions are, (4.32)

Interaction of glueball and scalar D 0 meson in holography
Since the lowest heavy-light meson is recognized as D-meson, let us identify the lowest heavylight meson field in our theory as the lowest D-meson field (i.e. the D 0 meson) and investigate the decay of glueball to them holographically in this section.

The mode eigenfunctions and the choice of parameters
The eigenfunctions φ m ,φ n (3.18) have to satisfy the normalization condition (3.19) which is exactly the same as the normalization of {ψ n } for n ≥ 1 presented in [12], From these, the functions {ψ n } for n ≥ 1 could be generated by the following eigen equation, Thus it is seen that we can choose φ m = ψm √ 2 ,φ n = 1 for all m, n ≥ 1. Note that there still exists a functionφ 0 orthogonal to φ n for all n ≥ 1 which is given as, On the other hand, the glueball mass in holography is determined by solving the eigen equations (2.11) (2.16) for exotic, dilatonic and tensor glueball respectively. For the reader's convenience, we collect the mass spectrum of various glueballs in Table 2. It is clear that the glueball decay occurs when the mass relation m glueball /m D−meson 2 is satisfied because the effective holographic action we discussed is quadratic of the heavy-light meson fields. Accordingly we will choose the excited mass of n = 3 as M E = 154.963/9M KK 3938MeV, M D = M T = 162.699/9M KK 4035MeV for the exotic, dilatonic and tensor glueball respectively. With these values for the parameters, the boundary condition for the eigen equations (2.11) (2.16) is numerically evaluated as, (5.5) And we will use the parameters above to investigate the decay of glueball the lowest D-meson.

Glueball decay to the lowest scalar D 0D0 meson
In this section, let us compute the decay width of the glueball to the lowest heavy-light mesons i.e. the (pseudo) scalar meson (D 0D0 ). The expression of the decay width for the scalar glueball into DD is given by 12 , where |p| = M E,D /2 represents the 3-momentum of one D 0 in the rest frame of the glueball. For the tensor glueball we however need the average over its polarizations. Hence let us integrate over the orientation of the coordinates by choosing a fixed polarization 11 = − 22 = 1. So it leads to the decay width as.
Note that (D 0D0 ) will be denoted as (DD) for simplicity in the following discussion of this section.

A. exotic scalar glueball
The relevant part of the action involving the exotic glueball and D 0D0 can be collected from (4.5) -(4.19) which is, where the coupling constants could be numerically evaluated as, could be identified as the heavy-light flavoured mesons so that the model is able to include their interactions with glueball. We numerically evaluate the coupling constants between the various glueballs (the exotic, dilatonic scalar and tensor glueballs) and the heavy-light flavoured mesons then compute the associated decay widths/rates in this model. We find the decay widths of glueballs are parametrically suppressed by the factor λ −1 N −2 c in the large N c limit while the results vary substantially for the different decay channels. So it could not give the picture of "universal narrowness" although the heavy flavour and large N c limitation are involved in the model. However the decay widths of the lightest glueball to the lowest light mesons in this model turn out to be consistent with the experimental data of the f 0 (1500) decay [25,26], thus our calculation with heavy flavour could be a further prediction of glueball-meson interaction although the experimental evidence are currently insufficient 13 . Besides, the decay rate of the exotic scalar glueball is turned out to have a larger width than that of the heavier dilatonic mode according to our calculation, so our results support partially the conjecture proposed in [26] that is the lowest glueball model begins with the dilatonic mode and the exotic scalar glueball should be discarded.
Noteworthily phenomenological models usually require very large gluon condensates in order to admit only narrow glueball states while the value of the gluon condensate is quite small in the WSS model. Nevertheless it could be interesting to introduce the gluon condensates into a top-down approach of holography as [39,40,41] so that it would be able to compare with the phenomenological models.
We finally comments that the mixing of glueballs with qq meson states would be also interesting to study since the signatures of glueball content can be strongly obscured by the mixing. But it might probably require more difficult stringy corrections which are not captured by the effective action following from the WSS model in holography. Nonetheless this would be more pivotal to identify the glueball in the future study.
where 2k 2 10 = 16G 10 /g 2 s = (2π) 7 l 8 s , G 10 is 10d Newton constant and F 4 = dC 3 is the field strength of the Romand-Romand (R-R) 3-form C 3 with, The equations of motion could be obtained by varying the action (A-1) with respect to the metric g M N , the dilaton field Φ and the R-R form C 3 which are, Note that the solution given in (2.7) exactly satisfies the above equations of motion. The Romand-Romand strength F 4 needs to be supported by N c units of flux on S 4 which satisfies, The factor N c arises because in the WSS model we consider a stack of N c D4-branes.

Appendix B: The action of the D-brane with non-Abelian excitation
The action of D-branes with non-Abelian excitations describes the dynamics of coincident N Dpbranes. It could be obtained by T-duality which is the standard technique in string theory. Let us consider a stack of Dp-branes in D dimensional spacetime parametrized by {X µ } , µ = 0, 1...D−1. Then we use the indices a, b = 0, 1...p and i, j, k = p + 1...D − 1 to denote respectively the directions parallel and vertical to the Dp-branes in the spacetime. The complete bosonic action of such Dp-branes is therefore collected as, S CS =µ p n=0,1ˆD p−branes C p−2n+1 ∧ (B + 2πα F ) n n! , Q i j =δ i j + 2πα ϕ i , ϕ k E kj , E µν = g µν + B µν . (B-2) Here g µν , B µν is the metric of the D dimensional spacetime and the 2-form field respectively. The "STr" refers to the "symmetric trace" and F is the non-Abelian gauge field strength defined as F ab = ∂ [a A b] + A [a A b] . The transverse modes of the Dp-branes are denoted as ϕ i 's which are in fact the "T-dualitized" coordinates given by 2πα ϕ i = X i . Note that we have chosen the "static gauge" throughout the manuscript in order to gauge away the 2-form field B i.e. B µν = 0 and defined ϕ i=9 ≡ Ψ in (3.14) since there is only one transverse coordinate for the case of D8/D8-branes. By keeping these in mind, the DBI action in (B-2) could be expanded as, (B-3) Note the gauge field A a and scalar field ϕ i 's are all in the adjoint representation of U (N ).
As a D-brane is usually supersymmetric, there may be superpartner terms to the actions given in (B-2). Nonetheless we do not attempt to give the explicit action of those fermionic superpartners since they are irrelevant to the calculations in this manuscript. However in the WSS model, one might find it remains to have supersymmetry on the N f flavour branes below the scale of M KK although the N c D4-branes are compactified (non-supersymmetric) [42]. Accordingly it forces us to include the superpartners of the mesons on the flavour branes. This becomes indeed a theoretical issue of this model because the meson spectrum containing superpartners would be unrealistic. In order to figure out this problem in this model, we are motivated by the mechanisms in the compactification of the N c D4-branes and the processes are as follows. First let us compact the time direction in 10d solution (2.7) on a circle with the period/mass scale δt ∼ M −1 S π−ρ G T =Trˆd 4 x 1 2 t 1 T µν ∂ µ π∂ ν π + 1 2 t 2 M 2 KK T µν ρ µ ρ ν + with the coefficients t i = √ 6d i .