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Holographic mesons in D4/D6 model revisited

We revisit holographic mesons in the D4/D6 model to study holographic light vector mesons and the properties of heavy quarkonium in confining and deconfining phases. To treat the light mesons and heavy quarkonium on the same footing, we use the same compactification scale MKK in both systems. We observe that like scalar and pseudo-scalar mesons the vector meson mass is linearly proportional to the square root of the quark mass, when the quark mass is large. With a MKK fixed by light meson masses, we calculate the mass of heavy quarkonium in confining and deconfining phases. We also obtain the inmedium dispersion relation for heavy quarkonium, which is important to understand the dissociation and the screening mass of heavy quarkonium in quark-gluon plasma.


Introduction
The properties of the heavy quarkonium both at zero and at finite temperature have been intensively studied, see [1] for a review. At zero temperature, the charmonium spectrum reflects detailed information about confinement and quark-antiquark potentials in QCD [2]. At finite temperature, melting of heavy quarkonia could be a signal of the formation of the quark-gluon plasma (QGP) in a relativistic heavy ion collision [3]. Moreover, lattice calculations suggest that the charmonium states will survive at finite temperature up to about 1.6 to 2 times the critical temperature T c [4,5]. This suggests that analyzing the charmonium data from heavy ion collision inevitably requires detailed information about the properties of charmonium states in QGP. Therefore, it is very important theoretical challenge to develop a consistent non-perturbative QCD picture for the heavy quark system both below and above the QCD phase transition temperature. In this respect, a promising attempt would be the holographic QCD (via AdS/CFT [6]).
In a stringy D4/D6 model, scalar and pseudo-scalar bound states have been studied at zero and finite temperature in [7,8]. In a more phenomenological approach, bottom-up model, the mass spectrum of a charmonium and its dissociation temperature have been investigated [9,10,11]. However, we note that in the bottom-up model, different infrared scales are introduced to describe light mesons and heavy quarkonia. For example, in the hard wall model, the location of the infrared cutoff z m varies from light mesons to heavy quarkonia: 1/z m ≃ 320 MeV [12,13] for the light meson and 1/z m ≃ 1315 MeV [9] for the charmonium.
In this paper we study the spectrum of the light vector meson and heavy quarkonium using the D4/D6 model [7] in high-temperature deconfining phase as well as in confining phase. To treat the light mesons and heavy quarkonium on the same footing, we use the same compactification scale M KK in both systems. For the light vector meson, the spectrum was discussed in the D4/D8/D8 model [14] where the chiral symmetry and its breaking are realized geometrically. However, in the Sakai-Sugimoto model [14] it is quite difficult to include the quark mass. Therefore it is still of worth to study the spectrum of the vector mesons in the D4/D6 model and study the effect of the quark mass. We observe that like scalar and pseudo-scalar mesons [7] the vector meson mass is proportional to a square root of the quark mass M 2 v ∼ m q for large quark masses. We also obtain the in-medium dispersion relation for heavy quarkonium, which is important to understand the dissociation and the screening mass of heavy quarkonium [15].

D4/D6 model
We briefly summarize a pioneering holographic QCD model, the D4/D6 system [7]. The model contains N c number of D4 branes and N f flavor D6 branes whose configuration is given in Table 1. In the probe limit, the N c D4 branes are replaced by their supergravity background, and N f D6 branes are treated as probes. In this model mesons of a QCD-like gauge theory are described by the fluctuations of the D6 brane in the D4 background. The geometry of confining D4 brane reads where string coupling constant g s and the period of τ are given by The parameter L is given by the string coupling constant g s and the string length l s , L 3 = πg s N c l 3 s , and the compactification scale M KK reads By introducing K(ρ), the metric is simplified to be The position of the D6 brane is described by r(λ) with φ=0 and τ =constant. Then the induced metric on D6 is Now the action for D6 brane becomes (2.7) By using the well-known identity up to quadratic order in fields, we obtain

Meson spectroscopy
In this section we are to compute the vector meson masses by considering gauge field fluctuations on the D6 brane. Although we have two different scale of meson masses, light mesons and heavy quarkonia, the origin of the difference is that of the quark mass rather than the interaction. So we need to introduce only a single interaction scale M KK . This scale is to be matched with the gauge theory scale Λ QCD , hence it is encoded in the background geometry while the quark masses are encoded in the geometry of the probe branes.

Embedding
We first find a D6 embedding geometry by solving the equation of motion for r(λ). From the DBI action (2.9), we obtain the equation of motion for r(λ) to be With the following dimensionless variables we rewrite it as For large λ, we can solve the equation of motion for r(λ) to obtain the asymptotic solution as where r ∞ , c are related to the quark mass and the chiral condensate, see [7] for details.

Scalar and pseudo-scalar fluctuations
We start with scalar and pseudo-scalar fluctuations. Though these were extensively studied in [7], we include them for completeness, not to improve the results in [7]. The transverse fluctuation of the D6 brane is given by where r v (λ) is the solution of the embedding equation. Inserting (3.5) into the induced metric (2.6) and the DBI action (2.9), we obtain the induced metric where a and b run over 0 to λ and the DBI action, up to quadratic order, Now we arrive at the linearized equation of motion for pseudo-scalar, and for scalar, These equations are numerically solved to get meson masses with proper boundary conditions [7].

Gauge field fluctuations
Now we move on to the gauge field fluctuation. The relevant part of the Lagrangian density for the gauge field is given by where r v is the embedding solution. We decompose the gauge fields in terms of the orthonormal basis ψ n , φ n as Then, the field strength takes the following form With the decomposition, the quadratic part of the Lagrangian for the B µ field reads where To recover the canonical kinetic term of the gauge field in 4D, we impose the normalization condition for the wave function ψ(λ) as whereT 6 = −T 6 V 2 /g s and V 2 = dΩ 2 √ g 66 g 77 . We will impose a similar condition for φ(λ).
The wave function ψ(λ) satisfies the following mode equation derived from the quadratic action where m 2 n = −q 2 . Then, we obtain where m n is the eigenvalue. From (3.14) and (3.16), we have now We rescale the coordinate by U K to obtain To solve this equation, we impose two boundary conditions: at the IR, either ψ n (0) oṙ ψ n (0) = 0, and at the UV, ψ n ∼ λ α with α ≤ 1/2 from the normalizability condition in (3.14). Now we consider φ n . The normalization of φ n is similar to (3.16) As explained below and also in [14], once we set the field as φ n =ψ n /m n for n≥1, it can be gauged away as a part of B µ field. However, zero mode is exceptional, which is orthogonal to the other modes Then, the constant C is given by The field strength is written as and therefore the action (3.13) becomes Note that for the heavy quarkonium system, and so ϕ 0 =0 due to (3.22). To impose the UV boundary condition for ψ n more precisely, we consider the mode equation (3.18) at large λ, With ψ n ∼ λ α , we obtain α(α − 3) = 0. Since the normalizability condition dictates α ≤ 1/2, we should choose α = 0.

Numerical results
We solve the mode equations for scalar, pseudo-scalar, and gauge field fluctuations numerically. We first compute the mass of light mesons to fix the model parameter r l ∞ and M KK . Since D4/D6 model has no non-Abelian chiral symmetry but for U(1) A , the pseudo-scalar meson in this model corresponds to η ′ in QCD [7]. In QCD, however, U(1) A symmetry is explicitly broken by the axial anomaly, and the observed mass of η ′ , m η ′ = 958 MeV, is much larger compared to the pion or kaon mass. Note that some portion of the η ′ mass comes from the anomaly effect which scales as N f /N c . Since we are working in the large N c limit, we may use the mass of η ′ with the anomaly contribution turned-off. So we use non-anomalous η ′ mass to obtain a rough number for the model input. To this end, we use the mass relation for Goldstone boson obtained in chiral perturbation theory at large N c [16]: The term with τ is from the axial anomaly. Now we take τ = 0 to estimate the η ′ mass from non-anomalous contribution. With m q = 7 MeV, m s = 150 MeV, f π = 93 MeV, and Σ = (230 MeV) 3 , we obtain m π ∼ 140 MeV and m ′ η ∼ 390 MeV. Note that the mass of qq bound state such as ρ meson mass stays almost constant as we increase N c : for instance the light meson mass at large N c is extensively studied in a unitarized chiral perturbation theory [17] and also in lattice QCD [18].
We use ρ-meson mass and η ′ mass in large N c limit as inputs to fix r ∞ = 0.191 and M KK = 1.039. Our fitting results are summarized in Table 2. In Fig. 1, we plot the masses of scalar, pseudo-scalar and vector as a function of r ∞ . As observed in [7], for large r ∞ the meson mass becomes degenerate and increases monotonically with r ∞ . This is simply because the equation of motion for scalar, pseudo-scalar and vector for heavy quark system, r ∞ ≫ U K , are degenerated. where Ψ can be real scalar δr, pseudo-scalar φ, or vector ψ, and y is a rescaled coordinate y=λ/r ∞ . In the equation (3.29), the only scale is This means that for the heavy quark system, the fluctuating field has mass being proportional to mqM K λt . As discussed in [7], the reason for the degeneracy is supersymmetry restoration. They are all in the same supermultiplet and for small m q limit supersymmetry is broken so their masses split. However, for the large quark mass or large separation between D4 and D6, the embedding is nearly flat, and so D6 brane restores supersymmetry.

Mode
Input (  Now we move on to the heavy quarkonium. To describe the light meson and heavy quarkonium systems on the same footing, we use the value of M KK determined in the light quark sector to calculate the mass of heavy quarkonium. We choose J/ψ mass as an input to fix r c ∞ . In Table 3, we list our results on the charmonium with r c ∞ = 11.92 , and in Table 4 we show bottomonium masses r b ∞ = 108.15. Note that r ∞ is related to the quark mass parameter of D4/D6 model as 1   using the same value of M KK for both cases. As long as the lowest lying KK mode masses are concerned, this unified approach seems working apart from the axial-vector meson. Note that this defect of degenerate vector and axial-vector mesons is the feature of D4/D6 model itself not that of unified approach. However, the way this attempted unified approach works seems almost guaranteed in the following sense. The light meson masses are mostly inputs to fix the r l ∞ and M KK , while for heavy quarkonia r l ∞ will be fixed to reproduce a heavy quarkonium mass. As in Fig. (1), the masses of heavy quarkonium (r h ∞ = 10.72) are almost degenerate regardless of their quantum numbers, while experimentally those masses are not very different from each other. Therefore, our results on the heavy quarkonium will be within experiments by roughly 10% deviations.

Heavy quarkonium in deconfined phase
We consider the meson spectroscopy in deconfining phase. By double Wick rotation, we get the black D4 background which is dual to the deconfining phase, Here the temperature of the black D4 brane is identified with the temperature of a dual gauge theory In terms of these periods, the metric parameters are given by As discussed in [7], there exists first order phase transition between confining and black D4 brane backgrounds; this phase transition is a deconfinement phase transition in field theory side. The critical temperature of the transition is fixed by the condition δτ = δt E Note that the temperature of the gauge theory is given as (4.5)

Embedding
The D6 brane embedding in the black D4 background is described by r(λ) The DBI action for r(λ) is then (4.7) From this, we obtain the equation of motion for the embedding function r(λ) The solution of the embedding equation was extensively studied in [7]. By comparing the energy density of various embedding solutions, the authors of [7] found that there is a first order phase transition at r ∞ ∼ U K . This transition, from Minkowski to black hole embedding, occurs when the D6 brane touches the black hole horizon at sufficiently large temperature. The transition temperature for charm quark is given by With the parameters fixed in confining phase, we obtain T dec ∼ 165 MeV and T f und ∼ 582 MeV. We summarize the different embedding solutions below.

T
Embedding Energy spectrum T < T dec Confining emb. discrete T dec < T < T f und Minkowski emb. discrete T f und < T Blackhole emb. continuous We note, however, that there is no priori reason for T dec < T f und , which is satisfied only when As a consequence, there is no Minkowski embedding for the light quark system, meaning that light mesons will be melted away immediately when the temperature exceeds the deconfinement temperature T > T dec . For the heavy quark system we expect that there are discrete meson spectra even in the deconfining phase since m h q is big enough to satisfy the condition.

Fluctuating fields as mesons
In this section we study temperature dependent meson masses. The temperature dependence of scalar and pseudo-scalar heavy quarkonia are already considered in [8], and now we extend the work by including vector mesons. In the high-temperature deconfining phase, there is only one relevant scale, T /M , which is nothing but the separation between D4 and D6 branes. Therefore varying temperature is equivalent to varying the D4-D6 separation with a fixed quark mass.

Scalar and pseudo-scalar fluctuations
Scalar and pseudo-scalar mesons are described by the D6 brane fluctuating on the transverse direction, r(λ) = r v (λ) + δr(t, λ), φ(t, λ). The quadratic part of Lagrangian after rescaling by U T is The equation of motion for the bulk scalar field (δr) and for pseudo-scalar (φ), , and the equation of motion for the pseudo-scalar is (4.12) Since there is no Lorentz symmetry due to the temperature, we define the thermal meson mass as ∂ 2 t φ = m 2 φ φ, i.e., at vanishing spatial momentum. The fields φ, δr have two linearly independent solutions at λ → ∞ as λ 0 , λ −1 . Here we choose the asymptotic solution as  Figure 2: The temperature dependent meson masses. Real scalar (left) and pseudo-scalar (right) masses in GeV unit. In confined phase the temperature dependence is trivially constant which is smooth at the deconfinement transition temperature, T dec ∼ 160 MeV. Note that the disjoint at T dec denotes that we use different backgrounds below and above T dec . Above the melting temperature, T f und ∼ 571 MeV, there is no discrete spectrum for the meson.
1/λ to have a normalizable mode. Note that The temperature dependent masses are shown in Fig. 2. Now we calculate the the disper- sion relation of heavy quarkonia, which is important to understand the dissociation and the screening mass of heavy quarkonium. The result in Fig. 3 shows that for sufficiently low temperature but larger than T dec , the group velocity v g is approaching one, and it is decreasing with increasing temperature. Note that in D3/D7 model the dispersion relations of mesons was studied [20].

Gauge field fluctuations
Now we move on to the gauge field to study vector mass in deconfined phase. The relevant Lagrangian is (4.14)

Transverse vector
The vector fluctuations at finite temperature are classified into transverse and longitudinal modes due to lack of Lorentz invariance. We first consider the transverse part. Assuming that the boundary gauge field propagates only in x direction, we obtain the Lagrangian for transverse vector fluctuations as The equation of motion for the transverse vector is where We calculate the dispersion relation and group velocity to obtain Fig. 4. Note that w ∼ k 2 for small k and w ∼ k in large k region.

Longitudinal vector
We decompose the gauge field into the ortho-normal basis to obtain Again, we define the thermal meson mass at zero momentum, ∂ i = 0 or k i = 0. Then the wave-function of the gauge field satisfies with the normalization condition From these, we obtain the equation of motion for the vector mesons The meson mass is given by We solve the mode equation numerically and show the result in Fig. 5. We impose ψ m ∼ 1/λ at large λ due to the normalizability. As shown in Fig. 2 and Fig. 5, the mass changes smoothly from confining to deconfining phase. This may be different from previous studies based on a bottom-up AdS/QCD model [9], QCD sum rule approach [21], and holographic heavy potential [22]. Note, however, that the maximum shift of charmonium mass at T = 1.05T c is about 200 MeV [21], which is roughly 6.7% of the mass of J/ψ and may be too small to be explained by our approach based on large N c approximation.

Remarks
Now, we discuss the relevance of the D4/D6 model for light mesons and heavy quarkonia. One of our basic goals is to study both light mesons and heavy quarkonia on the same footing based on a single D-brane model with a common energy scale M KK . Another one is to study heavy quarkonium properties in the high-temperature deconfining phase. Apart from non-Abelian chiral symmetry, the D4/D6 model might be a good candidate for these purposes since it has both confining and deconfining phases and includes quark mass through embeddings naturally. Especially for a heavy quark system, the D4/D6 model shows some similarity with heavy quarkonium physics. First, the non-Abelian chiral symmetry is not an issue for heavy quarks. Second, the D4/D6 model has one more transition at T = T f und other than the deconfinement transition, which may be associated with the dissociation of heavy quarkonia at high temperature. A positive clue for this is that T f und is proportional to the square root of a quark, meaning that the charmonium melts relatively at low temperature compared to the bottomonium. For instance, the dissociation temperature for a bottomonium Υ is ∼ 2.7T c , while that for a charmonium J/ψ is ∼ 1.3T c [23].
We finish this section with a summary of the discussion in [8] on the usefulness of Dq/Dp systems in studying meson bound states. A Dq/Dp system may be good for ss bound states at high temperature since the mesons in the Dq/Dp system are deeply bounded. Even above T f und , there still exists some broad peak in the spectral function of two-point correlators of mesons. Therefore T f und is slightly smaller than the dissociation temperature of the heavy quarkonium. The D4/D6 model may describe some exotic gauge theories. However, there exist certain properties of heavy quarkonia in the quark-gluon plasma that could be understood in the D4/D6 model.

Summary
We re-analyzed the D4/D6 model to study holographic light vector mesons and the properties of heavy quarkonium in confining and deconfining phases. To treat the light mesons and heavy quarkonium on the same footing, we used the same compactification scale M KK in both systems. In confined phase, we observed that the meson spectroscopy of the light meson and heavy quarkonium could be described with a single M KK . We found that like scalar and pseudo-scalar mesons the vector meson mass is linearly proportional to the square root of the quark mass, when the quark mass is large. With a M KK fixed by light meson masses, we calculated the mass of heavy quarkonium in confining and deconfining phases. Unlike previous studies based on a bottom-up AdS/QCD model [9], QCD sum rule approach [21], and holographic heavy potential [22], we found that the mass of heavy quarkonium changes smoothly from confining to deconfining phase. However, since the maximum shift of charmonium mass at T = 1.05T c is about 200 MeV [21], which is roughly 6.7% of the mass of J/ψ, it is not conclusive if our results are really different from a previous study. We also obtained the in-medium dispersion relation for heavy quarkonium, which is important to understand the dissociation and the screening mass of heavy quarkonium.
Certainly there are many things to be improved in our study to be more close to QCD. We list some of them here. Surely chiral symmetry should be the first one. As well known, non-Abelian chiral symmetry is essential to understand light mesons, and it is also important for heavy-light system due to the light quark. Second thing is how to include heavy-light meson in this picture with correct chiral symmetry and heavy quark spin symmetry.