Holographic Description of Heavy-Flavored Baryonic Matter Decay Involving Glueball

PHYSICAL REVIEW D 99, 046013 (2019)

Holographic description of heavy-flavored baryonic matter decay involving glueball

Si-wen Li* Department of Physics, School of Science, Dalian Maritime University, Dalian 116026, China

(Received 22 December 2018; published 20 February 2019)

We holographically investigate the decay of the heavy-flavored baryonic hadron involving a glueball by using the Witten-Sakai-Sugimoto model. Since the baryon in this model is recognized as the D4-brane wrapped on S4 and the glueball field is identified as the bulk gravitational fluctuations, the interaction of the bulk graviton and the baryon brane could be naturally interpreted as a glueball-baryon interaction through the holography which is nothing but the close-open string interaction in string theory. In order to take the heavy flavor into account, an extra pair of heavy-flavored branes separated from the other flavor branes with a heavy-light open string is embedded into the bulk. Due to the finite separation of the flavor branes, the heavy-light string creates massive multiplets which could be identified as the heavy-light meson fields in this model. As the baryon brane on the other hand could be equivalently described by the instanton configuration on the flavor brane, we solve the equations of motion for the heavy-light fields with the Belavin-Polyakov-Schwarz-Tyupkin (BPST) instanton solution for the Nf ¼ 2 flavored gauge fields. Then, with the solutions, we evaluate the soliton mass by deriving the flavored onshell action in the strong coupling limit and heavy quark limit. After the collectivization and quantization, the quantum mechanical system for the glueball and heavy-flavored baryon is obtained in which the effective Hamiltonian is time dependent. Finally, we use the standard technique for the time-dependent quantum mechanical system to analyze the decay of the heavy-flavored baryon involving the glueball, and we find one of the decay processes might correspond to the decay of the baryonic B-meson involving the glueball candidate f0ð1710Þ. This work is a holographic approach to studying the decay of the heavy-flavored hadron in nuclear physics.

DOI: 10.1103/PhysRevD.99.046013

I. INTRODUCTION that the scalar meson f0ð1710Þ could be considered as a glueball state. The glueball may be produced by the decay Quantum chromodynamics (QCD) as the fundamental of various hadrons in the heavy-ion collision [8–10], so the theory of nuclear physics predicts the bound state of pure dynamics of the glueball is very significant. However, gluons [1–3] because of its non-Abelian nature. Such a lattice QCD involving real-time quantities is extremely bound state is always named a “glueball,” which is believed complex, and phenomenological models usually include a to be the only possible composite particle state in the pure large number of parameters with some corresponding Yang-Mills theory. In general, glueball states could have uncertainties. Thus, it is still challenging to study the various Lorentz structures, e.g., a scalar, pseudoscalar, dynamics of glueball with traditional quantum field or a tensor glueball with either normal or exotic JPC theory. assignments. Although the glueball state has not been Fortunately, there is an alternatively different way to confirmed by the experiment, its spectrum has been studied – investigate the dynamics of glueball based on the famous by the simulation of lattice QCD [4 6]. According to the AdS/CFT correspondence or the gauge-gravity duality lattice calculations, it indicates that the lightest glueball pioneered in [11] where a top-down holographic approach state is a scalar with assignment of 0þþ and its mass is – from string theory by Witten [12] and Sakai and Sugimoto around 1500 2000 MeV [4,7]. These results also suggest [13] (i.e., the WSS model) is employed. Analyzing the AdS/CFT dictionary with the WSS model, the glueball *[email protected] field is identified as the bulk gravitational fluctuations carried by the close strings while the meson states are Published by the American Physical Society under the terms of created by the open strings on the N probe flavor branes. the Creative Commons Attribution 4.0 International license. f Further distribution of this work must maintain attribution to Hence this model naturally includes the interaction of the author(s) and the published article’s title, journal citation, glueball and meson through the holography which is and DOI. Funded by SCOAP3. nothing but the close-open string interaction in string

2470-0010=2019=99(4)=046013(13) 046013-1 Published by the American Physical Society SI-WEN LI PHYS. REV. D 99, 046013 (2019)

TABLE I. The brane configuration of the WSS model: “” denotes the world volume directions of the D-branes.

0123(4)5ðUÞ 6789

Colored NcD4 Flavored Nf D8=D8 Baryon vertex D40 theory. Along this direction, decay of glueball into mesons flavor brane. In Sec. III, we search for the analytical has been widely studied with this model e.g., in [14–16].1 solutions of the bulk gravitational fluctuations then explic- Keeping the above information in mind and partly itly compute the onshell action with these solutions. All the motivated by [8–10], in this work we would like to calculations are done in the limitation of large ’t Hooft holographically explore the glueball-baryon interaction coupling constant λ where the holography is exactly valid. particularly involving the heavy flavor as an extension to The final formulas of the effective Hamiltonian depend on the previous study in [19]. In the WSS model, baryon is the glueball field and the number of heavy-flavored quarks identified as the D40-brane2 wrapped on S4 [13,20] (namely so that it is time dependent. Therefore, the method for the the baryon vertex) which could be equivalently described time-dependent system in quantum mechanics would be by the instanton configuration on the flavored D8-branes suitable to describe the decay of heavy-flavored baryons according to the string theory [21,22]. The configuration of under the classical glueball field. Consequently, we obtain various D-branes is illustrated in Table I. In order to take several possible decay processes with the effective into account the heavy flavor, we embed an extra pair of Hamiltonian and pick out one of them which might flavored D8=D8-brane into the bulk geometry which is probably correspond to the decay of baryonic B-meson 1710 separated from the other N (light-flavored) D8=D8-branes involving the glueball candidate f0ð Þ as discussed f – with an open string (the heavy-light string) stretched in [8 10]. Since the WSS model has been presented in many between them [23,24] as illustrated in Fig. 1. In this ’ configuration, there would be additional multiplets created lectures, for reader s convenience we only collect some by the heavy-light (HL) string and they would acquire mass relevant information about this model in Appendixes A, B, and C which can be also reviewed in [13,22,25–27]. due to the finite separation of the flavor branes. Hence, we Respectively the gravitational polarization used in this could interpret these multiplets as the HL meson fields and paper are collected in Appendix A. Some useful formulas the instanton configuration on the D8-branes with the about the D-brane action and the embedding of the probe multiplets would include heavy flavor and, thus, can be branes and string in our setup can be found in Appendix B. identified as the heavy-flavored baryon [25,26]. So, sim- In Appendix C, it reviews the effective quantum mechani- ilarly to the case of the glueball and meson, there must be cal system for the collective modes of baryon. At the end of glueball-baryon interaction in holography as the close this manuscript, some messy but essential calculations string interacting with the D40-brane carrying the heavy about our main discussion have been summarized in flavor through the HL string or, namely, as the graviton Appendix D. interacting with the heavy-flavored instantons. Let us outline the content and the organization of this manuscript here. We consider the baryonic bound states II. BARYON AS INSTANTON created by the baryon vertex in this model with two flavors WITH HEAVY FLAVOR i.e., Nf ¼ 2. Following [22,24–26], baryons are identified The baryon spectrum with pure light flavors in this as Skyrmions in the WSS model and they can be described model is reviewed in Appendix C, so we only outline how by a quantum mechanical system for their collective modes to include the heavy flavor in this section. Some necessary in the moduli space. The effective Hamiltonian could be information about the embedding of the probe branes and obtained byR evaluating the classical mass of the soliton string in our setup could be reviewed in Appendix B. onshel − S ¼ dtMsoliton. So the main goal of this paper is to A simple way to involve the heavy flavor in this model is evaluate the effective Hamiltonian involving glueball- to embed an extra pair of flavored D8=D8-brane separated baryon interaction with heavy flavor. In Sec. II, we specify from the other Nf (light-flavored) D8=D8-branes with an the setup with the heavy flavor in this model and solve the open string (the heavy-light string) stretched between them classical equations of motion for the HL meson field on the as illustrated in Fig. 1. The HL string creates additional multiplets according to the string theory [27] since it

1 connects to the separated branes. And these multiplets Since the WSS model is based on AdS7=CFT6 correspondence, several previous work is also relevant to this model e.g., [17,18]. could be approximated by local vector fields near the 2We will use “D40-brane” to distinguish the baryon brane from worldvolume of the light flavor branes. Note that the those Nc D4-branes as color branes throughout the manuscript. multiplets acquire mass due to the finite length, or namely

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D8()L D8()L D8()H D ()L U U 8 D8()L D8()H

U = U UH

U = U 4 = KK X U U KK

Baryon vertex Baryon vertex

FIG. 1. The various D-brane configurations in the WSS model. Left: The configuration of the standard WSS model according to Table I. The bulk geometry is produced by Nc coincident D4-branes which represent “colors” in QCD. The flavors are introduced into the model by embedding Nf pairs of coincident D8=D8-branes at the antipodal position of the bulk geometry. U refers to the holographic direction and X4 is compactified on S1. The D40-brane as the baryon vertex looks like a point in the U − X4 plane. Mesons are created by the open string on the flavored D8=D8-branes while baryons are created by the wrapped D40-branes. Right: The WSS model with heavy flavor. An additional pair of flavored D8=D8-brane (denoted by the red line) as the heavy-flavored (H) brane separated from the other Nf pairs of light-flavored (L) D8=D8-branes with a HL string (denoted by the green line) is embedded. The baryon vertex contains heavy flavor in this configuration through the HL string. Z 1 ffiffiffiffiffiffi the nonzero vacuum expectation value (VEV) of the HL YM 0 2 9 −Φp ab cd S ¼ − ð2πα Þ T8Tr d xe −gg g F F string. Therefore, we could interpret the multiplets created DBI 4 ac bd Z D8=D8 by the HL string as the heavy-flavored mesons with 1 0 2 9 −Φ massive flavored (heavy-flavored) quarks. Actually, this ¼ − ð2πα Þ T8 d xe 4 D8=D8 mechanism to acquire mass is nothing but the “Higgs ffiffiffiffiffiffi p ab cd mechanism” in string theory. So let us replace the gauge × −g½g g TrðF acF bd − αacF bd − F acαbdÞ fields on the light flavor branes by its matrix-valued form to † − 2gabgcdf f ; ð2:3Þ involve the heavy flavor, ac bd

where Aa Φa A → A ¼ : ð2:1Þ a a −Φ† 0 a ∂ Φ Φ fab ¼ ½a b þ A½a b; A † −∂ Φ† − Φ† α Φ Φ† In our notation, a is an Nf × Nf matrix-valued 1-form fab ¼ ½a b ½aAb; ab ¼ ½a b: ð2:4Þ while Aa is an ðNf þ 1Þ × ðNf þ 1Þ matrix-valued Φ 1 1-form. a is an Nf × matrix-valued vector which We should notice that from the full formula of the DBI represents HL meson field and the index runs over the action, it would contain an additional term of the transverse A light flavor brane. Thus, the field strength of a also mode Ψ of D8=D8-branes as shown in Appendix B.And becomes matrix-valued as a 2-form, this term could be written as

† ! Z F − Φ Φ ∂ Φ þ A Φ pffiffiffiffiffiffiffiffiffiffiffiffiffi ab ½a b ½a b ½a b D8 ˜ 9 −Φ F → F S ¼ −T8Tr d xe − det g ab ab ¼ † † † ; Ψ −∂ Φ − Φ A −Φ Φ D8=D8 ½a b ½a b ½a b   1 1 : Ψ aΨ Ψ Ψ 2 ð2 2Þ × 2 Da D þ 4 ½ ; ; ð2:5Þ where F refers to the field strength of A . Imposing (2.1) ab a Ψ ∂ Ψ A Ψ ˜ 2πα0 2 (2.2) into D8-brane action (C1) and keep the quadric terms with Da ¼ a þ½ a; and T8 ¼ð Þ T8. In the 3 case of N pairs of light-flavored D8=D8 branes separated of Φa, it leads to a Yang-Mills (YM) action f from one pair of heavy-flavored D8=D8 branes, we can Ψ 3We do not given the explicit formula of the CS term with HL define the moduli solution of with a finite VEV v by the fields since it is independent on the metric; thus, it is irrelevant to extrema of the potential contribution or ½Ψ; ½Ψ; Ψ ¼ 0 the glueball-baryon interaction. [27,28] as

046013-3 SI-WEN LI PHYS. REV. D 99, 046013 (2019) − v 1 0 III. GLUEBALL-BARYON INTERACTION Nf Nf Ψ ¼ : ð2:6Þ WITH HEAVY FLAVOR 0 v The dynamic of the free glueball is reviewed in So the action (2.5) could be rewritten by plugging the Appendix A, so in this section we will focus on the solution (2.6) into (2.5) as interaction of the glueball and baryon with heavy flavor characterized by the collective Hamiltonian. As the glueball Z field is included by the metric fluctuations, the Chern- 1 2 ˜ 2 ðNf þ Þ 4 Simons (CS) term is independent on the metric; thus, it SΨ −T8v Tr d x ¼ 2 does not involve the glueball-baryon interaction. Hence, let Nf Z us start with the five-dimensional YM action plus the mass þ∞ pffiffiffiffiffiffiffiffiffiffiffiffiffi −Φ ab † × dZe − det gg ΦaΦb; ð2:7Þ term which are collected in (2.3) (2.7). The onshell form of −∞ (2.3) (2.7) corresponds to the effective interaction Hamiltonian of the glueball andR heavy-flavored baryon onshel which is exactly the mass term of the HL field. Then through the relation S ¼ − dtHG−B, accordingly we perform the rescaling (C3), we could obtain the classical first need to solve the eigenvalue equations for function λ equations of motions for Φa from (2.3) (2.7) as HE;D;T in large limit. The eigenvalue equations for HE;D;T are given in (A9) −1=2 −1 and (A13). In the rescaled coordinate Z → λ Z, the D D Φ − D D Φ þ 2F Φ þ Oðλ Þ¼0: M M N N M M NM M equations are written as Φ − Φ − F 0MΦ DMðD0 M DM 0Þ M 1 1 2Z 16 M2 1 − ϵ O λ−1 0 H00 ðZÞþ þ H0 ðZÞþ þ E H ðZÞ 2 MNPQKMNPQ þ ð Þ¼ ; ð2:8Þ E λ E 3λ 2 λ D 64π a Z MKK þ Oðλ−2Þ¼0; M i 1 where x ¼fx ;Zg, i ¼ , 2, 3 and the 4-form KMNPQ is 1 2 2 1 00 Z 0 MD given as HD;T ðZÞþ þ λ HD;T ðZÞþ 2 λ HD;T ðZÞ Z MKK þ Oðλ−2Þ¼0; ð3:1Þ KMNPQ ¼ ∂MAN∂PΦQ þ AMAN∂PΦQ 5 and they could be easily solved as þ ∂MANAPΦQ þ AMANAPΦQ: ð2:9Þ 6 3M2 þ 16M2 H z C 1 − E KK Z2 λ−1=2N−1M−1 Eð Þ¼ E 12 2 λ c KK Since the holographic approach is valid in the strong MKK λ → ∞ O λ−1 coupling limit , the contributions from ð Þ have þ Oðλ−3=2Þ; been dropped off. Note that the light flavored gauge field A M2 a satisfies the equations of motion obtained by varying H z C 1 − D;T Z2 λ−1=2N−1M−1 D;T ð Þ¼ D;T 4 2 λ c KK the action (C1), so their solution remains to be (C2) in the MKK 0 λ Φ ϕ imHx −3 2 large limit. And we could further define a ¼ ae þ Oðλ = Þ: ð3:2Þ in the heavy quark limit i.e., mH → ∞ as in [25,26] so that Φ ϕ “ ” D0 M ¼ðD0 imHÞ M where corresponds to quark Next we perform the rescaling as in (C3), then insert the and antiquark, respectively. By keeping these in mind, BPST solution (C2) for the gauge field A and (A9) for the altogether we find the full solution for (2.8) as heavy-light meson field Φa into the action (2.3) (2.7).   Afterwards by plugging the metric (A6) plus the dilaton 1 25ρ 7 (A7) with the solution (3.2) and various fluctuations which ϕ0 ¼ − þ χ; 1024aπ2 2ðx2 þ ρ2Þ5=2 ρðx2 þ ρ2Þ3=2 include the exotic scalar, dilatonic scalar and tensor glue- ρ ball field all given in Appendix A into the action (2.3) (2.7), ϕ σ χ M ¼ 2 2 3=2 M ; ð2:10Þ the onshell form of action (2.3) (2.7) could be obtained ðx þ ρ Þ μ μ with the dimensionless variable x → x =MKK; Aμ → AμMKK as χ M where is a spinor independent on x . Then in the double Z limit i.e., λ → ∞ followed by m → ∞, the Hamiltonian H onshell 4 1=2 E;D;T E;D;T S − ¼aC d xdZTr½λ L þL0 for the collective modes involving the heavy flavor could be GE;D;T B E;D;T 1=2 calculated as in (C7) by following the procedures in λ−1=2LE;D;T λ−1=2LE;D;T O λ−1m0 ; : Appendix C. þ −1=2 þ Ψ þ ð HÞ ð3 3Þ

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1 “ ” “ ðnÞ where a ¼ 216π3, E,D,T refers, respectively, to exotic TABLE II. The glueball mass spectrum ME;T in the WSS model scalar, dilatonic scalar, and tensor glueball.” Although the in the units of MKK is collected from [14] and the numerical C above calculation is very straightforward, the explicit forms values of the associated coefficients presented in (3.4) E;D;T are LE;D;T LE;D;T evaluated. of 1=2;0;−1=2 and Ψ are quite lengthy. So we summa- LE;D;T LE;D;T Excitation of glueball ðnÞ n ¼ 0 n ¼ 1 n ¼ 2 n ¼ 3 n ¼ 4 rize the full formulas of 1=2;0;−1=2 and Ψ with some ðnÞ essential instructions in Appendix D andR here skip to the Glueball mass ME 0.901 2.285 3.240 4.149 5.041 onshell − X s ðnÞ final results. Using the relation SG−B ¼ dtHG−Bðt; Þ, Glueball mass MD;T 1.567 2.485 3.373 4.252 5.124 the dimensionless interaction Hamiltonians are computed as,4 The coefficients n ¼ 0 n ¼ 1 n ¼ 2 n ¼ 3 n ¼ 4

CE 144.545 114.871 131.283 146.259 157.832 CD 29.772 36.583 42.237 47.220 51.724 C 15 T 72.927 89.609 103.46 115.664 126.696 s −1=2 −1 2 2 mH † H − ðt;X Þ¼−C λ M 5m π a þ G χ χ GE B E KK H 32ρ2 E Z −1 0 2 þ Oðλ m Þ Γ → 1 H B GþX s −iðEi−EjÞt ¼ dthijHG−Bðt;X Þjjie ; 3 mH mH s −1=2 −1 mH 2 2 † H − t;X C λ M − 6m π a G χ χ GD Bð Þ¼ D KK 2 H D 1 8ρ s 2 ¼ hijHG−BðX Þjji δðEj −Ei −ME;D;T Þ; ð3:5Þ þ Oðλ−1m0 Þ mH H 7 s −1=2 −1 2 2 mH † H − ðt;X Þ¼−C λ M m π a þ G χ χ jii, jji, Ei;j refers to the eigenstate and the associated GT B T KK 2 H 4ρ2 T eigenvalue of (C7). And the above decay occurs only if O λ−1 0 þ ð mHÞ; ð3:4Þ several physical quantities e.g., energy, total angular momentum J, are also conserved. Note that the interaction Hamiltonians in (3.4) are independent on Z,so s hijH − ðX Þjji would be vanished unless the states jii, The constants C are determined by the eigenvalue G B E;D;T jji take the same quantum number of n and l. The equations for H , and they depend on the mass of the Z E;D;T Hamiltonians in (3.4) can also describe the decay of an various glueballs. We numerically evaluate CE;D;T in antibaryon if we replace mH by −mH. Table II with the corresponding glueball mass. Notice that With the perturbed Hamiltonian in (3.4), this model the operator GE;D;T satisfies the equations of motion by includes various decays of heavy-flavored hadrons involving varying action (A10) (A14); thus, its classical solution is 1 the glueball. So we are going to examine the possible −iME;D;T t GE;D;T ¼ 2 ðe þ c:cÞ and it is time dependent. On transitions involving one glueball with the leading low- χ the other hand, the spinor has to be quantized by its energy excited baryon states nρ ≤ 5. Since our concern is the † anticommutation relation f χα; χβg¼δαβ in the full quan- situation of two-flavored meson, we could follow [22] by † l tum field theory so χ χ is the number operator of heavy setting 2 ¼ J ¼ 0, NQ ¼ 1, Nc ¼ 3 in order to fit the quarks. Therefore, in our theory, the glueball field could be experimental data of the (pseudo) scalar meson states with treated as the classical field while baryon is quantized in the one heavy flavor. Then let us first take account into the † moduli space and we can identify χ χ ¼ N as the number 0 Q energy conservation Eðnρ ¼nρ þΔnρ;l¼0;NB ¼1;nZÞ− of heavy quarks in a baryonic bound state. Moreover, 0 ðnÞ Eðnρ;l¼0;N ¼1;n Þ≡EðΔnρÞ¼M if the transition the Hamiltonians in (3.4) is definitely suitable to be B Z E;D;T ðnÞ perturbations to the quantum mechanics (C7) since they of hadron decay would happen, where ME;D;T refers to the −1=2 are all proportional to λ in the large λ limit. Then glueball mass given in Table II and Eðnρ;l;NB;nZÞ refers to the interaction of the glueball and heavy-flavored baryon the baryonic spectrum in (C9). By keeping these in mind, the could be accordingly described by using the method of following relations are picked out, time-dependent perturbation in the quantum mechanical system. Last but not least, the decay rates/width Γ can be ðn¼1Þ EðΔnρ ¼ 3Þ=M ≃ 0.986; evaluated by using the standard technique for the time- D;T dependent perturbation in quantum mechanics, which is E Δ 4 ðn¼2Þ ≃ 1 008 ð nρ ¼ Þ=ME . ; ð3:6Þ given as

while EðΔnρÞ;nρ ≤ 5 with Δnρ ¼ 0, 1, 2 does not match to 4The glueball field G in (3.4) is dimensional which E;D;T any MðnÞ . Hence we could find the following possible is in the unit of MKK while the other parameters are E;D;T dimensionless. decays involving the glueball according to (3.6),

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I∶ BaryonicjJ ¼ 0;nρ ¼ 3i glueball, while the pure scalar meson with even parity is less evident according to the current experimental data. On → ðn¼1Þ PC 0þþ 0 0 jGD ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i the other hand, as the glueball states we discussed in this II∶ BaryonicjJ ¼ 0;nρ ¼ 4i manuscript all have even parity, it implies that the parity of the transition I, III, IV, VI may be violated. We also notice ðn¼1Þ PC þþ → jG ;J ¼ 0 iþBaryonicjJ ¼ 0;nρ ¼ 1i l D that if 2 ¼ J is identified as the quantum number of the spin, III∶ BaryonicjJ ¼ 0;nρ ¼ 5i the decay processes IV, V, VI in (3.7) involving the tensor glueball JPC ¼ 2þþ may be probably forbidden since the → ðn¼1Þ PC 0þþ 0 2 jGD ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i initial and final baryonic states are all pure scalars i.e., the IV∶ BaryonicjJ ¼ 0;nρ ¼ 3i total angular momentum may not be conserved in these transitions,5 and this result would be in agreement with the → ðn¼1Þ PC 2þþ 0 0 jGT ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i previous discussion in [19]. Therefore we could conclude that only the decay process VIII in (3.7) might be realistic. V∶ BaryonicjJ ¼ 0;nρ ¼ 4i This transition describes the decay of the baryonic meson → ðn¼1Þ PC 2þþ 0 1 jGT ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i consisted of one heavy- and one light-flavored quark. So while the identification of the other transitions might be less VI∶ BaryonicjJ ¼ 0;nρ ¼ 5i clear, the transition VIII could be interpreted as the decay of → ðn¼1Þ PC 2þþ 0 2 the baryonic B-meson involving the glueball candidate jGT ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i f0 1710 – ∶ 0 4 ð Þ as discussed e.g., in [8 10] since the correspond- VII BaryonicjJ ¼ ;nρ ¼ i ing quantum numbers of the states could be identified. → ðn¼2Þ PC 0þþ 0 0 jGE ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i IV. SUMMARY VIII∶ BaryonicjJ ¼ 0;nρ ¼ 5i In this paper, with the top-down approach of WSS → ðn¼2Þ PC 0þþ 0 1 jGE ;J ¼ iþBaryonicjJ ¼ ;nρ ¼ i; model, we propose a holographic description of the decay ð3:7Þ of heavy-flavored meson involving the glueball. The HL field is introduced into the WSS model to describe the dynamics of heavy flavor, and it is created by the HL string where we have denoted the states by their quantum numbers with a pair of heavy-flavored D8=D8-brane separated from and the associated decay rates Γ are numerically evaluated in the other light flavored D8=D8-brane. Since baryon in this Table III by using the effective Hamiltonian in (3.4).Notice model could be equivalently represented by the instanton that the mass of the dilatonic and exotic scalar glueball in configurations on the light-flavored brane and the glueball ðn¼2Þ ðn¼1Þ ≃ 1 30 field is identified as the bulk gravitational waves, we solve (3.7) are given as ME =MD . which is close to the mass ratio of the glueball candidates f0ð1710Þ and f0ð1500Þ the classical equations of motion for the HL field with ≃ 1 14 instanton solution for the gauge fields. In the limitation of as Mf0ð1710Þ=Mf0ð1500Þ . , moreover all of them should λ m be the state of JPC ¼ 0þþ. Accordingly, we could identify large followed by large H, we derive the mass formula of the soliton as the onshell action of the flavor brane by the dilatonic and exotic scalar glueball in (3.7) as f0ð1500Þ taking account of the HL field and bulk gravitational and f0ð1710Þ, respectively, which are the two glueball waves. Then following the collectivization and quantiza- candidates discussed frequently in many lectures. tion of the soliton in [22,25,26], the effective Hamiltonian If we furthermore consider the parity of baryonic states for the collective modes of heavy-flavored baryons is as discussed in [22], the above states with odd n in this Z obtained which includes the interaction with the glueball. model would correspond to the meson states with odd Afterwards, we examine the possible decay processes and parity since the parity transformation is Z → −Z. In this compute the associated decay rates with the effective sense, the transition II, V, VII describes the decay of the Hamiltonian. We find these decay rates are in agreement heavy-flavored scalar (nonglueball) meson involving the with the previous works by using this model as in [14–16,19] since they are proportional to λ−1. Then by TABLE III. The corresponding decay rates in the units of mH to comparing the quantum numbers of the baryonic states 0 1 3 2 the transitions in (3.7) by setting l ¼ , NQ ¼ , Nc ¼ , Nf ¼ . with some experimental data and employing the identi- fication of baryonic states in [22], we find that one decay I II III IV process might be realistic and could be interpreted as the Γ 0.0392λ−1 0.0628λ−1 0.0785λ−1 0.1046λ−1 5For a tensor glueball, we suggest considering a tensor field V VI VII VIII dependent on the coordinates of the moduli space yI in order to Γ 0.1674λ−1 0.2093λ−1 0.6316λ−1 1.0527λ−1 obtain the correct decay process. We would like to leave it for future study and focus on the scalar glueball in the current work.

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1 decay of baryonic B-meson involving the glueball candi- 4 ∼ 4 2πδ 4 δ 4 1710 – X X þ X ; X ¼ : ðA3Þ date f0ð Þ as discussed in [8 10]. Noteworthily accord- MKK ing to lattice QCD f0ð1710Þ is an excited state in the glueball candidates which is just consistent with that the And the r, z, Z coordinate used in the paper is defined as glueball state discussed in transition VIII is also an excitation. 3 3 2 z As an improvement of [19], this work provides an U ¼ UKK þ UKKz ;Z¼ ; UKK alternative way to investigate the interaction of the glueball 6 2 and heavy-flavored baryons in the strong coupling system 1 2 r rKK þ Z ¼ 6 ;UKK ¼ 4 : ðA4Þ through the holographic approach of the underlying string rKK R theory. Although this approach is quite principal and 4 contains few parameters, it is actually valid in the large Note that ϵ4 represents a unit volume element on S . gs, ls Nc limit. So phenomenological theories or models are denotes the string coupling constant and the length always needed as a comparison with holography. of string. The indices μ, ν in (A2) run from 0 to 3. Additionally we could define the QCD variables in ACKNOWLEDGMENTS terms of

I would like to thank Anton Rebhan, Josef Leutgeb, and 2 2 λ ¼ g N ;g¼ 2πg l M ; ðA5Þ Chao Wu for valuable comments and discussions. S. W. L. YM c YM s s KK is supported by the research startup foundation of Dalian λ Maritime University in 2019. where gYM, respectively denotes the Yang-Mills and the ’t Hooft coupling constant. In this model, the glueball fields are identified as the APPENDIX A: THE BULK SUPERGRAVITY AND gravitational fluctuations to the bulk solution (A2); thus, → ð0Þ δ GLUEBALL DYNAMICS IN THE WSS MODEL we could rewrite the metric as GMN GMN þ GMN in order to involve the glueball field. The ten-dimensional The WSS model is based on the AdS7=CFT6 correspon- metric reduced from eleven-dimensional supergravity with dence of Nc M5-branes in string theory which can be gravitational fluctuations is reduced to N D4-branes compactified on S1 in the ten- c   dimensional bulk. So taking the large Nc limit, the bulk r3 L2 L2 dynamic is described by the ten-dimensional type IIA gμν ¼ 1 þ δG11 11 ημν þ δGμν ; L3 2r2 ; r2 supergravity action which is given as   r3f L2 L2 Z g44 1 δG11 11 δG44 ; 1 ffiffiffiffiffiffi 1 ¼ 3 þ 2 2 ; þ 2 10 p −2Φ M 2 L r r f S ¼ d x −ge R þ 4∇ Φ∇ Φ − jF4j ; IIA 2 2 M 2 2 2 k10 L L r f g ¼ 1 þ δG11 11 þ δG ; ðA1Þ rr rf 2r2 ; L2 rr r r L 2 L2 Φ 2 2 16 2 g μ ¼ δG μ;gΩΩ ¼ 1 þ δG11 11 ; ðA6Þ where denotes the dilaton field, k10 ¼ G10=gs ¼ r r 2 2 2 ; 7 8 L L r ð2πÞ ls. R, G10 is the ten-dimensional scalar curvature and Newton constant, respectively. F4 dC3 is the field ¼ with the dilaton strength of the Romand-Romand (R-R) 3-form C3. The geometrical solution for the bulk metric is given as r2 L2 4Φ=3 1 δ e ¼ 2 þ 2 G11;11 : ðA7Þ 3=2 2 U μ ν 4 2 L r ds ¼ ½ημνdX dX þ fðUÞðdX Þ R   δ 3=2 2 Since different formulas of GMN corresponds to various R dU 2 2 þ þ U dΩ4 ; glueball, in this paper we consider the following forms U fðUÞ δ of GMN: 3 3=4 1 − UKK Φ U fðUÞ¼ 3 ;e¼ ; U R 1. The exotic scalar glueball 2π Nc 3 3 The exotic scalar glueball corresponds to the exotic F4 ¼ ϵ4;R¼ πgsNcls ; ðA2Þ V4 polarizations of the bulk graviton whose quantum number is JCP ¼ 0þþ. The eleven-dimensional components of 4 with a periodic condition for X , δGMN are given as

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2 quantum number is JCP 2þþ. We can choose the δ − r ¼ G44 ¼ 2 fðrÞHEðrÞGEðxÞ; following components of the graviton polarizations as L   2 1 1 3 6 ∂ ∂ tensor glueball field, r rKK μ ν δGμν ¼ H ðrÞ ημν − þ G ðxÞ; 2 E 4 4 5 6 − 2 6 2 E 2 L r rKK ME r δGμν ¼ − H ðrÞTμνðxÞ; ðA12Þ r2 L2 T δG11 11 ¼ H ðrÞG ðxÞ; ; 4L2 E E where Tμν ≡ T μνGTðxÞ. T μν is a constant symmetric L2 1 3r6 δG − KK H r G x ; tensor satisfying the normalization and traceless condition rr ¼ 2 5 6 − 2 6 Eð Þ Eð Þ T T μν 1 ημνT 0 r fðrÞ r rKK μν ¼ , μν ¼ . The functions HD;T ðrÞ satisfies 90 7 6 the eigenvalue equation, r rKK δG μ H r ∂μG x ; r ¼ 2 2 5 6 −2 6 2 Eð Þ Eð Þ ðA8Þ   MEL ð r rKKÞ 1 d d rðr6 − r6 Þ H ðrÞ þ L4M2 H ðrÞ¼0: r3 dr KK dr D;T D;T D;T with the eigenvalue equation for function HEðrÞ as   ðA13Þ 1 d d rðr6 − r6 Þ H ðrÞ r3 dr KK dr E We can also obtain the kinetic action of the dilatonic scalar   2 12 and tensor glueball as 432r r 4 2 þ KK þ L M H ðrÞ¼0: ðA9Þ Z ð5r6 − 2r6 Þ2 E E KK 1 4 2 2 2 S ¼ − d x½ð∂μG Þ þ M G ; GDðxÞ 2 D D D In the ten-dimensional bulk, the above components Z 1 in (A8) satisfy the asymptotics δG44 ¼−4δG11 ¼−4δG22 ¼ − 4 ∂2 − 2 μν STðxÞ ¼ 4 d x½Tμνð MTÞT ; ðA14Þ −4δG33 ¼−4δG11;11 for r → ∞. Plugging the solution (A2) and the fluctuations (A8) with the eigenvalue equation (A9) into the action (A1), it leads to the kinetic term of the exotic once the solution (A2) and fluctuations (A11) (A12) with scalar glueball, eigenvalue equation (A13) are imposed on the action (A1) Z and the boundary value of HD;T has to been determined by 1 4 2 2 2 the normalization conditions in (A14). S ¼ − d x½ð∂μG Þ þ M G ; ðA10Þ GEðxÞ 2 E E E APPENDIX B: THE FULL Dp-BRANE where the pre-factor in (A10) has been normalized to −1=2 ACTION AND THE EMBEDDING OF by choosing the boundary value of H ðrÞ. E THE PROBE BRANES 2. The dilatonic and tensor glueball 1. The complete DBI action The fluctuations of the metric, We give the complete formula of the Dp-brane here and it

2 could also be reviewed in many textbooks of string theory, r Let us consider D dimensional spacetime parametrized by δG11 11 ¼ −3 H ðrÞG ðxÞ; ; 2 D D μ μ 0 1… − 1 L   fX g, ¼ ; D with a stack of Dp-branes. In this 2 ∂μ∂ν subsection, the indices a; b ¼ 0; 1…p and i; j; k ¼ p þ δ r ημν − Gμν ¼ 2 HDðrÞ 2 GDðxÞ; ðA11Þ 1… − 1 L M D denote respectively the directions parallel and D vertical to the Dp-branes. The complete bosonic action of a corresponds to another mode of the scalar glueball 0þþ.We Dp-branes is employ “dilatonic” for the upon mode since δG11;11 reduces to the ten-dimensional dilaton. SDp−branes ¼ SDBI þ SCS; ðB1Þ Besides the tensor glueball corresponds to the metric fluctuations with a transverse traceless polarization whose where [27]

Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − pþ1ξ −Φ − −1 − δ ij 2πα0 i SDBI ¼ TpSTr d e detf½Eab þ EaiðQ Þ Ejb þ FabQ jg; Z X 2πα0 n μ ∧ ðB þ FÞ SCS ¼ p Cp−2nþ1 ; − n! n¼0;1 Dp branes i δi 2πα0 φi φk Qj ¼ j þ ½ ; Ekj;Eμν ¼ gμν þ Bμν: ðB2Þ

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We have denoted the metric of the D dimensional space- Afterwards let us further introduce the coordinates ðr; ΘÞ time and the 2-form field as gμν, Bμν, respectively. F is the and ðy; zÞ which satisfy, gauge field strength defined on the D-brane and “STr” refers to the “symmetric trace.” We use φi ’s to represent the y ¼ r cos Θ;z¼ r sin Θ; transverse modes of the Dp-branes which are given by the 2π 3 1=2 T-duality relation 2πα0φi ¼ Xi. So the DBI action in (B2) 3 3 2 4 UKK U ¼ U þ UKKr ; Θ ¼ X ¼ : ðB8Þ could be expanded as KK β 2 R3=2 Z  ffiffiffiffiffiffi 1 1 −Φp 2 In the standard WSS model, the probe D8=D8-branes are S ¼ −T Tr dpþ ξe −g 1 þ ð2πα0Þ F Fab BDI p 4 ab Θ 1 π  embedded at ¼2 respectively i.e., the position of 1 1 y ¼ 0, which exactly corresponds to the antipodal D8=D8- φi φi φi φj 2 þ 2 Da Da þ 4 ½ ; þ high orders: ðB3Þ branes (blue) in Fig. 1. In this case, the solution for the 4 1 embedding function is X ðUÞ¼4 β and U0 ¼ UKK.In The 2-form field B has been gauged away. The gauge field addition, the (B7) also allows the nonantipodal solution if φi ’ 1 Aa and scalar field s are all in the adjoint representation we choose Θ ¼ΘH ≠ 2 π, U0 ¼ UH ≠ UKK which of UðNÞ. Note that there is only one transverse coordinate corresponds to the nonantipodal D8=D8-branes (red) in 9 for the D8=D8-brane which has been defined as Ψ ≡ φ in Fig. 1. On the other hand, while each end point of the HL the main text. string could move along the flavored branes, in our setup it is stretched between the heavy- (nonantipodal) and light- 2. Comments about the probe branes and strings flavored (antipodal) D8=D8-branes. So it connects the Here let us briefly outline the embedding of the probe positions respectively on the heavy- and light-flavored 8 8 D8=D8-brane and the HL string. Using the bulk metric D =D -branes which are most close to each other and − 4 0 (A2), the induced metric on the probe D8=D8-branes is in the U X plane, they are the positions of ðUKK; Þ on 0 obtained as the light-flavored branes and ðUH; Þ on the heavy-flavored branes. And this is the configuration of the HL string with   U 3=2 R 3 U02 minimal length i.e., the VEV. ds2 ¼ fðUÞþ ðdX4Þ2 D8=D8 R U fðUÞ 3=2 3=2 U μ ν R 2 2 APPENDIX C: THE COLLECTIVE MODES OF þ ημνdX dX þ U dΩ ; ðB4Þ R U 4 THE BARYON AND ITS QUANTIZATION 0 0 dU As the D4 -brane is identified as baryon in the WSS where U ¼ 4. Then insert the metric (B4) into the DBI dX model, it is equivalent to the instanton configuration on the 8 8 action of D =D -branes, it yields the formula D8-branes according to the string theory. So the dynamic of   the D8=D8-brane is given by the Dirac-Born-Infield (DBI) Z 2 R 3 U0 1=2 action plus the Chern-Simons (CS) action (B2) while the S ∝ d4xdUU4 fðUÞþ : ðB5Þ D8=D8 U fðUÞ baryonic D40-brane is identified as the instanton configu- ration of the gauge field strength on the D8=D8-brane. Hence we can obtain the equation of motion for the Altogether the action of the flavors with baryons can be 4 simplified as a five-dimensional Yang-Mills (YM) plus CS function UðX Þ as 4 action by integrating over the S which is given as 4 d U fðUÞ 0 4 3 1 2 1 2 ¼ : ðB6Þ dX R 0 = S ¼ SYM þ SCS: ½fðUÞþðUÞ fðUÞ U Z ffiffiffiffiffiffi −κ 4 −Φp− ab cdF F 4 SYM ¼ Tr d xdze gg g ac bd; Using the boundary condition in [13],asUðX ¼ 0Þ¼U0 4 Z and U0ðX ¼ 0Þ¼0, the generic solution for (B6) is 1 1 Nc 4 AF 2 − A3F − A5 computed as SCS ¼ 24π Tr d xdz 2 10 ; ðC1Þ Z R 3=2 U ðUÞð Þ μ 4 U where the indices α, β run over X and z. In particular, in X ðUÞ¼EðU0Þ dU 8 2 1=2 ; ðB7Þ − 0 U0 fðUÞ½U fðUÞ E ðU Þ the situation of two flavors i.e., Nf ¼ 2, the classical instanton configuration could be adopted as the Belavin- 4 1=2 where EðU0Þ¼U0f ðU0Þ and we have used U0 to Polyakov-Schwarz-Tyupkin (BPST) solution which is denotes the connected position of the D8=D8-branes. given as

046013-9 SI-WEN LI PHYS. REV. D 99, 046013 (2019) Z xN A ¼ −σ¯ ;M;N¼ 1; 2; 3;z; Sonshell ≃ Sonshell ¼ − dtUðX sÞ: ðC6Þ M MN x2 þ ρ2 D8=D8 YMþCS   i ρ4 A0 ¼ − 1 − ; ðC2Þ Using the solution (C2), the above integral is easy to 8π2 3=2 2 2 ρ2 2 ab x ðx þ Þ calculate in the case of pure light flavors while it becomes quite difficult if the heavy flavor is involved. Without loss where A is Uð2Þ and A0 is Uð1Þ gauge field. The gauge λ 6 of generality, let us consider the large limit followed by field strength is defined as F ¼ dA þ½A; A. And heavy mass limit of the heavy flavor. Hence the dimension- 2 2 x ¼ðxM − XMÞ , XM ’s are constants. Since the instanton less quantized Hamiltonian corresponding to (C5) for the −1 2 size ρ is of order λ = , it would be convenient to employ collective modes is calculated as the rescaling, O λ−1 0 H ¼ M0 þ Hy þ HZ þ ð mHÞ; 0 M 0 −1=2 M 1=2 ðx ;x Þ → ðx ; λ x Þ; ðA0; A Þ → ðA0; λ A Þ; M M 1 X4 ∂2 1 Q H ¼ − þ m ω2ρ2 þ ; ðC3Þ y 2m ∂ 2 2 y y ρ2 y I¼1 yI 1 ∂2 1 in order to obtain the explicit dependence of λ in the actions − ω2 2 HZ ¼ 2 þ mZ ZZ ; ðC7Þ in (C1). Inserting (C2) into the rescaled gauge field A, the 2mZ ∂Z 2 mass M ofR the classical soliton could be evaluated by onshell − where Scl ¼ dtM. Afterwards the baryon states could be identified as Skyrmions so that the characteristics of baryon 2 1 λ are reflected by their collective modes. Therefore we could 2 2 2 Nc M0 8π κ; ω ; ωρ ; κ ; ¼ Z ¼ 3 ¼ 6 ¼ 216π3 quantize the classical soliton in the moduli space to obtain the baryon spectrum. N N N 3 Q ¼ Q þQ ;Q¼ c ;Q¼ Q Q − : In the large λ limit, the topology of the moduli space for L H L 40π2a H 8π2a 3N 4 4 4 c N ¼ 2 case is given as R × R =Z2 since the contribution f ðC8Þ of Oðλ−1Þ could be neglected. Then the collective coor- M R4 ρ dinates fX g parametrize the first while the size and Q 4 The value of corresponds to the situation of a baryonic the SUð2Þ orientation of the instanton parametrize R =Z2. bound state consisting of NQ heavy flavored quarks. The Let us denote the SU 2 orientation as a yI , I 1,2,3, ð ÞP I ¼ ρ ¼ eigenfunctions and mass spectrum of (C7) can be evaluated 4 2 1 4 with the normalization pIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼1 aI ¼ so that the size of the by solving its Schrodinger equation, respectively they are 2 2 7 instanton satisfies ρ ¼ y1 þy4. The quantization obtained as procedures of the Lagrangian for the collective coordinates ðlÞ follows those in Ref. Specifically we need to assume that ψðyIÞ¼RðρÞT ðaIÞ; the moduli of the solution is time dependent. Thus, the −myωρρ2 ˜ ρ 2 ρl gauge transformation also becomes time dependent as Rð Þ¼e Hypergeometric1 ˜ 2 × F1ð−nρ; l þ 2; myωρρ Þ; A → Acl − ∂ −1 M Vð M i MÞV ; ˜ Eðl; nρ;nzÞ¼ωρðl þ 2nρ þ 2Þ F → F cl −1 → _ α∂ Acl − cl Φ −1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MN V MNV ;F0M VðX α M DM ÞV ; 2 ðl þ 1Þ 640 2ðnρ þ n Þþ2 ¼ þ a2π4Q2 þ pffiffiffiz : ðC4Þ 6 3 6 C9 The Lagrangian of the collective coordinates in such a ð Þ moduli space takes the form as ðlÞ ∇2 ðlÞ − 2 ðlÞ Notice that T ðaIÞ satisfies S3 T ¼ lðl þ ÞT which m is the function of the spherical part because H can be L ¼ X G X_s X_r −UðX sÞþOðλ−1Þ; ðC5Þ y 2 rs written with the radial coordinate ρ as   X s M where ¼fX ;aIg. The the kinetic term in (C5) 1 1 3 1 2 1 2 2 H − ∂ ρ ∂ ∇ 3 −2m Q m ω ρ : corresponds to the line element of the moduli space while y ¼ 3 ρð ρÞþ 2 ð S y Þ þ y ρ 2my ρ ρ 2 the potential corresponds to the onshell action of the soliton adopting the time-dependent gauge transformation, ðC10Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 † 7 ˜ ˜ 2 In our notation, A is anti-Hermitian which means A ¼ −A. l and l are related as l ¼ −1 þ ðl þ 1Þ þ 2myQ.

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LE;D;T LE;D;T APPENDIX D: EXPLICIT FORMULAS OF 1=2;0; − 1=2 AND Ψ LE;D;T LE;D;T Here we collect the explicit formulas of 1=2;0;−1=2 and Ψ . For the exotic scalar glueball,  1 5 3 5 LE ¼ − ∂i∂jG F F k þ G F F ij þ ∂2G F F ij 1=2 M 4M2 E ik j 16 E ij 16M2 E ij KK E E  5 7 5 − ∂i∂jG F F − ηijG F F þ ∂2G ηijF F ; 4M2 E iZ jZ 8 E iZ jZ 8M2 E iZ jZ  E E  1 20 5 5 E k ij 0 k ij 0 i L ¼ ∂ G η ZF F − ∂ ∂ G η F F 0 − ∂ ∂ G F F 0 ; 0 M2 3 E ik jZ 2M E ik j 2M E Zi Z E  KK KK Z2 5 15 5 LE ∂i∂jG F F k ∂i∂jG F F k − ∂2G F F ij −1=2 ¼ 16 2 E ik j þ 4 2 E ik j 64 2 E ij MKK MKK ME MKK 3M2 35 15 5 − E G F F ij − G F F ij − ∂2G F F ij ∂i∂jG F F 64 2 E ij 48 E ij 16 2 E ij þ 16 E iZ jZ MKK ME 25 M2 5 7 þ KK ∂i∂jG F F − ∂2G ηijF F þ M2 G ηijF F 12 M2 E iZ jZ 32 E iZ jZ 32 E E iZ jZ E   9 25 2 1 5 2 ij MKK 2 ij i j þ G M η F F − ∂ G η F F þ ∂ ∂ G F 0F 0 8 E KK iZ jZ 24 M2 E iZ jZ M2 M 4 E i j E E KK  3 5 5 − 2 ηijF F − ∂2 ηijF F − ∂0∂0 ηijF F 8 MEGE i0 j0 8 GE i0 j0 4 GE i0 j0   1 7 5 5 20 2 F F − ∂2 F F − ∂0∂0 F F − ∂0 ηijF F þ 2 8 GEME Z0 Z0 8 GE Z0 Z0 4 GE Z0 Z0 3 2 Z GE jZ i0; MEMKK ME 2  ðN þ 1Þ 5 † 5 † LE ¼ −v2 f − ∂i∂jG Φ Φ þ ∂2G δijΦ Φ Ψ N2 12M2 M E i j 24M2 M E i j f E KK  E KK 5 5 − G Φ† Φ ∂2G Φ† Φ : 12 E Z Z þ 24 2 E Z Z ðD1Þ MKK MEMKK

For the dilatonic scalar glueball,

∂i∂jG 3G ∂2G ∂i∂jG LD − D F F k D F F ij D F F ij − D F F 1=2 ¼ 2 ik j þ 4 ij þ 4 2 ij 2 iZ jZ MDMKK MKK MDMKK MDMKK 1 ∂2G −1 F F i D F F i þ 2 GDMKK Zi Z þ 2 2 Zi Z; MDMKK ∂i∂jG ∂i∂jG ∂2G 3G M2 G LD D Z2F F k D Z2F F k − D Z2F F ij − D D Z2F F ij − D Z2F F ij −1=2 ¼ 4 3 ik j þ 3 2 ik j 16 3 ij 16 3 ij 4 ij MKK MDMKK MKK MKK MKK ∂2G ∂i∂jG ∂i∂jG Z2 ∂2G Z2 G M2 Z2 − D Z2F F ij D Z2F F − D F F − D F F i − D D F F i 12 2 ij þ 4 3 iZ jZ 2 iZ jZ 8 3 Zi Z 8 3 Zi Z MDMKK MKK MDMKK MKK MKK 1 ∂2G Z2 ∂i∂jG 3 G ∂2G −1 2F F i D F F i D F F − D F F i − D F F i þ 2 GDMKKZ Zi Z þ 2 2 Zi Z þ 2 i0 j0 2 0i 0 2 2 0i 0 MDMKK MDMKK MKK MDMKK ∂0∂0 1 ∂2 ∂0∂0 GD i −1 2 GD 2 GD 2 D − F 0 F − G M F − F − F ; L ¼ 0; M2 M i 0 2 D KK 0Z 2M2 M 0Z M2 M 0Z 0 D KK  D KK D KK  D 2 i j 2 2 L ðN þ 1Þ ∂ ∂ G † 2G † ∂ G † G † ∂ G † Ψ v2 f − D Φ Φ D ηijΦ Φ D ηijΦ Φ D Φ Φ D Φ Φ : C ¼ 2 3 2 i j þ 3 i j þ 6 2 i j þ 3 Z Z þ 6 2 Z Z a D Nf MDMKK MKK MDMKK MKK MDMKK ðD2Þ

For the tensor glueball,

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kl ij μ 2 T T ∂μ∂ G M G which could greatly simplify LT − ηijF F − F F E;D;T ¼ E;D;T E;D;T 1=2 ¼ ik jl iZ jZ; E;D;T MKK MKK (D1) (D2) (D3). Since the LΨ refers to the mass term of 0 0 2T k 2T i the HL field, the mass of the heavy quarks mH must be LT − ηijF F − F F 0 ¼ ik j0 Zi Z0; related to the separation of the flavor branes i.e., the VEV MKK MKK Ψ 2 of . In the heavy quark limit, the explicit relation is given Tklηij M Tklηij – LT ¼ Z2F F þ T Z2F F as [24 26], −1=2 3M ik jl 4M3 ik jl KK KK Z 2 1 z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ij 2 MT ij 2 H p − M T Z F F þ T Z F F m ¼ lim dz −g00g ; KK iZ jZ 4 iZ jZ H 2 →∞ zz MKK πls zH 0 ij 00 00 T T T 1 1=3 2=3 0 F F − ηijF F − F F ≃ U z O z : þ i0 j0 i0 j0 Z0 Z0; π 2 KK H þ ð HÞ MKK MKK MKK ls 2 1 1 ðN þ 1Þ Nf þ LT − f 2 ijΦ†Φ ≡ pffiffiffi v; ðD4Þ Ψ ¼ 2 v T i i: ðD3Þ 3 6 Nf NfMKK

We assume the glueball field is onshell so that GE;D;T where zH refers to the position U ¼ UH. Then we further 1 − iME;D;T t O 2 O could be chosen as GE;D;T ¼ 2 ðe þ c:cÞ in the rest collect the terms of ðmHÞ and ðmHÞ then integral out the i frame of the glueball; hence, we have ∂ GE;D;T ¼ 0; part of z, it finally leads to the formulas in (3.4).

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