Correlation function of flavored fermion in holographic QCD

By using the gauge-gravity duality, we investigate the correlation function of flavored fermion in the \mathrm{D}_{p}/\mathrm{D}_{p+4} model as top-down approaches of holographic QCD for p=4,3. The bulk spinor, as the source of the flavored fermion in QCD, is identified to the worldvolume fermion on the flavor \mathrm{D}_{p+4}-branes and the standard form of its action can be therefore obtained by the T-duality rules in string theory. Keeping this in hand, we afterwards generalize the prescription for two-point correlation function in AdS/CFT dictionary into general D-brane backgrounds and apply it to the case of p=4,3, i.e. the D4/D8 and D3/D7 approach respectively. Resultantly, our numerical calculation with the bubble background always displays discrete peaks in the correlation functions which imply the bound states created by the flavored fermions as the confinement in QCD. With the black brane background, the onshell condition illustrated by the correlation function covers basically the dispersion curves of fermion obtained by the hard thermal loop approximation in the hot medium. Finally, we interpret the flavored fermions in the bubble background as baryons by taking into account a baryon vertex, then find the two-point correlation function is able to fit the lowest baryon spectrum. In this sense, we conclude remarkably that our top-down approach in this work could reveal the fundamental properties of QCD both in the confined and deconfined phase.


Introduction
The gauge-gravity duality and AdS/CFT correspondence have become a very powerful tool to investigate strongly coupled quantum field theory (QFT) by analyzing the corresponding classical gravity system [1,2,3] in holography.Particularly, the real-time two-point correlation functions of (composite) operators are significant to understand the QFT at finite temperature or with dense matter since the information about the collective behavior e.g. the properties of transport, existence of (quasi-) particles, are encoded in them.The prescription to compute the two-point retarded Green functions in AdS/CFT has been justified in many different ways e.g. in literature [4,5,6,7,8,9,10,11,12,13,14,15] which have instrumentally extracted many important insights about strongly coupled QFT systems from AdS/CFT.On the other hand, quantum chromodynamics (QCD) is the underlying theory to describe the property of strong interaction, however it is extremely complex to solve in the low-energy (strongly coupled) region due to its asymptotic freedom, especially at finite temperature with dense matter [15,16].Therefore investigating QCD through gauge-gravity duality naturally becomes an interesting topic.Although there are several bottom-up models e.g.[17,18,19] attempting to give a holographic version of QCD, the top-down approaches based on D4/D8 (the Witten-Sakai-Sugimoto model [20,21,22]) and D3/D7 [14] models are the most famous and successful achievements in holography since almost all the elementary features of QCD are included in these models in a very simple way [23,24,25].
Moreover, it is known that the self-energy of fermion has been studied with a very long history and attracted many interests since fermion is one of the fundamental elements in QFT as well as in nuclear physics.For example, in weakly coupled system, the self-energy of fermion can be analyzed by the perturbation method of QFT or be characterized by the thermal mass in the hot medium [26,27] while the results may not hold in the strong coupling regime [28,29].Hence there are many researches about fermionic correlations in order to investigate the properties of the strongly coupled fermionic system [11,12,30,31,32,33] in holography.As we can see, the presented works about holographic fermion are based on bottom-up approaches or with minimal coupled fermions, and the bulk fermionic field is identified to the fundamental representation of a U (N ) group when the chemical potential is taken into account especially.However, such a fermionic field in bulk is less clear in the top-down models e.g.D4/D8 and D3/D7 approaches.
Motivated by this issue, in this work we attempt to find a holographic correspondence to evaluate the two-point correlation function of the flavored fermion in D p /D p+4 model (for p = 4, 3, i.e. the D4/D8 and D3/D7 approach) as two top-down approaches to holographic QCD.Specifically, the D p /D p+4 model holographically dual to QCD consists of N c D p -branes as colors and N f probe D p+4 -branes as flavors which are intersected to each other.In the large N c limit, the dynamics of the bulk geometry is described approximately by the type II supergravity.Due to the supersymmetry on the N f probe D p+4 -branes, we identify the bulk spinor (as the source of the flavored fermion) to the worldvolume fermionic field on the probe D p+4 -branes and this may be the only consistent correspondence by analyzing the bulk spectrum in this holographic system.In this sense, the action of the bulk fermion can be obtained by the T-duality rules in string theory [34,35] and it reveals the bulk fermion is not minimal coupled.Keeping this in hand, we further generalize the prescription for two-point correlation function in AdS/CFT dictionary into general D-brane backgrounds with respect to the flavored fermion in the dual theory.Afterwards, we apply our method to the D4/D8 and D3/D7 models as tests in order to evaluate the two-point retarded Green functions variously.
Our numerical calculation displays that the retarded Green functions behave similarly in the D4/D8 and D3/D7 approaches.That is, with bubble background, the Green functions always contain many discrete peaks representing various bound states in QCD and it is consistent with the QCD confinement since the bubble background corresponds to confined phase of QCD in holography [14,20,21,23,25,36,40].Remarkably, the bound energies obtained in the Green function also agree with the numerical evaluation in [36] for p = 4 quantitatively.With the black brane background, the particle onshell condition obtained from the Green functions covers the dispersion curves obtained from the hard thermal loop approximation and the effective thermal mass of fermion can be also determined numerically while the chemical potential is not turned on in our current work.We notice the behaviors of confined Green functions in our topdown approaches are very different from them in the bottom-up approaches or it with minimal coupled fermion [30,31,32,33], while the deconfined Green functions (obtained from the black brane background) behave similarly.Thus it illustrates seemingly that the property of QCD confinement is less clear in the previous works with bottom-up models or with minimal coupled fermion e.g.[30,31,32,33].Finally, we discuss how to interpret the fermionic bound states in confined phase as baryon states by including a baryonic brane as baryon vertex [37] with the framework of D p /D p+4 model which may support that the open strings on the D-brane behaves somehow as baryons [38,39].
The outline of this manuscript is as follows.In Section 2, we review the D p /D p+4 model as holographic QCD briefly for the case of p = 4, 3.In Section 3, we discuss how to identify the bulk spinor, its associated action and the holographic prescription to compute the two-point correlation function in D-brane background.In Section 4, 5, we apply our method to D4/D8 and D3/D7 model respectively in order to evaluate the retarded Green functions for flavored fermions then analyze the numerical results.Summary and discussion are given in Section 6.
2 The D p /D p+4 model as holographic QCD In this section, let us review the holographic QCD through gauge-gravity duality based on the type II string theories.In order to obtain a dual field theory close to QCD, we start with the D-brane system consisted of N c coincident D p -branes and N f coincident D p+4 -branes.
Taking the near-horizon limit α ′ → 0 and the large N c limit by keeping N f finite, the bulk dynamic is described approximately by the type II supergravity (SUGRA) for N c coincident D p -branes whose action takes the following form as [40,41], where R, ϕ, C p+1 is respectively the 10-dimensional (10d) scalar curvature, the dilaton and the Ramond-Ramond (R-R) p + 1-form field with its strength F p+2 = dC p+1 .The constant κ 10 is the gravity coupling constant related to the 10d Newtonian constant G 10 as 2κ 2 10 = 16πG 10 = (2π) 7 l 8 s g 2 s where the string coupling constant and string length is denoted by g s , l s .Since we will discuss the D3/D7 and D4/D8 approach based on IIB and IIA string theory respectively, the cases for p = 3, 4 are only considered here.Vary the action (2.1) with respect to the metric, the dilaton and the R-R field, then the associated equations of motion can be obtained accordingly and they can be solved in general as the black brane solution [41], where the functions f (r) , H p (r) are given respectively as, Here r is the radial coordinate of the bulk which corresponds usually to the holographic direction.r H , r p are two constants related respectively to the position of the horizon and the R-R charge carried by the D p -branes 4 .We note that in the near-horizon limit, the harmonic function H p (r) becomes H p (r) → r 7−p p /r 7−p .The dual field theory associated to the bulk geometry (2.2) in the large N c limit can be examined by taking into account a probe D p -brane located at the holographic boundary r → ∞.And as it is known, the resultantly dual field theory is respectively the N = 4 super Yang-Mills (YM) theory [14] for p = 3 and the type N = (2, 0) super conformal field theory (SCFT) reduced on a circle [20,40] for p = 4, however both of them are less close to QCD due to the presence of supersymmetry.Thus Witten proposed a scheme [20] to construct the bulk geometry presented in (2.2) in order to obtain a dual field theory close to QCD with confinement.Specifically, the first step is to perform the double Wick rotation x 0 → −ix p , x p → −ix 0 on the D p -brane where x 0 = t, x p refer to the time and the p-th spacial direction of the D p -brane.So the geometry (2.2) becomes a bubble configuration as, which is defined only for r ∈ [r KK , ∞).We have replace r H by r KK in (2.4) because there is not a horizon in the bubble configuration.Notice that in the black brane solution (2.2), x 0 = t is compactified on a circle which implies in the bubble case x p becomes now periodic as, where M KK is the Klein-Kaluza (KK) energy scale.So the second step to found a QCD-like theory is to get rid of all massless field other than the gauge fields by imposing the periodic and 4 In our solution, the constant hp is defined as x µ x p x p+1 (r) x p+2 , ...x p+5 x p+6 , ...
Table 1: The D-brane configuration of the D p /D p+4 system for p = 3, 4. "-" represents that the D-branes extend along this direction and x p is the periodic direction.
anti-periodic boundary condition to bosons and supersymmetric fermions respectively along the periodic direction x p .Therefore the supersymmetric fermion acquires mass of order M KK which would be decoupled to the low-energy theory, while the gauge boson remains to be massless.Accordingly, the dual field theory on the D p -brane is p-dimensional pure U (N c ) Yang-Mills theory in the large N c limit below the energy scale M KK .We note that the wrap factor H p (r) in (2.4) never goes to zero, hence the behavior of the Wilson loop in this geometry obeys the area law which implies the dual field theory will exhibit confinement.Altogether, these D p -branes compactified on a circle correspond holographically to the color sector of QCD and its number N c is identified to the color number.The U (N c ) gauge field on the D p -branes is therefore identified as the gluon field and in the large N c limit, the bulk geometry is described by (2.4) with confinement.To further obtain a deconfined phase of QCD at finite temperature 5 , we can campactify the p-th direction in the black brane solution (2.2) as (2.5) then follow the same discussion as it is in the confined geometry to construct the dual theory [14,20,40].Afterwards the dual theory could be identified as deconfined QCD at finite temperature since there is a horizon in (2.2) introducing the Hawking temperature in the dual theory.
As QCD also includes fermions as flavors, the existing N f coincident D p+4 -branes in this holographic model can just play the role of flavor.The D-brane configuration of the D p /D p+4 system is given in Table 1 for the D3/D7 and D4/D8 approach in this manuscript.Since N f is set to be finite, the D p+4 -branes become the probes embedded into (2.2) (2.4) and N c → ∞ implies D p -brane dominates the bulk geometry.By analyzing the spectrum of the open string connecting the N c D p -branes and N f D p+4 -branes in R-sector, the low-energy theory includes fermions in the fundamental representation of U (N c ) and U (N f ) which can be identified nicely as the fundamental quarks in this model [14,21,22].In this sense, the dual theory contains both colors and flavors, thus it is expected to be the holographic version of QCD.

The holographic correspondence with flavored fermion
As our goal is to work out the two-point correlation function of the flavored fermions through gauge-gravity duality, in this section, let us attempt to found the associated holographic cor-respondence in the top-down approach first, then discuss how to generalize the prescription in AdS/CFT dictionary into a non-conformal background to compute the Green function.

Identification of the bulk spinor
To begin with, let us recall the principle of the AdS/CFT which means in general the partition function of the dual field theory Z QCD is equal to its gravitational partition function Z gravity in the bulk.Since the dual theory living in the boundary ∂M of (2.2) (2.4) would be expected to be QCD as it is reviewed in Section 1, we use Z QCD here to refer to its partition function.For a spinor ψ in the bulk M whose boundary value is ψ 0 = lim r→∞ ψ, we can write down, with And L ren gravity refers to the classical renormalized Lagrangian of the bulk field ψ.In our concern, χ refers to the flavored fermion in the dual field theory (QCD).Then the two-point retarded Green function G R for the spinor χ can be evaluated by following [10,13,14] as, and, Here ω, ⃗ k refers respectively to the frequency and 3-momentum of the associated Fourier modes.We note that the relation of the Euclidean Green function G E and the real-time Green function G R is given by The next step is to identify the bulk spinor field ψ in order to investigate its action S ren gravity [ψ] in holography.We notice that, in several bottom-up or phenomenological holographic approaches e.g.[12,30,31,32,33], the bulk field ψ is usually identified to a fundamental fermion of the U (N ) group whose action is the minimally coupled Dirac action.However such a bulk field ψ is less clear in the D p /D p+4 -brane system with a bosonic background (2.2) (2.4) since in general, the bulk modes created by the strings in the type II string theory do not include a U (N ) fundamental fermion.To figure out this problem, it is worth retaking into account the interaction of the open strings in the D p /D p+4 -brane system according to the string theory.For example, it is known the AdS/CFT dictionary for the approach of vector field exp illustrates that the correlator of the flavored current operator J µ living on a probe D p -brane at the boundary could be evaluated by varying the action of the worldvolume vector A µ on the D p+4 -branes as the bulk action S bulk [A µ ] [5, 14, 10], since the D p+4 -branes as flavors extend along the holographically radial direction of the bulk according to Table 1.In this sense, the source term µ presented in (3.5) implies the interaction of the (p, p) and (p + 4, p + 4) strings 6 .
Keeping the above in mind and turning to our case, once the spinor χ in (3.1) (3.2) is identified to the flavored fermion living on the D p -brane at boundary, we must find its fermionic dual field ψ by analyzing the low-energy spectrum of type II string theories.Among the fields in the massless spectrum of the D p /D p+4 model, the only possible choice for ψ should be the fermionic field on the D p+4 -branes 7 .In this sense the source term χψ 0 + ψ0 χ presented in (3.2) corresponds to the interaction of (p, p + 4) and (p + 4, p + 4) strings at boundary, which reveals the interaction to boundary theory involving the fermionic part to A µ presented in (3.5), as it is displayed in Figure 1.We note that, in the D-brane setup given in Table 1 on the bubble background (2.4), the (p + 4, p + 4) string remains to be supersymmetric [39,40,44] since there is not any mechanism to break down the supersymmetry on the D p+4 -branes in principle while the D p -brane is not supersymmetric due to the boundary condition on its compactified direction.Therefore the fermionic field ψ in the worldvolume of D p+4 -branes must exist in the low-energy theory, and in our D p /D p+4 model, it is seemingly to be the suitable candidate uniquely as the source of the flavored fermion χ living in the boundary which may also be the only consistent interpretation in this system.However in the black brane solution (2.2), the worldvolume fermion on the D p+4 -branes may acquire thermal mass due to the contribution from loop diagrams.This means the supersymmetry may break down so that the supersymmetric fermion could decouple to the low-energy theory.Nonetheless, it would be interesting to investigate the fermionic correlation in the black brane background (as the deconfined phase of QCD) as a comparison to several bottom-up approaches with deconfined fermions [30,31,32,33].Therefore we will identify the bulk field ψ to the supersymmetric fermion on worldvolume of the D p+4 -brane in order to continue our discussion.In addition, we also need an interpretation of the dual operator χ in terms of hadron physics because it is presented in the dual theory which is very close to QCD in holography.As the bulk field ψ and its dual operator χ are usually gauge-invariant operators and share same quantum numbers in gauge-gravity duality or AdS/CFT [14,40,41], it implies χ must be a color singlet in the language of QCD.On the other hand, the bulk field ψ is in the adjoint representation of the flavor group (just as its bosonic superpartner), it means dual operator χ is also the adjoint representation of the flavor group thus it must be a fermionic hadronic field.Mathematically, χ is not the fundamental representation of the color group, instead it must be the singlet irreducible representation obtained by decomposing the tensor product of the fundamental representation of the color group.So χ could be a mesino (the supersymmetric fermionic meson) [36,38,39] or baryon produced by multiple fundamental quark fields, or equivalently by multiple (p, p + 4) strings.This would be clear by recalling the construction of the hadronic state in QCD with SU (3) color group.For example, the two-quarks color singlet can be obtained by decomposing the tensor product of irreducible representation of SU (3) in QCD as 3 ⊗ 3 = 8 ⊕ 1, where 1 refers to the two-quarks color singlet i.e. meson.Similarly, for the three-quarks color singlet, we have 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 * ⊕ 1 where 1 refers to the three-quarks color singlet i.e. baryon.Therefore χ refers to such a color singlet as a hadronic operator in our holographic approach with the D p /D p+4 model.However, interpreting χ as mesino is less close to the realistic hadron physics since mesino is always absent in QCD, thus we attempt to interpret χ as baryon by taking into account a baryon vertex in this work and we will discuss it in details in the end of this paper.

Action for the bulk spinor
In the last section, since the bulk field dual to the flavored fermion is identified to the worldvolume fermion on the N f coincident D p+4 -branes, let us take a look at the action for such a fermionic field on the worldvolume of the D-brane.
As it is known that the action for the worldvolume fields on a D-brane is in principle obtained under the rule of T-duality [40,41] in string theory which includes a bosonic part consisted of Dirac-Born-Infeld (DBI) term plus Wess-Zumino (WZ) term, and a fermionic part.Since our concern is the worldvolume fermion on the D-brane, let us focus the fermionic part of the D p -brane action which on the bosonic background is given by [34,35], where, for the type IIA string theory (p is even number), for the type IIB string theory (p is odd number), Since the two-point Green function in holography is our concern, only the quadratic term in fermionic action is given in (3.6).Then let us clarify the notation presented in (3.6) -(3.8).T p refers to the tension of a D p -brane given as . The indices denoted by capital letters K, L, M, N... run over the 10d spacetime and indices denoted by lowercase letters a, b, ... run over the tangent space of the 10d spacetime.We use Greek alphabet α, β, λ to denote the indices running over the worldvolume of the D p -brane.The metric is written in terms of elfbein as g M N = e a M η ab e b N , so the gamma matrices are defined by with e a M Γ M = γ a .ω αab refers to the spin connection and ∇ α = ∂ α + 1 4 ω αab γ ab is the covariant derivative for fermion.The gamma matrix with multiple indices is defined by ranking alternate anti-symmetrically or symmetrically the indices e.g.
Γ M N K... shares the same definition as γ abc... .γ is defined as γ = γ 01...9 and σ 2 refers to the associated Pauli matrix.The worldvolume field F is given as F = B + (2πα ′ ) f where B is the NS-NS 2-form in type II string theory with H = dB and f is the Yang-Mills field strength.F M , F M N , F KLM ... refer to the associated field strength of the massless R-R fields presented in IIA and IIB string theory.We note that, in our D p /D p+4 approach, p should be replaced by p+4 in the above formulas since the worldvolume fermion on the probe D p+4 -brane is our concern.

The prescription for two-point correlation function
In this section, let us attempt to generalize the prescription in AdS/CFT dictionary for twopoint correlation function into the background presented in (2.2) and (2.4).We first summarize the steps as follows then give some comments.
1. Since one dimension of the D p -brane is compactified on a circle, the dual theory is effectively p dimensional below the energy scale provided by the circle size.So we need to simplify the T-dualitized action (3.6) with respect to the background fields given in (2.2) (2.4)8 by integrating out the dependence on S 8−p in order to obtain an effective p + 1 dimensional bulk action involving the bulk spinor ψ (x, r) as 2. Derive the equation of motion for ψ (x, r) by varying the p + 1 dimensional action obtained in Step 1, then solve it by using the Fourier mode as the ansatz for the ψ (x, r) as, 3. Impose the solution for ψ (x, r) obtained in Step 2 in to the p + 1 dimensional action obtained in Step 1, so the p + 1 dimensional fermionic action becomes onshell S cl p+1 , then define the boundary value of ψ as ψ 0 , hence the two-point correlation function is obtained as, where Π 0 is defined in (3.4).Accordingly, one can obtained the correlation function by solving ψ 0 and Π 0 in gravity side.
The above steps are based on the prescription in AdS/CFT dictionary for two-point correlation function [10,14,40].Note that the last term in (3.11) would become vanished once the classical solution (3.12) is imposed since (as we will see) this term is nothing but the Dirac equation.Accordingly, in the actual calculation, we will introduce a ratio function defined as where so that the boundary value of ξ is the correlation function.Therefore, our goal is to derive and solve the equations for ξ with the in-falling boundary conditions usually used in AdS/CFT.Besides, in the AdS/CFT correspondence, the onshell action (3.11) would always include a finite part due to the isometry of AdS or equivalently the conformal symmetry in the dual theory [14].Hence the holographic renormalization could work consistently to remove the divergence presented in action (3.11).However, in a general background, the onshell action (3.11) might not include any finite parts which could lead to a non-renormalizable dual theory.Nevertheless, it is possible to define a finite Green function consistently by rescaling the boundary value ψ 0 with respect to ψ which, as we will see, is equivalent to exact the finite part of ξ given in (3.15).Therefore in the following sections, we will test our prescription in the top-down D4/D8 and D3/D7 approaches by holography.

Approach to the D4/D8 model
In this section, let us apply our prescription to the D4/D8 model (Witten-Salai-Sugimoto model) to compute the two-point Green function.We will first introduce the D4/D8 model briefly, then test our prescription and analyze the results numerically in the 1+3 dimensional QCD.In Section 4.1, we perform the Step 1 in the prescription which is to obtain a 5d effective bulk action for fermion with respect to the bubble and black D4 background.In Section 4.2, we follow the Step 2 and Step 3, that is to solve the associated equations of motion for the bulk field, define its boundary value then derive its onshell action to evaluate the correlation function.

The 5d action for the bulk spinor
The D4/D8 model is based on type IIA string theory and it corresponds to the case of p = 4 as it is discussed in Section 2. For the readers' convenience, let us give its supergravity solution which is to set p = 4 in the background geometry (2.4) for the confined phase as [21], Here R relates to the radius of the spacetime, ϵ 4 is the volume form of a unit S 4 and we have taken the near-horizon limit.In the language of QCD, the parameters presented in (4.1) can be expressed as, where g YM refers to the Yang-Mills coupling constant in the dual theory.For the deconfined phase, the bulk geometry in the near-horizon limit is obtained from (2.2) with p = 4 as, As it is discussed in Section 2, the bulk spinor is identified to the fermionic field on the worldvolume of the probe D8-branes, we need the induced metric on the worldvolume of the D8-branes which according to Table 1 is given as, for the confined and deconfined phase respectively.Considering the most simple configuration to reveal the chirality in the dual theory, we note that the D8-and anti D8-branes are embedded at the antipodal position of x 4 i.e. x 4 = const in the bubble D4-brane background (2.4).For the black brane background (2.2), the D8-and anti D8-branes are embedded parallel which also implies x 4 = const as it is illustrated in Figure 2. Then the action for the worldvolume fermion on D8-branes can be collected from (3.6) (3.7) as, Since our concern is the fermionic part, we have turned off the irrelative bosonic fields by setting F = 0. Imposing the solution for the dilaton ϕ and the R-R form F 4 presented in (4.1) or (4.3), the action (4.5), after some algebraic calculations, can be simplified in the confined background as, with, In the deconfined background, the action (4.5) becomes, where Here T, β t refers to the Hawking temperature and the period of the time direction in (2.2).We have used Ψ to denote the worldvolume fermion whose boundary value relates to ψ 0 .The index m runs over S 4 and the unit volume of S 4 is given as Ω 4 = 8π 2 /3.We recall that in the D4/D8 model, the flavor branes are introduced as N f pairs of probe D8-and anti D8-branes located at the antipodal points of x 4 .Hence the Cartesian coordinates Z is usually used as, so that the D8-and anti D8-branes are located respectively at Z → ±∞ representing approximately the chiral symmetry U (N f ) R × U (N f ) L at boundary.And the dual theory on the D4-branes at boundary takes the chirally symmetric action as QCD as [44], where we use Weyl spinor χ R,L to denote the fundamental chiral fermion and F M N is the gauge field strength of the gluon A M .Therefore the source term of spinor presented in (3.2) can be written in terms of Weyl spinors as, exp once we focus on the Green function of chiral fermion, as it is illustrated in Figure 3.
In order to compute the two-point Green function, let us reduce the fermionic action (4.6) (4.8) to a 5d form as most works about AdS/CFT.First, we decompose the 10d spinor into a 3+1 dimensional part ψ (x, Z) with holographic coordinate Z, an S 4 part φ and a remaining 2d part β as 9 , ψ (x, Z) ⊗ φ S 4 ⊗ β, (4.13) so that the worldvolume fermion on D8-brane should be Ψ = ψ ⊗ φ.Then the associated 10d gamma matrices are chosen as, where we use bold to denote the 4 × 4 gamma matrices.γ m , m = 6, 7, 8, 9 refers to the gamma matrix on tangent space of S 4 .We note that, the 10d chirality matrix takes a very simple form as γ = σ 3 ⊗ 1 ⊗ 1 in this decomposition.Choosing the σ 3 representation, β can therefore be decomposed by the eigenstates of σ 3 with where we use β ± to denote the two eigenstates of σ 3 .Since the kappa symmetry fixes the condition γΨ = Ψ10 , we have to chose β = β + on the D8-brane.In addition, as φ must satisfy the Dirac equation on S 4 , we can decompose it by using the spherical harmonic function with the eigenstates of Γ m ∇ S 4 m as [45], Here s, l represents the angular quantum numbers of the spherical harmonic function.Put (4.13) into (4.6)(4.8) with the decomposition (4.14) -(4.16) for φ and β, we could obtain a 5d effective action for the fermion on the worldvolume of the D8-branes after integrating the S 4 part as, and where Λ l = Λ ± l .As we are going to focus on the Green function in the dual theory according to the Section 3.3, the action (4.17) can be further simplified by rescaling ψ → 2πα ′ K −13/24 ψ as, while the action (4.18) is rescaled by ψ → 2πα ′ K −7/24 Z −1/2 ψ as, And we will use these 5d fermionic actions (4.19) (4.20) to evaluate the two-point Green function.

Use the prescription
Since the 5d effective action is obtained in Section 4.
Picking up the ansatz in Weyl basis as, the equation (4.21) becomes where we have chosen the 5d gamma matrices as, with σ µ = 1, −σ i , σµ = 1, σ i .Solving (4.23), we can further obtain the decoupled equations for ψ R,L as, where "′" refers to the derivative with respect to Z.The above equations can be analytically solve at Z → +∞ as, where A, B, C, D are constant spinors depended on ω, ⃗ k, Λ l and furthermore one can find the following relations as, once the analytical solution (4.26) is plugged back into (4.23).Besides, we notice that the solution at Z → −∞ is equivalent to interchange ψ R,L in the solution (4.26).Λ l is the eigenvalue of the S 4 part and it satisfies Λ l ≥ 2 according to (4.16).So we obtain the boundary behavior of ψ as,

.28)
This can be nicely interpreted as the source coupling to the chiral fermion in each boundary theory, as it is illustrated in Figure 3. Hence we will focus on the calculation with respect to the N f D8-branes since the discussion would be exactly same on the anti D8-branes.Afterwards, let us take care of the boundary value ψ 0 of ψ at Z → +∞ by picking up its dominantly finite part in (4.26) as [13,14], Next, the conjugate momentum Π 0 associated to ψ 0 can be evaluated by using (3.4).To this goal, let us take a look at the action (4.19), where Notice that the term in the third line of (4.30) vanishes since it is nothing but the conjugate equation to (4.21).So once we impose the solution (4.26) to (4.30), it implies the action (4.30) includes divergence at boundary Z → ∞ as 11 , Therefore we can see, it requests for a holographic boundary counterterm S ct as, to remove the divergence in (4.30).We note that action (4.32) contains a finite part which will survive after holographic renormalization.And this should relate to the residual isometry of AdS 7 since the D4/D8 model is based on the holographic correspondence on AdS 7 × S 4 [20,40].
Then the conjugate momentum on the D8-branes to ψ 0 is evaluated as which implies the Green function G R ω, ⃗ k is obtained by according to (3.3) (3.4).Using the representation and introducing the ratio the Green function can therefore been written as, satisfying the equation of motion, according to (4.23).Here we have set k µ = (−ω, k, 0, 0) with, On the other hand, since the dual theory is chirally symmetric, we obtain which implies [36] ψ R (0) = ±ψ L (0) .(4.42) Therefore we can set h = ±1 in (4.40) for ξ 1,2 respectively.Altogether, (4.39) could be numerically solved with the incoming wave boundary condition ξ 1,2 (0) = ±1.

Deconfined phase
In the deconfined phase, the equation of motion for ψ reads from (4.20) as, Using the ansatz which can be solved analytically at Z → ∞ as, The asymptotics (4.46) leads to that the onshell action (4.20) takes form at Z → ∞ as, which however does not include any finite part for Λ l ≥ 2. Although this behavior might imply the deconfined hot QCD is probably non-renormalizable in holography, it is possible to evaluate its correlation function with the prescription in Section 3.3 if we take the boundary value ψ 0 of ψ carefully as Thus conjugate momentum for ψ 0 is which reduces to finite Green function as G R ∼ D/A.Then let us use the ratio given in (4.37) so that the Green function can be written as, And the equation for ξ 1,2 can be obtained from (4.45) as, which can be solved numerically by using the incoming wave boundary condition ξ 1,2 (0) = ±i on the horizon.

The numerical analysis
In this section, we evaluate numerically the retarded Green functions by solving (4.39) (4.51) with incoming wave boundary conditions.
In confined phase, the results are collected in Figure 4 and 5.The behavior of the peaks in the Green function displays discreteness as n basically where M n represents various constants.According to the general form of the fermionic propagator here m n , Σ (k) refers respectively to the eigenvalue of the bare mass and self-energy, M n corresponds to the onshell energy of the various bound states consisted of the flavored fermions (quarks).And we believe this result is consistent with the property of the bubble background (4.1) since its dual theory is expected to be QCD with confinement states, e.g.meson, baryon states.We will further discuss these bound states at the end of this manuscript.Moreover, for ω > 0, the location of the peak in the Green function can be read approximately once we set k = 0, so that the peaks correspond to ω = m n − Σ (k) as it is illustrated in Figure 5.In this sense, we find quantitatively, for Λ l = 2, l = 0, the bound energy (the position of the peak) is located at ω ≃ 1.6, 2.7, 3.7... in G R and at ω ≃ 2.2, 3.1, 4.2... in G (2) R .Furthermore, the location of the peaks also depends on the value of Λ l as it is displayed in Figure 6.Accordingly, if we identify h = ∓1 as the parity of the bound states, the onshell energy evaluated by the Green function agrees consistently to the spectrum of the worldvolume fermion on the D8-branes with various Λ l in this model as it is calculated in [36].
Next, let us take a look at the Green function as the spectral function obtained in the deconfined phase illustrated in Figure 7 and 8 where the peaks in the Green function have been fitted by green lines.Since the relation of R (ω, k) for convenience.First, we can see the behaviors of the particle onshell (peaks in the Green function) is very different from them in the confined case which satisfies ω ≃ k for k ≫ 1 and ω ≃ k 2 + ... for k ≪ 1.While we discuss the positive frequency mode, it is same to the negative frequency mode.This behavior is in qualitative agreement with the dispersion curves of fermions obtained by the hard thermal loop (HTL) approximation in hot QCD, up to one-loop calculation, as [46], where m f is the effective mass of fermion generated by the medium effect given by We note that chemical potential can also contribute to mass of fermion while it is not turned on in the current holographic setup and C F is suggested to be C F = 4/3 for fundamental quarks.According to our numerical calculations presented in Figure 8, the effective mass of fermion is evaluated to be m f = ω (k = 0) ≃ 2.9 × (2πT ) which is expected to be the non-perturbative results predicted by holography.However, we must keep in mind that the holographic approach is valid in strong coupling region i.e. the 't Hooft coupling constant λ = g 2 YM N c is finite and satisfies λ ≫ 1 for N c → ∞.Hence the consistent interpretation here should be that, in QFT side, the high order contribution of Yang-Mills coupling constant g YM by HTL perturbation is suppressed in the large N c limit since in this limit we have g YM ≪ 1.Second, our current numerical evaluation of the the dispersion curves implies the minus sign should be picked up in (4.53) for k ≪ 1 while there are two branches given by HTL approximation in (4.53).And following the discussion in [30,31], this branch could correspond to the dispersion curves of the fermionic plasmino at high temperature.Nonetheless, since the numerical calculation always displays a width of the peaks in the spectral function, the dispersion curves of quarks (which is expected to be ω (k may also be included in the spectral function.However, distinguishing exactly the dispersion curves of fundamental quark from plasmino would very depend on the trick of the numerical calculation we chose.In this sense, it would be less necessary to further compare our current numerical results with the HTL approximation.Overall, we believe our results, based on the holography with the principle in string theory, reveal mostly the fundamental properties of both confinement and deconfinement in QCD, while they are very different from some bottom-up approaches or phenomenological approaches in this model as [30,31] where the features of QCD confinement is less clear.

Approach to the D3/D7 model
In order to further test the prescription in Section 3.3, let us attempt to use it in the D3/D7 approach in this section, which is a holographic version of 1+2 dimensional QCD.Similarly as the case of the D4/D8 approach, in Section 5.1, we perform the Step 1 in the prescription to obtain a 4d effective fermionic action with respect to the bubble and black D3 background.In Section 4.2, we perform Step 2 and Step 3 which is to solve the equations of motion for the bulk field, define its boundary value and derive its onshell action in order to evaluate the correlation function.

The 4d action for the bulk spinor
We start with the SUGRA solutions (2.4) and (2.2) in the case of p = 3 for the IIB SUGRA.So in the near horizon limit, the bubble D3-brane background is summarized as 12 , which corresponds to the confined phase of the dual theory and the black D3-brane background is given as, which corresponds to the deconfined phase of the dual theory.Here we use ϵ 5 to refer to the volume form of a unit S 5 and R 4 = 4πg s N c l 4 s .Note that the dilaton field vanishes for p = 3 due to (2.2) and since x 3 is compactified on a circle, the dual field theory is effectively the 1+2 dimensional Yang-Mills theory below the energy scale M KK = 2πβ −1 3 where β 3 refers to the period of x 3 [14,40].
Then let us introduce the probe D7-branes as flavors embedded into the background (5.1) (5.2) in order to investigate its worldvolume fermions.According to Table 1, the D7-branes are located at x 3,9 = const, hence we impose the coordinate transformation, to the radial and S 5 part in (5.1) (5.2), then they can be rewritten as, where ρ 2 = 8 m=4 x m x m and dΩ 2 4 refers to the metric on S 4 whose solid angle is defined by parametrizing x 4,5...8 .We note that due to r ≥ r KK,H , it reduces to two branches for ζ equivalently as 0 < ζ ≤ 1 and ζ ≥ 1.In our setup, we chose the branch of ζ ≥ 1. Therefore the induced metric on the flavor D7-branes reads, We note that u = L as a constant represents the separation of the D3-and D7-branes and it is proportional to the bare mass of fundamental quarks in the D3/D7 approach [14] as the vacuum expectation value (VEV) of a (3, 7) string.For chirally symmetric dual theory, we may simply set L = 0 which implies the fundamental quarks are massless.Keeping these in mind, the action for the worldvolume fermions on the D7-brane is given by (3.6) (3.8) as, where we have imposed γ Ψ = Ψ and F KLM N P refers to the component of the R-R 5-form presented in (5.1) (5.2).After some straightforward calculations, the action (5.6) can be written as, where we have used γ 0123ρ Ψ = Ψ and with respect to the bubble (5.1) and black brane (5.2) background.Follow the decomposition presented in (4.13) -(4.15) with the Eigen equation on S 4 (4.16) and integrate the S 4 part, the action (5.7) can be further written as a 4d form as, by rescaling ψ → (2πα ′ ) ψ ζ 2 ρ 2 1 r 3/2 and action (5.8) can be written as, by rescaling ψ → (2πα ′ ) ψ (5.12) We will use these 4d fermionic actions (5.10) (5.11) to compute the two-point correlation function in the 2+1 dimensional dual theory.

Use the prescription
Let us use the prescription in Section 3.3 to study the two-point Green function for the 1+2 dimensional QCD in the D3/D7 model.

Confined phase
For the confined phase, the equation of motion for the worldvolume ψ can be obtained by varying (5.10) as, Using the ansatz given in (4.22), we can obtain the coupled equations as, Setting k µ = (−ω, k, 0, 0), the equations presented in (5.14) can be written as decoupled second order differential equations, which, at boundary r → ∞ (ρ → ∞), take the asymptotic form as, And they can be solved exactly as, where A, B, C, D are integration constant.As we have outlined Λ l ≥ 2, so the boundary value ψ 0 of ψ should be defined as, (5.17) Then the onshell action (5.10) will include a finite part as, thus it request for a holographic counterterm as, The existing finite part in (5.18) is due to the residual isometry of AdS 5 since the D3-brane solution (5.1) (5.2) as bulk geometry is AdS 5 × S 5 asymptotically.Follow the discussion in Section 3, 4, the retarded Green function is given by, Using the representation (4.36) and introducing the ratio (4.37), the Green function can be written as, where the equation for ξ 1,2 can be obtained from (5.14) as, which will be evaluated numerically with the incoming wave boundary condition and h = ±1 for ξ 1,2 respectively.

Deconfined phase
For the deconfined phase, the equation of motion for the worldvolume ψ is given by varying action (5.11) which is, as the spectral function of the 1+2 dimensional QCD from the D3/D7 approach.The parameters are set as which can reduce to two second-order differential equations.At boundary r → ∞ (ρ → ∞), they are, which can be solved as, (5.26) While the boundary value of ψ remains to be ψ R , the onshell action (5.11) taking the form as, does not include any finite part which implies the deconfined phase of 3d QCD is also nonrenormalizable in holography.In order to obtain a finite Green function, we can follow the same discussion as it is in the D4/D8 model.Therefore, let us first take the boundary value ψ 0 of ψ carefully as, Then the conjugate momentum Π 0 for ψ 0 is given as, Accordingly, it leads to a finite Green function as, where the equation for ξ 1,2 defined in (4.37) can be obtained from (5.24), as (5.31)And we will evaluate the Green function with the incoming wave boundary condition on the horizon and h = ±1 for ξ 1,2 respectively.

The numerical analysis
In this section, let us summarize the numerical results by solving (5.22) and (5.31) in the D3/D7 approach.The behavior of two-point Green function in the confined phase is illustrated in Figure 9 which is basically similar to the approach in the D4/D8 model.The peaks in the R which is only quantitatively different from the results in the D4/D8 model.To specify the position of the onshell energy, we have plotted out the relation of the Green function G R as a function of ω which is illustrated in Figure 10.Note that, while we have set L = 0 for the chirally symmetric theory, the position of the peaks in the Green function depends on the quark mass L as it is illustrated in Figure 11.The dependence on L in Green function is different from the D4/D8 approach, since there is not a direction perpendicular to both D4-and D8-branes in the D4/D8 model (which means the VEV of a (4, 8) string is vanished i.e. the bare mass of fundamental quark is vanished.).Furthermore, our numerical calculation reveals that the onshell mass of the bound fermion is suppressed by the quark bare mass due to the correction of the self-energy Σ (k) in (4.52).In the deconfined phase, we plot out the imaginary part of the Green function as the spectral function in Figure 12 and 13 in which the onshell condition fitted by green lines is in qualitative agreement with the dispersion curves obtained by the hard thermal loop approximation presented in (4.53).The effective mass of fermion in hot medium is evaluated as m f ≃ 4.8 (2πT ) by the D3/D7 model which also implies the high order contribution in HTL approximation might be suppressed in the large N c limit.Altogether, the D3/D7 approach also reveals mostly the fundamental properties of QCD as it is expected.from the black D3-brane background.The white regions represent the protrusion or pit, i.e. the peaks, in the Green functions which is fitted by green lines.The dashed lines refer to ω = ±k as the light cone and the parameters are chosen as Λ l = 2, l = 0, T = 1 in the unit of 2πT = 1.

Summary and discussion
In this work, we study the correlation function of the flavored fermion in the D p /D p+4 model (p = 4, 3) as the holographic top-down approaches to QCD.Since our concern is the fermionic correlator, the bulk spinor is identified to the worldvolume fermion created by the (p + 4, p + 4) string on the D p+4 -brane.Then we pick up the action for the worldvolume fermion on the D-brane which is obtained by T-duality in string theory, and generalize the prescription for two-point correlation function in AdS/CFT dictionary into general D-brane background with respect to the flavored fermion in the dual theory.Afterwards we apply our viewpoint to the case of p = 4, 3 respectively.The numerical calculation and result show that the Green functions in both D4/D8 and D3/D7 approaches reveal the discrete peaks with the bubble background which represents the bound states as confinement in QCD.In particular, the various bound energies agree basically with the numerical evaluation in [36] for p = 4 quantitatively.With the black brane background, the onshell condition given by the Green function agrees qualitatively with the dispersion curves obtained by the hard thermal loop approximation which displays the behavior of the deconfined fermion in hot medium.Overall our approaches to the D4/D8 and D3/D7 model illustrate the fundamental properties of QCD in holography which would therefore be remarkable.
Besides, let us give some comments about this work.First, although the onshell condition in the deconfined phase is in qualitative agreement with several bottom-up approaches involving minimal coupled fermion e.g.[30,31], the behavior of Green function in confined phase is totally different.The reasons are mostly that, as the dependence of the inner sphere S 4 is neglected and the probe brane is absent in the bottom-up approaches, the action for the bulk fermion would take a different form so that the asymptotic behavior of fermion, related to the Green function, is also different.And notice that the action (3.6) for worldvolume fermion on a Dbrane is not minimal coupled in general, thus it is not surprised that the results in our top-down approach could be different from those in the bottom-up approaches.However, the property of confinement in QCD is less clear in the previous works with bottom-up models or with minimal coupled fermion.
Second, in the deconfined phase, while our analysis is consistent with the hard thermal loop approximation, we must keep in mind that the QFT approach is only valid in weak coupled field theory.Therefore our current result in the deconfined phase (which is valid in the strong coupling region) may imply the high order contribution from the hard thermal loop approximation at large N c limit is suppressed.On the other hand, the hard thermal loop approximation implies the chemical potential may also contribute to the effective thermal mass of fermion while it is not included in this work.However, if the chemical potential (relating to the gauge field potential on the worldvolume of the D p+4 -brane) is taken into account in this setup, its equation of motion coupling to bulk fermion would be totally different from the bottom-up approaches with minimal coupled fermion according to action (3.6) -(3.8), because the bulk fermion is identified to be the fundamental representation of a U (N ) group in most bottom-up approaches while it is not in our top-down approach.Accordingly, we believe the Green function with non-vanished chemical potential in our top-down approach will display a very different behavior from it in the bottom-up approaches and we will leave this for the future work.Last but not least, let us discuss the interpretation of the dual operator χ to the bulk fermion in terms of hadron physics and outline how to interpret it as a baryon field in the confined phase.As we have specified that to interpret the dual field χ as baryon is more close to the realistic QCD, so it is necessary to let χ produced by multiple fundamental quarks takes baryon number in the side of string theory.Fortunately, this can be achieved by following Witten's [37], that is to introduce a D 8−p -brane wrapped on S 8−p [37,47,48] as a baryon vertex located at r = r KK in the bubble background.Since the N c fundamental quarks created by the (p, p + 4) string must always take baryon numbers, there must be N c (p, p + 4) strings as fundamental quarks connect the wrapped D 8−p -brane and stretch to the bulk boundary totally inside the D p+4 -branes thus they are flavored, colored, baryonic strings and are also (p + 4, p + 4) strings.By taking into account a probe D p -brane at the holographic boundary r → ∞, the D-brane configuration with a baryon vertex is illustrated in Figure 14.In this configuration, the (p, p + 4) strings as fundamental quarks are colored, flavored and take baryon numbers, hence the dual field χ produced by multiple N c (p, p + 4) string points is a baryonic hadron field i.e. a baryon field.On the other hand, since the (p, p + 4) strings are also (p + 4, p + 4) strings, the bulk field ψ dual to χ created by the multiple quarks is also baryonic.So it provides a nicely holographic correspondence with respect to baryon field.In this sense, the bound states as the various peaks presented in the confined Green function refer to the baryon spectrum in holography.And it could be possible to fit the experimental data if we further identify the quantum number l, s in (4.16) to the isospin and spin of the baryon.In this sense, we can identify, for examples, the first three lowest states with same parity presented in Figure 6 as proton, N (1440) and N (1710) for l = 1 as [49].Thus our approach with D4/D8 model gives the mass ratios of N (1440) and proton, N (1710) and proton as M N (1440) /M proton ≃ 1.51, M N (1710) /M proton ≃ 1.95, which are very close to the experimental data M exp N (1440) /M exp proton ≃ 1.53, M exp N (1440) /M exp proton ≃ 1.82 [50].And this viewpoint may also support that the open strings on the D-brane behaves somehow as baryons in the D p /D p+4 model as it is discussed in [38].We further comment at last, to interpret χ as quark field, i.e. produced by a single (p, p + 4) string, may be possible in the deconfined background since in hadron physics a free quark as a free color charge could be observables theoretically in the deconfinement phase of QCD and baryon will be dissolved in the deconfinement phase.

Figure 1 :
Figure 1: The correspondence of the bulk field ψ and boundary operator χ in the D p /D p+4 system.Here r refers to the radial direction as the holographic direction.χ is a hadronic field as a gauge-invariant operator produced by multiple fundamental quark fields, or equivalently by multiple (p, p + 4) strings.

Figure 3 :
Figure 3: The correspondence of the bulk fields ψ R,L and flavored fermions χ R,L consisted of chiral quarks in the D4/D8 model.The N f pairs of D8-and anti D8-branes are located at Z → ±∞ at boundary providing U (N f ) R × U (N f ) L chiral symmetry in the dual theory.The bulk field ψ R,L as source couples to the chiral quarks in χ R,L respectively as it is in QCD.

Figure 4 :
Figure 4: Density plot of the confined retarded Green function G (1,2) R as the spectral function of 1+3 dimensional QCD from the D4/D8 model.The white regions refer to the peaks in the Green function and the dashed lines refer to ω = k as the light cones.The parameters are chosen as Λ l = 2, l = 0, T c = 1 in the unit of M KK = 1.

Figure 5 :
Figure 5: The confined retarded Green function G (1,2) R as the spectral function of 1+3 dimensional QCD from the D4/D8 model.The parameters are chosen as Λ l = 2, k = 0, T c = 1 in the unit of M KK = 1.

Figure 6 :
Figure 6: The imaginary part of confined retarded Green function G (1) R as the spectral function from the D4/D8 model with various Λ l for k = 0.

Figure 7 :Figure 8 :
Figure 7: The 3d plot of the imaginary part of the Green function G (1,2) R from the black D4-brane background.The parameters are chosen as Λ l = 2, l = 0, T d = 1 in the unit of 2πT = 1.

Figure 9 :
Figure 9: Density plot of the confined retarded Green function G (1,2) R as the spectral function of the 1+2 dimensional QCD from the D3/D7 model.The parameters are chosen as M KK = 1, Λ l = 2, L = 0.The white regions refer to the peaks in the Green function and dashed lines refer to the light cones.

Figure 12 :Figure 13 :
Figure 12: The 3d plot of the imaginary part of the Green function G (1,2) R from the black D3brane background.The parameters are chosen as Λ l = 2, l = 0, T = 1 in the unit of 2πT = 1.

Figure 14 :
Figure 14: The D-brane configuration of D p /D p+4 system with a baryon vertex.The bulk field ψ is created by N c (p, p + 4) strings as quarks and the dual field χ at boundary is produced by N c quark points as a baryon field.