# Correlation function of fundamental fermion in holographic QCD

###### Abstract

By using the gauge-gravity duality, we study the correlation function of
fundamental fermion in the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model as top-down
approach of holographic QCD for $p=4,3$. The bulk spinor, as the source of the
chiral quark 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 as the
confinement in QCD. With the black brane background, the onshell condition
illustrated by the correlation function covers basically the dispersion curves
obtained by the hard thermal loop approximation in the hot medium. 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.

Si-wen Li111Email<EMAIL_ADDRESS>Yi-peng Zhang222Email:
<EMAIL_ADDRESS>Hao-qian Li333Email<EMAIL_ADDRESS>
_Department of Physics, School of Science,_

_Dalian Maritime University,_

_Dalian 116026, China_

###### Contents

  1. 1 Introduction
  2. 2 The $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model as holographic QCD
  3. 3 The holographic correspondence with fundamental fermion
    1. 3.1 Identification of the bulk spinor
    2. 3.2 Action for the bulk spinor
    3. 3.3 The prescription for two-point correlation function
  4. 4 Approach to the D4/D8 model
    1. 4.1 The 5d action for the bulk spinor
    2. 4.2 Use the prescription
    3. 4.3 The numerical analysis
  5. 5 Approach to the D3/D7 model
    1. 5.1 The 4d action for the bulk spinor
    2. 5.2 Use the prescription
    3. 5.3 The numerical analysis
  6. 6 Summary and discussion

##  1 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] model 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 $U\left(N\right)$ 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
fundamental fermion in $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model (for $p=4,3$,
i.e. the D4/D8 and D3/D7 approach) as two top-down approaches to holographic
QCD. Specifically, the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model holographically
dual to QCD consists of $N_{c}$ $\mathrm{D}_{p}$-branes as colors and $N_{f}$
probe $\mathrm{D}_{p+4}$-brane 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. Accordingly, we identify the bulk
spinor, as the source of the fundamental fermion, to the worldvolume fermionic
field on the probe $\mathrm{D}_{p+4}$-branes, thus its action can be obtained
by the T-duality rules in string theory [34, 35] which 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 fundamental 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, 38]. 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 top-down approaches is very different from it
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. Finally, we discuss how to interpret the
fermionic bound states in confined phase as baryon states with the framework
of $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model which may support that the open
strings on the D-brane behaves somehow as baryons [37].

The outline of this manuscript is as follows. In Section 2, we review the
$\mathrm{D}_{p}/\mathrm{D}_{p+4}$ 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 fundamental fermions then analyze the numerical results.
Summary and discussion are given in Section 6.

##  2 The $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ 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 $\mathrm{D}_{p}$-branes and $N_{f}$ coincident
$\mathrm{D}_{p+4}$-branes.

Taking the near-horizon limit $\alpha^{\prime}\rightarrow 0$ 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
$\mathrm{D}_{p}$-branes whose action takes the following form as [38, 39],

$S_{\mathrm{II}}=\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\sqrt{-g}\left[e^{-2\phi}\left(\mathcal{R}+4\partial_{M}\phi\partial^{M}\phi\right)-\frac{g_{s}^{2}}{2}\left|F_{p+2}\right|^{2}\right],$
(2.1)

where $\mathcal{R},\phi,C_{p+1}$ 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 $\kappa_{10}$ is the
gravity coupling constant related to the 10d Newtonian constant $G_{10}$ as
$2\kappa_{10}^{2}=16\pi G_{10}=\left(2\pi\right)^{7}l_{s}^{8}g_{s}^{2}$ 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 [39],

$\displaystyle ds^{2}$
$\displaystyle=H_{p}^{-\frac{1}{2}}\left[f\left(r\right)dt^{2}+d\vec{x}\cdot
d\vec{x}\right]+H_{p}^{\frac{1}{2}}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{8-p}^{2}\right],$
$\displaystyle e^{\phi}$
$\displaystyle=H_{p}^{-\frac{p-3}{4}},C_{01...p}=g_{s}^{-1}H_{p}^{-1},F_{r01...p}=\frac{\left(7-p\right)g_{s}^{-1}h_{p}^{7-p}}{r^{8-p}H_{p}^{2}},$
(2.2)

where the functions $f\left(r\right),H_{p}\left(r\right)$ are given
respectively as,

$f\left(r\right)=1-\frac{r_{H}^{7-p}}{r^{7-p}},H_{p}\left(r\right)=1+\frac{r_{p}^{7-p}}{r^{7-p}}.$
(2.3)

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
$\mathrm{D}_{p}$-branes. We note that in the near-horizon limit, the harmonic
function $H_{p}\left(r\right)$ becomes $H_{p}\left(r\right)\rightarrow
r_{p}^{7-p}/r^{7-p}$ .

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
$\mathrm{D}_{p}$-brane located at the holographic boundary
$r\rightarrow\infty$. And as it is known, the resultantly dual field theory is
respectively the $\mathcal{N}=4$ super Yang-Mills (YM) theory [14] for $p=3$
and the type $\mathcal{N}=\left(2,0\right)$ super conformal field theory
(SCFT) reduced on a circle [20, 38] 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 $\left\\{x^{0}\rightarrow-
ix^{p},x^{p}\rightarrow-ix^{0}\right\\}$ on the $\mathrm{D}_{p}$-brane where
$x^{0}=t,x^{p}$ refer to the time and the $p$-th spacial direction of the
$\mathrm{D}_{p}$-brane. So the geometry (2.2) becomes a bubble configuration
as,

$\displaystyle ds^{2}$
$\displaystyle=H_{p}^{-\frac{1}{2}}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+f\left(r\right)\left(dx^{p}\right)^{2}\right]+H_{p}^{\frac{1}{2}}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{8-p}^{2}\right],$
$\displaystyle f\left(r\right)$
$\displaystyle=1-\frac{r_{KK}^{7-p}}{r^{7-p}},H_{p}\left(r\right)=1+\frac{r_{KK}^{7-p}}{r^{7-p}},\
\ \mu,\nu=0,1...p-1.$ (2.4)

which is defined only for $r\in\left[r_{KK},\infty\right)$. 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,

$x^{p}\sim x^{p}+\delta x^{p},\delta x^{p}=\frac{2\pi}{M_{KK}},$ (2.5)

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 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
$\mathrm{D}_{p}$-brane is $p$-dimensional pure $U\left(N_{c}\right)$ Yang-
Mills theory in the large $N_{c}$ limit below the energy scale $M_{KK}$. We
note that the wrap factor $H_{p}\left(r\right)$ 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 $\mathrm{D}_{p}$-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 $U\left(N_{c}\right)$ gauge field on the
$\mathrm{D}_{p}$-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
temperature444Identifying the black brane solution (2.2) to the deconfined
phase of QCD may contain some issues [40, 41]. Nonetheless we will continue
our discussion by this identification as most works with the
$\mathrm{D}_{p}/\mathrm{D}_{p+4}$ system. The reason is that the black brane
solution corresponds to the dual theory at finite temperature which may
include some properties of hot QCD, close to the deconfinement in QCD., 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, 38]. 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 fundamental fermions as flavors, the existing $N_{f}$
coincident $\mathrm{D}_{p+4}$-branes in this holographic model can just play
the role of flavor. The D-brane configuration of the
$\mathrm{D}_{p}/\mathrm{D}_{p+4}$ system is given in Table 1 for the D3/D7 and
D4/D8 approach in this manuscript.

| $x^{\mu}$ | $x^{p}$ | $x^{p+1}\left(r\right)$ | $x^{p+2},...x^{p+5}$ | $x^{p+6},...x^{9}$
---|---|---|---|---|---
$\mathrm{D}_{p}$ | - | - |  |  |
$\mathrm{D}_{p+4}$ | - |  | - | - |
Table 1:  The D-brane configuration of the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$
system for $p=3,4$. “-” represents that the D-branes extend along this
direction and $x^{p}$ is the periodic direction.

Since $N_{f}$ is set to be finite, the $\mathrm{D}_{p+4}$-branes become the
probes embedded into (2.2) (2.4) and $N_{c}\rightarrow\infty$ implies
$\mathrm{D}_{p}$-brane dominates the bulk geometry. By analyzing the spectrum
of the open string connecting the $N_{c}$ $\mathrm{D}_{p}$-branes and $N_{f}$
$\mathrm{D}_{p+4}$-branes in R-sector, the low-energy theory includes fermions
in the fundamental representation of $U\left(N_{c}\right)$ and
$U\left(N_{f}\right)$ 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 QCD.

##  3 The holographic correspondence with fundamental fermion

As our goal is to work out the two-point correlation function of fundamental
fermions through gauge-gravity duality, in this section, let us attempt to
found the associated holographic correspondence 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.

###  3.1 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 $\partial\mathcal{M}$ 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 $\psi$ in the bulk $\mathcal{M}$
whose boundary value is $\psi_{0}=\lim_{r\rightarrow\infty}\psi$, we can write
down,

$Z_{QCD}\left[\bar{\psi}_{0},\psi_{0}\right]=Z_{gravity}\left[\bar{\psi},\psi\right],$
(3.1)

with

$\displaystyle Z_{QCD}\left[\bar{\psi_{0}},\psi_{0}\right]$
$\displaystyle=\left\langle\exp\left\\{\int_{\partial\mathcal{M}}\left(\bar{\chi}\psi_{0}+\bar{\psi}_{0}\chi\right)d^{D}x\right\\}\right\rangle,$
$\displaystyle Z_{gravity}\left[\bar{\psi},\psi\right]$
$\displaystyle=\exp\left\\{\int_{\mathcal{M}}\mathcal{L}_{gravity}^{ren}\left[\bar{\psi},\psi\right]d^{D+1}x\right\\}.$
(3.2)

And $\mathcal{L}_{gravity}^{ren}$ refers to the classical renormalized
Lagrangian of the bulk field $\psi$. In our concern, $\chi$ refers to the
fundamental fermion (quark) in the dual field theory. Then the two-point
retarded Green function $G_{R}$ for the spinor $\chi$ can be evaluated by
following [10, 13, 14] as,

$\left\langle\chi\left(\omega,\vec{k}\right)\right\rangle=G_{R}\left(\omega,\vec{k}\right)\psi_{0}\left(\omega,\vec{k}\right),$
(3.3)

and,

$\displaystyle\left\langle\bar{\chi}\left(\omega,\vec{k}\right)\right\rangle$
$\displaystyle=-\frac{\delta
S_{gravity}^{ren}}{\delta\psi_{0}}=\Pi_{0}\left(\omega,\vec{k}\right),$
$\displaystyle S_{gravity}^{ren}$
$\displaystyle=\int_{\mathcal{M}}\mathcal{L}_{gravity}^{ren}\left[\bar{\psi}_{0},\psi_{0}\right]d^{D+1}x.$
(3.4)

Here $\omega,\vec{k}$ 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
$G_{E}\left(\omega_{E},\vec{k}\right)=G_{R}\left(\omega,\vec{k}\right)$ with
$\omega_{E}=-i\omega$.

The next step is to identify the bulk spinor field $\psi$ in order to
investigate its action $S_{gravity}^{ren}\left[\psi\right]$ in holography. We
notice that, in several bottom-up or phenomenological holographic approaches
e.g. [12, 30, 31, 32, 33], the bulk field $\psi$ is usually identified to a
fundamental fermion of the $U\left(N\right)$ group whose action is the
minimally coupled Dirac action. However such a bulk field $\psi$ is less clear
in the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$-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 $U\left(N\right)$ fundamental
fermion. To figure out this problem, it is worth retaking into account the
interaction of the open strings in the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$-brane
system according to the string theory. For example, it is known the AdS/CFT
dictionary for the approach of vector field

$\left\langle\exp\left\\{\int_{\partial\mathcal{M}}J^{\mu}A_{\mu}^{\left(0\right)}d^{D}x\right\\}\right\rangle=\exp\left\\{-S_{bulk}\left[A_{\mu}\right]\right\\},\
\ A_{\mu}^{\left(0\right)}=\lim_{r\rightarrow\infty}A_{\mu}.$ (3.5)

illustrates that the correlator of the flavored current operator $J^{\mu}$
living on a probe $\mathrm{D}_{p}$-brane at the boundary could be evaluated by
varying the action of the worldvolume vector $A_{\mu}$ on the
$\mathrm{D}_{p+4}$-branes as the bulk action $S_{bulk}\left[A_{\mu}\right]$
[5, 14, 10], since the $\mathrm{D}_{p+4}$-branes as flavors extend along the
holographically radial direction of the bulk according to Table 1. In this
sense, the source term $J^{\mu}A_{\mu}^{\left(0\right)}$ presented in (3.5)
implies the interaction of the $\left(p,p\right)$ and $\left(p+4,p+4\right)$
strings555Here we use $\left(p,q\right)$ to refer to an open string connecting
$\mathrm{D}_{p}$\- and $\mathrm{D}_{q}$-brane..

Keeping the above in mind and turning to our case, once the spinor $\chi$ in
(3.1) (3.2) is identified to the fundamental fermion living on the
$\mathrm{D}_{p}$-brane at boundary, we must find its fermionic dual field
$\psi$ by analyzing the low-energy spectrum of type II string theories. Among
the fields in the massless spectrum of the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$
model, the only possible choice for $\psi$ should be the fermionic field on
the $\mathrm{D}_{p+4}$-branes666Since we are working in the bosonic background
given in (2.2) (2.4) , the fermions in the sector of close string can be
neglected.. In this sense the source term
$\bar{\chi}\psi_{0}+\bar{\psi}_{0}\chi$ presented in (3.2) corresponds to the
interaction of $\left(p,p+4\right)$ and $\left(p+4,p+4\right)$ strings at
boundary, which reveals the interaction to boundary theory involving the
fermionic part to $A_{\mu}$ presented in (3.5), as it is displayed in Figure
1.

Figure 1:  The correspondence of the bulk field $\psi$ and boundary operator
$\chi$ in the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ system. Here $r$ refers to the
radial direction as the holographic direction.

We note that, in the D-brane setup given in Table 1 on the bubble background
(2.4), the $\left(p+4,p+4\right)$ string remains to be supersymmetric [42]
since there is not any mechanism to break down the supersymmetry on the
$\mathrm{D}_{p+4}$-branes in principle while the $\mathrm{D}_{p}$-brane is not
supersymmetric due to the boundary condition on its compactified direction.
Therefore the fermionic field $\psi$ in the worldvolume of
$\mathrm{D}_{p+4}$-branes must exist in the low-energy theory, and in our
$\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model, it is seemingly to be the suitable
candidate uniquely as the source of the fundamental fermion $\chi$ 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 $\mathrm{D}_{p+4}$-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 $\psi$ to the supersymmetric fermion on the
$\mathrm{D}_{p+4}$-brane in order to continue our discussion.

###  3.2 Action for the bulk spinor

In the last section, since the bulk field dual to the fundamental fermion is
identified to the worldvolume fermion on the $N_{f}$ coincident
$\mathrm{D}_{p+4}$-branes, let us take a look at the action for such a
fermionic field.

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 [38, 39] 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
$\mathrm{D}_{p}$-brane action which on the bosonic background is given by [34,
35],

$S_{f}^{\mathrm{D}_{p}}=\frac{iT_{p}}{2}\int
d^{p+1}xe^{-\phi}\sqrt{-\left(g+\mathcal{F}\right)}\bar{\Psi}\left(1-\Gamma_{\mathrm{D}_{p}}\right)\left(\Gamma^{\alpha}\hat{D}_{\alpha}-\Delta+\mathrm{L}_{\mathrm{D}_{p}}\right)\Psi,$
(3.6)

where, for the type IIA string theory ($p$ is even number),

$\displaystyle\hat{D}_{\alpha}$ $\displaystyle=\nabla_{\alpha}+\frac{1}{4\cdot
2!}H_{\alpha
NK}\Gamma^{NK}\bar{\gamma}+\frac{1}{8}e^{\phi}\left(\frac{1}{2!}F_{NK}\Gamma^{NK}\Gamma_{\alpha}\bar{\gamma}+\frac{1}{4!}F_{KLNP}\Gamma^{KLNP}\Gamma_{\alpha}\right),$
$\displaystyle\Delta$
$\displaystyle=\frac{1}{2}\left(\Gamma^{M}\partial_{M}\phi+\frac{1}{2\cdot
3!}H_{MNK}\Gamma^{MNK}\bar{\gamma}\right)+\frac{1}{8}e^{\phi}\left(\frac{3}{2!}F_{MN}\Gamma^{MN}\bar{\gamma}+\frac{1}{4!}F_{KLNP}\Gamma^{KLNP}\right),$
$\displaystyle\Gamma_{\mathrm{D}_{p}}$
$\displaystyle=\frac{1}{\sqrt{-\left(g+\mathcal{F}\right)}}\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}\mathcal{F}_{\alpha_{1}\alpha_{2}}...\mathcal{F}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}\bar{\gamma}^{\frac{p-2q+2}{2}},$
$\displaystyle\mathrm{L}_{\mathrm{D}_{p}}$
$\displaystyle=\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}\frac{\left(-\bar{\gamma}\right)^{\frac{p}{2}-q+1}}{\sqrt{-\left(g+\mathcal{F}\right)}}\mathcal{F}_{\alpha_{1}\alpha_{2}}...\mathcal{F}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}^{\
\ \ \ \ \ \ \ \ \ \ \ \ \lambda}\hat{D}_{\lambda},$ (3.7)

for the type IIB string theory ($p$ is odd number),

$\displaystyle\hat{D}_{\alpha}=$ $\displaystyle\nabla_{\alpha}+\frac{1}{4\cdot
2!}H_{\alpha
NK}\Gamma^{NK}\bar{\gamma}-\frac{1}{8}e^{\phi}\big{(}F_{N}\Gamma^{N}\Gamma_{\alpha}\bar{\gamma}+\frac{1}{3!}F_{KLN}\Gamma^{KLN}\Gamma_{\alpha}$
$\displaystyle+\frac{1}{2\cdot
5!}F_{KLMNP}\Gamma^{KLMNP}\Gamma_{\alpha}\bar{\gamma}\big{)},$
$\displaystyle\Delta=$
$\displaystyle\frac{1}{2}\left(\Gamma^{M}\partial_{M}\phi+\frac{1}{2\cdot
3!}H_{MNK}\Gamma^{MNK}\bar{\gamma}\right)+\frac{1}{2}e^{\phi}\left(F_{M}\Gamma^{M}\bar{\gamma}+\frac{1}{2\cdot
3!}F_{KLN}\Gamma^{KLN}\right),$ $\displaystyle\Gamma_{\mathrm{D}_{p}}=$
$\displaystyle\frac{1}{\sqrt{-\left(g+\mathcal{F}\right)}}\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}\mathcal{F}_{\alpha_{1}\alpha_{2}}...\mathcal{F}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}\bar{\gamma}^{\frac{p-2q+1}{2}},$
$\displaystyle\mathrm{L}_{\mathrm{D}_{p}}$
$\displaystyle=\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}\frac{\left(-i\sigma_{2}\right)\left(\bar{\gamma}\right)^{\frac{p-2q+1}{2}}}{\sqrt{-\left(g+\mathcal{F}\right)}}\mathcal{F}_{\alpha_{1}\alpha_{2}}...\mathcal{F}_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}^{\
\ \ \ \ \ \ \ \ \ \ \ \ \lambda}\hat{D}_{\lambda}.$ (3.8)

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
$\mathrm{D}_{p}$-brane given as
$T_{p}=g_{s}^{-1}\left(2\pi\right)^{-p}l_{s}^{-\left(p+1\right)}$. 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 $\alpha,\beta,\lambda$ to denote the indices
running over the worldvolume of the $\mathrm{D}_{p}$-brane. The metric is
written in terms of elfbein as $g_{MN}=e_{M}^{a}\eta_{ab}e_{N}^{b}$, so the
gamma matrices are defined by

$\left\\{\gamma^{a},\gamma^{b}\right\\}=2\eta^{ab},\left\\{\Gamma^{M},\Gamma^{N}\right\\}=2g^{MN},$
(3.9)

with $e_{M}^{a}\Gamma^{M}=\gamma^{a}$. $\omega_{\alpha ab}$ refers to the spin
connection and $\nabla_{\alpha}=\partial_{\alpha}+\frac{1}{4}\omega_{\alpha
ab}\gamma^{ab}$ 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.

$\gamma^{ab}=\frac{1}{2}\left[\gamma^{a},\gamma^{b}\right],\gamma^{abc}=\frac{1}{2}\left\\{\gamma^{a},\gamma^{bc}\right\\},\gamma^{abcd}=\frac{1}{2}\left[\gamma^{a},\gamma^{bcd}\right]...$
(3.10)

$\Gamma^{MNK...}$ shares the same definition as $\gamma^{abc...}$.
$\bar{\gamma}$ is defined as $\bar{\gamma}=\gamma^{01...9}$ and $\sigma_{2}$
refers to the associated Pauli matrix. The worldvolume field $\mathcal{F}$ is
given as $\mathcal{F}=B+\left(2\pi\alpha^{\prime}\right)f$ where $B$ is the
NS-NS 2-form in type II string theory with $H=dB$ and $f$ is the Yang-Mills
field strength. $F_{M},F_{MN},F_{KLM}...$ refer to the associated field
strength of the massless R-R fields presented in IIA and IIB string theory. We
note that, in our $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ approach, $p$ should be
replaced by $p+4$ in the above formulas since the worldvolume fermion on the
probe $\mathrm{D}_{p+4}$-brane is our concern.

###  3.3 The prescription for two-point correlation function

In this section, let us attempt to generalize the prescription in AdS/CFT
dictionary for two-point correlation function into the background presented in
(2.2) and (2.4). We first summarize the steps as follows then give some
comments.

  1. 1.

Simplify the T-dualitized action (3.6) with respect to the background fields
given in (2.2) (2.4)777When we recall the background (2.2) (2.4), it means we
will use their near-horizon version for holography. in order to obtain an
effective $D+1$ dimensional action involving the bulk spinor
$\psi\left(x,r\right)$, for a $D$ dimensional dual theory, as

$\displaystyle S_{D+1}$ $\displaystyle\propto i\int
d^{D}xdr\bar{\psi}\gamma^{r}\partial_{r}\psi+...$ $\displaystyle=i\int
d^{D}x\left(\bar{\psi}\gamma^{r}\psi\right)\big{|}_{r=r_{KK,H}}^{r=\infty}-i\int
d^{4}xdr\partial_{r}\bar{\psi}\gamma^{r}\psi+...$ (3.11)

where we define

$\Pi=-i\bar{\psi}\gamma^{r}.$ (3.12)

  2. 2.

Use the equation of motion for $\psi\left(x,r\right)$ and the Fourier mode as
the ansatz for the solution $\psi\left(x,r\right)$ as,

$\psi\left(x,r\right)=e^{ik\cdot
x}\beta\left(\omega,\vec{k},r\right),k_{\mu}=\left(-\omega,\vec{k}\right),$
(3.13)

to derive the equation for $\xi$ which is given by

$\bar{\Pi}=\xi\psi.$ (3.14)

  3. 3.

Solve the the equation for $\xi$ with incoming wave boundary condition and the
two-point retarded Green function $G_{R}$ is given by extracting the finite
part of $\xi$.

The above steps are based on the prescription in AdS/CFT dictionary for two-
point correlation function [10, 14]. We notice that 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
$\psi_{0}$ with respect to $\psi$ which, as we will see, is equivalent to
exact the finite part of $\xi$ given in (3.14). Therefore in the following
sections, we will test our prescription in the top-down D4/D8 and D3/D7
approaches in holography.

##  4 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 to calculate the
two-point Green function and analyze the results numerically in the 1+3
dimensional QCD.

###  4.1 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],

$\displaystyle ds_{c}^{2}$
$\displaystyle=\left(\frac{r}{R}\right)^{3/2}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+f\left(r\right)\left(dx^{4}\right)^{2}\right]+\left(\frac{R}{r}\right)^{3/2}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{4}^{2}\right],$
$\displaystyle e^{\phi}$ $\displaystyle=\left(\frac{r}{R}\right)^{3/4},\
F_{4}=dC_{3}=3R^{3}g_{s}^{-1}\epsilon_{4},\
f\left(r\right)=1-\frac{r_{KK}^{3}}{r^{3}},\mu,\nu=0,..3.$ (4.1)

Here $R$ relates to the radius of the spacetime, $\epsilon_{4}$ 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,

$R^{3}=\frac{1}{2}\frac{g_{\mathrm{YM}}^{2}N_{c}l_{s}^{2}}{M_{KK}},\
r_{KK}=\frac{2}{3}g_{\mathrm{YM}}^{2}N_{c}M_{KK}l_{s}^{2},\
g_{s}=\frac{1}{2\pi}\frac{g_{\mathrm{YM}}^{2}}{M_{KK}l_{s}},$ (4.2)

where $g_{\mathrm{YM}}$ 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,

$\displaystyle ds_{d}^{2}$
$\displaystyle=\left(\frac{r}{R}\right)^{3/2}\left[-f_{T}\left(r\right)dt^{2}+\delta_{ij}dx^{i}dx^{j}+\left(dx^{4}\right)^{2}\right]+\left(\frac{R}{r}\right)^{3/2}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{4}^{2}\right]$
$\displaystyle e^{\phi}$ $\displaystyle=\left(\frac{r}{R}\right)^{3/4},\
F_{4}=dC_{3}=3R^{3}g_{s}^{-1}\epsilon_{4},\
f_{T}\left(r\right)=1-\frac{r_{H}^{3}}{r^{3}},i,j=1,2,3.$ (4.3)

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,

$\displaystyle ds_{\mathrm{D8},c}^{2}$
$\displaystyle=\left(\frac{r}{R}\right)^{3/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\left(\frac{R}{r}\right)^{3/2}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{4}^{2}\right],$
$\displaystyle ds_{\mathrm{D8},d}^{2}$
$\displaystyle=\left(\frac{r}{R}\right)^{3/2}\left[-f_{T}\left(r\right)dt^{2}+\delta_{ij}dx^{i}dx^{j}\right]+\left(\frac{R}{r}\right)^{3/2}\left[\frac{dr^{2}}{f_{T}\left(r\right)}+r^{2}d\Omega_{4}^{2}\right],$
(4.4)

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. $x^{4}=\mathrm{const}$ 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 $x^{4}=\mathrm{const}$ as it is illustrated in Figure 2.

Figure 2:  The configuration of the D4/D8 model in the bubble (confined)
background (a) and black brane (deconfined) background (b) on $r-x^{4}$ plane.
Red line refers to the D8-branes ($\mathrm{D8}$) and anti D8-branes
($\mathrm{\overline{D8}}$).

Then the action for the worldvolume fermion on D8-branes can be collected from
(3.6) (3.7) as,

$\displaystyle S_{f}^{\mathrm{D8}}$ $\displaystyle=\frac{iT_{8}}{2}\int
d^{9}xe^{-\phi}\sqrt{-g}\bar{\Psi}P_{-}\big{[}\Gamma^{\alpha}\nabla_{\alpha}+\frac{1}{8\cdot
4!}e^{\phi}F_{KLNP}\left(\Gamma^{\alpha}\Gamma^{KLNP}\Gamma_{\alpha}-\Gamma^{KLNP}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{1}{2}\Gamma^{M}\partial_{M}\phi\big{]}\Psi.$ (4.5)

Since our concern is the fermionic part, we have turned off the irrelative
bosonic fields by setting $\mathcal{F}=0$. Imposing the solution for the
dilaton $\phi$ 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,

$\displaystyle S_{f,c}^{\mathrm{D8}}=$
$\displaystyle\frac{i\mathcal{T}_{c}}{\left(2\pi\alpha^{\prime}\right)^{2}\Omega_{4}}\int
d^{4}xdZd\Omega_{4}\bar{\Psi}P_{-}\big{(}\frac{2}{3}M_{KK}K^{\frac{7}{12}}\gamma^{m}\nabla_{m}^{S^{4}}+K^{\frac{5}{12}}\gamma^{\mu}\partial_{\mu}+M_{KK}K^{\frac{13}{12}}\gamma^{Z}\partial_{Z},$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{13}{12}M_{KK}ZK^{\frac{1}{12}}\gamma^{4}\big{)}\Psi,$ (4.6)

with,

$K\left(Z\right)=1+Z^{2},\mathcal{T}_{c}=\frac{1}{2}\left(\frac{2}{3}\right)^{13/2}T_{8}\Omega_{4}\left(2\pi\alpha^{\prime}\right)^{2}\left(M_{KK}R\right)^{11/2}R^{5},P_{-}=\frac{1}{2}\left(1-\Gamma_{\mathrm{D8}}\right).$
(4.7)

In the deconfined background, the action (4.5) becomes,

$\displaystyle S_{f,d}^{\mathrm{D8}}=$
$\displaystyle\frac{i\mathcal{T}_{d}}{\left(2\pi\alpha^{\prime}\right)^{2}\Omega_{4}}\int
d^{4}xdZd\Omega_{4}\bar{\Psi}P_{-}\bigg{[}\frac{4}{3}\pi
TZK^{1/12}\gamma^{m}\nabla_{m}^{S^{4}}+K^{5/12}\gamma^{0}\partial_{0}+ZK^{-1/12}\gamma^{i}\partial_{i}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ +2\pi
TZK^{7/12}\gamma^{Z}\partial_{Z}+\pi
T\left(\frac{13}{6}Z^{2}K^{-5/12}+K^{7/12}\right)\gamma^{Z}\bigg{]}\Psi,$
(4.8)

where

$\mathcal{T}_{d}=\frac{1}{2}\left(\frac{2}{3}\right)^{13/2}T_{8}\Omega_{4}\left(2\pi\alpha^{\prime}\right)^{2}\left(2\pi
TR\right)^{11/2}R^{5},r_{H}=\frac{16}{9}\pi^{2}R^{3}T^{2},T=\beta_{t}^{-1}.$
(4.9)

Here $T,\beta_{t}$ refers to the Hawking temperature and the period of the
time direction in (2.2). We have used $\Psi$ to denote the worldvolume fermion
whose boundary value relates to $\psi_{0}$. The index $m$ runs over $S^{4}$
and the unit volume of $S^{4}$ is given as $\Omega_{4}=8\pi^{2}/3$. 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,

$K\left(Z\right)=1+Z^{2}=\frac{r^{3}}{r_{KK,H}^{3}},$ (4.10)

so that the D8- and anti D8-branes are located respectively at
$Z\rightarrow\pm\infty$ representing approximately the chiral symmetry
$U\left(N_{f}\right)_{R}\times U\left(N_{f}\right)_{L}$ at boundary. And the
dual theory on the D4-branes at boundary takes the chirally symmetric action
as QCD as [42],

$\displaystyle S=$
$\displaystyle\int_{\mathrm{D4}}d^{5}x\sqrt{-g}\left[\delta\left(Z-Z_{+\infty}\right)\chi_{L}^{\dagger}\bar{\sigma}^{\mu}\left(i\nabla_{\mu}+A_{\mu}\right)\chi_{L}+\delta\left(Z-Z_{-\infty}\right)\chi_{R}^{\dagger}\bar{\sigma}^{\mu}\left(i\nabla_{\mu}+A_{\mu}\right)\chi_{R}\right]$
$\displaystyle-\frac{1}{4g_{\mathrm{YM}}^{2}}\int_{\mathrm{D4}}d^{5}x\sqrt{-g}\mathrm{Tr}F_{\mu\nu}F^{\mu\nu},\
\mu,\nu=0,1...3,$ (4.11)

where we use Weyl spinor $\chi_{R,L}$ to denote the fundamental chiral fermion
and $F_{MN}$ 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,

$\left\langle\exp\left\\{\int_{\partial\mathcal{M}}\left(\chi_{L}^{\dagger}\psi_{L}+\chi_{R}^{\dagger}\psi_{R}+h.c.\right)d^{D}x\right\\}\right\rangle,$
(4.12)

once we focus on the Green function of chiral fermion, as it is illustrated in
Figure 3.

Figure 3:  The correspondence of the bulk fields $\psi_{R,L}$ and chiral
fermions $\chi_{R,L}$ in the D4/D8 model. The $N_{f}$ pairs of D8- and anti
D8-branes are located at $Z\rightarrow\pm\infty$ at boundary providing
$U\left(N_{f}\right)_{R}\times U\left(N_{f}\right)_{L}$ chiral symmetry in the
dual theory. The bulk field $\psi_{R,L}$ as source couples to the chiral
fermion $\chi_{R,L}$ respectively as it is in QCD.

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 $\psi\left(x,Z\right)$
with holographic coordinate $Z$, an $S^{4}$ part $\varphi$ and a remaining 2d
part $\beta$ as888We note that in $D$ dimension, a spinor has
$\left[D/2\right]$ components where $\left[D/2\right]$ refers to the integer
part of $D/2$.,

$\psi\left(x,Z\right)\otimes\varphi\left(S^{4}\right)\otimes\beta,$ (4.13)

so that the worldvolume fermion on D8-brane should be
$\Psi=\psi\otimes\varphi$. Then the associated 10d gamma matrices are chosen
as,

$\displaystyle\gamma^{\mu}$
$\displaystyle=\sigma_{1}\otimes\bm{\gamma}^{\mu}\otimes\bm{1},\mu=0,1,2,3$
$\displaystyle\gamma^{Z}$
$\displaystyle=\sigma_{1}\otimes\bm{\gamma}\otimes\bm{1},$
$\displaystyle\gamma^{4}$
$\displaystyle=\sigma_{2}\otimes\bm{1}\otimes\tilde{\bm{\gamma}},$
$\displaystyle\gamma^{m}$
$\displaystyle=\sigma_{2}\otimes\bm{1}\otimes\bm{\gamma}^{m},m=6,7,8,9,$
$\displaystyle\bm{\gamma}$
$\displaystyle=i\bm{\gamma}^{0}\bm{\gamma}^{1}\bm{\gamma}^{2}\bm{\gamma}^{3},$
$\displaystyle\tilde{\bm{\gamma}}$
$\displaystyle=i\bm{\gamma}^{6}\bm{\gamma}^{7}\bm{\gamma}^{8}\bm{\gamma}^{9},$
(4.14)

where we use bold to denote the $4\times 4$ gamma matrices.
$\bm{\gamma}^{m},m=6,7,8,9$ refers to the gamma matrix on tangent space of
$S^{4}$. We note that, the 10d chirality matrix takes a very simple form as
$\bar{\gamma}=\sigma_{3}\otimes\bm{1}\otimes\bm{1}$ in this decomposition.
Choosing the $\sigma_{3}$ representation, $\beta$ can therefore be decomposed
by the eigenstates of $\sigma_{3}$ with

$\sigma_{3}\beta_{\pm}=\beta_{\pm},\sigma_{1}\beta_{\pm}=\beta_{\mp},\sigma_{2}\beta_{\pm}=\pm
i\beta_{\mp},$ (4.15)

where we use $\beta_{\pm}$ to denote the two eigenstates of $\sigma_{3}$.
Since the kappa symmetry fixes the condition $\bar{\gamma}\Psi=\Psi$999This
condition could be opposite on the anti D8-branes., we have to chose
$\beta=\beta_{+}$ on the D8-brane. In addition, as $\varphi$ must satisfy the
Dirac equation on $S^{4}$, we can decompose it by using the spherical harmonic
function with the eigenstates of $\Gamma^{\underline{m}}\nabla_{m}^{S^{4}}$ as
[43],

$\bm{\gamma}^{m}\nabla_{m}^{S^{4}}\varphi^{\pm
l,s}=i\Lambda_{l}^{\pm}\varphi^{\pm
l,s};\Lambda_{l}^{\pm}=\pm\left(2+l\right),l=0,1...$ (4.16)

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
$\varphi$ and $\beta$, we could obtain a 5d effective action for the fermion
on the worldvolume of the D8-branes after integrating the $S^{4}$ part as,

$\displaystyle S_{f,c}^{\mathrm{D8}}=$
$\displaystyle\frac{i\mathcal{T}_{c}}{\left(2\pi\alpha^{\prime}\right)^{2}}\int
d^{4}xdZ\bar{\psi}\big{(}-\frac{2}{3}M_{KK}\Lambda_{l}K^{\frac{7}{12}}+K^{\frac{5}{12}}\bm{\gamma}^{\mu}\partial_{\mu}+M_{KK}K^{\frac{13}{12}}\bm{\gamma}\partial_{Z}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
+\frac{13}{12}M_{KK}ZK^{\frac{1}{12}}\bm{\gamma}\big{)}\psi,$ (4.17)

and

$\displaystyle S_{f,d}^{\mathrm{D8}}=$
$\displaystyle\frac{i\mathcal{T}_{d}}{\left(2\pi\alpha^{\prime}\right)^{2}}\int
d^{4}xdZ\bar{\psi}\bigg{[}-\frac{4}{3}\pi
TZK^{1/12}\Lambda_{l}+K^{5/12}\bm{\gamma}^{0}\partial_{0}+ZK^{-1/12}\bm{\gamma}^{i}\partial_{i}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ +2\pi
TZK^{7/12}\bm{\gamma}\partial_{Z}+\pi
T\left(\frac{13}{6}Z^{2}K^{-5/12}+K^{7/12}\right)\bm{\gamma}\bigg{]}\psi,$
(4.18)

where $\Lambda_{l}=\left|\Lambda_{l}^{\pm}\right|$. 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 $\psi\rightarrow
2\pi\alpha^{\prime}K^{-13/24}\psi$ as,

$S_{f,c}^{\mathrm{D8}}=i\mathcal{T}_{c}\int
d^{4}xdZ\bar{\psi}\left(-\frac{2}{3}M_{KK}\Lambda_{l}K^{-1/2}+K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}+M_{KK}\bm{\gamma}\partial_{Z}\right)\psi,$
(4.19)

while the action (4.18) is rescaled by $\psi\rightarrow
2\pi\alpha^{\prime}K^{-7/24}Z^{-1/2}\psi$ as,

$\displaystyle S_{f,d}^{\mathrm{D8}}$ $\displaystyle=i\mathcal{T}_{d}\int
d^{4}xdZ\bar{\psi}\bigg{(}-\frac{4}{3}\pi
TK^{-1/2}\Lambda_{l}+Z^{-1}K^{-1/6}\bm{\gamma}^{0}\partial_{0}+K^{-2/3}\bm{\gamma}^{i}\partial_{i}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ +2\pi T\bm{\gamma}\partial_{Z}+\pi
TZK^{-1}\bm{\gamma}\bigg{)}\psi.$ (4.20)

And we will use these 5d fermionic actions (4.19) (4.20) to evaluate the two-
point Green function.

###  4.2 Use the prescription

Let us attempt to use the prescription in Section 3.3 to evaluate the two-
point Green function with confined (2.4) and deconfined (2.2) geometry by
using action (4.19) (4.20) respectively.

#### Confined phase

First the equation of motion associated to action (4.19) is derived as,

$\left(-\frac{2}{3}M_{KK}\Lambda_{l}K^{-1/2}+K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}+M_{KK}\bm{\gamma}\partial_{Z}\right)\psi=0.$
(4.21)

Picking up the ansatz in Weyl basis as,

$\psi=e^{ik\cdot x}\left(\begin{array}[]{c}\psi_{R}\\\
\psi_{L}\end{array}\right),\ k\cdot x=k_{\mu}x^{\mu}=-\omega
t+\vec{k}\cdot\vec{x},$ (4.22)

the equation (4.21) becomes

$\displaystyle-\frac{2}{3}M_{KK}\Lambda_{l}K^{-1/2}\psi_{R}-K^{-2/3}\left(\sigma\cdot
k\right)\psi_{L}+M_{KK}\partial_{Z}\psi_{R}$ $\displaystyle=0,$
$\displaystyle-\frac{2}{3}M_{KK}\Lambda_{l}K^{-1/2}\psi_{L}-K^{-2/3}\left(\bar{\sigma}\cdot
k\right)\psi_{R}-M_{KK}\partial_{Z}\psi_{L}$ $\displaystyle=0,$ (4.23)

where we have chosen the 5d gamma matrices as,

$\bm{\gamma}^{\mu}=i\left(\begin{array}[]{cc}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{array}\right),\bm{\gamma}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (4.24)

with
$\sigma^{\mu}=\left(1,-\sigma^{i}\right),\bar{\sigma}^{\mu}=\left(1,\sigma^{i}\right)$.
Solving (4.23), we can further obtain the decoupled equations for $\psi_{R,L}$
as,

$\displaystyle\partial_{Z}^{2}\psi_{L}+\frac{2K^{\prime}}{3K}\partial_{Z}\psi_{L}+\left(\frac{\lambda
K^{\prime}}{9K^{3/2}}-\frac{4}{9}\frac{\Lambda_{l}^{2}}{K}-\frac{k^{2}}{M_{KK}^{2}}\frac{1}{K^{4/3}}\right)\psi_{L}$
$\displaystyle=0,$
$\displaystyle\partial_{Z}^{2}\psi_{R}+\frac{2K^{\prime}}{3K}\partial_{Z}\psi_{R}-\left(\frac{\lambda
K^{\prime}}{9K^{3/2}}+\frac{4}{9}\frac{\Lambda_{l}^{2}}{K}+\frac{k^{2}}{M_{KK}^{2}}\frac{1}{K^{4/3}}\right)\psi_{R}$
$\displaystyle=0,$ (4.25)

where “$\prime$” refers to the derivative with respect to $Z$. The above
equations can be analytically solve at $Z\rightarrow+\infty$ as,

$\displaystyle\psi_{R}$
$\displaystyle=A\left(k,\Lambda_{l}\right)Z^{\frac{2}{3}\Lambda_{l}}+B\left(k,\Lambda_{l}\right)Z^{-\frac{1}{3}-\frac{2}{3}\Lambda_{l}},$
$\displaystyle\psi_{L}$
$\displaystyle=C\left(k,\Lambda_{l}\right)Z^{-\frac{1}{3}+\frac{2}{3}\Lambda_{l}}+D\left(k,\Lambda_{l}\right)Z^{-\frac{2}{3}\Lambda_{l}},$
(4.26)

where $A,B,C,D$ are constant spinors and furthermore one can find the
following relations as,

$C\left(k,\Lambda_{l}\right)=\frac{3}{\left(1-4\lambda\right)M_{KK}}\left(\bar{\sigma}\cdot
k\right)A\left(k,\Lambda_{l}\right),\
B\left(k,\Lambda_{l}\right)=\frac{3}{\left(1+4\lambda\right)M_{KK}}\left(\sigma\cdot
k\right)D\left(k,\Lambda_{l}\right),$ (4.27)

once the analytical solution (4.26) is plugged back into (4.23). Besides, we
notice that the solution at $Z\rightarrow-\infty$ is equivalent to interchange
$\psi_{R,L}$ in the solution (4.26). $\Lambda_{l}$ is the eigenvalue of the
$S^{4}$ part and it satisfies $\Lambda_{l}\geq 2$ according to (4.16). So we
obtain the boundary behavior of $\psi$ as,

$\lim_{Z\rightarrow+\infty}\psi\rightarrow\left(\begin{array}[]{c}\psi_{R}\\\
0\end{array}\right),\
\lim_{Z\rightarrow-\infty}\psi\rightarrow\left(\begin{array}[]{c}0\\\
\psi_{L}\end{array}\right).$ (4.28)

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 $\psi_{0}$ of $\psi$ at $Z\rightarrow+\infty$ by picking up
its dominantly finite part in (4.26) as [13, 14],

$\psi_{0}=\lim_{Z\rightarrow+\infty}Z^{-\frac{2}{3}\Lambda_{l}}\psi=A\left(k,\Lambda_{l}\right).$
(4.29)

Next, the conjugate momentum $\Pi_{0}$ associated to $\psi_{0}$ can be
evaluated by using (3.4). To this goal, let us take a look at the action
(4.19),

$\displaystyle S_{f,c}^{\mathrm{D8}}=$ $\displaystyle\ i\mathcal{T}_{c}\int
d^{4}xdZ\bar{\psi}\left(-\frac{2}{3}M_{KK}\lambda
K^{-1/2}+K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}+M_{KK}\bm{\gamma}\partial_{Z}\right)\psi,$
$\displaystyle=$ $\displaystyle\ i\mathcal{T}_{c}\int
d^{4}x\left(\bar{\psi}M_{KK}\bm{\gamma}\psi\right)|_{-\infty}^{+\infty}$
$\displaystyle-i\mathcal{T}_{c}\int
d^{4}xdZ\left(\partial_{Z}\bar{\psi}M_{KK}\bm{\gamma}+\frac{2}{3}M_{KK}\lambda
K^{-1/2}\bar{\psi}-K^{-2/3}\partial_{\mu}\bar{\psi}\bm{\gamma}^{\mu}\right)\psi,$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}x\left(\Pi\psi\right)|_{-\infty}^{+\infty}=\int
d^{4}x\left(\Pi_{R}\psi_{R}+\Pi_{L}\psi_{L}\right)|_{-\infty}^{+\infty},$
(4.30)

where

$\Pi_{R}=-\mathcal{T}_{c}M_{KK}\psi_{L}^{\dagger},\
\Pi_{L}=\mathcal{T}_{c}M_{KK}\psi_{R}^{\dagger}.$ (4.31)

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
$Z\rightarrow\infty$ as101010Since the solution at $Z\rightarrow-\infty$ can
be obtained by interchanging the roles of $\psi_{R,L}$ in (4.26), the action
(4.30) has same asymptotic behavior at $Z\rightarrow-\infty$ on the anti
D8-branes. ,

$\displaystyle S_{f,c}^{\mathrm{D8}}$
$\displaystyle\supseteq-\mathcal{T}_{c}M_{KK}\int
d^{4}x\left[C\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)Z^{-\frac{1}{3}+\frac{4}{3}\Lambda_{l}}+D\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)+h.c\right]|_{Z\rightarrow+\infty}+...$
(4.32)

Therefore we can see, it requests for a holographic boundary counterterm
$S_{ct}$ as,

$S_{ct}=\mathcal{T}_{c}M_{KK}\int
d^{4}x\left[C\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)Z^{-\frac{1}{3}+\frac{4}{3}\Lambda_{l}}+h.c\right]|_{Z\rightarrow+\infty},$
(4.33)

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 $\mathrm{AdS}_{7}$ since the D4/D8
model is based on the holographic correspondence on $\mathrm{AdS}_{7}\times
S^{4}$ [20, 38]. Then the conjugate momentum on the D8-branes to $\psi_{0}$ is
evaluated as

$\Pi_{0}=-\frac{\delta
S^{ren}}{\delta\psi_{0}}=\mathcal{T}_{c}M_{KK}D\left(k,\Lambda_{l}\right)^{\dagger},\
S^{ren}=S_{f,c}^{\mathrm{D8}}+S_{ct},$ (4.34)

which implies the Green function $G_{R}\left(k,\Lambda_{l}\right)$ is obtained
by

$\mathcal{T}_{c}M_{KK}D\left(k,\Lambda_{l}\right)=G_{R}\left(k,\Lambda_{l}\right)A\left(k,\Lambda_{l}\right),$
(4.35)

according to (3.3) (3.4). Using the representation

$\psi_{R}=\left(\begin{array}[]{c}\psi_{R}^{\left(1\right)}\\\
\psi_{R}^{\left(2\right)}\end{array}\right),\psi_{L}=\left(\begin{array}[]{c}\psi_{L}^{\left(1\right)}\\\
\psi_{L}^{\left(2\right)}\end{array}\right),$ (4.36)

and introducing the ratio

$\xi_{1}=\frac{\psi_{L}^{\left(1\right)}}{\psi_{R}^{\left(1\right)}},\xi_{2}=\frac{\psi_{L}^{\left(2\right)}}{\psi_{R}^{\left(2\right)}},$
(4.37)

the Green function can therefore been written as,

$G_{R}^{\left(1,2\right)}\left(k,\Lambda_{l}\right)=\mathcal{T}_{c}M_{KK}\lim_{Z\rightarrow\infty}Z^{\frac{4}{3}\Lambda_{l}}\xi_{1,2}\left(Z\right).$
(4.38)

satisfying the equation of motion,

$\xi_{1,2}^{\prime}=-\frac{4}{3}\Lambda_{l}K^{-1/2}\xi_{1,2}-\frac{K^{-2/3}}{M_{KK}}\left(-\omega+h\cdot\mathrm{k}\right)-\frac{K^{-2/3}}{M_{KK}}\left(-\omega-h\cdot\mathrm{k}\right)\xi_{1,2}^{2},$
(4.39)

according to (4.23). Here we have set
$k_{\mu}=\left(-\omega,\mathrm{k},0,0\right)$ with,

$\frac{\psi_{R}^{\left(2\right)}}{\psi_{R}^{\left(1\right)}}=\frac{\psi_{L}^{\left(2\right)}}{\psi_{L}^{\left(1\right)}}=h.$
(4.40)

On the other hand, since the dual theory is chirally symmetric, we obtain

$\psi_{R}|_{Z\rightarrow+\infty}=\psi_{L}|_{Z\rightarrow-\infty},$ (4.41)

which implies [36]

$\psi_{R}\left(0\right)=\pm\psi_{L}\left(0\right).$ (4.42)

Therefore we can set $h=\pm 1$ in (4.40) for $\xi_{1,2}$ respectively.
Altogether, (4.39) could be numerically solved with the incoming wave boundary
condition $\xi_{1,2}\left(0\right)=\pm 1$.

Figure 4:  Density plot of the confined retarded Green function
$G_{R}^{\left(1,2\right)}$ 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 white/black dashed lines refer to $\omega=\mathrm{k}$ as the
light cones. The parameters are chosen as
$\Lambda_{l}=2,l=0,\mathcal{T}_{c}=1$ in the unit of $M_{KK}=1$.

Figure 5:  The confined retarded Green function $G_{R}^{\left(1,2\right)}$ as
the spectral function of 1+3 dimensional QCD from the D4/D8 model. The
parameters are chosen as $\Lambda_{l}=2,\mathrm{k}=0,\mathcal{T}_{c}=1$ in the
unit of $M_{KK}=1$.

#### Deconfined phase

In the deconfined phase, the equation of motion for $\psi$ reads from (4.20)
as,

$\left(-\frac{4}{3}\pi
TK^{-1/2}\Lambda_{l}+Z^{-1}K^{-1/6}\bm{\gamma}^{0}\partial_{0}+K^{-2/3}\bm{\gamma}^{i}\partial_{i}+2\pi
T\bm{\gamma}\partial_{Z}+\pi TZK^{-1}\bm{\gamma}\right)\psi=0.$ (4.43)

Using the ansatz

$\psi=e^{ik\cdot x}\left(\begin{array}[]{c}\psi_{R}\\\
\psi_{L}\end{array}\right),\psi_{R,L}=\left(\begin{array}[]{c}\psi_{R,L}^{\left(1\right)}\\\
\psi_{R,L}^{\left(2\right)}\end{array}\right),$ (4.44)

with $k_{\mu}=\left(-\omega,\mathrm{k},0,0\right)$, (4.43) reduces to

$\displaystyle\partial_{Z}\psi_{R}^{\left(1,2\right)}+\left(\frac{ZK^{-1}}{2}-\frac{2}{3}K^{-1/2}\Lambda_{l}\right)\psi_{R}^{\left(1,2\right)}+\left(\frac{\omega}{2\pi
T}Z^{-1}K^{-1/6}+\frac{h\cdot\mathrm{k}}{2\pi
T}K^{-2/3}\right)\psi_{L}^{\left(1,2\right)}$ $\displaystyle=0$
$\displaystyle\partial_{Z}\psi_{L}^{\left(1,2\right)}+\left(\frac{ZK^{-1}}{2}+\frac{2}{3}K^{-1/2}\Lambda_{l}\right)\psi_{L}^{\left(1,2\right)}+\left(\frac{h\cdot\mathrm{k}}{2\pi
T}K^{-2/3}-\frac{\omega}{2\pi
T}Z^{-1}K^{-1/6}\right)\psi_{R}^{\left(1,2\right)}$ $\displaystyle=0,$ (4.45)

which can be solved analytically at $Z\rightarrow\infty$ as,

$\displaystyle\psi_{R}^{\left(1,2\right)}$
$\displaystyle=A\left(k,\Lambda_{l}\right)Z^{\frac{2}{3}\Lambda_{l}-\frac{1}{2}}+B\left(k,\Lambda_{l}\right)Z^{-\frac{2}{3}\Lambda_{l}-\frac{5}{6}},$
$\displaystyle\psi_{L}^{\left(1,2\right)}$
$\displaystyle=C\left(k,\Lambda_{l}\right)Z^{-\frac{2}{3}\Lambda_{l}-\frac{1}{2}}+D\left(k,\Lambda_{l}\right)Z^{\frac{2}{3}\Lambda_{l}-\frac{5}{6}}.$
(4.46)

The asymptotics (4.46) leads to that the onshell action (4.20) takes form at
$Z\rightarrow\infty$ as,

$\displaystyle S_{f,d}^{\mathrm{D8}}$ $\displaystyle\supseteq 2\pi T\int
d^{4}xdZi\bar{\psi}\bm{\gamma}\partial_{Z}\psi=-2\pi T\int
d^{4}x\psi_{L}^{\dagger}\psi_{R}+...$ $\displaystyle=-2\pi T\int
d^{4}xA\left(k,\Lambda_{l}\right)D\left(k,\Lambda_{l}\right)Z^{\frac{4}{3}\Lambda_{l}-\frac{4}{3}},$
(4.47)

which however does not include any finite part for $\Lambda_{l}\geq 2$.
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
$\psi_{0}$ of $\psi$ carefully as

$\psi_{0}=\lim_{Z\rightarrow+\epsilon^{-1}}\epsilon^{\frac{1}{6}}\psi\rightarrow\epsilon^{\frac{1}{6}}\left(\begin{array}[]{c}\psi_{R}\\\
0\end{array}\right)=\epsilon^{-\frac{2}{3}\Lambda_{l}+\frac{2}{3}}\left(\begin{array}[]{c}A\\\
0\end{array}\right),\ \epsilon\rightarrow 0$ (4.48)

Thus conjugate momentum for $\psi_{0}$ is

$\Pi_{0}=-\frac{\delta
S_{f,d}^{\mathrm{D8}}}{\delta\psi_{0}}=\mathcal{T}_{d}\left(2\pi
T\right)\left(0,D^{\dagger}\right)\epsilon^{-\frac{2}{3}\Lambda_{l}+\frac{2}{3}},$
(4.49)

which reduces to finite Green function as $G_{R}\sim D/A$. Then let us use the
ratio given in (4.37) so that the Green function can be written as,

$G_{R}^{\left(1,2\right)}\left(k,\Lambda_{l}\right)=\mathcal{T}_{d}\left(2\pi
T\right)\lim_{Z\rightarrow+\infty}Z^{1/3}\xi_{1,2}\left(Z\right).$ (4.50)

And the equation for $\xi_{1,2}$ can be obtained from (4.45) as,

$\xi_{1,2}^{\prime}=-\frac{4}{3}K^{-1/2}\Lambda_{l}\xi_{1,2}-\frac{h\cdot\mathrm{k}}{2\pi
T}K^{-2/3}+\frac{\omega}{2\pi T}Z^{-1}K^{-1/6}+\left(\frac{\omega}{2\pi
T}Z^{-1}K^{-1/6}+\frac{h\cdot\mathrm{k}}{2\pi T}K^{-2/3}\right)\xi_{1,2}^{2},$
(4.51)

which can be solved numerically by using the incoming wave boundary condition
$\xi_{1,2}\left(0\right)=\pm i$ on the horizon.

###  4.3 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
$\omega^{2}-k^{2}=M_{n}^{2}$ basically where $M_{n}$ represents various
constants. According to the general form of the fermionic propagator

$G_{R}\left(k\right)=\frac{1}{ik_{\mu}\bm{\gamma}^{\mu}-M_{n}},\
M_{n}=m_{n}-\Sigma\left(k\right),$ (4.52)

here $m_{n},\Sigma\left(k\right)$ 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 fundamental 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. Moreover, for
$\omega>0$, the location of the peak in the Green function can be read
approximately once we set $\mathrm{k}=0$, so that the peaks correspond to
$\omega=m_{n}-\Sigma\left(k\right)$ as it is illustrated in Figure 5. In this
sense, we find quantitatively, for $\Lambda_{l}=2,l=0$, the bound energy (the
position of the peak) is located at $\omega\simeq 1.6,2.7,3.7...$ in
$G_{R}^{\left(1\right)}$ and at $\omega\simeq 2.2,3.1,4.2...$ in
$G_{R}^{\left(2\right)}$. Furthermore, the location of the peaks also depends
on the value of $\Lambda_{l}$ as it is displayed in Figure 6.

Figure 6:  The imaginary part of confined retarded Green function
$G_{R}^{\left(1\right)}$ as the spectral function from the D4/D8 model with
various $\Lambda_{l}$ for $\mathrm{k}=0$.

Accordingly, if we identify $h=\mp 1$ 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
$\Lambda_{l}$ in this model as it is calculated in [36].

Next, let us take a look at the Green function obtained in the deconfined
phase which is illustrated in Figure 7.

Figure 7:  Density plot of the imaginary part of the Green function
$G_{R}^{\left(1,2\right)}$ from the black D4-brane background. The white
region refers to the peaks in the Green functions and the dashed lines refer
to $\omega=\pm\mathrm{k}$ as the light cone. The parameters are chosen as
$\Lambda_{l}=2,l=0,\mathcal{T}_{d}=1$ in the unit of $2\pi T=1$.

Since the relation of $G_{R}^{\left(1,2\right)}$ is given as
$G_{R}^{\left(2\right)}\left(\omega,\mathrm{k},\Lambda_{l}\right)=-G_{R}^{\left(1\right)}\left(\omega,-\mathrm{k},\Lambda_{l}\right)$,
let us focus on
$G_{R}^{\left(2\right)}\left(\omega,\mathrm{k},\Lambda_{l}\right)$ for
convenience. First, we can see the particle onshell is very different from the
confined case which satisfies $\omega\simeq\mathrm{k}$ for $\mathrm{k}\gg 1$
and $\omega\simeq\mathrm{k}^{2}+...$ for $\mathrm{k}\ll 1$. This behavior is
in qualitative agreement with the dispersion curves obtained by the hard
thermal loop (HTL) approximation in hot QCD, up to one-loop calculation, as
[44],

$\displaystyle\omega\left(\mathrm{k}\right)$ $\displaystyle\simeq
m_{f}+\frac{1}{3}\mathrm{k}+\frac{1}{3m_{f}}\mathrm{k}^{2},\ \mathrm{k}\ll 1;$
$\displaystyle\omega\left(\mathrm{k}\right)$ $\displaystyle\simeq\mathrm{k},\
\mathrm{k}\gg 1,$ (4.53)

where $m_{f}$ is the effective mass of fermion generated by the medium effect
given by

$m_{f}=\sqrt{\frac{C_{F}}{8}}g_{\mathrm{YM}}T.$ (4.54)

We note that chemical potential can also contribute to $m_{f}$ 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 7, the effective mass of quarks are evaluated to be
$m_{f}=\omega\left(\mathrm{k}=0\right)\simeq 2.9\times\left(2\pi T\right)$
which is expected to be the non-perturbative results predicted by holography.
However, we must keep in mind that the holographic approach is only valid in
strong coupling region while it covers the result from the perturbation method
in HTL. The only consistent interpretation is seemingly that the high order
contribution by HTL perturbation is suppressed in the large $N_{c}$ limit.
Second, since the peaks have a width, our numerical calculation may also
include the dispersion curves of plasmino which is expected to be
$\omega\left(\mathrm{k}\right)\simeq
m_{f}-\frac{1}{3}\mathrm{k}+\frac{1}{3m_{f}}\mathrm{k}^{2}$ for $\mathrm{k}\ll
1$. However, it is out of reach to distinguish exactly the dispersion curves
of fundamental quarks from plasmino due to the inevitable random error in our
numerical calculation while this may be indeed interesting. Overall, we
believe our current 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.

##  5 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.

###  5.1 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 as111111In the IIB SUGRA, the R-R 5-form is usually defined to be
self-dual which is replaced as $F_{5}\rightarrow\frac{1}{2}\left(F_{5}+\star
F_{5}\right)$.,

$\displaystyle ds_{c}^{2}$
$\displaystyle=\frac{r^{2}}{R^{2}}\left[\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}+f\left(r\right)\left(dx^{3}\right)^{2}\right]+\frac{R^{2}}{r^{2}}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{5}^{2}\right],$
$\displaystyle f\left(r\right)$
$\displaystyle=1-\frac{r_{KK}^{4}}{r^{4}},\\!F_{5}=dC_{4}=4R^{4}g_{s}^{-1}\epsilon_{5},\
\alpha,\beta=0,1,2,$ (5.1)

which corresponds to the confined phase of the dual theory and the black
D3-brane background is given as,

$\displaystyle ds_{d}^{2}$
$\displaystyle=\frac{r^{2}}{R^{2}}\left[-f_{T}\left(r\right)dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}+\left(dx^{3}\right)^{2}\right]+\frac{R^{2}}{r^{2}}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{5}^{2}\right],$
$\displaystyle f_{T}\left(r\right)$
$\displaystyle=1-\frac{r_{H}^{4}}{r^{4}},\\!F_{5}=dC_{4}=4R^{4}g_{s}^{-1}\epsilon_{5},$
(5.2)

which corresponds to the deconfined phase of the dual theory. Here we use
$\epsilon_{5}$ to refer to the volume form of a unit $S^{5}$ and $R^{4}=4\pi
g_{s}N_{c}l_{s}^{4}$. Note that the dilaton field vanishes for $p=3$ due to
(2.2) and since $x^{3}$ is conpactified on a circle, the dual field theory is
effectively the 1+2 dimensional Yang-Mills theory below the energy scale
$M_{KK}=2\pi\beta_{3}^{-1}$ where $\beta_{3}$ refers to the period of $x^{3}$
[14, 38].

Then let us introduce the probe D7-branes as flavor 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 $x^{3,9}=\mathrm{const}$,
hence we impose the coordinate transformation,

$r=\frac{r_{KK,H}}{\sqrt{2}}\left(\frac{1}{\zeta^{2}}+\zeta^{2}\right)^{1/2},\zeta^{2}=\rho^{2}+u^{2},u\equiv
x^{9},$ (5.3)

to the radial and $S^{5}$ part in (5.1) (5.2), then they can be rewritten as,

$\frac{R^{2}}{r^{2}}\left[\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{5}^{2}\right]=\frac{R^{2}}{\zeta^{2}}\left(d\rho^{2}+\rho^{2}d\Omega_{4}^{2}+du^{2}\right),$
(5.4)

where $\rho^{2}=\sum_{m=4}^{8}x^{m}x_{m}$ and $d\Omega_{4}^{2}$ refers to the
metric on $S^{4}$ whose solid angle is defined by parametrizing $x^{4,5...8}$.
We note that due to $r\geq r_{KK,H}$, it reduces to two branches for $\zeta$
equivalently as $0<\zeta\leq 1$ and $\zeta\geq 1$. In our setup, we chose the
branch of $\zeta\geq 1$. Therefore the induced metric on the flavor D7-branes
reads,

$\displaystyle ds_{\mathrm{D7},c}^{2}$
$\displaystyle=\frac{r^{2}}{R^{2}}\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}+\frac{R^{2}}{\zeta^{2}}\left(d\rho^{2}+\rho^{2}d\Omega_{4}^{2}\right),\
\alpha,\beta=0,1,2,$ $\displaystyle ds_{\mathrm{D7},d}^{2}$
$\displaystyle=\frac{r^{2}}{R^{2}}\left[-f_{T}\left(r\right)dt^{2}+\left(dx^{1}\right)^{2}+\left(dx^{2}\right)^{2}\right]+\frac{R^{2}}{\zeta^{2}}\left(d\rho^{2}+\rho^{2}d\Omega_{4}^{2}\right).$
(5.5)

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
$\left(3,7\right)$ 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,

$S_{f}^{\mathrm{D7}}=\frac{iT_{7}}{2}\int
d^{8}x\sqrt{-g}\bar{\Psi}P_{-}\left(\Gamma^{\alpha}\nabla_{\alpha}-\frac{1}{2\cdot
8\cdot 5!}F_{KLMNP}\Gamma^{\alpha}\Gamma^{KLMNP}\Gamma_{\alpha}\right)\Psi,$
(5.6)

where we have imposed $\bar{\gamma}\bar{\Psi}=\bar{\Psi}$ and $F_{KLMNP}$
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,

$\displaystyle S_{f,c}^{\mathrm{D7}}=$ $\displaystyle\frac{i}{2}T_{7}R^{2}\int
d^{3}xd\rho
d\Omega_{4}\bar{\Psi}P_{-}\bigg{(}\frac{\rho^{4}r^{2}R}{\zeta^{5}}\gamma^{\alpha}\partial_{\alpha}+\frac{3\rho^{5}r^{3}}{2\zeta^{6}}\sqrt{f}\gamma^{\rho}+\frac{\rho^{4}r^{3}}{\zeta^{4}R}\gamma^{\rho}\partial_{\rho}$
$\displaystyle+\frac{\rho^{3}r^{3}}{\zeta^{4}R}\gamma^{m}D_{m}+\frac{\rho^{4}r^{3}}{\zeta^{5}R}\bigg{)}\Psi,$
(5.7) $\displaystyle S_{f,d}^{\mathrm{D7}}=$
$\displaystyle\frac{i}{2}T_{7}R^{2}\int d^{3}xd\rho
d\Omega_{4}\bar{\Psi}P_{-}\bigg{[}\frac{\rho^{4}r^{2}R}{\zeta^{5}}\gamma^{0}\partial_{0}+\frac{\rho^{4}\sqrt{f_{T}}r^{3}}{2\zeta^{5}R}\left(\frac{\zeta
f^{\prime}}{2f}+\frac{3\rho\sqrt{f_{T}}}{\zeta}\right)\gamma^{4}$
$\displaystyle+\frac{\rho^{4}\sqrt{f_{T}}r^{2}R}{\zeta^{5}}\gamma^{\alpha}\partial_{\alpha}+\frac{\rho^{4}\sqrt{f_{T}}r^{3}}{\zeta^{4}R}\gamma^{\rho}\partial_{\rho}+\frac{\rho^{3}\sqrt{f_{T}}r^{3}}{\zeta^{4}R}\gamma^{m}D_{m}+\frac{\rho^{4}\sqrt{f_{T}}r^{3}}{\zeta^{5}R}\bigg{]}\Psi,$
(5.8)

where we have used $\gamma^{0123\rho}\Psi=\Psi$ and

$P_{-}=\frac{1}{2}\left(1-\Gamma_{\mathrm{D7}}\right),$ (5.9)

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 5d form as,

$\displaystyle S_{f,c}^{\mathrm{D7}}=$ $\displaystyle i\mathcal{T}\int
d^{3}xd\rho\bar{\psi}\left[\frac{1}{\zeta}-\frac{1}{\rho}\Lambda_{l}+\frac{R^{2}}{r\zeta}\bm{\gamma}^{\alpha}\partial_{\alpha}+\left(\frac{2\rho}{\zeta^{2}}-\frac{2}{\rho}\right)\bm{\gamma}+\bm{\gamma}\partial_{\rho}\right]\psi,$
(5.10)

by rescaling
$\psi\rightarrow\left(2\pi\alpha^{\prime}\right)\psi\frac{\zeta^{2}}{\rho^{2}}\frac{1}{r^{3/2}}$
and action (5.8) can be written as,

$\displaystyle S_{f,d}^{\mathrm{D7}}=$ $\displaystyle i\mathcal{T}\int
d^{3}xd\rho\bar{\psi}\bigg{[}\frac{1}{\sqrt{f_{T}}}\frac{R}{r\zeta}\bm{\gamma}^{0}\partial_{0}+\frac{R}{r\zeta}\bm{\gamma}^{\alpha}\partial_{\alpha}+\frac{1}{R}\bm{\gamma}\partial_{\rho}+\left(\frac{3\rho\sqrt{f_{T}}}{2R\zeta^{2}}+\frac{f_{T}^{\prime}}{4f_{T}R}\right)\bm{\gamma}$
$\displaystyle+\frac{1}{R}\left(\frac{1}{\zeta}-\frac{f_{T}^{\prime}}{4f_{T}}\right)-\frac{1}{R\rho}\left(\Lambda_{l}+2\right)+\frac{\rho}{R\zeta^{2}}\left(\frac{5}{2}-\frac{3}{2}\sqrt{f_{T}}\right)\bigg{]}\psi,$
(5.11)

by rescaling
$\psi\rightarrow\left(2\pi\alpha^{\prime}\right)\psi\frac{\zeta^{2}}{\rho^{2}r^{3/2}f_{T}^{1/4}}$
where

$\mathcal{T}=\frac{1}{2}T_{7}R^{2}\Omega_{4}\left(2\pi\alpha^{\prime}\right)^{2}.$
(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.

###  5.2 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 $\psi$ can
be obtained by varying (5.10) as,

$\left[\frac{1}{\xi}-\frac{1}{\rho}\Lambda_{l}+\frac{R^{2}}{r\xi}\bm{\gamma}^{\alpha}\partial_{\alpha}+\left(\frac{2\rho}{\xi^{2}}-\frac{2}{\rho}\right)\bm{\gamma}+\bm{\gamma}\partial_{\rho}\right]\psi=0.$
(5.13)

Using the ansatz given in (4.22), we can obtain the coupled equations as,

$\displaystyle\left(\frac{1}{\xi}-\frac{1}{\rho}\Lambda_{l}+\frac{2\rho}{\xi^{2}}-\frac{2}{\rho}+\partial_{\rho}\right)\psi_{R}+i\frac{R^{2}}{r\xi}\sigma^{\alpha}\partial_{\alpha}\psi_{L}$
$\displaystyle=0,$ $\displaystyle
i\frac{R^{2}}{r\xi}\bar{\sigma}^{\alpha}\partial_{\alpha}\psi_{R}+\left(\frac{1}{\xi}-\frac{1}{\rho}\Lambda_{l}-\frac{2\rho}{\xi^{2}}+\frac{2}{\rho}-\partial_{\rho}\right)\psi_{L}$
$\displaystyle=0.$ (5.14)

Setting $k_{\mu}=\left(-\omega,\mathrm{k},0,0\right)$, the equations presented
in (5.14) can be written as decoupled second order differential equations,
which, at boundary $r\rightarrow\infty$ ($\rho\rightarrow\infty$), take the
asymptotic form as,

$\displaystyle\psi_{R}^{\prime\prime}+\frac{2}{\rho}\psi_{R}^{\prime}-\frac{\left(\Lambda_{l}^{2}+3\Lambda_{l}+2\right)}{\rho^{2}}\psi_{R}$
$\displaystyle=0,$
$\displaystyle\psi_{L}^{\prime\prime}+\frac{2}{\rho}\psi_{L}^{\prime}-\frac{\left(\Lambda_{l}^{2}+\Lambda_{l}\right)}{\rho^{2}}\psi_{L}$
$\displaystyle=0.$ (5.15)

And they can be solved exactly as,

$\displaystyle\psi_{R}$
$\displaystyle=A\left(k,\Lambda_{l}\right)\rho^{1+\Lambda_{l}}+B\left(k,\Lambda_{l}\right)\rho^{-2-\Lambda_{l}},$
$\displaystyle\psi_{L}$
$\displaystyle=C\left(k,\Lambda_{l}\right)\rho^{\Lambda_{l}}+D\left(k,\Lambda_{l}\right)\rho^{-1-\Lambda_{l}},$
(5.16)

where $A,B,C,D$ are integration constant. As we have outlined $\Lambda_{l}\geq
2$, so the boundary value $\psi_{0}$ of $\psi$ should be defined as,

$\psi_{0}=\lim_{\rho\rightarrow\infty}\rho^{-1-\Lambda_{l}}\psi=\rho^{-1-\Lambda_{l}}\left(\begin{array}[]{c}\psi_{R}\\\
0\end{array}\right)=\left(\begin{array}[]{c}A\\\ 0\end{array}\right).$ (5.17)

Then the onshell action (5.10) will include a finite part as,

$\displaystyle S_{f,c}^{\mathrm{D7}}$ $\displaystyle\supseteq-\mathcal{T}\int
d^{3}x\left[\left(\psi_{L}^{\dagger}\psi_{R}\right)|_{\rho\rightarrow\infty}+h.c+...\right]$
$\displaystyle=-\mathcal{T}\int
d^{3}x\left[C\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)\rho^{2\Lambda_{l}+1}+D\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)+h.c\right]|_{\rho\rightarrow\infty},$
(5.18)

thus it request for a holographic counterterm as,

$S_{ct}=\mathcal{T}\int
d^{3}x\left[C\left(k,\Lambda_{l}\right)^{\dagger}A\left(k,\Lambda_{l}\right)\rho^{2\Lambda_{l}+1}+h.c\right]|_{\rho\rightarrow\infty}.$
(5.19)

The existing finite part in (5.18) is due to the residual isometry of
$\mathrm{AdS}_{5}$ since the D3-brane solution (5.1) (5.2) as bulk geometry is
$\mathrm{AdS}_{5}\times S^{5}$ asymptotically. Follow the discussion in
Section 3, 4, the retarded Green function is given by,

$\mathcal{T}D\left(k,\Lambda_{l}\right)=G_{R}\left(k,\Lambda_{l}\right)A\left(k,\Lambda_{l}\right).$
(5.20)

Using the representation (4.36) and introducing the ratio (4.37), the Green
function can be written as,

$G_{R}^{\left(1,2\right)}=\lim_{\rho\rightarrow\infty}\rho^{2\Lambda_{l}+2}\xi_{1,2},$
(5.21)

where the equation for $\xi_{1,2}$ can be obtained from (5.14) as,

$\xi_{1,2}^{\prime}=\frac{R^{2}}{r\zeta}\left(\omega-h\cdot\mathrm{k}\right)+2\left(\frac{1}{\zeta}-\frac{1}{\rho}\Lambda_{l}\right)\xi_{1}+\frac{R^{2}}{r\zeta}\left(\omega+h\cdot\mathrm{k}\right)\xi_{1,2}^{2},$
(5.22)

which will be evaluated numerically with the incoming wave boundary condition
and $h=\pm 1$ for $\xi_{1,2}$ respectively.

Figure 8:  Density plot of the confined retarded Green function
$G_{R}^{\left(1,2\right)}$ as the spectral function of the 1+2 dimensional QCD
from the D3/D7 model. The parameters are chosen as
$M_{KK}=1,\Lambda_{l}=2,L=0$. The white regions refer to the peaks in the
Green function and dashed lines refer to the light cones.

Figure 9:  The confined retarded Green function $G_{R}^{\left(1,2\right)}$ as
the spectral function of the 1+2 dimensional QCD from the D3/D7 approach. The
parameters are set as $M_{KK}=1,\Lambda_{l}=2,L=0$.

#### Deconfined phase

For the deconfined phase, the equation of motion for the worldvolume $\psi$ is
given by varying action (5.11) which is,

$\displaystyle\bigg{[}\frac{1}{\sqrt{f_{T}}}\frac{R}{r\zeta}\bm{\gamma}^{0}\partial_{0}+\frac{R}{r\zeta}\bm{\gamma}^{\alpha}\partial_{\alpha}+\frac{1}{R}\bm{\gamma}\partial_{\rho}+\left(\frac{3\rho\sqrt{f_{T}}}{2R\zeta^{2}}+\frac{f_{T}^{\prime}}{4fR}\right)\bm{\gamma}$
$\displaystyle+\frac{1}{R}\left(\frac{1}{\zeta}-\frac{f_{T}^{\prime}}{4f_{T}}\right)-\frac{1}{R\rho}\left(\Lambda_{l}+2\right)+\frac{\rho}{R\zeta^{2}}\left(\frac{5}{2}-\frac{3}{2}\sqrt{f_{T}}\right)\bigg{]}\psi$
$\displaystyle=0.$ (5.23)

Using the ansatz (4.22), the coupled equations for $\psi_{L,R}$ are,

$\displaystyle\bigg{[}\frac{1}{R}\partial_{\rho}+\frac{3\rho\sqrt{f_{T}}}{2R\zeta^{2}}+\frac{f_{T}^{\prime}}{4f_{T}R}+\frac{1}{R}\left(\frac{1}{\zeta}-\frac{f_{T}^{\prime}}{4f_{T}}\right)-\frac{1}{R\rho}\left(\Lambda_{l}+2\right)$
$\displaystyle+\frac{\rho}{R\zeta^{2}}\left(\frac{5}{2}-\frac{3}{2}\sqrt{f_{T}}\right)\bigg{]}\psi_{R}+i\frac{R}{r\zeta}\left(\frac{1}{\sqrt{f_{T}}}\partial_{0}+\sigma^{\alpha}\partial_{\alpha}\right)\psi_{L}$
$\displaystyle=0,$
$\displaystyle\bigg{[}-\frac{1}{R}\partial_{\rho}-\frac{3\rho\sqrt{f_{T}}}{2R\zeta^{2}}-\frac{f_{T}^{\prime}}{4f_{T}R}+\frac{1}{R}\left(\frac{1}{\zeta}-\frac{f_{T}^{\prime}}{4f_{T}}\right)-\frac{1}{R\rho}\left(\Lambda_{l}+2\right)$
$\displaystyle+\frac{\rho}{R\zeta^{2}}\left(\frac{5}{2}-\frac{3}{2}\sqrt{f_{T}}\right)\bigg{]}\psi_{L}+i\frac{R}{r\zeta}\left(\frac{1}{\sqrt{f_{T}}}\partial_{0}+\bar{\sigma}^{\alpha}\partial_{\alpha}\right)\psi_{R}$
$\displaystyle=0,$ (5.24)

which can reduce to two second-order differential equations. At boundary
$r\rightarrow\infty$ ($\rho\rightarrow\infty$), they are,

$\displaystyle\psi_{R}^{\prime\prime}+\frac{5}{\rho}\psi_{R}^{\prime}-\frac{\left(5+2\Lambda_{l}\right)\left(2\Lambda_{l}-3\right)}{4\rho^{2}}$
$\displaystyle=0,$
$\displaystyle\psi_{R}^{\prime\prime}+\frac{5}{\rho}\psi_{R}^{\prime}-\frac{\left(2\Lambda_{l}-5\right)\left(2\Lambda_{l}+3\right)}{4\rho^{2}}$
$\displaystyle=0,$ (5.25)

which can be solved as,

$\displaystyle\psi_{R}$
$\displaystyle=\rho^{\Lambda_{l}-\frac{3}{2}}A\left(k,\Lambda_{l}\right)+\rho^{-\Lambda_{l}-\frac{3}{2}}B\left(k,\Lambda_{l}\right),$
$\displaystyle\psi_{L}$
$\displaystyle=\rho^{-\Lambda_{l}-\frac{3}{2}}C\left(k,\Lambda_{l}\right)+\rho^{-\frac{5}{2}+\Lambda_{l}}D\left(k,\Lambda_{l}\right).$
(5.26)

While the boundary value of $\psi$ remains to be $\psi_{R}$, the onshell
action (5.11) taking the form as,

$\displaystyle S_{f,c}^{\mathrm{D7}}$ $\displaystyle\supseteq-\mathcal{T}\int
d^{3}x\left[\left(\psi_{L}^{\dagger}\psi_{R}\right)|_{\rho\rightarrow\infty}+h.c+...\right]$
$\displaystyle=-\mathcal{T}\int
d^{3}x\left[\rho^{2\Lambda_{l}-4}D^{\dagger}\left(k,\Lambda_{l}\right)A\left(k,\Lambda_{l}\right)+h.c\right]|_{\rho\rightarrow\infty}.$
(5.27)

does not include any finite part which implies the deconfined phase of 3d QCD
is also non-renormalizable 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 $\psi_{0}$ of $\psi$ carefully
as,

$\psi_{0}=\lim_{\rho\rightarrow\epsilon^{-1}}\epsilon^{\frac{1}{2}}\psi=\epsilon^{-\Lambda_{l}+2}\left(\begin{array}[]{c}A\\\
0\end{array}\right),\epsilon\rightarrow 0.$ (5.28)

Then the conjugate momentum $\Pi_{0}$ for $\psi_{0}$ is given as,

$\Pi_{0}=-\frac{\delta
S_{f,d}^{\mathrm{D8}}}{\delta\psi_{0}}=\mathcal{T}\left(0,D^{\dagger}\right)\epsilon^{-\Lambda_{l}+2}.$
(5.29)

Accordingly, it leads to a finite Green function as,

$\bar{\Pi}_{0}=G_{R}\left(k,\Lambda_{l}\right)\psi_{0},\
G_{R}^{\left(1,2\right)}=\lim_{\epsilon\rightarrow 0}\epsilon^{-1}\xi_{1,2},$
(5.30)

where the equation for $\xi_{1,2}$ defined in (4.37) can be obtained from
(5.24), as

$\displaystyle\xi_{1,2}^{\prime}=$
$\displaystyle\frac{R^{2}}{r\zeta}\left(\frac{\omega}{\sqrt{f}}-h\cdot\mathrm{k}\right)\xi_{2}+\frac{R^{2}}{r\zeta}\left(\frac{\omega}{\sqrt{f}}+h\cdot\mathrm{k}\right)\xi_{1,2}^{2}$
$\displaystyle+2\left[\left(\frac{1}{\zeta}-\frac{f^{\prime}}{4f}\right)-\frac{1}{\rho}\left(\Lambda_{l}+2\right)+\frac{\rho}{\zeta^{2}}\left(\frac{5}{2}-\frac{3}{2}\sqrt{f}\right)\right]\xi_{1,2}.$
(5.31)

And we will evaluate the Green function with the incoming wave boundary
condition on the horizon and $h=\pm 1$ for $\xi_{1,2}$ respectively.

Figure 10:  Imaginary part of the confined retarded Green function
$G_{R}^{\left(1,2\right)}$ with $L=0$ (solid blue line), $L=1$ (Dashed red
line).

###  5.3 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 8 which is basically similar to the
approach in the D4/D8 model. The peaks in the Green function display the
discreteness representing the onshell bound state as confinement in 3d QCD, as
$\omega\simeq 1.6,2.6,4.1...$ in $G_{R}^{\left(1\right)}$ and $\omega\simeq
1.9,3.4,4.8...$ in $G_{R}^{\left(2\right)}$ 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 $\omega$ which is illustrated in Figure 9. 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 10. 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 $\left(4,8\right)$
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 $\Sigma\left(k\right)$ in (4.52). In the deconfined phase, we
plot out the imaginary part of the Green function as the spectral function in
Figure 11 in which the onshell condition 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
$m_{f}\simeq 4.8\left(2\pi T\right)$ by the D3/D7 model which also implies the
high order contribution in HTL approximation might be suppressed in the large
$N_{c}$ limit.

Figure 11:  Density plot of the imaginary part of the Green function
$G_{R}^{\left(1,2\right)}$ from the black D3-brane background. The white
region refers to the peaks in the Green functions and the dashed lines refer
to $\omega=\pm\mathrm{k}$ as the light cone. The parameters are chosen as
$\Lambda_{l}=2,l=0,\mathcal{T}=1$ in the unit of $2\pi T=1$.

Altogether, the D3/D7 approach also reveals mostly the fundamental properties
of QCD as it is expected.

##  6 Summary and discussion

In this work, we study the correlation function of the fundamental fermion in
the $\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model ($p=4,3$) as the holographic top-
down approach to QCD. Since our concern is the fermionic correlator, the bulk
spinor is identified to the worldvolume fermion created by the
$\left(p+4,p+4\right)$ string on the $\mathrm{D}_{p+4}$-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 fundamental 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 D-brane 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. Hence 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, when the chemical potential (relating to the gauge field potential on
the worldvolume of the $\mathrm{D}_{p+4}$-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 $U\left(N\right)$ group in most bottom-up
approaches while it is not in our top-down approach. Therefore, 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.

Finally, in the confined phase, the interpretation of the bound states
presented in the Green function also need to be considered. According to
AdS/CFT, the bulk and dual theory should share the same Hilbert space, hence
both of them would include such fermionic bound states. In the side of bulk
theory, these bound states should correspond to the supersymmetrically
fermionic meson [36] in holography since they are created by the
supersymmetric fermion on the $\mathrm{D}_{p+4}$-brane. However the fermionic
meson is absent in QCD as the dual theory. Nonetheless, it may be possible to
interpret these fermionic bound states as baryon states if we take into
account a baryon vertex in the bulk. For example, consider a baryon vertex,
which is identified as a $\mathrm{D}_{8-p}$-brane wrapped on $S^{8-p}$ [45,
46, 47], located at $r=r_{KK}$ in the bubble background and the
$\left(p,p+4\right)$ string is totally inside the $\mathrm{D}_{p+4}$-brane
terminating on the baryon vertex. In this configuration, the
$\left(p,p+4\right)$ string is also a $\left(p+4,p+4\right)$ string connecting
to a baryon vertex, therefore the associated fermionic bound states are
equivalently created by the baryon vertex which can be identified as baryon
states in the dual theory. This viewpoint may support that the open strings on
the D-brane behaves somehow as baryons in the
$\mathrm{D}_{p}/\mathrm{D}_{p+4}$ model as it is discussed in [37].

## Acknowledgements

We would like to thank Konstantinos Rigatos, Xin-li Sheng and Yan-qing Zhao
for helpful discussion. This work is supported by the National Natural Science
Foundation of China (NSFC) under Grant No. 12005033 and the Fundamental
Research Funds for the Central Universities under Grant No. 3132023198.

## References

  * [1] E. Witten, “Anti-de Sitter space and holography”, Adv.Theor.Math.Phys. 2 (1998) 253-291, arXiv:hep-th/9802150.
  * [2] J. Maldacena, “The Large N limit of superconformal field theories and supergravity”, Adv.Theor.Math.Phys. 2 (1998) 231-252, arXiv: hep-th/9711200.
  * [3] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, “Large N field theories, string theory and gravity”, Phys.Rept. 323 (2000) 183-386, arXiv:hep-th/9905111.
  * [4] S. Gubser, I. Klebanov, A. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys.Lett.B 428 (1998) 105-114, arXiv:hep-th/9802109.
  * [5] D. Son, A. Starinets, “Minkowski-space correlators in AdS/CFT correspondence: recipe and applications”, JHEP 09 (2002) 042, arXiv:hep-th/0205051.
  * [6] C. Herzog, D. Son, “Schwinger-Keldysh propagators from AdS/CFT correspondence,” JHEP 03 (2003) 046, arXiv:hep-th/0212072.
  * [7] D. Marolf, “States and boundary terms: Subtleties of Lorentzian AdS / CFT”, JHEP 05 (2005) 042, arXiv: hep-th/0412032.
  * [8] S. Gubser, S. Pufu, F. Rocha, “Bulk viscosity of strongly coupled plasmas with holographic duals”, JHEP 05 (2005) 042, arXiv: hep-th/0412032.
  * [9] Balt C. van Rees, “Real-time gauge/gravity duality and ingoing boundary conditions”, Nucl.Phys.B Proc.Suppl. 192-193 (2009) 193-196, arXiv: 0902.4010.
  * [10] N. Iqbal, H. Liu, “Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm”, Phys.Rev.D 79 (2009) 025023, arXiv: 0809.3808.
  * [11] S. Lee, “A Non-Fermi Liquid from a Charged Black Hole: A Critical Fermi Ball”, Phys.Rev.D 79 (2009) 086006, arXiv: 0809.3402.
  * [12] H. Liu, J. McGreevy, D. Vegh, “Non-Fermi liquids from holography”, Phys.Rev.D 83 (2011) 065029, arXiv: 0903.2477.
  * [13] N. Iqbal, H. Liu, “Real-time response in AdS/CFT with application to spinors”, Fortsch.Phys. 57 (2009) 367-384, arXiv: 0903.2596.
  * [14] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, U. Wiedemann, “Gauge/String Duality, Hot QCD and Heavy Ion Collisions”, Cambridge University Press, 2014, arXiv: 1101.0618.
  * [15] D. J. Gross, F. Wilczek, “Ultraviolet Behavior of Nonabelian Gauge Theories”, Phys.Rev.Lett. 30 (1973) 1343-1346.
  * [16] H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?”, Phys.Rev.Lett. 30 (1973) 1346-1349.
  * [17] J. Erlich, E. Katz, D. T. Son, M. A. Stephanov, “QCD and a holographic model of hadrons”, Phys.Rev.Lett. 95 (2005) 261602, arXiv: hep-ph/0501128.
  * [18] A. Karch, E. Katz, D. T. Son, M. A. Stephanov, “Linear Confinement and AdS/QCD”, arXiv:hep-ph/0602229.
  * [19] L. Rold, A. Pomarol, “Chiral symmetry breaking from five dimensional spaces”, Nucl.Phys.B 721 (2005) 79-97, arXiv: hep-ph/0501218.
  * [20] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories”, Adv. Theor. Math. Phys. 2 (1998), 505-532, arXiv:hep-th/9803131.
  * [21] T. Sakai, S. Sugimoto, “Low energy hadron physics in holographic QCD”, Prog.Theor.Phys. 113 (2005) 843-882, arXiv:hep-th/0412141.
  * [22] T. Sakai, S. Sugimoto, “More on a holographic dual of QCD”, Prog.Theor.Phys. 114 (2005) 1083-1118, arXiv: hep-th/0507073.
  * [23] S. Li, X. Zhang, “The D4/D8 model and holographic QCD”, Symmetry 2023, 15(6), 1213, arXiv: 2304.10826.
  * [24] S. Li, A. Schmitt, Q. Wang, “From holography towards real-world nuclear matter”, Phys.Rev.D 92 (2015) 2, 026006, arXiv: 1505.04886.
  * [25] S. Li, S. Luo, Y. Hu, “Holographic QCD3 and Chern-Simons theory from anisotropic supergravity”, JHEP 08 (2022) 206, arXiv: 2205.01885.
  * [26] M. Le Bellac, “Thermal Field Theory” , Cambridge University Press, 2000.
  * [27] J. Kapusta, C. Gale, ”Finite-Temperature Field Theory: Principles and Applications”, Cambridge University Press, 2006.
  * [28] M. Harada, Y. Nemotoo, “Quasi-fermion spectrum at finite temperature from coupled Schwinger-Dyson equations for a fermion-boson system”, Phys.Rev.D 78 (2008) 014004, arXiv:0803.3257.
  * [29] H. Nakkagawa, H. Yokota, K. Yoshida, “Vanishing Thermal Mass in the Strongly Coupled QCD/QED medium”, Phys.Rev.D 85 (2012) 031902, arXiv:1111.0117.
  * [30] Y. Seo, S. Sin, Y. Zhou, “Thermal Mass and Plasmino for Strongly Interacting Fermions”, JHEP 06 (2013) 076, arXiv: 1305.1446.
  * [31] Y. Seo, S. Sin, Y. Zhou, “Self-energy of Strongly Interacting Fermions in Medium: a Holographic Approach”, Phys.Lett.B 723 (2013) 207-212, arXiv: 1205.3377.
  * [32] L. Fang, X. Ge, X. Kuang, “Holographic fermions in charged Lifshitz theory”, Phys.Rev.D 86 (2012) 105037, arXiv: 1201.3832.
  * [33] L. Fang, X. Ge, J. Wu, H. Leng, “Anisotropic Fermi surface from holography”, Phys.Rev.D 91 (2015) 12, 126009, arXiv: 1409.6062.
  * [34] D. Marolf, L. Martucci, P. Silva, “Fermions, T duality and effective actions for D-branes in bosonic backgrounds”, JHEP 04 (2003) 051, arXiv: hep-th/0303209.
  * [35] D. Marolf, L. Martucci, P. Silva, “Actions and Fermionic symmetries for D-branes in bosonic backgrounds”, JHEP 07 (2003) 019, arXiv: hep-th/0306066.
  * [36] R. Heise, H. Svendsen, “A note on fermionic mesons in holographic QCD”, JHEP 08 (2007) 065, arXiv: 0706.2253.
  * [37] T. Nakas, K. Rigatos, “Fermions and baryons as open-string states from brane junctions”, JHEP 12 (2020) 157, arXiv: 2010.00025.
  * [38] K. Becker, M. Becker, J.H. Schwarz, “String theory and M-theory, A Modern Introduction”, Cambridge University Press 2007\.
  * [39] C. Johnson, “D-Branes (Cambridge Monographs on Mathematical Physics)”, Cambridge University Press, 2006.
  * [40] G. Mandal, T. Morita, “Gregory-Laflamme as the confinement/deconfinement transition in holographic QCD”, JHEP 09 (2011) 073, arXiv: 1107.4048.
  * [41] G. Mandal, T. Morita, “What is the gravity dual of the confinement/deconfinement transition in holographic QCD?”, J.Phys.Conf.Ser. 343 (2012) 012079, arXiv: 1111.5190.
  * [42] E. Antonyan, J.A. Harvey, S. Jensen, D. Kutasov, “NJL and QCD from String Theory”, arXiv:hep-th/0604017.
  * [43] R. Camporesi, A. Higuchi, “On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces”, J.Geom.Phys. 20 (1996) 1-18, arXiv: gr-qc/9505009.
  * [44] V. Klimov, “Spectrum of Elementary Fermi Excitations in Quark Gluon Plasma”, Sov.J.Nucl.Phys. 33 (1981) 934-935, Yad.Fiz. 33 (1981) 1734-1736.
  * [45] E. Witten, “Baryons and branes in anti-de Sitter space”, JHEP 07 (1998) 006, arXiv: hep-th/9805112.
  * [46] S. Li, T. Jia, “Matrix model and Holographic Baryons in the D0-D4 background”, Phys.Rev.D 92 (2015) 4, 046007, arXiv: 1506.00068.
  * [47] K. Hashimoto, N. Iizuka, P. Yi, “A Matrix Model for Baryons and Nuclear Forces”, JHEP 10 (2010) 003, arXiv: 1003.4988.