# The worldvolume fermion as baryon in holographic QCD with a theta angle

###### Abstract

In this work, we investigate the worldvolume fermion on the flavor brane in
the D0-D4/D8 model which is holographically equivalent to the four-dimensional
QCD with a theta term. The action for the worldvolume fermion is obtained by
the T-duality rules in the string theory and its effectively five-dimensional,
canonical four-dimensional forms are derived by the systematical dimensional
reduction and decomposition of spinor. Afterwards, we employ the AdS/CFT
dictionary to evaluate the two-point correlation function as the spectral
function for the worldvolume fermion and interpret the fermion as baryon by
analyzing its quantum number with the baryon vertex in holography. In this
sense, we finally derive the formulas of the interacted action involving the
worldvolume fermion and gauge field on the flavor brane, in order to describe
the various interaction of meson and baryon with the theta angle through
gauge-gravity duality.

Si-wen Li111Email<EMAIL_ADDRESS>Hao-qian Li222Email:
<EMAIL_ADDRESS>Yi-peng Zhang333Email<EMAIL_ADDRESS>
_Department of Physics, School of Science,_

_Dalian Maritime University,_

_Dalian 116026, China_

###### Contents

  1. 1 Introduction
  2. 2 The D0-D4/D8 model as theta-dependent QCD in holography
    1. 2.1 The color sector
    2. 2.2 The flavor sector
    3. 2.3 The bosonic meson tower
  3. 3 The flavored fermionic spectroscopy on the worldvolume of the D8-branes
    1. 3.1 The fermionic action and dimensional reduction
    2. 3.2 The canonical four-dimensional action
    3. 3.3 The holographic Green function as spectral function
    4. 3.4 The numerical analysis
  4. 4 The worldvolume fermion as baryon
    1. 4.1 The holographic interpretation
    2. 4.2 The interaction of meson and baryon
  5. 5 Summary and discussion

##  1 Introduction

The spontaneous parity violation in QCD have been studied for a significant
amount of time with the running of the relativistic heavy-ion collision (RHIC)
[1, 2]. In theory, the effects related to the spontaneous parity violation can
usually be described by the QCD or Yang-Mills action with an non-vanished
theta term as,

$S=-\frac{1}{4g_{\mathrm{YM}}^{2}}\int
d^{4}x\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}+\frac{\theta}{8\pi^{2}}\mathrm{Tr}\int
F\wedge F,$ (1.1)

where $g_{\mathrm{YM}}$ is the Yang-Mills coupling constant and $\theta$
refers to the concerned theta angle. Although the exactly experimental value
of the theta angle may be very small ($\left|\theta\right|\leq 10^{-10}$), it
attracts great interests in the theoretical and phenomenological
investigations in Yang-Mills theory or QCD, e.g. the deconfinement phase
transition [3, 4], the glueball spectrum [5], the large N behavior [6] with
the theta angle. And the summary of the theta term in Yang- Mills theory or
QCD can be detailedly reviewed in the excellent literature [7]. Note that the
chiral magnet effect (CME) in heavy-ion collisions has also become one of the
important focus to confirm the theta dependence in QCD in recent years [8, 9,
10, 11, 12].

However, since the asymptotic freedom is one of the characteristic features of
QCD, it implies QCD is strongly coupled and non-analytical in the low-energy
region. That means the standard analytical technique in quantum field theory
(QFT) based on the perturbation method is powerless to analyze the QCD
matters, e.g. meson and baryon, in the low-energy region. Fortunately, the
framework of AdS/CFT and gauge-gravity duality based on string theory [13, 14]
could offer an alternatively analytical approach to investigate the aspects of
the strongly coupled gauge theory. Significantly in 2004, Sakai and Sugimoto
proposed a concrete model [15] (i.e. the D4/D8 model or named as Witten-Sakai-
Sugimoto model) by using the construction of the D4-brane in Witten’s [16]
which successfully includes almost all the elementary ingredients of QCD e.g.
quark, gluon, meson [17, 18, 19], baryon [20, 21, 22, 23, 24, 25], glueball
[26, 27, 28, 29, 30, 31], chiral/deconfinement transitions [32, 33, 34].
Moreover, in order to include the theta term presented in (1.1) in the dual
theory of the D4/D8 model, the authors of [35] suggest to introduce $N_{0}$
smeared D0-branes into the background geometry produced by $N_{c}$ D4-branes
in the D4/D8 model. By keeping the ratio of $N_{0}/N_{c}$ fixed in the large
$N_{c}$ limit, the background geometry determined by D0- and D4- branes
together can be obtained by solving the IIA supergravity in which the number
density of D0-branes (as we will see in the following sections) corresponds to
the theta angle in QCD [36, 37, 38]. Therefore it is possible to use this
D0-D4/D8 system to study systematically the properties of QCD with a theta
term in holograph e.g. [38, 39, 40, 41, 42].

While the above framework of gauge-gravity duality achieves many successes,
there may be an issue in the approach of D4/D8 or D0-D4/D8 system by imposing
the project of the compactification in [16]. That is the D8-branes (as the
flavor branes) remain to be supersymmetric in principle in the low-energy
theory, since the project of the compactification as the mechanism to break
down the supersymmetry in the model works only for the D4-branes instead of
for the D8-branes [43]. So the remaining supersymmetry on the flavor branes
leads to the existence of the superpartner of the bosonic meson (named as
mesino) in the dual theory which however is always absent in QCD and hadron
physics. And there is no reason to neglect these worldvolume fermions as
mesinos on the flavor branes without the mechanism to further break down the
supersymmetry.

Motivated by this issue, in this work we study systematically the worldvolume
fermion on the flavor branes in the D0-D4/D8 model by analyzing its action,
dimensional reduction, spectrum strictly through the string theory and gauge-
gravity duality, in order to explore the possibly valid interpretation of
these fermions in terms of hadron physics. Our numerical evaluation
illustrates if the worldvolume fermion is identified to the superpartner of
the bosonic mesons, they are too heavy to arise in the low-energy theory while
they can not be neglected in principle. Thus it means below the compactified
energy scale, the meson sector of the dual theory is purely bosonic without
mesino. Moreover, when the baryon vertex described as [20, 21] is introduced
in this model, we analyze the associated quantum numbers of the worldvolume
fermion and its dual operator by gauge-gravity duality, then find it may be
possible to interpret this worldvolume fermion as baryon in holography which
leads to a nicely natural description of the fermionic baryon in term of
hadron physics. Accordingly, we finally derive the interacted action of meson
and baryon in holography by using the dimensional reduction for the coupling
terms in the worldvolume action involving the fermions. Since the previous
works e.g. [39, 40, 44, 45, 46] do not reveal exactly that baryon in this
model is fermion, this work may fulfill this blank. Besides, our numerical
evaluation also displays the metastable states of baryon in the presence of
the theta angle and this conclusion is in agreement with the existing works
[39, 40, 44, 45, 46] describing the metastabilization in the theta-dependent
QCD [1, 2].

The outline of this work is given as follows. In Section2, we review the
D0-D4/D8 model as 4d QCD with a theta angle in holography. In Section 3, we
derive the 5d effective action and the associated 4d canonical form for the
worldvolume fermion, then evaluate numerically the fermionic spectrum by
analyzing the holographic correlation function. In Section 5, we specify how
to interpret the worldvolume fermion as baryon with the baryon vertex, then
derive the interacted action for the various interaction of meson and baryon
in holography. Summary and discussion are given in the last section as Section
6.

##  2 The D0-D4/D8 model as theta-dependent QCD in holography

###  2.1 The color sector

In this section, let us briefly review the D0-D4/D8 model as holographic QCD
with a theta term, the details can be found in [35, 36, 37, 38]. In this
model, the gravity background is produced by $N_{c}$ coincident D4-branes as
colors with $N_{0}$ smeared D0-branes as D-instantons. In the large $N_{c}$
limit, the dynamics of the gravity background is described by the IIA
supergravity whose bosonic action is given as,

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

where $\mathcal{R},\phi,G$ refers respectively to the ten-dimensional (10d)
scalar curvature, dilaton, determinant of the metric and
$2\kappa_{10}^{2}=16\pi G_{10}=\left(2\pi\right)^{7}l_{s}^{8}g_{s}^{2}$ is the
10d gravity coupling constant. $F_{2,4}=dC_{1,3}$ denotes the field strength
of the Ramond-Ramond 1- and 3-form $C_{1,3}$. We note that, in order to take
into account the back reaction of the D0-branes, in the large $N_{c}$ limit,
we keep $N_{0}/N_{c}\ll 1$ but is finite since, as we will see, $N_{0}$
relates to the theta term of QCD in this model. In this sense, the equations
of motion obtained by action (2.1) can be solved by a D4 bubble solution with
$N_{0}$ smeared D0-branes [35, 36, 37, 38] as it is in the D4/D8 model [15].
Taking the near-horizon limit, in string frame, the supergravity solution is
given as [38],

$\displaystyle ds^{2}=$
$\displaystyle\left(\frac{U}{R}\right)^{3/2}\left[H_{0}^{1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H_{0}^{-1/2}f\left(U\right)\left(dx^{4}\right)^{2}\right]$
$\displaystyle+H_{0}^{1/2}\left(\frac{R}{U}\right)^{3/2}\left[\frac{dU^{2}}{f\left(U\right)}+U^{2}d\Omega_{4}^{2}\right],$
(2.2)

and

$e^{\phi}=\left(\frac{U}{R}\right)^{3/4}H_{0}^{3/4},\ F_{2}=\frac{\left(2\pi
l_{s}\right)^{7}g_{s}N_{0}}{\Omega_{4}V_{4}}\frac{1}{U^{4}H_{0}^{2}}dU\wedge
dx^{4},\ F_{4}=\frac{\left(2\pi
l_{s}\right)^{3}N_{c}g_{s}}{\Omega_{4}}\epsilon_{4},$ (2.3)

where

$\displaystyle H_{0}$ $\displaystyle=1+\frac{U_{Q_{0}}^{3}}{U^{3}},\
f\left(U\right)=1-\frac{U_{KK}^{3}}{U^{3}},$ $\displaystyle U_{Q_{0}}^{3}$
$\displaystyle=\frac{1}{2}\left(-U_{KK}^{3}+\sqrt{U_{KK}^{6}+\left(\left(2\pi\right)^{5}l_{s}^{7}g_{s}\kappa
N_{c}\right)^{2}}\right),$ $\displaystyle R^{3}$ $\displaystyle=\pi
g_{s}l_{s}^{3}N_{c},\ \kappa=\frac{N_{0}}{N_{c}V_{4}}.$ (2.4)

We not that, $U$ is the radial coordinate perpendicular to the $N_{c}$
D4-branes, thus the holographic boundary is located at $U\rightarrow\infty$.
The parameter $l_{s}$ refers to the length of string and $V_{4}$ denotes the
worldvolume of the D4-brane. $\Omega_{4}$ refers the volume of a unit $S^{4}$
which means $\Omega_{4}=8\pi^{2}/3$. $\kappa$ relates to the density of the
D0-branes presented in the worldvolume of the D4-branes. Since the $N_{0}$
D0-branes are considered to be homogeneously smeared, $\kappa$ is also a
constant. Overall, the supergravity solution (2.2) - (2.4) describes the
bubble geometry produced by $N_{c}$ coincident D4-branes in which the $N_{0}$
D0-branes are homogeneously smeared along the direction $x^{4}$ as it is
illustrated in Table 1.

| 0 | 1 | 2 | 3 | 4 | 5($U$) | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
$N_{c}$ D4-branes | - | - | - | - | - |  |  |  |  |
$N_{0}$ D0-branes | ▫ | ▫ | ▫ | ▫ | - |  |  |  |  |
$N_{f}\ \mathrm{D8/\overline{D8}}$ -branes | - | - | - | - |  | - | - | - | - | -
Baryon vertex (D4) | - |  |  |  |  |  | - | - | - | -
Table 1:  The D-brane configuration of the D0-D4/D8 model. “-” represents that
the D-brane extends along this direction. “▫” denotes the smeared directions
of the D0-branes inside the $N_{c}$ D4-branes.

The bubble geometry means there is not an event horizon located at $U=U_{KK}$,
instead the bulk shrinks at $U=U_{KK}$ which implies it must be defined in
$U>U_{KK}$. In order to obtain a dual theory close to QCD, we need further to
eliminate the supersymmetry on the worldvolume of the D4-branes in the low-
energy theory. A simple way for this goal is to follow Witten’s [16] as it is
used in the D4/D8 model, that is to compactly the $x^{4}$-direction on a
circle $S^{1}$, then impose the periodic and anti-periodic boundary condition
to the gauge field and supersymmetric fermion respectively. Hence, below the
energy scale $M_{KK}=2\pi/\beta$, where $\beta$ refers to the size of the
$S^{1}$, the dual theory on the D4-brane is effectively 4d pure Yang-Mills
theory. In addition, since the wrap factor $\left(U/R\right)^{3/2}H_{0}^{1/2}$
in (2.1) can never go to zero, the dual theory would also exhibit confinement
due to the behavior of the Wilson loop in this geometry [16, 37].

Next, let us take a close look at the dual theory. First, in order to avoid
the conical singularity in the dual theory, we have to impose the following
condition,

$\beta=\frac{4\pi}{3}U_{KK}^{-1/2}R^{3/2}b^{1/2},\
b=H_{0}\left(U_{KK}\right).$ (2.5)

Then Yang-Mills coupling constant $g_{\mathrm{YM}}$ for the dual theory must
be given by following the dimensional reduction as,

$g_{\mathrm{YM}}^{2}=\frac{g_{5}^{2}}{\beta}=\frac{4\pi^{2}g_{s}l_{s}}{\beta},$
(2.6)

where $g_{5}$ refers the 5d Yang-Mills coupling and $g_{s}$ is the string
coupling constant. Accordingly, the relation of $b$ and $R^{3}$ is evaluated
as,

$b=\frac{1}{2}\left[1+\left(1+\mathcal{C}\beta^{2}\right)^{1/2}\right],\
\mathcal{C}\equiv\left(2\pi l_{s}^{2}\right)^{6}\lambda\kappa^{2}/U_{KK}^{6},\
R^{3}=\frac{\beta\lambda l_{s}^{2}}{4\pi},$ (2.7)

where $\lambda$ is the t’ Hooft coupling constant given by
$\lambda=g_{\mathrm{YM}}^{2}N_{c}$ and $b\geq 1$. Afterwards, the dual theory
can be examined by taking into account a probe D4-brane at the holographic
boundary and its action is given as,

$S_{\mathrm{D4}}=-g_{s}^{-1}\mu_{4}\mathrm{Tr}\int
d^{4}xdx^{4}e^{-\phi}\sqrt{-\det\left(G+2\pi\alpha^{\prime}\mathcal{F}\right)}+\mu_{4}\int
C_{5}+\frac{1}{2}\left(2\pi\alpha^{\prime}\right)^{2}\mu_{4}\int
C_{1}\wedge\mathcal{F}\wedge\mathcal{F},$ (2.8)

where $\alpha^{\prime}=l_{s}^{2}$ and $G$ is the induced metric on the
D4-brane. $\mathcal{F}$ refers to the Yang-Mills gauge field strength on the
D4-brane. $C_{1,5}$ is the Romand-Romand 5- and 1-form. While the field
strength of $C_{1}$ is given in (2.3), $C_{5}$ satisfies
${}^{\star}dC_{5}=dC_{3}=F_{4}$ where $F_{4}$ is given in (2.3). Keeping these
in hand, we can find at the holographic boundary $U\rightarrow\infty$ and in
the low-energy limit $\alpha^{\prime}\rightarrow 0$, the leading order action
of the first term in (2.8) is the 4d Yang-Mills action, the second term in
(2.8) is a constant by inserting the solution for $C_{4}$, the last term
reduces to a theta term of the Yang-Mills theory. Altogether, the action (2.8)
reduces to

$S_{\mathrm{D4}}\simeq-\frac{N_{c}}{4\lambda}\mathrm{Tr}\int
d^{4}x\mathcal{F}^{2}+\frac{\theta}{8\pi^{2}}\mathrm{Tr}\int\mathcal{F}\wedge\mathcal{F}+O\left(\mathcal{F}^{4}\right),$
(2.9)

where

$C_{1}=\sqrt{\frac{b-1}{b}\lambda}\frac{f\left(U\right)}{H_{0}\left(U\right)}dx^{4},\theta+2\pi
k=l_{s}^{-1}\int_{S^{1}}C_{1}|_{U\rightarrow\infty}=\frac{\beta}{l_{s}}\sqrt{\frac{b-1}{b}\lambda},\
k\in\mathbb{Z}.$ (2.10)

Therefore, we can see in a given branch, if the D0-branes vanish i.e.
$N_{0}=0,b=1$, the theta angle $\theta$ vanishes as well. In this sense, we
can obtain a confining Yang-Mills theory with a theta term which relates to
the number density of the D0-branes by all the above holographic construction
in the D0-D4 system. And the glue condensate
$\left\langle\mathrm{Tr}\mathcal{F}\wedge\mathcal{F}\right\rangle$ is possible
to be evaluated in this model as
$\left\langle\mathrm{Tr}\mathcal{F}\wedge\mathcal{F}\right\rangle=8\pi^{2}N_{c}\kappa$
[38].

###  2.2 The flavor sector

Since QCD also has flavors, in the D0-D4 background, it is possible to
introduce a stack of coincident $N_{f}$ pairs of probe D8- and (anti-D8)
$\overline{\mathrm{D8}}$-branes as flavors by following the discussion in the
D4/D8 model. The $N_{f}$ pairs of the probe $\mathrm{D8/\overline{D8}}$-branes
are located at the antipodal position of $S^{1}$ perpendicular to the $N_{c}$
D4-branes. The relevant configuration of the
$\mathrm{D8/\overline{D8}}$-branes is given in Table 1 and illustrated in
Figure 1. The fundamental fermion in the low-energy theory is identified to
the fermionic zero modes of the $4-8$ or $4-\bar{8}$ string in R-sector
(Ramond-sector) 444The $4-8$ string denotes the open string connecting the
$N_{c}$ D4-brane and $N_{f}$ D8-branes. And it is similar for e.g. the $8-8$
or $4-\bar{8}$ string. since such strings take both colors and flavors whose
fermionic zero mode is in the fundamental representation of
$U\left(N_{c}\right)$ and $U\left(N_{f}\right)$. Besides, in string theory,
the GSO (Gliozzi-Scherk-Olive) projection will remove the fundamental fermion
with one of the chiralities, so it is possible to choose the fundamental
fermion with positive and negative chirality as the massless fermionic modes
of $4-8$ and $4-\bar{8}$ string respectively. Thus the flavor symmetry on the
$\mathrm{D8/\overline{D8}}$-branes can be denoted as
$U\left(N_{f}\right)_{L}\times U\left(N_{f}\right)_{R}$ as the chiral
symmetry. Note that these chirally fermions are all complex spinors since the
$4-8$ and $4-\bar{8}$ strings have two orientations.

As the case in the D4/D8 model, the disconnected and connected configuration
of the $\mathrm{D8/\overline{D8}}$-branes represents respectively the chirally
symmetric and broken phase in the dual theory. To test the dual theory, let us
introduce a probe D4-brane located at $U=U_{\Lambda}$ as before, then the
effective action for the fundamental fermions denoted by $q_{L,R}$ on the
D4-branes intersected with $N_{f}$ $\mathrm{D8/\overline{D8}}$-branes is,

$\displaystyle S=$
$\displaystyle\int_{\mathrm{D4}}d^{4}xdx^{4}\sqrt{-g}\left[\delta\left(x^{4}-X_{L}\right)q_{L}^{\dagger}\bar{\sigma}^{\mu}\left(i\nabla_{\mu}+A_{\mu}\right)q_{L}+\delta\left(x^{4}-X_{R}\right)q_{R}^{\dagger}\bar{\sigma}^{\mu}\left(i\nabla_{\mu}+A_{\mu}\right)q_{R}\right],$
(2.11)

where $X_{L,R}$ denotes respectively the intersection of the D4- and
D8-branes, D4- and $\overline{\mathrm{D8}}$-branes. $A_{\mu}$ refers to the
gauge field on the D4-branes i.e. the gluon field. As all the fields depend on
$\left\\{x^{\mu},x^{4}\right\\}$, it means $q_{L}$ would be identified to
$q_{R}$ if $X_{L}=X_{R}$ which leads to an action with a single flavor group
$U\left(N_{f}\right)$. So for the connected configuration of the
$\mathrm{D8/\overline{D8}}$-branes given in Figure 1, we can see the D8- and
$\overline{\mathrm{D8}}$-branes are separated at very high energy
($U_{\Lambda}\rightarrow\infty$, $X_{L}\neq X_{R}$), which leads to an
approximated $U\left(N_{f}\right)_{L}\times U\left(N_{f}\right)_{R}$ chiral
symmetry. However, at low energy $(U_{\Lambda}\rightarrow
U_{KK},X_{L}\rightarrow X_{R})$, the flavored D8- and
$\overline{\mathrm{D8}}$-branes are joined into a single pair of D8-branes at
$U_{\Lambda}=U_{KK}$ ($X_{L}=X_{R}$) which means the
$U\left(N_{f}\right)_{L}\times U\left(N_{f}\right)_{R}$ symmetry breaks down
to a single $U\left(N_{f}\right)$. Accordingly, this configuration of
$\mathrm{D8/\overline{D8}}$-branes displays nicely a geometric interpretation
of chiral symmetry in holography.

Figure 1:  The configuration of the $\mathrm{D8/\overline{D8}}$-branes in the
D0-D4 background. Figure (a) illustrates that
$\mathrm{D8/\overline{D8}}$-branes are located at the antipodal position of
$S^{1}$. Figure (b) illustrates this configuration in the large $N_{c}$ limit
i.e. in the D4 bubble background with D0-branes.

###  2.3 The bosonic meson tower

The meson in this model is identified to the zero modes of the bosonic states
created by the open strings on the flavor branes since such states are the
gauge fields in the adjoint representation of the flavor group
$U\left(N_{f}\right)_{L}\times U\left(N_{f}\right)_{R}$. Accordingly, let us
consider the bosonic action for the gauge fields on the flavor D8-branes as,

$\displaystyle S_{\mathrm{D8}}=$ $\displaystyle-
T_{\mathrm{D8}}\int_{\mathrm{D8}}d^{9}xe^{-\phi}\mathrm{Tr}\sqrt{-\det\left[g_{ab}+\left(2\pi\alpha^{\prime}\right)\mathcal{F}_{ab}\right]}+S_{WZ},$
$\displaystyle=$ $\displaystyle-
T_{\mathrm{D8}}\int_{\mathrm{D8}}d^{9}x\sqrt{-g}e^{-\phi}\left[1+\frac{1}{4}\left(2\pi\alpha^{\prime}\right)^{2}\mathcal{F}_{MN}\mathcal{F}^{MN}+\mathcal{O}\left(\mathcal{F}^{4}\right)\right],$
(2.12)

where $S_{WZ}$ refers to the Wess-Zumino term for the D8-brane and we have
expanded the action up to the leading order of $\alpha^{\prime}$. Following
the discussion in [15, 19] and assuming the non-zero components of the gauge
field are denoted as
$\mathcal{A}_{M}=\left\\{\mathcal{A}_{\mu}\left(x,z\right),\mathcal{A}_{z}\left(x,z\right)\right\\},\mu=0,1...3$,
then the Yang-Mills part of the action (2.12) becomes,

$\displaystyle S_{\mathrm{YM}}=$ $\displaystyle-
T_{\mathrm{D8}}\int_{\mathrm{D8}}d^{9}x\sqrt{-g}e^{-\phi}\frac{1}{4}\left(2\pi\alpha^{\prime}\right)^{2}\mathcal{F}_{MN}\mathcal{F}^{MN}$
$\displaystyle=$ $\displaystyle-T\left(2\pi\alpha^{\prime}\right)^{2}\int
d^{4}xdZH_{0}^{1/2}\mathrm{Tr}\left(\frac{1}{2}K^{-1/3}\eta^{\mu\rho}\eta^{\nu\sigma}\mathcal{F}_{\mu\nu}\mathcal{F}_{\rho\sigma}+KM_{KK}^{2}\eta^{\mu\nu}\mathcal{F}_{\mu
Z}\mathcal{F}_{\nu Z}\right),$ (2.13)

where

$T=\frac{2}{3}R^{3/2}U_{KK}^{1/2}T_{\mathrm{D8}}\Omega_{4},K\left(Z\right)\equiv
1+Z^{2}=\frac{U^{3}}{U_{KK}^{3}},$ (2.14)

and we have used the induced metric for the antipodal D8-branes which is
computed as,

$\displaystyle ds_{\mathrm{D8}}^{2}$
$\displaystyle=\left(\frac{U}{R}\right)^{3/2}H_{0}^{1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H_{0}^{1/2}\left(\frac{R}{U}\right)^{3/2}\left[\frac{dU^{2}}{f\left(U\right)}+U^{2}d\Omega_{4}^{2}\right]$
$\displaystyle=\frac{2}{3}M_{KK}R^{3}b^{1/2}H_{0}^{1/2}\left(\frac{4}{9}M_{KK}^{2}b\eta_{\mu\nu}K^{1/2}dx^{\mu}dx^{\nu}+\frac{4}{9}K^{-5/6}dZ^{2}+K^{1/6}d\Omega_{4}^{2}\right).$
(2.15)

To obtain a 4d canonical action for mesons, we expand
$\mathcal{A}_{\mu}\left(x,z\right),\mathcal{A}_{z}\left(x,z\right)$ by a
complete set of basis functions
$\left\\{\psi_{n}\left(z\right),\phi_{n}\left(z\right)\right\\}$ as,

$\mathcal{A}_{\mu}\left(x,z\right)=\sum_{n}B_{\mu}^{\left(n\right)}\left(x\right)\psi_{n}\left(z\right),\mathcal{A}_{z}\left(x,z\right)=\sum_{n}\varphi^{\left(n\right)}\left(x\right)\phi_{n}\left(z\right),$
(2.16)

where the basis functions are expected to satisfy the normalization condition

$\displaystyle T\left(2\pi\alpha^{\prime}\right)^{2}R^{3}\int
dZH_{0}^{1/2}K^{-1/3}\psi_{n}\psi_{m}$ $\displaystyle=\delta_{mn},$
$\displaystyle
T\left(2\pi\alpha^{\prime}\right)^{2}R^{3}M_{KK}^{2}H_{0}\left(U_{KK}\right)U_{KK}^{2}\int
dZH_{0}^{1/2}K\phi_{n}\phi_{m}$ $\displaystyle=\delta_{mn},$ (2.17)

and the eigenvalue equation

$-H_{0}^{-1/2}K^{1/3}\partial_{Z}\left(H_{0}^{1/2}K\partial_{Z}\psi_{m}\right)=\Lambda_{m}\psi_{m}.$
(2.18)

These conditions implies the eigenfunctions can be chosen as

$\phi_{n}=\frac{1}{M_{n}U_{KK}}\partial_{Z}\psi_{n},M_{n}=\Lambda_{n}^{1/2}M_{KK}H_{0}^{1/2},$
(2.19)

for $n>0$ and for $n=0$,

$\phi_{0}=\frac{c}{H_{0}^{1/2}K},$ (2.20)

where $c$ is a numerical number given by

$c=\left[T\left(2\pi\alpha^{\prime}\right)^{2}R^{3}M_{KK}^{2}H_{0}\left(U_{KK}\right)U_{KK}^{2}\int
dZH_{0}^{-1/2}K^{-1}\right]^{-1/2}.$ (2.21)

Then impose (2.16) – (2.21) into (2.13), we can obtain the canonical form of
the action as,

$S_{\mathrm{YM}}=-\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}\varphi^{\left(0\right)}\partial^{\mu}\varphi^{\left(0\right)}+\sum_{n=1}^{\infty}\left[\frac{1}{4}F_{\mu\nu}^{\left(n\right)}F^{\left(n\right)\mu\nu}+\frac{1}{2}M_{n}^{2}V_{\mu}^{\left(n\right)}V^{\left(n\right)\mu}\right]\right),$
(2.22)

where

$\displaystyle F_{\mu\nu}^{\left(n\right)}$
$\displaystyle=\partial_{\mu}V_{\nu}^{\left(n\right)}-\partial_{\nu}V_{\mu}^{\left(n\right)},$
$\displaystyle V_{\mu}^{\left(n\right)}$
$\displaystyle=B_{\mu}^{\left(n\right)}-M_{n}^{-1}\partial_{\mu}\varphi^{\left(n\right)}.$
(2.23)

giving the infinite meson tower. We note that there is an alternative gauge
condition $A_{z}=0$ which can be obtained by a gauge transformation

$\mathcal{A}_{M}\rightarrow\mathcal{A}_{M}-\partial_{M}\Lambda.$ (2.24)

In this case, the components of $A_{M}$ become

$\displaystyle\mathcal{A}_{\mu}\left(x,z\right)=$
$\displaystyle-\partial_{\mu}\varphi^{\left(0\right)}\left(x\right)\psi_{0}\left(z\right)+\sum_{n=1}^{\infty}\left[B_{\mu}^{\left(n\right)}\left(x\right)-M_{n}^{-1}\partial_{\mu}\varphi^{\left(n\right)}\right]\psi_{n}\left(z\right)$
$\displaystyle=$
$\displaystyle-\partial_{\mu}\varphi^{\left(0\right)}\left(x\right)\psi_{0}\left(z\right)+\sum_{n=1}^{\infty}V_{\mu}^{\left(n\right)}\left(x\right)\psi_{n}\left(z\right),$
$\displaystyle\mathcal{A}_{z}\left(x,z\right)=$ $\displaystyle 0.$ (2.25)

##  3 The flavored fermionic spectroscopy on the worldvolume of the D8-branes

While the supersymmetry on the D4-branes is broken down due to the
compactification on $S^{1}$ and the method used in [16], the flavored
D8-branes remain to be supersymmetric since the
$\mathrm{D8/\overline{D8}}$-branes are perpendicular to $S^{1}$ thus is not
compactified. The same issue also arises in the D4/D8 model [43]. That means
the supersymmetric fermion additional to the gauge boson will also arise in
the low-energy theory and there is no reason to neglect them in principle.
Therefore, let us investigate the spectroscopy of the worldvolume fermion on
the D8-branes first, then attempt to find an reasonable interpretation in
terms of hadron physics in holography.

###  3.1 The fermionic action and dimensional reduction

In string theory, the action for the worldvolume field on a D-brane is in
principle obtained under the rule of T-duality [19] which includes
supersymmetrically the bosonic and fermionic part. The bosonic action of a
D-brane can be reviewed in many textbooks e.g. [18, 47, 48]. In particular,
the bosonic feature of D0-D4/D8 model can be completely reviewed in [35, 36,
37, 38, 39]. However the full formula of the action for a worldvolume fermion
on D-brane is quite complexed in general. Since our concern is the fermionic
spectroscopy in holography, let us focus on the quadratic part of the
fermionic action which can be collected as [19, 49, 50],

$S_{f}^{\mathrm{D}_{p}}=\frac{iT_{p}}{2}\int
d^{p+1}xe^{-\phi}\sqrt{-\left(g+f\right)}\bar{\Psi}\left(1-\Gamma_{\mathrm{D}_{p}}\right)\left(\Gamma^{\alpha}\hat{D}_{\alpha}-\Delta+\mathrm{L}_{\mathrm{D}_{p}}\right)\Psi,$
(3.1)

where for IIA string theory,

$\displaystyle\hat{D}_{\alpha}$ $\displaystyle=\nabla_{\alpha}+\frac{1}{4\cdot
2!}H_{\alpha
NK}\Gamma^{NK}\bar{\gamma}+\frac{1}{8}e^{\phi}\left(\frac{1}{2!}F_{NK}\Gamma^{NK}\Gamma_{\alpha}\bar{\gamma}+\frac{1}{4!}F_{KLNP}\Gamma^{KLNP}\Gamma_{\alpha}\right),$
$\displaystyle\Delta$
$\displaystyle=\frac{1}{2}\left(\Gamma^{M}\partial_{M}\phi+\frac{1}{2\cdot
3!}H_{MNK}\Gamma^{MNK}\bar{\gamma}\right)+\frac{1}{8}e^{\phi}\left(\frac{3}{2!}F_{MN}\Gamma^{MN}\bar{\gamma}+\frac{1}{4!}F_{KLNP}\Gamma^{KLNP}\right),$
$\displaystyle\Gamma_{\mathrm{D}_{p}}$
$\displaystyle=\frac{1}{\sqrt{-\left(g+f\right)}}\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}f_{\alpha_{1}\alpha_{2}}...f_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}\bar{\gamma}^{\frac{p-2q+2}{2}},$
$\displaystyle\mathrm{L}_{\mathrm{D}_{p}}$
$\displaystyle=\sum_{q}\frac{\epsilon^{\alpha_{1}...\alpha_{2q}\beta_{1}...\beta_{p-2q+1}}}{q!2^{q}\left(p-2q+1\right)!}\frac{\left(-\bar{\gamma}\right)^{\frac{p}{2}-q+1}}{\sqrt{-\left(g+f\right)}}f_{\alpha_{1}\alpha_{2}}...f_{\alpha_{2q-1}\alpha_{2q}}\Gamma_{\beta_{1}...\beta_{p-2q+1}}^{\
\ \ \ \ \ \ \ \ \ \ \ \ \lambda}\hat{D}_{\lambda}.$ (3.2)

Let us clarify the notations used in (3.1) and (3.2). First, $\Psi$ denotes
the worldvolume fermion on the a $\mathrm{D}_{p}$-brane and $T_{p}$ denotes
the tension of the $\mathrm{D}_{p}$-brane which is given as
$T_{p}=g_{s}^{-1}\left(2\pi\right)^{-p}l_{s}^{-\left(p+1\right)}$. Then the
capital letters $K,L,M,N...$ denoting the index run over the 10d spacetime and
the lowercase letters $a,b,...$ denoting the index run over the tangent space
of the 10d spacetime. The Greek alphabet $\alpha,\beta,\lambda$ refers to the
indices running over the worldvolume of the $\mathrm{D}_{p}$-brane. For a
gravity theory with fermion, the metric should be written in terms of elfbein
$e_{M}^{a}$ as $g_{MN}=e_{M}^{a}\eta_{ab}e_{N}^{b}$, and the gamma matrices
are given through

$\left\\{\gamma^{a},\gamma^{b}\right\\}=2\eta^{ab},\left\\{\Gamma^{M},\Gamma^{N}\right\\}=2g^{MN},$
(3.3)

with $e_{M}^{a}\Gamma^{M}=\gamma^{a}$. Note that $\omega_{\alpha ab}$ refers
to the spin connection and the covariant derivative for fermion is given by
$\nabla_{\alpha}=\partial_{\alpha}+\frac{1}{4}\omega_{\alpha ab}\gamma^{ab}$.
By ranking alternate anti-symmetrically or symmetrically the indices, we can
define gamma matrix with multiple indices e.g.

$\gamma^{ab}=\frac{1}{2}\left[\gamma^{a},\gamma^{b}\right],\gamma^{abc}=\frac{1}{2}\left\\{\gamma^{a},\gamma^{bc}\right\\},\gamma^{abcd}=\frac{1}{2}\left[\gamma^{a},\gamma^{bcd}\right]...$
(3.4)

and $\gamma^{abc...}$ shares the same definition with $\Gamma^{MNK...}$.
$\bar{\gamma}$ is $\bar{\gamma}=\gamma^{01...9}$ and $\sigma_{2}$ is the
associated Pauli matrix. The worldvolume field $f$ is a sum as
$f=B+\left(2\pi\alpha^{\prime}\right)\mathcal{F}$ where $\mathcal{F}$ is the
Yang-Mills field strength and $B$ refers to the NS-NS 2-form in type IIA
string theory with its field strength $H=dB$. All the fields denoted by $F$
e.g. $F_{M},F_{MN},F_{KLM}...$ refer to the field strength of the R-R forms.
Note that, $p$ should be chosen as $p=8$ in the above formulas for the
D8-brane.

For the convenience in the following discussion by holography, let us simplify
the kinetic part of the action (3.1) to be a 5d form by setting $f=0,p=8$. In
this case, the action (3.1) becomes,

$\displaystyle S_{f}^{\mathrm{D}_{8}}=$ $\displaystyle\frac{iT_{8}}{2}\int
d^{9}x\sqrt{-g}\bar{\Psi}\bigg{[}e^{-\phi}\Gamma^{\alpha}\nabla_{\alpha}+\frac{1}{8\cdot
4!}F_{KLNP}\left(\Gamma^{\alpha}\Gamma^{KLNP}\Gamma_{\alpha}-\Gamma^{KLNP}\right)$
$\displaystyle-\frac{1}{2}e^{-\phi}\Gamma^{M}\partial_{M}\phi+\frac{1}{8\cdot
2!}F_{NK}\left(\Gamma^{\alpha}\Gamma^{NK}\Gamma_{\alpha}-3\Gamma^{MN}\right)\bar{\gamma}\bigg{]}\Psi.$
(3.5)

Plugging the metric (2.15) into (3.5) then after some straightforward
calculations, we can obtain the Dirac operator as

$\displaystyle\Gamma^{\alpha}\nabla_{\alpha}=$
$\displaystyle\left(\frac{2}{3}b^{1/2}M_{KK}R\right)^{-3/2}H_{0}^{-1/4}\times\bigg{[}K^{-1/4}\gamma^{\mu}\partial_{\mu}+\frac{2}{3}M_{KK}b^{1/2}K^{-1/12}\cancel{D}_{S^{4}}$
$\displaystyle+M_{KK}b^{1/2}\gamma^{Z}\left(ZK^{-7/12}+\frac{1}{2}K^{5/12}\frac{H_{0}^{\prime}}{H_{0}}+K^{5/12}\partial_{Z}\right)\bigg{]},$
(3.6)

where $\cancel{D}_{S^{4}}=\gamma^{m}D_{m}$ is the covariant derivative
operator for a spinor on $S^{4}$. Further substitute the solution (2.3) for
the R-R fields and dilaton, it leads to,

$\displaystyle\frac{1}{8\cdot
2!}F_{NK}\left(\Gamma^{\alpha}\Gamma^{NK}\Gamma_{\alpha}-3\Gamma^{MN}\right)\bar{\gamma}\Psi$
$\displaystyle=$
$\displaystyle\left(\frac{2}{3}b^{1/2}M_{KK}R\right)^{-3/2}H_{0}^{-1/4}\times$
$\displaystyle\left(\frac{2}{3}b^{1/2}M_{KK}R\right)^{-15/2}\left(2\pi
l_{s}\right)^{4}\kappa M_{KK}b^{1/2}H_{0}^{-7/4}K^{-4/3}\gamma^{Z}\Psi,$ (3.7)

and

$\frac{1}{2}\Gamma^{M}\partial_{M}\phi=\left(\frac{2}{3}b^{1/2}M_{KK}R\right)^{-3/2}H_{0}^{-1/4}M_{KK}b^{1/2}\gamma^{Z}\left(\frac{1}{4}ZK^{-7/12}+\frac{3}{8}K^{5/12}\frac{H_{0}^{\prime}}{H_{0}}\right),$
(3.8)

where we have imposed the project $\bar{\gamma}\Psi=\gamma^{4}\Psi=\Psi$. Note
that the contribution of R-R $C_{4}$ form is vanished in the presented setup.
Then the fermionic action for the D8-branes can be written as,

$\displaystyle S_{f}^{\mathrm{D}_{8}}=$
$\displaystyle\frac{i\mathcal{T}}{\left(2\pi\alpha^{\prime}\right)^{2}\Omega_{4}}b^{11/4}\int
d^{4}xdZd\Omega_{4}H_{0}^{5/4}\bar{\Psi}P_{-}\bigg{\\{}K^{5/12}\gamma^{\mu}\partial_{\mu}+\frac{2}{3}M_{KK}b^{1/2}K^{7/12}\cancel{D}_{S^{4}}$
$\displaystyle+M_{KK}b^{1/2}\gamma^{Z}\left[\sqrt{\lambda
b\left(b-1\right)}H_{0}^{-1}K^{-5/12}+\frac{3}{4}ZK^{1/12}+\frac{1}{4}ZK^{1/12}\frac{b-1}{Z^{2}+b}+K^{13/12}\partial_{Z}\right]\bigg{\\}}\Psi,$
(3.9)

where

$\mathcal{T}=\frac{1}{2}\left(\frac{2}{3}\right)^{13/2}T_{8}\left(2\pi\alpha^{\prime}\right)^{2}\Omega_{4}\left(M_{KK}R\right)^{11/2}R^{5},P_{-}=\frac{1}{2}\left(1-\Gamma_{\mathrm{D8}}\right).$
(3.10)

Keeping these in hand, let us impose the decomposition for the spinor in this
model by following the steps in [15, 19, 43] to (3.9). Specifically, we first
decompose the worldvolume 9d spinor into a 1+3 dimensional part
$\psi\left(x,Z\right)$, an $S^{4}$ part $\varphi$ and a remaining 2d part
$\beta$ as555In $D$ dimension, a Dirac spinor usually has $\left[D/2\right]$
components where $\left[D/2\right]$ refers to the integer part of $D/2$ [47,
48].,

$\Psi=\psi\left(x,Z\right)\otimes\varphi\left(S^{4}\right)\otimes\beta.$
(3.11)

Second, the associated 10d gamma matrices can be chosen as,

$\displaystyle\gamma^{\mu}$
$\displaystyle=\sigma_{1}\otimes\bm{\gamma}^{\mu}\otimes\bm{1},\mu=0,1,2,3$
$\displaystyle\gamma^{Z}$
$\displaystyle=\sigma_{1}\otimes\bm{\gamma}\otimes\bm{1},$
$\displaystyle\gamma^{4}$
$\displaystyle=\sigma_{2}\otimes\bm{1}\otimes\tilde{\bm{\gamma}},$
$\displaystyle\gamma^{m}$
$\displaystyle=\sigma_{2}\otimes\bm{1}\otimes\bm{\gamma}^{m},m=6,7,8,9,$
$\displaystyle\bm{\gamma}$
$\displaystyle=i\bm{\gamma}^{0}\bm{\gamma}^{1}\bm{\gamma}^{2}\bm{\gamma}^{3},$
$\displaystyle\tilde{\bm{\gamma}}$
$\displaystyle=i\bm{\gamma}^{6}\bm{\gamma}^{7}\bm{\gamma}^{8}\bm{\gamma}^{9},$
(3.12)

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

$\sigma_{3}\beta_{\pm}=\beta_{\pm},\sigma_{1}\beta_{\pm}=\beta_{\mp},\sigma_{2}\beta_{\pm}=\pm
i\beta_{\mp},$ (3.13)

where $\beta_{\pm}$ denotes the two eigenstates of $\sigma_{3}$. Since the
condition $\bar{\gamma}\Psi=\Psi$ is fixed by the kappa symmetry, we need to
chose $\beta=\beta_{+}$ on the D8-brane. Besides, the spinor $\varphi$ as the
component on $S^{4}$ must satisfy the Dirac equation on $S^{4}$, it is
possible to use the spherical harmonic function with the eigenstates of
$\Gamma^{\underline{m}}\nabla_{m}^{S^{4}}$ as [51, 52],

$\bm{\gamma}^{m}\nabla_{m}^{S^{4}}\varphi^{\pm
l,s}=i\Lambda_{l}^{\pm}\varphi^{\pm
l,s};\Lambda_{l}^{\pm}=\pm\left(2+l\right),l=0,1...$ (3.14)

where the angular quantum number $s,l$ represents the angular quantum numbers
on $S^{4}$. Altogether, by imposing the decomposition (3.11) \- (3.14) into
the action (3.9) and rescaling
$\psi\rightarrow\left(2\pi\alpha^{\prime}\right)H_{0}^{-5/8}K^{-13/24}\psi$,
we can reach to the following 5d effective action as
($\Lambda_{l}\equiv\Lambda_{l}^{+}$),

$\displaystyle S_{f}^{\mathrm{D}_{8}}=$ $\displaystyle
i\mathcal{T}b^{11/4}\int
d^{4}xdZ\bar{\psi}\bigg{\\{}K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}-\frac{2}{3}M_{KK}b^{1/2}\Lambda_{l}K^{-1/2}$
$\displaystyle+M_{KK}b^{1/2}\bm{\gamma}\partial_{Z}+M_{KK}b^{1/2}\bm{\gamma}\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right]\bigg{\\}}\psi.$
(3.15)

In the following sections, we will study the fermionic spectroscopy in this
model with this 5d fermionic action (3.15) and attempt to find its holographic
interpretation in terms of hadron physics.

###  3.2 The canonical four-dimensional action

In order to obtain the mass term in the 4d dual theory, we need to rewrite the
action (3.15) in the canonical form. To this goal, let us work with the Weyl
basis by

$\psi=\left(\begin{array}[]{c}\psi_{+}\\\ \psi_{-}\end{array}\right),$ (3.16)

with the choice of gamma matrices,

$\bm{\gamma}^{\mu}=i\left(\begin{array}[]{cc}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{array}\right),\bm{\gamma}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (3.17)

where
$\sigma^{\mu}=\left(1,-\sigma^{i}\right),\bar{\sigma}^{\mu}=\left(1,\sigma^{i}\right)$.
Then we decompose the spinor by the basis functions
$\left\\{f_{\pm}^{\left(n\right)}\left(Z\right)\right\\},n=0,1,2...$ as a
complete set as

$\psi_{+}=\sum_{n}\psi_{+}^{\left(n\right)}\left(x\right)f_{+}^{\left(n\right)}\left(Z\right),\psi_{-}=\sum_{n}\psi_{-}^{\left(n\right)}\left(x\right)f_{-}^{\left(n\right)}\left(Z\right),$
(3.18)

where the functions $\left\\{f_{\pm}^{\left(n\right)}\left(Z\right)\right\\}$
are real eigenfunctions for the coupled eigen equations,

$\pm
M_{KK}b^{1/2}\partial_{Z}f_{\pm}^{\left(n\right)}+M_{KK}b^{1/2}V_{\pm}f_{\pm}^{\left(n\right)}=M_{n}^{f}K^{-2/3}f_{\mp}^{\left(n\right)},$
(3.19)

with

$V_{\pm}=-\frac{2}{3}\Lambda_{l}K^{-1/2}\pm\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right],$
(3.20)

and satisfy the normalization condition

$\mathcal{T}b^{11/4}\int
dZK^{-2/3}f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}=\delta^{mn}.$
(3.21)

Here $M_{n}^{f}$ refers to the $n$-th eigenvalue of the equations in (3.19).
Imposing (3.19) - (3.21) into the action (3.15), it reduces to the following
4d canonical action as,

$S_{f}^{\mathrm{D}_{8}}=-\sum_{n}\int
d^{4}x\left[i\psi_{-}^{\left(n\right)\dagger}\sigma^{\mu}\partial_{\mu}\psi_{-}^{\left(n\right)}+i\psi_{+}^{\left(n\right)\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{+}^{\left(n\right)}+M_{n}^{f}\psi_{+}^{\left(n\right)\dagger}\psi_{-}^{\left(n\right)}+M_{n}^{f}\psi_{-}^{\left(n\right)\dagger}\psi_{+}^{\left(n\right)}\right],$
(3.22)

which can be further written as,

$S_{f}^{\mathrm{D}_{8}}=i\sum_{n}\int
d^{4}x\left[\bar{\psi}^{\left(n\right)}\bm{\gamma}^{\mu}\partial_{\mu}\psi^{\left(n\right)}+M_{n}^{f}\bar{\psi}^{\left(n\right)}\psi^{\left(n\right)}\right],$
(3.23)

once the Dirac spinor

$\psi^{\left(n\right)}=\left(\begin{array}[]{c}\psi_{+}^{\left(n\right)}\\\
\psi_{-}^{\left(n\right)}\end{array}\right),$ (3.24)

is introduced. We note that the equations in (3.19) can be rewritten as two
decoupled second order differential equations in Sturm-Liouville form as,

$\displaystyle-\partial_{Z}\left[\frac{\partial_{Z}f_{+}^{\left(n\right)}+V_{+}f_{+}^{\left(n\right)}}{K^{-2/3}}\right]+V_{-}\left[\frac{\partial_{Z}f_{+}^{\left(n\right)}+V_{+}f_{+}^{\left(n\right)}}{K^{-2/3}}\right]$
$\displaystyle=m_{n}^{2}K^{-2/3}f_{+}^{\left(n\right)}$
$\displaystyle\partial_{Z}\left[\frac{-\partial_{Z}f_{-}^{\left(n\right)}+V_{-}f_{-}^{\left(n\right)}}{K^{-2/3}}\right]+V_{+}\left[\frac{-\partial_{Z}f_{-}^{\left(n\right)}+V_{-}f_{-}^{\left(n\right)}}{K^{-2/3}}\right]$
$\displaystyle=m_{n}^{2}K^{-2/3}f_{-}^{\left(n\right)}.$ (3.25)

where $m_{n}=\frac{M_{n}^{f}}{M_{KK}b^{1/2}}$ and their eigenvalue can be
evaluated numerically. Since $V_{\pm}$ depends on the density of the
D0-branes, the eigenvalue $m_{n}$ also relates to the charge of the D0-brane
which means it depends on the theta term in the language of QCD. In the next
section, we will evaluate numerically the mass spectrum of the fermion by the
two-point Green function in AdS/CFT as its spectral function.

###  3.3 The holographic Green function as spectral function

In this section, we will evaluate numerically the mass spectrum of the
fermions on the D8-branes by using the prescription for the correlation
function in AdS/CFT dictionary. Let us take into account a fermionic operator
$\chi$ in the dual theory described by the QCD action (2.11) in which the bulk
operator of $\chi$ is the worldvolume fermion $\psi$ on the D8-branes
$\mathcal{M}$ presented in action (3.15). Recall the AdS/CFT dictionary with
spinor [53, 54], it is known that the partition function associated to $\psi$
in the bulk is equivalent to the average value of the generating function
associated to $\chi$, as

$\displaystyle\exp\left\\{\int_{\mathcal{M}}\mathcal{L}_{f,\mathrm{ren}}^{\mathrm{D}_{8}}\left[\bar{\psi},\psi\right]d^{D+1}x\right\\}$
$\displaystyle=\left\langle\exp\left\\{\int_{\partial\mathcal{M}}\left(\bar{\chi}\psi_{0}+\bar{\psi}_{0}\chi\right)d^{D}x\right\\}\right\rangle,$
(3.26)

where $\omega,\vec{p}$ denotes the frequency and 3-momentum of the associated
Fourier modes, $\mathcal{L}_{f,\mathrm{ren}}^{\mathrm{D}_{8}}$ refers to the
renormalized onshell Lagrangian associated to the action (3.15), $\psi_{0}$
denotes the boundary value of $\psi$ and $D$ refers to the dimension of the
dual theory. Hence the retarded two-point correlation function
$G_{R}\left(\omega,\vec{p}\right)$ of $\chi$ can be obtained by,

$\left\langle\chi\left(\omega,\vec{p}\right)\right\rangle=G_{R}\left(\omega,\vec{p}\right)\psi_{0}\left(\omega,\vec{p}\right).$
(3.27)

And we on the other hand have

$\displaystyle\left\langle\bar{\chi}\left(\omega,\vec{p}\right)\right\rangle$
$\displaystyle=-\frac{\delta
S_{f,\mathrm{ren}}^{\mathrm{D}_{8}}}{\delta\psi_{0}}=\Pi_{0}\left(\omega,\vec{p}\right),$
$\displaystyle S_{f,\mathrm{ren}}^{\mathrm{D}_{8}}$
$\displaystyle=\int_{\mathcal{M}}\mathcal{L}_{f,\mathrm{ren}}^{\mathrm{D}_{8}}\left[\bar{\psi},\psi\right]d^{D+1}x.$
(3.28)

Therefore it is possible to evaluate the two-point correlation function by
imposing (3.27) (3.28) with action (3.15).

To our goal, we need first to evaluate the renormalized onshell action
$S_{f,\mathrm{ren}}^{\mathrm{D}_{8}}$by solving the Dirac equation associated
to (3.15) which is derived as,

$\displaystyle\bigg{\\{}K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}-\frac{2}{3}M_{KK}b^{1/2}\Lambda_{l}K^{-1/2}+M_{KK}b^{1/2}\bm{\gamma}\partial_{Z}$
$\displaystyle+M_{KK}b^{1/2}\bm{\gamma}\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right]\bigg{\\}}\psi$
$\displaystyle=0.$ (3.29)

Using the ansatz of the Fourier mode of (3.16) in Weyl basis,

$\psi=e^{ip\cdot x}\left(\begin{array}[]{c}\psi_{+}\\\
\psi_{-}\end{array}\right),\ p\cdot x=p_{\mu}x^{\mu}=-\omega
t+\vec{p}\cdot\vec{x},$ (3.30)

the equation (3.29) can be rewritten as,

$\displaystyle\partial_{Z}f_{+}+V_{+}f_{+}-K^{-2/3}\frac{\left(\sigma\cdot
p\right)}{M_{KK}b^{1/2}}f_{-}$ $\displaystyle=0,$
$\displaystyle-K^{-2/3}\frac{\left(\bar{\sigma}\cdot
p\right)}{M_{KK}b^{1/2}}f_{+}-\partial_{Z}f_{-}+V_{-}f_{-}$ $\displaystyle=0,$
(3.31)

where

$V_{\pm}=-\frac{2}{3}\Lambda_{l}K^{-1/2}\pm\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right].$
(3.32)

The (3.31) can be further written as two decoupled second order differential
equations which are nothing but (3.25). With simplification, they are

$\displaystyle\partial_{Z}^{2}f_{+}+\left(V_{+}-V_{-}+\frac{4}{3}\frac{Z}{K}\right)\partial_{Z}f_{+}+\left(\frac{4}{3}\frac{Z}{K}V_{+}-V_{+}V_{-}+\partial_{Z}V_{+}-\frac{p^{2}}{K^{4/3}M_{KK}^{2}b}\right)f_{+}$
$\displaystyle=0,$
$\displaystyle\partial_{Z}^{2}f_{-}+\left(V_{+}-V_{-}+\frac{4}{3}\frac{Z}{K}\right)\partial_{Z}f_{-}-\left(\frac{4}{3}\frac{Z}{K}V_{-}+V_{+}V_{-}+\partial_{Z}V_{-}-\frac{p^{2}}{K^{4/3}M_{KK}^{2}b}\right)f_{-}$
$\displaystyle=0,$ (3.33)

which can be solved analytically at the holographic boundary i.e.
$Z\rightarrow\pm\infty$ since in AdS/CFT their boundary values contribute to
$\psi_{0}$ (the boundary value of the bulk fermion $\psi$). Here we can assume
that the $N_{f}$ D8-branes stretch to the boundary $Z\rightarrow\infty$ and
the $N_{f}$ anti D8-branes stretch to the boundary $Z\rightarrow-\infty$ while
they are connected at $Z=0$. So although we will discuss the solution for
(3.33) on the D8-branes, it would be same to the solution on the anti
D8-branes. In this sense, we can obtain the analytical solution at
$Z\rightarrow\infty$ as,

$\displaystyle f_{+}$
$\displaystyle=AZ^{\frac{2}{3}\Lambda_{l}}+BZ^{-\frac{1}{3}-\frac{2}{3}\Lambda_{l}},$
$\displaystyle f_{-}$
$\displaystyle=CZ^{-\frac{1}{3}+\frac{2}{3}\Lambda_{l}}+DZ^{-\frac{2}{3}\Lambda_{l}}.$
(3.34)

Due to $\Lambda_{l}\geq 2$, the finite boundary value $\psi_{0}$ of $\psi$ can
be defined as [53, 54],

$\psi_{0}=\lim_{Z\rightarrow+\infty}Z^{-\frac{2}{3}\Lambda_{l}}\psi=\left(\begin{array}[]{c}A\\\
0\end{array}\right).$ (3.35)

Then the onshell value of the action (3.15) is computed as

$\displaystyle S_{f}^{\mathrm{D}_{8}}=$ $\displaystyle
i\mathcal{T}b^{11/4}\int
d^{4}xdZ\bar{\psi}\bigg{\\{}K^{-2/3}\bm{\gamma}^{\mu}\partial_{\mu}-\frac{2}{3}M_{KK}b^{1/2}\Lambda_{l}K^{-1/2}$
$\displaystyle+M_{KK}b^{1/2}\bm{\gamma}\partial_{Z}+M_{KK}b^{1/2}\bm{\gamma}\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right]\bigg{\\}}\psi.$
$\displaystyle=$ $\displaystyle i\mathcal{T}b^{11/4}\int
d^{4}x\left(\bar{\psi}M_{KK}\bm{\gamma}\psi\right)|_{0}^{+\infty}$
$\displaystyle-i\mathcal{T}_{c}\int
d^{4}xdZ\bar{\psi}\bigg{\\{}-K^{-2/3}\bm{\gamma}^{\mu}\overleftarrow{\partial}_{\mu}+\frac{2}{3}M_{KK}b^{1/2}\Lambda_{l}K^{-1/2}$
$\displaystyle+M_{KK}b^{1/2}\bm{\gamma}\overleftarrow{\partial}_{Z}+M_{KK}b^{1/2}\bm{\gamma}\left[\frac{\sqrt{b\left(b-1\right)\lambda}}{H_{0}K^{3/2}}+\frac{3}{2}\frac{\left(b-1\right)Z}{H_{0}K^{2}}\right]\bigg{\\}}\psi$
$\displaystyle\equiv$ $\displaystyle\int
d^{4}x\left(\Pi\psi\right)|_{0}^{+\infty}=\int
d^{4}x\Pi_{+}\psi_{+}|_{Z\rightarrow+\infty}+...,$ (3.36)

where we have imposed the conjugate equation associated to (3.15). Using the
solution (3.34), the boundary action in (3.36) is given as,

$S_{f}^{\mathrm{D}_{8}}\supseteq-\mathcal{T}M_{KK}b^{13/4}\int
d^{4}x\left[C^{\dagger}AZ^{-\frac{1}{3}+\frac{4}{3}\Lambda_{l}}+D^{\dagger}A\right]|_{Z\rightarrow+\infty},$
(3.37)

which leads to the holographic boundary counterterm $S_{ct}$ to the fermionic
action as

$S_{ct}=\mathcal{T}M_{KK}b^{13/4}\int
d^{4}x\left[C^{\dagger}AZ^{-\frac{1}{3}+\frac{4}{3}\Lambda_{l}}\right]|_{Z\rightarrow+\infty}.$
(3.38)

Therefore the renormalized onshell action for the fermions on the D8-branes is
obtained as,

$\displaystyle S_{f,\mathrm{ren}}^{\mathrm{D}_{8}}$
$\displaystyle=S_{f}^{\mathrm{D}_{8}}+S_{ct}$
$\displaystyle=-\mathcal{T}M_{KK}b^{13/4}\int d^{4}xD^{\dagger}A,$ (3.39)

and the Green function is given by

$\mathcal{T}M_{KK}b^{13/4}D=G_{R}\left(\omega,\vec{p}\right)A,$ (3.40)

by using (3.27) (3.28) (3.35) and (3.39). In order to evaluate numerically the
Green function, we can write the concerned function $f_{\pm}$ as

$f_{+}=\left(\begin{array}[]{c}F_{+}^{\left(1\right)}\\\
F_{+}^{\left(2\right)}\end{array}\right),f_{-}=\left(\begin{array}[]{c}F_{-}^{\left(1\right)}\\\
F_{-}^{\left(2\right)}\end{array}\right),$ (3.41)

and without loss of generality set momentum along $x^{1}$ as
$p_{\mu}=\left(-\omega,\mathrm{p},0,0\right)$, then define the ratio,

$\xi_{1}=\frac{F_{-}^{\left(1\right)}}{F_{+}^{\left(1\right)}},\xi_{2}=\frac{F_{-}^{\left(2\right)}}{F_{+}^{\left(2\right)}}.$
(3.42)

According to (3.34), we can see the boundary value of $f_{\pm}$ gives the
boundary spinor $D,A$, so using (3.35) (3.40), the Green function can be
rewritten as,

$G_{R}^{\left(1,2\right)}=\mathcal{T}M_{KK}b^{13/4}\lim_{Z\rightarrow\infty}Z^{\frac{4}{3}\Lambda_{l}}\xi_{1,2}\left(Z\right).$
(3.43)

And the ratios satisfy the equation (“$\prime$” is the derivative with respect
to $Z$)

$\xi_{1,2}^{\prime}=\left(V_{+}+V_{-}\right)\xi_{1,2}+\frac{K^{-2/3}}{M_{KK}b^{1/2}}\left[\left(\omega+\mathrm{p}\cdot
h\right)\xi_{1,2}^{2}+\left(\omega-\mathrm{p}\cdot h\right)\right],$ (3.44)

which is derived from the Dirac equation (3.31). Here

$\frac{F_{+}^{\left(2\right)}}{F_{+}^{\left(1\right)}}=\frac{F_{-}^{\left(2\right)}}{F_{-}^{\left(1\right)}}=h,$
(3.45)

can be simply set as $h=\pm 1$ for the periodic and anti-periodic fermion also
as the boundary condition of $\xi_{1,2}$. Altogether, it is possible to solve
numerically the equation (3.44) with incoming wave boundary condition
$\xi_{1,2}\left(0\right)=\pm 1$.

###  3.4 The numerical analysis

Figure 2:  The real and imaginary part of the fermionic Green function
$G_{R}\left(\omega,\mathrm{p}\right)$ as the spectral function in the D0-D4/D8
model. The parameters are chosen as $\Lambda_{l}=2,b=2,M_{KK}=1$. The peaks of
the Green function is represented by the white corrugation.

In this section, let us analyze numerically the fermionic spectrum by solving
(3.44). As a first overlook, we plot out numerically the Green function as a
dense function of $\omega,\mathrm{p}$ as it is illustrated in Figure 2. We can
see the peaks in the Green function basically display the dispersion curves
for the onshell relation as $\omega^{2}-k^{2}=\left(M_{n}^{f}\right)^{2}$.
Since the fermionic action (3.15) can be written as the canonical 4d form
(3.23), the general form of the fermionic propagator for the operator $\chi$
in the dual theory should be

$G_{R}\left(\omega,\vec{p}\right)=\frac{1}{ip_{\mu}\bm{\gamma}^{\mu}-M_{n}^{f}}.$
(3.46)

Therefore the poles in the Green function denotes the onshell energy of the
states created by $\chi$ and in this sense, the two-point Green function is
the spectral function of $\chi$. Besides, recall the relation given in (2.10)
(3.19) and (3.21), that means $M_{n}^{f}$ must depend on $b$ which is related
to the theta term in the Yang-Mills theory, thus we also plot out the relation
of $M_{n}^{f}$ and $b$ in Figure 3 to confirm the relation by setting
$\mathrm{p}=0$.

Figure 3:  The real and imaginary part of the fermionic Green function
$G_{R}\left(\omega,\mathrm{p}\right)$ as the spectral function in the D0-D4/D8
model with $\mathrm{p}=0$. The position of the peaks refers to the onshell
energy of a fermionic bound state and the green, blue, red colors represent
respectively the Green function with $b=1,1.5,2$. The other parameters are
fixed as $\Lambda_{l}=2,M_{KK}=1$.

According to our numerical calculation, the relation of $M_{n}$ and $b$ can be
simply fitted as,

$M_{n}^{f}\propto
M_{KK}b^{1/2}\simeq\sqrt{\frac{\lambda}{\lambda-\frac{l_{s}^{2}}{\beta^{2}}\theta^{2}}}M_{KK}.$
(3.47)

The above formula means that the fermionic mass spectrum increases by $b$
(i.e. it relates to the theta angle in QCD), hence it can describe the
metastable fermionic states for $b>1$. And this conclusion is also in
agreement with several present works [39, 40, 44, 45, 46] which reveals that
the hadron could be metastable in the presence of the theta term.

Figure 4:  The lowest spectrum with $n=0,1,2,3$ of fermions as a function of
$b$ from the position of the poles in the spectral function
$G_{R}\left(\omega,\vec{p}\right)$. The index “(1), (2)” denote that the mass
spectrum is obtained from $G_{R}^{\left(1,2\right)}$ respectively.

##  4 The worldvolume fermion as baryon

###  4.1 The holographic interpretation

As we have discussed, although the $N_{c}$ D4-branes are non-supersymmetric by
following the proposal of the compactification in Witten’s [16], there is no
mechanism to break down the supersymmetry on the $N_{f}$ D8-branes in
principle, so they remain to be supersymmetric and its low-energy theory
contains the supersymmetric fermion additional to the gauge boson. Usually,
this supersymmetric fermion is interpreted as mesino (the supersymmetric
partner of meson) in terms of hadron physics [43], since its bosonic partner
(the gauge field) is identified to the meson in the low-energy theory as we
have illustrated in Section 2. However, such mesino is always absent in QCD or
hadron physics.

Figure 5:  The D0-D4/D8 model with baryon vertex. In the bulk side, the
$N_{c}$ open strings inside the flavor branes are 8 - 8 strings creating
$\psi$ as the worldvolume fermion on the D8-branes. In the boundary theory,
the $N_{c}$ open strings are 4 - 8 strings producing the baryon field $\chi$
as the dual operator of $\psi$.

In this work, we attempt to interpret the supersymmetric fermion as baryon by
including a baryon vertex in this model due to the following reasons. First,
it is well-known that all the baryons are fermionic. Second, the
supersymmetric fermion on the worldvolume of D8-branes takes flavors and is
color singlet since it is also the adjoint representation of the flavor group
and singlet of color group. So the feature of this fermion is in agreement
with the current understanding of baryon. On the other hand, the baryon number
is usually conserved when the baryon is concerned, thus the baryon vertex is
essential in our holographic construction. Recall that the baryon vertex in
AdS/CFT is identified to the D-brane wrapped on the spherical part of the bulk
geometry with $N_{c}$ open strings ending on it and stretching to the
holographic boundary [20, 21], therefore in this model the baryon vertex is a
D4-brane wrapped on $S^{4}$ with $N_{c}$ open strings as it is illustrated in
Table 1 and in Figure 5.

Keeping these in hand, when the $N_{f}$ D8-branes are introduced into this
model, the baryon vertex lives totally inside the $N_{f}$ D8-branes as it is
displayed in Table 1 and in Figure 5, hence the $N_{c}$ open strings as 8 - 8
strings are flavored and take the baryon number from the baryon vertex, the
worldvolume fermion $\psi$ created by these open strings is flavored and takes
baryon number as well. In the side of the boundary theory which lives on a
probe D4-brane, the $N_{c}$ open strings play as 4 - 8 strings as the
fundamental quarks i.e. fundamental representation of the color group.
Therefore the fermionic color singlet operator $\chi$ must be obtained by the
decomposition of direct products of irreducible representations of the color
and flavor symmetry group i.e. it could be a baryon field consisted of $N_{c}$
quarks. In the side of the bulk theory, the $N_{c}$ open strings are also 8 -
8 strings, so the worldvolume fermionic field $\psi$ created by the $N_{c}$
open strings is naturally the dual field of $\chi$ which must share exactly
the same quantum numbers. This is consistent with the AdS/CFT dictionary and
the analysis of the symmetries in the D0-D4/D8 system. In this sense, we
construct a holographic correspondence of a baryon field $\chi$ in the
boundary theory and the worldvolume fermionic field $\psi$ as the bulk field
dual to $\chi$, which is an holographic interpretation of (3.26) in terms of
hadron physics.

Figure 6:  The imaginary part of the Green function and lowest mass spectrum
with $l=1,\Lambda_{l}=3$ in the unit of $M_{KK}=1$. Blue and red lines in the
upper figures denotes the Green function with $b=1,1.5$ respectively.

While the above discussion may be less clear to the decomposition of the
unitary group in the large $N_{c}$ limit with generic $N_{f}$, it could be
easy to understand when we take into account the realistic case of baryon in
QCD with $N_{c},N_{f}=3$. Recall the $SU\left(3\right)$ decomposition of
direct products of irreducible representations,

$\bm{3}\otimes\bm{3}\otimes\bm{3}=\bm{10}\otimes\bm{8}\otimes\bm{8^{*}}\otimes\bm{1},$
(4.1)

as it is known, in the color sector, baryon is the color singlet thus it is
the $\bm{1}$ of $SU\left(3\right)_{c}$. In the flavor sector, since baryon
usually consists of three flavored quarks, so it could be the $\bm{10}$
(decuplet), $\bm{8}$ (octet) or $\bm{8^{*}}$ (anti-octet) of
$SU\left(3\right)_{f}$. Keeping these in mind, let us turn to the D0-D4/D8
model with $N_{c},N_{f}=3$. In the bulk side, the worldvolume fermion $\psi$
takes baryon number from the baryon vertex which is also the adjoint
representation of $SU\left(3\right)_{f}$ and color singlet, thus it could be
baryonic $\bm{8}$ (octet) or $\bm{8^{*}}$ (anti-octet) of
$SU\left(3\right)_{f}$ and $\bm{1}$ of $SU\left(3\right)_{c}$. In the boundary
theory, the dual operator $\chi$ must share the same quantum numbers of $\psi$
according to AdS/CFT [18, 19, 48], therefore it implies $\chi$ is
holographically a baryonic field of octet or anti-octet in QCD due to its
quantum numbers. In addition, to include the contribution of the $N_{c}$ open
strings to the worldvolume field $\psi$ as a baryonic field (i.e it means in
holography its dual field $\chi$ is a baryon field of $N_{c}$ quarks), we need
to further rescale $\psi$ by $\psi\rightarrow\sqrt{N_{c}}\psi$ as it is
usually done in large $N_{c}$ theory [55], so that the fermionic mass in
(3.23) rescales with an overall factor $N_{c}$ to be $M_{f}\rightarrow
M_{f}N_{c}$ as it is expected as the baryon mass in the large $N_{c}$ limit.
In this sense, the fermionic spectrum discussed in Section 3 could therefore
be identified to the baryon states and the quantum numbers of the angular
momentum $l,s$ presented in (3.14) can represent the isospin and spin of a
baryon. The baryon spectrum from $G_{R}^{\left(1,2\right)}$ corresponds to the
baryons with different parity. Altogether, it is possible to compare the
fermionic spectrum with the experimental data. For example, the lowest octets
with same parity can be identified to proton, $N\left(1440\right)$ and
$N\left(1710\right)$, in our model, their mass data can be read from Figure 6
for $l=1$, it leads to a numerical evaluation as
$M_{N\left(1440\right)}/M_{proton}\simeq
1.49,M_{N\left(1710\right)}/M_{proton}\simeq 1.87$ which are very close to the
existing experimental data
$M_{N\left(1440\right)}^{\mathrm{exp}}/M_{proton}^{\mathrm{exp}}\simeq
1.53,M_{N\left(1440\right)}^{\mathrm{exp}}/M_{proton}^{\mathrm{exp}}\simeq
1.82$ [56].

###  4.2 The interaction of meson and baryon

By interpreting the worldvolume fermion as baryon, this model naturally
includes the various interaction of meson and baryon with the influence of the
theta angle. Recall the action (3.1) for the worldvolume fermion on the flavor
brane, we can see the coupling terms of gauge boson and fermion which refers
to the interaction of meson and baryon. Let us expand the action (3.1) up to
the linear order of $\mathcal{F}_{MN}$, it includes the interaction terms in
the action as

$\displaystyle S_{int}=$ $\displaystyle i\frac{T_{8}}{4}\int
d^{9}x\sqrt{-g}e^{-\phi}\bar{\Psi}\frac{1}{2}\gamma^{Z}\bar{\gamma}\Gamma^{mn}\mathcal{F}_{mn}\left(\Gamma^{\rho}\hat{D}_{\rho}-\Delta\right)\Psi$
$\displaystyle-i\frac{T_{8}}{4}\int
d^{9}x\sqrt{-g}e^{-\phi}\bar{\Psi}\left(1-\gamma^{Z}\right)\gamma^{Z}\bar{\gamma}\Gamma^{m}\mathcal{F}_{mn}g^{nl}\hat{D}_{l}\Psi,$
(4.2)

where the index $m,n$ run over $0,1,2,3,Z$. When the decomposition project of
the spinor as it is discussed in Section 3.1 is imposed, the term in the first
line of (4.2) vanishes since it is the Dirac equation that the basis functions
$f_{\pm}^{\left(n\right)}$satisfy. Thus only the second line contributes to
$S_{int}$ which can be written as,

$S_{int}=-i\frac{T_{8}}{4}\int
d^{9}x\sqrt{-g}e^{-\phi}\bar{\Psi}\left(1-\gamma^{Z}\right)\gamma^{Z}\bar{\gamma}\left(\Gamma^{\mu}\mathcal{F}_{\mu
Z}g^{ZZ}\hat{D}_{Z}+\Gamma^{\rho}\mathcal{F}_{\rho\nu}g^{\nu\mu}\hat{D}_{\mu}+\Gamma^{Z}\mathcal{F}_{Z\nu}g^{\nu\mu}\hat{D}_{\mu}\right)\Psi,$
(4.3)

where the indices run over $0,1,2,3$. Plugging the induced metric (2.15), the
solution for dilaton and the R-R fields (2.3), then after a series of
straightforward calculation, the action (4.3) becomes,

$\displaystyle S_{int}=$
$\displaystyle-i\frac{27\pi}{4\lambda}\mathcal{T}b^{9/4}\int
d^{4}xdZH_{0}^{-1/2}K^{1/6}\bar{\psi}\bm{\gamma}^{\mu}\mathcal{F}_{\mu
Z}\left[\partial_{Z}+\mathcal{U}_{1}\left(Z,b\right)\right]\psi$
$\displaystyle-i\frac{27\pi}{4\lambda}\frac{\mathcal{T}b^{5/4}}{M_{KK}^{2}}\int
d^{4}xdZH_{0}^{-1/2}K^{-7/6}\bar{\psi}\bm{\gamma}^{\mu}\mathcal{F}_{\mu\nu}\left[\partial^{\nu}+\mathcal{U}_{2}\left(Z,b\right)\bm{\gamma}^{Z}\bm{\gamma}^{\nu}\right]\psi$
$\displaystyle-i\frac{27\pi}{4\lambda}\frac{\mathcal{T}b^{7/4}}{M_{KK}}\int
d^{4}xdZH_{0}^{-1/2}K^{-1/2}\bar{\psi}\mathcal{F}_{Z\mu}\bm{\gamma}^{Z}\left[\partial^{\mu}+\mathcal{U}_{2}\left(Z,b\right)\bm{\gamma}^{Z}\bm{\gamma}^{\mu}\right]\psi,$
(4.4)

where we have imposed the dimensional reduction as it is discussed in Section
3.1 and,

$\displaystyle\mathcal{U}_{1}\left(Z,b\right)$
$\displaystyle=-\frac{\sqrt{\left(b-1\right)b\lambda}}{4H_{0}K^{3/2}}-\frac{13Z}{12K}-\frac{5H_{0}^{\prime}}{8H_{0}}+\frac{1}{4K^{1/2}},$
$\displaystyle\mathcal{U}_{2}\left(Z,b\right)$
$\displaystyle=-\frac{M_{KK}b^{1/2}}{8H_{0}K^{1/3}}\left(2ZH_{0}+KH_{0}^{\prime}\right)-\frac{M_{KK}b\sqrt{\left(b-1\right)\lambda}}{4H_{0}K^{5/6}}+\frac{1}{4}M_{KK}b^{1/2}K^{1/6}.$
(4.5)

Further recall the meson tower given in (2.25), it is easy to obtain

$\displaystyle\mathcal{F}_{\mu Z}=$
$\displaystyle-\partial_{Z}\mathcal{A}_{\mu}=\partial_{\mu}\pi\phi_{0}^{\prime}-\sum_{n=1}^{\infty}V_{\mu}^{\left(n\right)}\psi_{n}^{\prime},$
$\displaystyle\mathcal{F}_{\mu\nu}=$
$\displaystyle\partial_{\nu}\mathcal{A}_{\mu}-\partial_{\mu}\mathcal{A}_{\nu}+i\left[A_{\mu},A_{\nu}\right]$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}W_{\mu\nu}^{\left(n\right)}\psi_{n}+i\left[\partial_{\mu}\pi,\partial_{\nu}\pi\right]\phi_{0}+i\phi_{0}\sum_{n=1}^{\infty}\psi_{n}\left(\left[V_{\mu}^{\left(n\right)},\partial_{\nu}\pi\right]+\left[\partial_{\mu}\pi,V_{\nu}^{\left(n\right)}\right]\right)$
$\displaystyle+i\sum_{n,m=1}^{\infty}\psi_{n}\psi_{m}\left[V_{\mu}^{\left(n\right)},V_{\nu}^{\left(m\right)}\right],$
(4.6)

where

$W_{\mu\nu}^{\left(n\right)}=\partial_{\mu}V_{\nu}^{\left(n\right)}-\partial_{\nu}V_{\mu}^{\left(n\right)},$
(4.7)

and we have denote the scalar $\varphi^{\left(0\right)}$ as $\pi$ meson. Then
substitute (4.6) to (4.4), the action (4.4) describes the $n$-th baryon decays
to light mesons e.g scalar meson $\pi$, vector meson $V_{\mu}$ ($\rho$ meson).
Specifically, after a series of tedious but straightforward calculations, the
exact form of $S_{int}$ (up to linear term of light mesons) is given as,

$\displaystyle S_{int}$ $\displaystyle=\int
d^{4}x\left[a_{+,+}^{\left(m,n,0\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\pi\psi_{+}^{\left(n\right)}-a_{+,+}^{\left(m,n,l\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}V_{\mu}^{\left(l\right)}\psi_{+}^{\left(n\right)}\right]$
$\displaystyle+\int
d^{4}x\left[a_{-,-}^{\left(m,n,0\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}\partial_{\mu}\pi\psi_{-}^{\left(n\right)}-a_{-,-}^{\left(m,n,l\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}V_{\mu}^{\left(l\right)}\psi_{-}^{\left(n\right)}\right]$
$\displaystyle+\int
d^{4}x\sum_{m,n}\left[b_{-,+}^{\left(m,n,0\right)}\psi_{-}^{\left(m\right)\dagger}\partial_{\mu}\pi\partial^{\mu}\psi_{+}^{\left(n\right)}+b_{+,-}^{\left(m,n,0\right)}\psi_{+}^{\left(m\right)\dagger}\partial_{\mu}\pi\partial^{\mu}\psi_{-}^{\left(n\right)}\right]$
$\displaystyle+\int
d^{4}x\sum_{m,n}\left[c_{-,-}^{\left(m,n,0\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}\partial_{\mu}\pi\psi_{-}^{\left(n\right)}-c_{+,+}^{\left(m,n,0\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\pi\psi_{+}^{\left(n\right)}\right]$
$\displaystyle-\int
d^{4}x\sum_{m,n,l}\left[b_{-,+}^{\left(m,n,l\right)}\psi_{-}^{\left(m\right)\dagger}V_{\mu}^{\left(l\right)}\partial^{\mu}\psi_{+}^{\left(n\right)}+b_{+,-}^{\left(m,n,l\right)}\psi_{+}^{\left(m\right)\dagger}V_{\mu}^{\left(l\right)}\partial^{\mu}\psi_{-}^{\left(n\right)}\right]$
$\displaystyle-\int
d^{4}x\sum_{m,n,l}\left[c_{-,-}^{\left(m,n,l\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}V_{\mu}^{\left(l\right)}\psi_{-}^{\left(n\right)}-c_{+,+}^{\left(m,n,l\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}V_{\mu}^{\left(l\right)}\psi_{+}^{\left(n\right)}\right]$
$\displaystyle+\int
d^{4}x\sum_{m,n,l}\left[d_{+,+}^{\left(m,n,l\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}W_{\mu\nu}^{\left(l\right)}\partial^{\nu}\psi_{+}^{\left(n\right)}+d_{-,-}^{\left(m,n,l\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}W_{\mu\nu}^{\left(l\right)}\partial^{\nu}\psi_{-}^{\left(n\right)}\right]$
$\displaystyle+\int
d^{4}x\sum_{m,n,l}\left[f_{-,+}^{\left(m,n,l\right)}\psi_{-}^{\left(m\right)\dagger}\sigma^{\mu}\bar{\sigma}^{\nu}W_{\mu\nu}^{\left(l\right)}\psi_{+}^{\left(n\right)}-f_{+,-}^{\left(m,n,l\right)}\psi_{+}^{\left(m\right)\dagger}\bar{\sigma}^{\mu}\sigma^{\nu}W_{\mu\nu}^{\left(l\right)}\psi_{-}^{\left(n\right)}\right],$
(4.8)

where we have imposed the rescaling $\psi\rightarrow\sqrt{N_{c}}\psi$ and the
associated coupling constants are given by the following numerical integrals
with the basis functions $\psi_{n},\phi_{0}$ and $f_{\pm}^{\left(n\right)}$as,

$\displaystyle g=$
$\displaystyle\frac{3^{9/2}\pi^{5/2}N_{c}^{1/2}}{2\lambda^{3/2}},\
\mathcal{N}=\mathcal{T}b^{11/4}T^{1/2}\left(2\pi\alpha^{\prime}\right)R^{3/2}$
$\displaystyle a_{\pm,\pm}^{\left(m,n,l\right)}=$
$\displaystyle\frac{g}{b^{3/4}}\mathcal{N}\int
d^{4}xdZH_{0}^{-1/2}K^{1/6}\bigg{[}f_{\pm}^{\left(m\right)}\partial_{Z}f_{\pm}^{\left(n\right)}+\mathcal{U}_{1}\left(Z,b\right)f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}\bigg{]}\psi_{l}^{\prime},$
$\displaystyle b_{\pm,\pm}^{\left(m,n,l\right)}=$
$\displaystyle\frac{ig}{M_{KK}b^{5/4}}\mathcal{N}\int
d^{4}xdZH_{0}^{-1/2}K^{-1/2}f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}\psi_{l}^{\prime},$
$\displaystyle c_{\pm,\pm}^{\left(m,n,l\right)}=$
$\displaystyle-\frac{g}{M_{KK}b^{5/4}}\mathcal{N}\int
d^{4}xdZH_{0}^{-1/2}K^{-1/2}\mathcal{U}_{2}\left(Z,b\right)f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}\psi_{l}^{\prime}$
$\displaystyle d_{\pm,\pm}^{\left(m,n,l\right)}$
$\displaystyle=\frac{g}{M_{KK}^{2}b^{7/4}}\mathcal{N}\int
dZH_{0}^{-1/2}K^{-7/6}f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}\psi_{l},$
$\displaystyle f_{\pm,\pm}^{\left(m,n,l\right)}$
$\displaystyle=-\frac{ig}{M_{KK}^{2}b^{7/4}}\mathcal{N}\int
dZH_{0}^{-1/2}K^{-7/6}\mathcal{U}_{2}\left(Z,b\right)f_{\pm}^{\left(m\right)}f_{\pm}^{\left(n\right)}\psi_{l}.$
(4.9)

We note that the function $\psi_{l}$ is the basis function given in (2.16) -
(2.18) and we have defined $\psi_{0}=\phi_{0}$ in order to write (4.9)
compactly. $\mathcal{N}$ is the combination of the normalization factors of
basis function given in (2.17) and (3.21), so all the integrals in (4.9) with
factor $\mathcal{N}$ are purely numerical numbers. And it is easy to verify
that all the integrand functions in (4.9) converge at $Z\rightarrow\infty$ due
to the asymptotics of the basis functions

$\psi_{0}\sim\frac{1}{Z^{2}},\psi_{n}\sim\frac{1}{Z},f_{+}^{\left(m\right)}\sim
Z^{\frac{4}{3}},f_{-}^{\left(n\right)}\sim Z,$ (4.10)

according to the eigen equations (2.18) and (3.25). Therefore all the coupling
constants listed in (4.9) are finite and depend obviously on the parameter $b$
i.e. they are affected by the presence of the theta angle in QCD according to
(2.10). Besides, the unit of the coupling constant $g$ presented in (4.9) is
of $\mathcal{O}\left(N_{c}^{1/2}\right)$ whose large $N_{c}$ behavior agrees
with coupling of the meson-baryon in the the large $N_{c}$ theory [55].
Altogether, the holographic action (4.8) with the coupling constants listed in
(4.9) describes various interaction of baryon and meson in the D0-D4/D8 model.

##  5 Summary and discussion

In this work, we first demonstrate the dimensional reduction with respect to
the fermionic action for the flavor brane which is obtained by the T-duality
rules in string theory used in the D0-D4/D8 model, derive its 5d effective
form and 4d canonical action which illustrates the essential conditions for
the dimensional reduction. Then, we compute the two-point Green function for
the worldvolume fermions on the flavor brane through the standard technique in
AdS/CFT in order to evaluate the fermionic mass spectrum. Afterwards, we
analyze holographically the quantum number of the bulk field and its dual
field in the D0-D4/D8 system and accordingly interpret the worldvolume fermion
as baryon with a baryon vertex. In this sense, our fermionic mass spectrum is
recognized to be the baryon spectrum and we find it could fit well the
experimental data of the lowest baryon spectrum. Finally, we derive the linear
interaction term involving the gauge field in the fermionic action and
interpret it as the various interactions of baryons and light mesons. In the
presence of a theta term, the mass spectrum and interacted coupling constants
all depend on the theta angle which implies the metastabilization in theta-
dependent QCD as it is discussed in [39, 40, 44, 45, 46]. Overall, this work
investigates the worldvolume fermion on D-brane and find a holographic way to
describe the interaction of baryon and meson, hence it is also an extension
and supplement to our previous work [57].

Finally, we would like to give some comments to close this work. First, below
the energy scale $M_{KK}$, as the D0-D4/D8 model is holographically dual to 4d
QCD with a theta term, so if we interpret the worldvolume fermions without a
baryon vertex on the flavor branes as the supersymmetric mesons (mesino), they
remain to be absent in the low-energy theory due to their over heavy mass.
Because the lowest mass of the fermionic meson is about $1.67M_{KK}$ thus it
is larger than $M_{KK}$ according to our numerical calculation. In this sense,
the issue about the D4/D8 model proposed in [43] is figured out automatically.
Second, the baryon vertex is indispensable when the fundamental quark and
baryon are concerned in this model since the baryon number would not be
conserved without the baryon vertex as it is commented in [20]. In this sense,
baryon vertex is the essential element to make that open strings on the flavor
branes behave like a baryon, so that our work is also a supplement to [58].
Last but not least, the classical mass of a baryon in this model can be
obtained by evaluating the total energy of a wrapped D4-brane as baryon vertex
as $m_{B}=\frac{\lambda N_{c}}{27\pi}b^{3/2}$ [39, 40] which is however a
little deviated from our numerical evaluation $m_{B}\propto b^{1/2}$. To match
exactly the mass of the baryon vertex, we can further rescale the basis
functions presented in (3.18) as $f_{\pm}^{\left(m\right)}\rightarrow
b^{1/2}f_{\pm}^{\left(m\right)}$. As a result, all the coupling constants
given in (4.9) will pick up a factor $b$. This may determine completely the
dependence of $b$ and the associated basis functions in this model. With all
the above, to interpret the worldvolume fermions as baryon with a baryon
vertex in this work is seemingly reasonable and workable, thus it could
provide us a new way to study baryon in holography.

## Acknowledgements

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.

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