Dynamical holographic QCD model for glueball and light meson spectra
In this work, we offer a dynamical soft-wall model to describe the gluodynamics and chiral dynamics in one systematical framework. We firstly construct a quenched dynamical holographic QCD (hQCD) model in the graviton-dilaton framework for the pure gluon system, then develop a dynamical hQCD model for the two flavor system in the graviton-dilaton-scalar framework by adding light flavors on the gluodynamical background. For two forms of dilaton background field and , the quadratic correction to dilaton background field at infrared encodes important non-perturbative gluodynamics and naturally induces a deformed warp factor of the metric. By self-consistently solving the deformed metric induced by the dilaton background field, we find that the scalar glueball spectra in the quenched dynamical model is in very well agreement with lattice data. For two flavor system in the graviton-dilaton-scalar framework, the deformed metric is self-consistently solved by considering both the chiral condensate and nonperturbative gluodynamics in the vacuum, which are responsible for the chiral symmetry breaking and linear confinement, respectively. It is found that the mixing between the chiral condensate and gluon condensate is important to produce the correct light flavor meson spectra. The pion form factor and the vector couplings are also investigated in the dynamical hQCD model. Besides, we give the criteria for the existence of linear quark potential from the metric structure, and show a negative quadratic dilaton background field is not favored in the graviton-dilaton framework.
Graviton-dilaton-scalar system, gluon condensate, linear confinement, chiral condensate, chiral symmetry breaking
The local quantum field theoretical description has made great success since it was firstly developed in the quantization of electrodynamics and further been developed and applied to the descriptions of elementary particles. Nowadays, quantum chromodynamics (QCD) is accepted as the fundamental theory of the strong interaction. In the ultraviolet (UV) or weak coupling regime of QCD, the perturbative calculations agree well with experiment. However, in the infrared (IR) regime, the description of QCD vacuum as well as hadron properties and processes in terms of quark and gluon still remains as outstanding challenge in the formulation of QCD as a local quantum field theory. In order to derive the low-energy hadron physics and understand the deep-infrared sector of QCD from first principle, various non-perturbative methods have been employed, in particular lattice QCD [1, 2, 3, 4], Dyson-Schwinger equations (DSEs)[5, 6], and functional renormalization group equations (FRGs)[7, 8, 9].
In recent decades, an entirely new method based on the anti-de Sitter/conformal field theory (AdS/CFT) correspondence and the conjecture of the gravity/gauge duality [10, 11, 12] provides a revolutionary method to tackle the problem of strongly coupled gauge theories. Though the original discovery of holographic duality requires supersymmetry and conformality, the holographic duality has been widely used in investigating hadron physics, strongly coupled quark gluon plasma and condensed matter. It is widely believed that the duality between the quantum field theory and quantum gravity is an unproven but true fact. In general, holography relates quantum field theory (QFT) in d-dimensions to quantum gravity in (d + 1)-dimensions, with the gravitational description becoming classical when the QFT is strongly-coupled. The extra dimension can be interpreted as an energy scale or renormalization group (RG) flow in the QFT [13, 14, 15, 16, 17, 18, 19, 20].
Many efforts have been invested in applying holography duality to describe the real QCD world, e.g. for mesons [21, 22, 23], baryons  and glueballs , see Refs. [26, 27] for reviews. It is well-known that the QCD vacuum is characterized by spontaneous chiral symmetry breaking and color charge confinement. The chiral symmetry breaking can be read from the mass difference between the chiral partners of the hadron spectra, and the spontaneous chiral symmetry breaking is well understood by the dimension-3 quark condensate  in the vacuum. The understanding of confinement remains a challenge . From the hadron spectra, confinement can be read from the Regge trajectories of hadrons [30, 31], which suggests that the color charge can form the string-like structure inside hadrons. Confinement can also manifest itself by the linear potential between two quarks at large distances .
A successful holographic QCD model should describe chiral symmetry breaking, and at the same time should describe both the Regge trajectories of hadron spectra and linear quark potential, two aspects in the manifestation of color confinement. Nonetheless, these important nonperturbative features haven’t been successfully accommodated in a unique hQCD model.
The current achievements of AdS/QCD models for hadron spectra are the hard-wall AdS/QCD model  and the soft-wall AdS/QCD or KKSS model . In the hard-wall model , the chiral symmetry breaking can be realized by chiral condensation in the vacuum, however, the resulting mass spectra for the excited mesons behave as , which is different from the linear Regge behavior . In order to generate the linear Regge behavior, the authors of Ref. introduced a quadratic dilaton background, one can obtain the desired mass spectra for the excited vector mesons, while the chiral symmetry breaking phenomenon cannot be consistently realized .
Interesting progress was made in Refs. [34, 35], where a quartic interaction term in the bulk scalar potential was introduced to incorporate linear trajectories and chiral symmetry breaking. However, such a term was shown  to result in a negative mass square for the lowest lying scalar meson state, which might cause an instability of the background. In Ref., a deformed warp factor is introduced, which can cure the instability and maintain the linear behavior of the spectra.
With AdS metric in the soft-wall model and its extended versions [34, 35, 36, 39, 40, 41, 42, 43] (except ), only Coulomb potential between the two quarks can be produced . On the other hand, the linear quark potential can be realized in the Andreev-Zakharov model , where a positive quadratic correction in the deformed warp factor of geometry was introduced. The linear heavy quark potential can also be obtained by introducing other deformed warp factors as in Refs. [46, 47]. The positive quadratic correction in the deformed warp factor in some sense behaves as a negative dilaton background in the 5D action, which motivates the proposal of the negative dilaton soft-wall model [48, 49]. More discussions on the sign of the dilaton correction can be found in [51, 50].
It is noticed that the quadratic correction, whether appears in the 5D action or in the deformed warp factor, indeed plays an important role to realize the linear confinement, though only partly. Since both the Regge trajectories of hadron spectra and linear quark potential are two aspects in the manifestation of color confinement, they should share the same dynamical origin and should be realized in the same holographic QCD model.
In the soft-wall model and its improved versions, the dilaton field or the deformed warp factor are introduced by hand. In Ref., we have successfully described the chiral symmetry breaking, the Regge trajectories of hadron spectra and linear quark potential in the graviton-dilaton-scalar coupling framework, in which the metric, the field(s) and the potential(s) of the field(s) are self-consistently determined by field equations, and one can self-consistently solve out the other two with one input. There are at least three different ways to deal with the system in the literature: 1) Input the form of the field(s) to solve the metric structure and the potential(s) of the field(s) [53, 52]; 2) Input the potential(s) of the field(s) to solve the metric and the field(s) [54, 55]; 3) Input the form of the metric structure to solve the field(s) and the potential(s) of the field(s) .
This work is an extension of Ref., and we intend to establish the relation between the QCD dynamics including at IR and its induced geometry. The paper is organized as follows: In Sec. 2, we establish a quenched dynamical hQCD model in the graviton-dilaton framework to describe the pure gluon system, and by selfconsistently solve the deformed warp factor induced by the dilaton field, we get the scalar glueball spectra; We introduce the meson spectra in the KKSS model and its improved versions in Sec.3; We then develop the graviton-dilaton-scalar coupling framework for two flavor system and investigate the hadron spectra in Sec.4, and we also investigate pion form factors and vector couplings in Sec. 5. We give discussion and summary in Sec.6.
2 Pure gluon system: quenched dynamical soft-wall holographic model
At the classical level, QCD is a scale invariant theory, which is broken by quantum fluctuations. The pure gluon part of QCD Lagrangian in 4-dimension is described by
Where is the gluon field with the color indices.
In the vacuum, the scale invariance is explicitly broken, and the relevant degrees of freedom of QCD at infrared are still poorly understood. Varies of vacuum condensates provide important information to understand the non-perturbative dynamics of QCD. For example, the gauge invariant dimension-4 gluon condensate has been widely investigated in both QCD sum rules and lattice calculations [57, 58, 59], and the non-vanishing value of the condensate does not signal the breaking of any symmetry directly, but rather the non-perturbative dynamics of strongly interacting gluon fields. In last decade, there have been growing interests in dimension-2 gluon condensates in SU gauge theory and its possible relation to confinement [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76].
On the other hand, the effective Lagrangian for pure gluon system can also be constructed in terms of lightest glueball [77, 78, 79] or one scalar particle - dilaton [80, 81, 82, 83, 84, 85]. The dilaton field is an hypothetical scalar particle predicted by string theory and Kaluza-Klein type theories, and its expectation value probes the strength of the gauge coupling. In Ref., an effective coupling of a massive dilaton to the 4-dimensional gauge fields provides an interesting mechanism which accommodates both the Coulomb and confining potentials between heavy quarks.
Csaki and Reece in Ref. proposed to model the pure gluon system in the graviton-dilaton framework by considering the correspondence between the dilaton background field and the non-perturbative gluon condensate, which provides a natural IR cut-off. They investigated the case of dilaton field coupling with dimension-4 gluon operator and higher dimension-6 gluon condensation , and found that such IR correction cannot generate Regge spectra of glueball. They also discussed a tachyon-dilaton-graviton system, where the tachyon corresponds to a dimension-2 gluon condensate to realize the linear confinement. However, the local dimension-2 gluon operator is not gauge invariant.
In the following, we construct a 5-dimension dynamical hQCD model in the graviton-dilaton coupled system for the pure gluodynamics, and investigate three different forms for the dilaton background field.
2.1 Quenched dynamical soft-wall holographic model and gluodynamics
We construct a 5D effective model for pure gluon system by introducing one scalar dilaton field in the bulk. It is not known how the dilaton field should couple with the gauge field in 4-dimension. The 5D graviton-dilaton coupled action in the string frame is given below:
Where is the 5D Newton constant, , and are the 5D metric, the dilaton field and dilaton potential in the string frame, respectively. The metric ansatz is often chosen to be
In this paper, the capital letters like ”M,N” would stand for all the coordinates(0,1,..,4), and the greek indexes would stand for the 4D coordinates(0,…,3). We would use the convention .)
Under the frame transformation
The equations of motion can be easily derived by doing functional variation with respective to the corresponding fields. It takes the familiar form in the Einstein frame,
Under the metric ansatz Eq.(4), the above Einstein equations has two independent equations,
in the new variables of
In the string frame, the above two equations of motion become
Dimension-4 dilaton background field
As Csaki and Reece proposed proposed in  to model the pure gluon system in the graviton-dilaton framework by considering the correspondence between the dilaton background field and the non-perturbative and gauge invariant dimension-4 gluon condensate, which provides a natural IR cut-off.
Assuming the dimension-4 gluon condensate dominant in the IR region, we take the quartic dilaton field as
and from Eq.(10), we can solve out the metric background and the dilaton potential as follows:
The UV expansion of the above potential is
which means the 5D mass is zero. From the AdS/CFT dictionary , one can derive its dimension , so it could be dual to the gauge invariant dimension-4 gluon condensate .
Dimension-2 dilaton background field
To realize the Regge behavior for the vector meson, it has been shown in Ref.  that a quadratic dilaton background is essential. The simplest dimension-2 dilaton background field has the quadratic form as
The positive quadratic dilaton background is the same as the one introduced in the KKSS model . We will show in Section 2.2 and 2.3 that only positive quadratic correction can generate the linear confinement.
In the original soft-wall model or the KKSS model , the dilaton field is introduced to generate the linear Regge spectra of vector meson but the metric remains as AdS. In the graviton-dilaton coupled framework, the quadratic dilaton field is introduced dynamically in correspondence with non-perturbative gluodynamics, and the metric structure is automatically deformed by selfconsistently solving the Einstein equations. With the quadratic dialton background given in Eq.(18), we can solve the metric and the dilaton potential in the Einstein frame as
with the hypergeometric function. It is noticed that in the Einstein frame, both the positive and negative quadratic correction give the same results. However, in the string frame, the positive quadratic dilaton background field gives:
and the negative dilaton background field gives
With the normalized variable which is defined as
the dilaton potential in the Einstein frame takes the form of
here the radius of AdS and the hypergeometric function. In the ultraviolet limit,
with the 5D mass for the dilaton field
From the AdS/CFT dictionary , one can derive its dimension .
It will be shown in Sec. 2.2.3 and 4.4 that, with positive quadratic correction to the dilaton background filed at IR, by self-consistently solving the graviton-dilaton framework for the pure gluon system and the graviton-dilaton-scalar framework for two-flavor system, one can produce the scalar glueball spectra and meson spectra in good agreement with lattice/experiment data. This indicates that some form of dimension-2 gluon operator plays important role in QCD vaccum. Indeed, in last decade, there have been growing interests in dimension-2 gluon condensates in SU gauge theory and its possible relation to confinement [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76].
Dilaton field with quartic form at UV and quadratic form at IR
To build a holographic dual to the pure gluon system, we have to find the dual bulk dilaton field which encodes the non-perturbative QCD gluodynamics. The natural candidate is the quartic dilaton field which is dual to the gauge invariant dimension-4 gluon condensate. Unfortunately, as shown in  as well as in Sec. 2.2.2, one cannot produce confinement property of the glueball spectra with quartic dilaton field. On the other hand, the studies in  and in Secs. 2.2.3 and 4.4 show that the quadratic correction to the dilaton field at IR is essential to produce the glueball and meson spectra as well as to realize the linear confinement. However, the gluon operator corresponding to the dimension-2 dilaton field is not well defined.
The dimension-2 dilaton field might be dual to the dimension-2 gluon condensate [63, 66, 67], which has been discussed in some literatures, e.g. Refs.[53, 45, 35, 41]. The simplest dimension-2 gluon operator is the zero momentum mode of , i.e. , the Bose-Einstein condensation (BEC) of the “pairing” of two gluons in the vacuum due to the strong interaction [60, 70]. The BEC of the ”pairing” of two gluons spontaneously generates an effective gluon mass and breaks scale invariance, and in this scenario, the dimension-4 gluon condensation is proportional to the dimension-2 gluon condensation. Recent lattice results support a gluon mass at IR [96, 97, 98] which was proposed by Cornwall in 1981  and recently developed in . However, the dimension-2 gluon condensate encounters the gauge invariant problem.
Motivated by Refs. and , one might introduce the holography dictionary as dual to the gauge invariant dimension-4 gluon condensation . In this case, though the dilaton field itself has dimension of 2, the action is always in terms of thus there is no gauge invariant problem. However, a composite bulk operator is not consistent gauge/gravity duality.
The dimension-2 dilaton field might also correspond to the gauge invariant but non-local operator related to topological defects in the QCD vacuum . However, gauge/gravity duality requires to map a local bulk field to a local operator at the boundary.
To avoid the gauge non-invariant problem and to meet the requirement of gauge/gravity duality, we take the dilaton field in the form of
In this way, the dilaton field at UV behaves
and is dual to the dimension-4 gauge invariant gluon condensate , while at IR it takes the quadratic form
from the constraint of the linear confinement.
2.2 Scalar glueball spectrum in quenched dynamical soft-wall model
The glueball spectrum has attracted much attention more than three decades . The study of particles like glueballs where the gauge field plays a more important dynamical role than that in the standard hadrons, offers a good opportunity of understanding the nonperturbative aspects of QCD, e.g. see reviews . In Table 1, we list the scalar glueball spectra from several lattice groups [88, 89, 90, 91].
The glueball has been studied in the holographic QCD models [92, 93, 94, 95]. The scalar glueball is associated with the local gauge-invariant QCD operator defined on the boundary spacetime, which has dimension . From the AdS/CFT dictionary, the scalar glueball has zero 5D mass, i.e. .
We assume the glueball can be excited from the QCD vacuum described by the quenched dynamical holographic model in Section 2.1, and the 5D action for the scalar glueball in the string frame takes the form as that in the original soft-wall model [92, 93]
The only difference is that the metric structure in the original soft-wall model is , but in our dynamical soft-wall model the metric structure is selfconsistently solved from Section 2.1.
The Equation of motion for has the form of
After the transformation , we get the schrodinger like equation of motion for the scalar glueball
with the 5D effective schrodinger potential
Glueball spectra in original soft-wall model
which implying that the Regge slope for the scalar glueball is , and the lightest glueball mass square is . In Table 2, we list some numerical results for the scalar glueball based on Eq. (34) with , respectively.
From the lattice data for the scalar glueball as given in Table 1, one can read that the slope of the Regge spectra is around , which means . From Eq. (34), the lightest scalar glueball mass square in the soft-wall model should be around , which is too large comparing with the lattice result . If one fixes the lightest scalar glueball mass square , which gives , then the slope for the Regge spectra will be around , which is too small comparing with the lattice results .
In summary, by using the metric, the soft-wall model with the quadratic dilaton background field cannot accommodate both the lightest scalar glueball mass and the Regge slope.
Glueball spectra with quartic dilaton background
In the previous subsection we have shown that the positive quadratic dilaton background can generate the linear Regge behavior of glueball spectra, which agrees well with the Lattice data [88, 89, 90, 91]. However, dimension-4 gluon condensate is one of the most important gauge invariant non-perturbative quantity in the QCD vacuum, it is worthwhile to investigate how much the dimension-4 gluon condensate contribute to the linear Regge behavior of the glueball spectra. Actually, in the dynamical hard-wall model, Csaki and Reece have studied the effect of dimension-4 gluon condensate dual to a quartic dilaton field to mimic the IR brane effect, and they have found that the glueball spectrum is non-linear with .
In this subsection, we follow the approach introduced in previous subsections and study the effect of dimension-4 gluon condensate on the glueball spectra. Then with the metric warp factor as given in Eq.(15), we can get the effective potential in Eq.(33) for the glueball in this background. By solving the schrodinger-like equation with this potential, we can get the scalar glueball spectra as shown in Table 3 and in Fig.1. We have chosen two sets of parameters and corresponding to a ground state scalar glueball mass of and , which are around the lightest and heaviest glueball ground state mass in Table.1, respectively. It is shown in Fig.1 that for both cases, higher excitation states deviate from the linear behavior. Our result is consistent with the result in , i.e. the spectra are non-linear and behave as for high excitation states. Both Ref. and our results show that the quartic dilaton field which dual to the dimension-4 gluon condensate would induce the nonlinear excitation spectra for scalar glueball.
Glueball spectra with quadratic dilaton background
For the quadratic dilaton background field, we firstly investigate the scalar glueball spectra with the positive quadratic dilaton background Eq.(21).
Under the boundary condition and , we get the scalar glueball spectra as shown in Table 4. It is observed that with , the scalar glueball spectra in the dynamical soft-wall model with positive quadratic dilaton background can fit lattice results quite well.
We would like to emphasize that the dynamical soft-wall model has the same parameters as the original soft-wall model, i.e. the radius which is taken to be , and the quadratic coefficient of the dilaton background field . As we have shown in Sec.2.2.1, the original soft-wall model cannot accommodate both the ground state and the Regge slope. However, if one self-consistently solves the metric background under the dynamical dilaton field, it gives the correct ground state and at the same time gives the correct Regge slope. This is a surprise result! To explicitly see the difference, we show the scalar glueball spectra in the soft-wall model (blue dash-dotted line) and the dynamical soft-wall model (red solid line) in Fig. 2 for the case of .
It is observed from Fig. 2 that the glueball spectra in the dynamical soft-wall model is parallel to that in the soft-wall model, and the separation is about . This indicates the ground sate of the scalar glueball has mass square around , and has mass around . From numerical results, we extract the Regge spectra in the dynamical soft-wall (DSW) model:
In order to understand the difference between the soft-wall model and the dynamical soft-wall model, we plot the effective schrodinger potentials of the two models and their difference in Fig. 3. It is observed that the schrodinger potential (red solid line) in the dynamical soft-wall model has a lower minimum than that in the soft-wall model (blue dashed line), the difference is about , which is the same as the difference of the mass square in these two models, i.e. .
If the dynamical soft-wall model takes the negative quadratic dilaton background , the metric structure has the form of Eq.(22), and the scalar glueball spectra is shown in Fig. 4 with , respectively.
It is observed that the negative quadratic dilaton background can also generate the Regge spectra. However, like the soft-wall model, the dynamical soft-wall model with negative dilaton cannot accommodate both the ground state and the Regge slope.
Glueball spectra for dilaton field with quartic form at UV and quardratic form at IR
For the dilaton background field Eq.(27) with quartic form at UV and quardratic form at IR, we can solve the background metric under this dilaton field from the equation of motion Eq.(10), and the numerical result is shown in Fig.5.
Then from Eq. (32), we can solve the scalar glueball spectra as in the previous sections and the result is shown in Fig.6. It is found that the glueball spectra is not sensitive to the value of as long as . For , the scalar glueball spectra for the dilaton field is almost the same as that for the quadratic dilaton field with .
2.3 Linear quark potential in quenched dynamical soft-wall model
We follow the standard procedure [44, 101] to derive the static heavy quark potential in the dynamical soft-wall holographic model under the general metric background Eq.(4). In gauge theory, the interaction potential for infinity massive heavy quark antiquark is calculated from the Wilson loop
where is the gauge field, the trace is over the fundamental representation, stands for path ordering. denotes a closed loop in space-time, which is a rectangle with one direction along the time direction of length and the other space direction of length .
The Wilson loop describes the creation of a pair with distance at time and the annihilation of this pair at time . For , the expectation value of the Wilson loop behaves as . According to the holographic dictionary, the expectation value of the Wilson loop in four dimensions should be equal to the string partition function on the modified space, with the string world sheet ending on the contour at the boundary of
where is the classical world sheet Nambu-Goto action
with the 5D string tension which has dimension of , and is the induced worldsheet metric with the two indices of the world sheet coordinates (). Without loss of generality, we can choose the , and the position of one quark is and the other is . Under the background (4), the Nambu-Goto action Eq.(38) becomes
with the prime denotes the derivative with respective to .
Since there’s no dependence on , we can easily obtain the equation of motion:
for the minimum world-sheet surface configuration.
Here the is dependent on which is the maximal value of and . For the configuration mentioned above and the given equation of motion, we impose the following boundary condtions . Following the standard procedure, one can derive the interquark distance as a function of
The heavy quark potential can be worked out from the Nambu-Goto string action:
with . It is noticed that the integral in Eq.(42) in principle include a pole in the UV region (), which induces . The infinite energy should be extracted through certain regularization procedure. The divergence of is related to the vacuum energy for two static quarks. Generally speaking, the vacuum energy of two static quarks will be different in various background. In our latter calculations, we will use the regularized , which means the vacuum energy has been subtracted. A minimal subtracted result related to the background solution Eq.(19) is as following,
The integrate kernel in Eq.(43) has a pole at , and by expanding the integral kernel at one has
then the integral is dominated by region, one can obtain the string tension
Fig.7 shows the metric structure as functions of for the metric (blue dash-dotted line), and for the solutions of the quenched dynamical soft-wall model with dilaton background fields (red solid line), (black dashed line) and (cyan solid line), respectively. We can see that only for the case of positive dilaton background and , the metric solution Eq.(21) has a minimum point . Therefore, the quark-antiquark potential should have a linear part for positive quadratic dilaton background and for , which can be seen explicitly from Fig.8. While for the pure case as well as for the dynamical soft-wall model with negative dilaton background field , there doesn’t exist a where , and correspondingly the heavy quark potential does not show a linear behavior at large .
2.4 Short summary
In this section, we have modeled the pure gluon system by using the quenched dynamical soft-wall model in the graviton-dilaton framework. Comparing with the original soft-wall model with metric, here the metric background at IR is self-consistently deformed by the gluon condensate. The quartic dilaton field effect should be negligible in the confinement issue.
It is found that the positive quadratic dilaton background can give the correct glueball spectra including the Regge slope and ground state, as well as the linear quark potential, and the negative quadratic dilaton background field can be safely excluded. In the following study, we will only focus on the case of IR positive quadratic dilaton background.
3 Two flavor system: KKSS model and improved KKSS model
We now turn to the the light flavor system with chiral symmetry . As we have mentioned in the Introduction, the current achievements of AdS/QCD models for hadron spectra are the hard-wall AdS/QCD model  and the soft-wall AdS/QCD or KKSS model  and its extended version [34, 35, 36, 37]. In the hard-wall model , the chiral symmetry breaking can be realized by chiral condensation in the vacuum, however, the resulting mass spectra for the excited mesons behave as , which is different from the linear Regge behavior . In order to generate the linear Regge behavior, the authors of Ref. introduced a quadratic dilaton background, one can obtain a desired mass spectra for the excited vector mesons, while the chiral symmetry breaking phenomenon cannot consistently be realized. In the following, we firstly give a brief introduction on the KKSS model and review the meson spectra in this model.
3.1 The KKSS model
The KKSS model  has two background fields: the positive quadratic dilaton background and the metric background . Note, in the following, we will use instead of to distinguish from the pure gluon system. The background geometry is not dynamically generated but assumed to be space with the metric structure
which gives .
The mesons are described by 5D fields propagating on the background with the action given by
with . The scalar field is dual to the dimension-3 operator, and is the 5D scalar mass. According to AdS/CFT dictionary, the dimension-3 scalar has 5D mass . The field is actually a complex field to incorporate the scalar and the pseudoscalar fields,
where are in the isospin space, are the generator index. The scalar field takes a nonzero vacuum expectation value (VEV) , which is expected to realize the chiral symmetry breaking.
The Gauge fields and model the SU(2) SU(2) global chiral symmetry of QCD for two flavors of quarks, which are defined as
where and Tr. The covariant derivative becomes
To describe the vector and axial-vector fields, we simply transform the and gauge fields into the vector () and axial-vector () fields with and , one can have , with
In terms of the vector and axial-vector fields, the KKSS action Eq.(49) can be rewritten as
where the covariant derivative now becomes
3.2 Degeneration of chiral partners in KKSS model
The scalar field takes a nonzero vacuum expectation value (VEV) , which is expected to realize the chiral symmetry breaking as in the hard wall model. We will show in the following that the chiral symmetry breaking is not realized in the soft-wall model or KKSS model, and we will analyze the reason.
Scalar vacuum expectation value
The equation of motion for the scalar vacuum expectation value (VEV) defined in Eq.(50) can be deduced and takes the following form,
In the hard wall model, , the scalar VEV has the exact solution