$a_1$ meson-baryon coupling constants in the framework of soft-wall and hard-wall AdS/QCD models

TL;DR

This paper computes the $a_1$ meson-nucleon coupling constants using soft-wall and hard-wall AdS/QCD models, finding results closer to experimental data than previous studies, and also calculates the $a_1 \, \Delta\Delta$ coupling in the hard-wall model.

Contribution

It provides a detailed calculation of $a_1$ meson-baryon coupling constants within AdS/QCD models, improving agreement with experimental values and extending to $\, \Delta$-baryons.

Findings

Numerical $g_{a_1 NN}$ values are closer to experimental data.

Calculated $g_{a_1 \Delta \Delta}$ coupling in hard-wall model.

AdS/QCD models effectively describe meson-baryon interactions.

Abstract

We calculate the $a_1$ meson-nucleon coupling constant in the framework of soft-wall and hard-wall AdS/QCD models. In the bulk of AdS space were written bulk interaction Lagrangians for a minimal gauge coupling, a magnetic gauge coupling and a triple coupling. To use AdS/CFT correspondence and these bulk interaction Lagrangians we calculate the $a_1$ axial-vector meson-nucleon coupling constant in the boundary of AdS space within both models. We observe that the numerical values for the $g_{a_1 NN}$ coupling constant in the framework of both models are more close to the experimental value than results in other works. We also calculate the $g_{a_1 \Delta \Delta}$ axial-vector meson-$\Delta$-baryon coupling constant in the framework of hard-wall AdS/QCD model.

Tables

Table 1. Table 1: Numerical results for g a 1 ​ N ​ N subscript 𝑔 subscript 𝑎 1 𝑁 𝑁 g_{a_{1}NN} coupling constant in hard-wall model for m a 1 = 1.230 subscript 𝑚 subscript 𝑎 1 1.230 m_{a_{1}}=1.230  GeV, k 1 = − 0.98 subscript 𝑘 1 0.98 k_{1}=-0.98 , g Y = 9.182 subscript 𝑔 𝑌 9.182 g_{Y}=9.182 , σ = ( 0.213 ) 3 𝜎 superscript 0.213 3 \sigma=(0.213)^{3}  GeV 3 and m q = 0.0083 subscript 𝑚 𝑞 0.0083 m_{q}=0.0083  GeV.

$z_m^{-1}$	$g_{a_{1}NN}^{\left(0\right)nm}+g_{a_{1}NN}^{\left(1\right)nm}$	$g_{a_{1}NN}^{\left(2\right)nm}$	$g_{a_{1}NN}^{hw}$	$g_{a_{1}NN}^{phen}\left[27\right]$	$g_{a_{1}NN}^{sw}\left[20\right]$	$g_{a_{1}NN}^{hw}\left[11\right]$	$g_{a_1\Delta \Delta }^{\left(0\right)nm}+g_{a_1\Delta \Delta }^{\left(1\right)nm}$	$g_{a_1\Delta \Delta }^{\left(2\right)nm}$	$g_{a_1\Delta \Delta }^{hw}$
0.286	1.541	-0.0204	1.520	4.7$\pm$0.6	0.14	-2.93 (0.42)	31.641	11.074	42.715
0.205	7.306	-54.486	-47.18	4.7$\pm$0.6	0.14	-2.93 (0.42)	62.756	123	186
0.33	0.654	0.80	1.453	4.7$\pm$0.6	0.14	-2.93 (0.42)	7.214	11.949	19.163
0.4	-0.01	0.573	0.564	4.7$\pm$0.6	0.14	-2.93 (0.42)	0.194	4.43	4.623
0.5	-0.362	0.266	-0.096	4.7$\pm$0.6	0.14	-2.93 (0.42)	0.64	1.031	0.104
0.6	-0.505	0.133	-0.372	4.7$\pm$0.6	0.14	-2.93 (0.42)	-0.132	0.283	-0.457
0.7	-0.578	0.073	-0.505	4.7$\pm$0.6	0.14	-2.93 (0.42)	-0.212	0.091	-0.411
0.8	-0.622	0.044	-0.578	4.7$\pm$0.6	0.14	-2.93 (0.42)	-0.176	0.033	-0.302
0.9	-0.649	0.028	-0.621	4.7$\pm$0.6	0.14	-2.93 (0.42)	-0.131	0.013	-0.214
1	-0.668	0.018	-0.649	4.7$\pm$0.6	0.14	-2.93 (0.42)	-0.095	0.006	-0.152

Table 2. Table 2: Numerical results for g a 1 ​ N ​ N subscript 𝑔 subscript 𝑎 1 𝑁 𝑁 g_{a_{1}NN} coupling constant in soft-wall model for k = 0.383 𝑘 0.383 k=0.383 , k 1 = − 0.98 subscript 𝑘 1 0.98 k_{1}=-0.98 , g Y = 9.182 subscript 𝑔 𝑌 9.182 g_{Y}=9.182 , σ = ( 0.213 ) 3 𝜎 superscript 0.213 3 \sigma=(0.213)^{3}  GeV 3 and m q = 0.0083 subscript 𝑚 𝑞 0.0083 m_{q}=0.0083  GeV.

$g_{a_{1}NN}^{\left(0\right)nm}+g_{a_{1}NN}^{\left(1\right)nm}$	$g_{a_{1}NN}^{\left(2\right)nm}$	$g_{a_{1}NN}^{sw}$	$g_{a_{1}NN}^{phen}\left[27\right]$	$g_{a_{1}NN}^{sw}\left[20\right]$	$g_{a_{1}NN}^{hw}\left[11\right]$
-1.678	26.032	24.354	4.7$\pm$0.6	0.14	-2.93 (0.42)

Equations

$$S={\int }_0^{z_m}{d}^{4}xdz\sqrt{g}\mathcal{L}\left(x,z\right),$$

$$d{s}^2=\frac{1}{z^2}\left(-d{z}^2+{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }\right),\mu ,\nu =0,1,2,3,$$

$${\eta }_{\mu \nu }=diag(1,-1,-1,-1).$$

$$S={\int }_0^{z_m}{d}^{5}x\sqrt{g}Tr\{|DX{|}^2+3|X{|}^2-\frac{1}{4g_5^2}Tr\left[F_V^2+F_A^2\right]\}$$

$$z^3{\partial }_z\left(\frac{1}{z}{\partial }_z{\tilde{A}}_{\mu }^a(p,z)\right)+p^2{z}^2{\tilde{A}}_{\mu }^a(p,z)-g_5^2{v}^2{\tilde{A}}_{\mu }^a(p,z)=0.$$

$$\left[\frac{-m_n^2}{z}-{\partial }_z\left(\frac{1}{z}\right){\partial }_z+\frac{2g_5^2{v}^2}{z^3}\right]A_n\left(z\right)=0,$$

$$0=\left({\partial }_z+\frac{2g_5^2{v}^2}{z^2}\right)A_n\left(z\right){\mid }_{z=z_m},$$

$$A\left(z\right)=\frac{z{J}_1\left(m_{a_1}z\right)}{\sqrt{{\int }_0^{z_m}dzz{\left[J_1\left(m_{a_1}z\right)\right]}^2}}$$

$$S_{F_1}={\int }_0^{z_m}{d}^{4}xdz\sqrt{g}\left[\frac{i}{2}{\overline{\Psi }}_1{e}_A^M{\Gamma }^A{D}_M{\Psi }_1-\frac{i}{2}{\left(D_M{\Psi }_1\right)}^{†}{\Gamma }^0{e}_A^M{\Gamma }^A{\Psi }_1-m_5{\overline{\Psi }}_1{\Psi }_1\right],$$

$$\left(i{\Gamma }^A{D}_A\mp m_5\right){\Psi }_{1,2}=0,$$

$$\left(\delta {\overline{\Psi }}_1{e}_A^5{\Gamma }^A{\Psi }_1\right){|}_{\epsilon }^{z_m}=0.$$

$${\Psi }_{L,R}\left(x,z\right)=\int d^{4}p{e}^{-ip\cdot x}{f}_{L,R}\left(p,z\right){\psi }_{L,R}\left(p\right)$$

$$/p\psi \left(p\right)=\left|p\right|\psi \left(p\right),$$

$$\left({\partial }_z^2-\frac{4}{z}{\partial }_z+\frac{6\pm m_5-m_5^2}{z^2}\right)f_{L,R}=-p^2{f}_{L,R}.$$

$$f_{1L}^{(n)}(z)=c_1^n{\left(z\right)}^{\frac{5}{2}}{J}_2\left(m_{n}z\right),$$

$$f_{1R}^{(n)}(z)=c_1^n{\left(z\right)}^{\frac{5}{2}}{J}_3\left(m_{n}z\right)$$

$$|c_{1,2}|$$

$$f_{1L}=f_{2R},f_{1R}=-f_{2L},$$

$${\Psi }_{1R}\left(x,z_m\right)=0.$$

$$M_n=\frac{{\alpha }_n^{(3)}}{z_m}.$$

$$S={\int }_0^{\infty }{d}^{4}xdz\sqrt{g}{e}^{-\Phi (z)}\mathcal{L}\left(x,z\right),$$

$$S_{gauge}=-\frac{1}{4g_5^2}{\int }_0^{\infty }{d}^{5}x\sqrt{g}{e}^{-\Phi (z)}Tr\left[F_L^2+F_R^2\right]=-\frac{1}{4g_5^2}{\int }_0^{\infty }{d}^{5}x\sqrt{g}{e}^{-\Phi (z)}Tr\left[F_V^2+F_A^2\right],$$

$${\partial }_z\left[\frac{1}{z}{e}^{-k^2{z}^2}{\partial }_z{\tilde{A}}_{\mu }^{aT}(p,z)\right]+p^2\frac{1}{z}{e}^{-k^2{z}^2}{\tilde{A}}_{\mu }^{aT}(p,z)=0$$

$${\partial }_z\left(e^{-B(z)}{\partial }_z{A}_n\right)+m_n^2{e}^{-B(z)}{A}_n=0,$$

$$A_n(z)=e^{B(z)/2}{\psi }_n(z)$$

$$A_n^{soft}(z)=e^{-k^2{z}^2/2}{\left(kz\right)}^{m+1/2}\sqrt{\frac{2n!}{\left(m+n\right)!}}{L}_n^m\left(k^2{z}^2\right).$$

$$A_n^{soft}(z)=k^2{z}^2\sqrt{\frac{2}{n+1}}{L}_n^1\left(k^2{z}^2\right).$$

$$S_{F_1}^{soft}={\int }_0^{\infty }{d}^{4}xdz\sqrt{g}{e}^{-\Phi (z)}\left[\frac{i}{2}{\overline{\Psi }}_1{e}_A^N{\Gamma }^A{D}_N{\overline{\Psi }}_1-\frac{i}{2}{\left(D_N{\Psi }_1\right)}^{†}{\Gamma }^0{e}_A^N{\Gamma }^A{\Psi }_1-(M+\Phi (z)){\overline{\Psi }}_1{\Psi }_1\right],$$

$$\left[i{e}_A^N{\Gamma }^A{D}_N-\frac{i}{2}({\partial }_N\Phi )e_A^N{\Gamma }^A-(M+\Phi (z))\right]{\Psi }_1=0.$$

$$\left({\partial }_z-\frac{d/2-\Delta +\Phi +\left(M+\Phi \right)}{z}\right)f_{1R}=-p{f}_{1L},$$

$$\left({\partial }_z-\frac{d/2-\Delta +\Phi -\left(M+\Phi \right)}{z}\right)f_{1L}=p{f}_{1R}.$$

Full text

$a_{1}$-meson-baryon coupling constants in the framework of soft-wall and hard-wall AdS/QCD models

Narmin Huseynova a111e-mail : nerminh236@gmail.com, Shahin Mamedov a,b,c222e-mail : sh.mamedov62@gmail.com (corresponding author)

${}^{a}\,$Institute for Physical Problems, Baku State University,

Z. Khalilov 23, Baku, AZ-1148, Azerbaijan,

b**Theoretical Physics Department, Physics Faculty, Baku State University,

*Z.Khalilov 23, Baku, AZ-1148, Azerbaijan,

cInstitute of Physics, Azerbaijan National Academy of Sciences,*

*H. Cavid ave. 33, Baku, AZ-1143 Azerbaijan,

We calculate the $a_{1}$-meson-nucleon ($\Delta$-baryon) coupling constant in the framework of soft-wall and hard-wall AdS/QCD models. Interaction Lagrangian in the bulk of AdS space was written for a minimal gauge coupling, a magnetic type coupling and a triple coupling of bulk fields. Applying AdS/CFT correspondence from the bulk interaction Lagrangian we obtain holographic expressions for the $a_{1}$-axial-vector meson-nucleon ($\Delta$ -baryon) coupling constant in the boundary QCD theory. Numerical results for the $g_{a_{1}NN}$ coupling constant in the framework of both models are close to the known phenomenological estimates. The $g_{a_{1}\Delta\Delta}$- axial-vector meson-$\Delta$-baryon coupling constant was considered in the framework of hard-wall model.

Keywords: Holographic QCD, meson-nucleon coupling

PACS numbers: 11.25.Wx, 11.25.Tq

I introduction

Theoretical and experimental studies of the coupling constants and form-factors are one of the most important problems of hadron physics. Some theoretical approaches such as chiral quark model, QCD sum rules and etc. are used for solving this problem. The holographic principle also has important consequence in order to solve phenomenological problems of strong interaction, such as calculation of coupling and decay constants, form-factors, mass spectrum and etc.. In QCD at a low energy limit the ordinary perturbation theory does not work because of the strong coupling constant gets a high value in a small values of the transferred momentum. Thus, direct application of perturbative methods at a low energy limit to QCD is impossible. Holographic QCD does not run into such difficulties and is used to solve QCD problems without restrictions to the transferred momentum and energy region. So, the holographic QCD is considered to be very effective method in QCD at a low energies.

There are two approaches in holographic QCD: top-down and bottom-up approaches. In the top-down approach the model for QCD is based on the string and $D$-brane theories. The bottom-up approach is constructed according to direct application of the AdS/CFT principle to the theory of strongly interacting particles and named AdS/QCD models. The AdS/CFT principle is a correspondence between the fields in the 5-dimensional bulk of an anti-de Sitter (AdS) space with the field theory operators defined on the 4-dimensional ultraviolet (UV) boundary of AdS space 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 . There are two main models of AdS/QCD: hard-wall and soft-wall models. These models include nonperturbative aspects of QCD such as, chiral symmetry breaking and confinement and were constructed under the finiteness condition of the 5D action at the infrared (IR) boundary of AdS space. In the hard-wall model this condition is provided by cutoff the AdS space at this boundary 7 ; 11 ; 12 , while in the soft-wall model it is ensured by multiplying Lagrangian to extra exponential factor in the action  2 ; 6 .

After building the AdS/QCD models, the coupling constants and form-factors were calculated in the framework of these models 2 ; 6 ; 7 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 20 ; 21 .

In this paper, we calculate the $g_{a_{1}NN}$ $a_{1}$-axial-vector meson-nucleon coupling constant within the soft-wall and hard-wall AdS/QCD models and the $g_{a_{1}\Delta\Delta}$-axial-vector meson-spin 3/2 $\Delta$-baryon coupling constant in the framework of the hard-wall AdS/QCD model. Note, that this coupling constant was investigated in the framework of a hard-wall model in  11 and in the framework of a soft-wall model in 20 . In both cases, this constant was calculated using only two kinds of interaction Lagrangian terms, the minimal coupling term and Pauli type interaction term. In  21 it was added new kind of interaction term to bulk lagrangian, which produces axial-vector current of nucleons in the boundary QCD. However, this term does not possess 5D Lorentz and parity covariances, which should obey bulk interaction Lagrangian and for this reason was not considered as a physical interaction, when the axial-vector current of nucleons was obtained in the framework of AdS/QCD in  14 . Here we take into account a Lagrangian term of triple interaction of bulk fields, which characterizes the interaction between the scalar, axial-vector and fermion fields in the bulk of AdS space  14 . This term is next order small quantity and changes the chirality of nucleons similarly Yukava interaction. We have calculated numerical values of this constant in the framework of both-hard-wall and soft-wall models and within the different parameters of $z_{m}$ to determine the accuracy of the model and the effect of $z_{m}$ parameter.

We also calculate the $g_{a_{1}\Delta\Delta}$-axial-vector meson-spin 3/2 $\Delta$-baryon coupling constant taking into account triple interaction Lagrangian in the framework of hard-wall AdS/QCD model and predict the numerical values for this coupling constant. This paper is arranged as follows. First, we describe the basic features of AdS space, then we write profile functions for the axial-vector, nucleon and spin 3/2 $\Delta$-baryons in the framework of hard-wall AdS/QCD model. In Sec. 3, we describe profile functions for the axial-vector and nucleon in the framework of soft-wall AdS/QCD model. In Sec. 4, we write a Lagrangian for the axial-vector meson-nucleon and the axial-vector meson-spin 3/2 $\Delta$-baryon interactions in the bulk of AdS space and derive the boundary $g_{a_{1}NN}$ coupling constant within the soft-wall and hard-wall AdS/QCD models and $g_{a_{1}\Delta\Delta}$ coupling constant in the framework of hard-wall AdS/QCD model from the bulk Lagrangians. In Sec. 5, we present a numerical results for these coupling constants in a Table form and make comparison of the obtained values with the existing ones.

II The hard-wall model

Finiteness of the 5D action in the hard-wall model is provided by cutting off the AdS space at infrared boundary and it contains integration in cutoff AdS space  1 ; 7 ; 11 ; 22 :

[TABLE]

where $g=|\det g_{MN}|$ $(M,N=0,1,2,3,5)$ and the fifth coordinate $z$ varies in the range $\epsilon\leq z\leq z_{m}\left(\epsilon\rightarrow 0\right)$. The metric of AdS space is chosen in Poincare coordinates:

[TABLE]

where $\eta_{\mu\nu}$ is a 4-dimensional Minkowski metric  7 ; 11 ; 12 ; 22 :

[TABLE]

II.1 $a_{1}$ meson in hard-wall model

According to the holographic QCD there is a correspondence between 4-dimensional axial-vector current in the boundary theory and the 5-dimensional axial-vector field in the bulk theory. So, to obtain axial-vector-nucleon coupling constant in the boundary QCD, one needs to introduce in the bulk of AdS space two gauge fields $A_{L}^{M}$ and $A_{R}^{M}$, which transform as a left and right chiral fields under $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry group of the model 11 ; 23 . In the bulk of AdS space the scalar $X$ field is also introduced, which transform under the bifundamental representation of this symmetry group. Due to the interaction of the bulk gauge fields with the scalar field $X$ the chiral symmetry group is broken to the $SU(2)_{V}$ group. In accordance to the AdS/CFT correspondence the bulk $SU(2)_{V}$ symmetry group is the symmetry group of the dual boundary theory and the $a_{1}$ meson is described by this representation of the $SU(2)_{V}$ group. From the $A_{L}^{M}$ and $A_{R}^{M}$ gauge fields the bulk vector $V^{M}=\frac{1}{\sqrt{2}}\left(A_{L}^{M}+A_{R}^{M}\right)$ and axial-vector $A^{M}=\frac{1}{\sqrt{2}}\left(A_{L}^{M}-A_{R}^{M}\right)$ fields are constructed. Axial- vector meson states are the KK modes of the transverse part of the axial-vector gauge field.

Action for the scalar and gauge field sector has been written in terms of bulk vector and axial-vector fields is as follows  23 :

[TABLE]

where $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}-i\left[A_{M},A_{N}\right]$ is a field stress tensor, $A_{M}=A_{M}^{a}t^{a}$, $t^{a}=\sigma^{a}/2$ and $\sigma^{a}$ are Pauli matrices. The 5-dimensional coupling constant $g_{5}$ is related to the number of colors $N_{c}$ in the dual theory as $g_{5}^{2}=12\pi^{2}/N_{c}$ and for the dual boundary $SU(2)$ gauge group it has the value $g_{5}=2\pi$. Transverse part of axial-vector field at UV boundary $\left(A_{\mu}(x,z=0)\right)$ is a source for the axial-vector current and fluctuations of this part corresponds to the axial-vector mesons on the boundary. The $X\left(x,z\right)=v\left(z\right)\frac{U\left(x,z\right)}{2}$ field is a bulk scalar field, which is written in the $U\left(x,z\right)=\exp\left(2it^{a}\pi^{a}\left(x,z\right)\right)$ form and $v(z)=\frac{1}{2}am_{q}z+\frac{1}{2a}\sigma z^{3}$, where the coefficient $m_{q}$ is the mass of $u$ and $d$ quarks, the $\sigma$ is the value of chiral condensate and $a$ is the constant $a=\sqrt{N_{c}}/2\pi=\sqrt{3}/2\pi$ 24 . The coefficients $m_{q}$ and $\sigma$ were established from the UV and IR boundary conditions on the solution for the $X$ field. It is useful to write the transverse part of bulk axial-vector gauge field $A_{\perp\mu}^{a}(x,z)$ in momentum space. Equation of motion for Fourier components $\widetilde{A}_{\perp\mu}^{a}(p,z)$ is easily obtained from the action (4)  11 ; 23 and has the form:

[TABLE]

Here $\widetilde{A}_{\perp\mu}^{a}(p,z)$ can be written as $\widetilde{A}_{\perp\mu}^{a}(p,z)=A_{\mu}^{a}(p)A(p,z)$ and at IR boundary $A(p,z)$ satisfies the condition $A(p,0)=1$, $A^{\prime}(p,z_{0})=0$  11 ; 23 . For the $n$-th mode $A_{n}(z)$ in the Kaluza-Klein decomposition $A(p,z)=\sum\limits_{n=0}^{\infty}A_{n}(z)f_{n}(p)$ with $p^{2}=m_{n}^{2}$ the equation (5) has the form  11 :

[TABLE]

This equation of motion for the axial-vector field has a z-dependent mass term, so that it can not be analytically solved. In the approximation the EOM is the same with vector meson, where the bulk mass is supposed to be the brane localized mass at QCD brane$\left(z=z_{m}\right)$. The boundary condition at $\left(z=z_{m}\right)$ is as follow 11 :

[TABLE]

by using from the boundary condition $\left(8\right)$ normalized wave function for the axial vector meson $a_{1}$ was found as  11 :

[TABLE]

II.2 Nucleons in hard-wall model

In the framework of hard-wall model nucleons were introduced in 7 ; 22 . In order to represent nucleon operators in the boundary of the AdS space it is necessary to introduce $\Psi_{1}(x,z)$ and $\Psi_{2}(x,z)$ two spinor fields in order to describe the left- and right-handed components of nucleons in boundary QCD, which transform differently under the $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry group. Extra two chiral components vanish applying boundary conditions at IR boundary  7 ; 11 ; 22 . The action for the bulk $\Psi_{1}(x,z)$ spinor field is written as follows:

[TABLE]

where $D_{B}=\partial_{B}-\frac{i}{4}\omega_{B}^{MN}\Sigma_{MN}-i\left(A_{L}^{a}\right)_{B}T^{a}$ is the covariant derivative, $\Sigma_{MN}=\frac{1}{i}\Gamma_{MN}$ and $\Gamma_{MN}=\frac{1}{2}\left[\Gamma_{M},\Gamma_{N}\right]$, $e_{M}^{A}$ is a vielbein and chosen as $e_{M}^{A}=\frac{1}{z}\eta_{M}^{A}$ and the non-zero components of spin connection $\omega_{B}^{MN}$ are $\omega_{\mu}^{5A}=-\omega_{\mu}^{A5}=\frac{1}{z}\delta_{\mu}^{A}$ $\left(\mu=0,1,2,3\right)$. From the action (9) the equation of motion is obtained as a Dirac equations in the AdS space:

[TABLE]

The boundary term arising on obtaining the equation of motion is written as

[TABLE]

Here, $\Psi_{z}=0$ is a condition to eliminate an extra $\Psi_{z}$ degree of freedom  7 ; 22 . $\gamma^{5}\Psi_{L}=\Psi_{L}$ and $\gamma^{5}\Psi_{R}=-\Psi_{R}$ is a properties the left- and right-handed components of the spinor fields. Fourier transformation for $\Psi_{L,R}$ is written as follow:

[TABLE]

where $\psi\left(p\right)$ is a 4D spinor field and obeys the 4D Dirac equation

[TABLE]

where, $\left|p\right|=\sqrt{p^{2}}$ for a time-like four-momentum $p$. The 5D Dirac equation (10) will lead to equations over the fifth coordinate $z$ for the $f_{L,R}$ amplitudes:

[TABLE]

The $n$-th normalized Kaluza-Klein mode $f^{(n)}_{L,R}\left(z\right)$ of the solutions $f_{L,R}$ with $p^{2}=m_{n}^{2}$ can be expressed in terms of Bessel functions:

[TABLE]

The constants $c_{1}^{n}$ are found from the normalization condition and are equal to

[TABLE]

There exist the following relations between the profile functions of first and second bulk fermion fields 7 ; 11 ; 22 :

[TABLE]

For obtaining only a left-handed component of the nucleon from $\Psi_{1}$ the right-handed component of this spinor is eliminated by the boundary condition at $z=z_{m}$:

[TABLE]

This condition gives the Kaluza-Klein mass spectrum $M_{n}$ of excited states, which is expressed in terms of zeros $\alpha_{n}^{(3)}$ of the Bessel function $J_{3}$:

[TABLE]

The quantum number $n$ corresponds to the excitation number of a nucleon in the dual boundary theory.

III The soft-wall model

As was noted in introduction, in the soft-wall model the finiteness of the 5-dimensional action is provided by introducing exponential factor of the extra dimension  2 ; 6 ; 12 and the action for the soft-wall model is written in the form:

[TABLE]

where $\Phi\left(z\right)=k^{2}z^{2}$ is the dilaton field  2 ; 6 ; 12 . Let us present profile functions of axial-vector and spinor fields in this model.

III.1 ${a_{1}}$ meson in soft-wall model

In the framework of this model an action for the gauge fields $A_{L}^{M}$ and $A_{R}^{M}$ is written in terms of bulk vector and axial-vector fields as  2 ; 6 ; 12 :

[TABLE]

Equation of motion for Fourier components of transversal part $\widetilde{A}_{\mu}^{aT}(p,z)$ of axial-vector field is easily obtained from the action (21) 20 ; 25 :

[TABLE]

and the $\widetilde{A}_{\mu}^{aT}(p,z)$ can be written as $\widetilde{A}_{\mu}^{aT}(p,z)=A_{\mu}^{a}(p)A(p,z)$. The $A(p,z)$ profile function satisfies the condition $A(p,\epsilon)=1$ at UV boundary. In the Kaluza-Klein decomposition $A(p,z)=\sum\limits_{n=0}^{\infty}A_{n}(z)f_{n}(p)$ for the $n$-th mode $A_{n}(z)$ with mass $m_{n}^{2}=p^{2}$ the equation (22) is written as follows:

[TABLE]

where $B(z)=\Phi(z)-A(z)=k^{2}z^{2}+\ln z$. After substitution

[TABLE]

the equation (23) is converted to the Schroedinger equation form and has a solution given in terms of Laguerre polynomials $L_{m}^{n}$  20 ; 25 :

[TABLE]

For the eigenvalues $m_{n}^{2}$ there is a linear dependence on the number $n$: $m_{n}^{2}=4k^{2}(n+2)$, which enables us to fix the free parameter $k$. In the AdS/CFT correspondence $m_{n}^{2}$ is identified with the mass spectrum of the vector mesons in the dual boundary QCD. For the $a_{1}$-meson we have $m=1$ and the $A_{n}(z)$ becomes

[TABLE]

Notice, that solution (26) was obtained for the free axial vector field case and does not take into account the backreaction of bulk spinor field describing nucleons.

III.2 Nucleons in soft-wall model

In the framework of the soft-wall model nucleons were introduced in 21 and their excited states within this model were considered in 28 . We present here some formulas for the spinor field in the soft-wall model. Within soft-wall model the Lagrangian for the spinor field contains additional term $\Phi\overline{\Psi}\Psi$, which describe coupling of a dilaton field with the bulk fermion fields  2 and the sign of this term for the second fermion field is chosen oppositely to the one for the first fermion 7 . The action for the first free spinor field is written as follows:

[TABLE]

The equation of motion for the spinor field obtained from the action (27) is written as follows:

[TABLE]

In order to solve equation (28) for $\Psi$ the left- and right-handed components $\Psi_{L,R}=\left(1/2\right)\left(1\mp\gamma^{5}\right)\Psi$ in momentum space $\Psi_{L,R}$ were written like this: $\Psi_{L,R}(p,z)=z^{\Delta}\Psi_{L,R}^{0}(p)f_{L,R}(p,z)$. The equation  (28) now is written as follows in 2 :

[TABLE]

Using relation $\Delta=\frac{d}{2}-M$ ($d=4$ in our model) in (29), the differential equations were obtained for the profile functions $f_{L,R}(p,z)$ 2

[TABLE]

After solving the second order equation of motions (30) with $p^{2}=m_{n}^{2}$, we get the expression for the $f^{(n)}_{L,R}\left(z\right)$ $n$-th normalized Kaluza-Klein modes which correspond to the nucleons state:

[TABLE]

where the $L_{n}^{(\alpha)}$ are the Laguerre polynomials, $\alpha$ is related to the 5-dimensional mass $M$ as $\alpha=M+\frac{1}{2}$, the mass of the $n$-th mode $m_{n}^{2}=4k^{2}\left(n+\alpha\right)$ and the constants $n_{L,R}$ are found from the normalization condition $\int dz\frac{e^{-k^{2}z^{2}}}{z^{2M}}f_{1L}^{(n)}f_{1L}^{(m)}=\delta_{nm}$ and are equal to

[TABLE]

where $\alpha=2$, because $M=\frac{3}{2}$ and $f_{1L}=f_{2R},\quad f_{1R}=-f_{2L}$, is a relations between the profile functions of the first and second bulk fermion fields 7

IV Bulk interaction Lagrangians and the $g_{a_{1}NN}$ and the $g_{a_{1}\Delta\Delta}$ coupling constants

In this section we derive the $g_{a_{1}NN}$ coupling constant in both models using AdS/CFT correspondence and then the $g_{a_{1}\Delta\Delta}$ coupling constant in the framework of hard-wall model of AdS/QCD.

Let us consider the $g_{a_{1}NN}$ constant at first. Then we can repeat all calculation steps for the $g_{a_{1}\Delta\Delta}$ constant and give final expressions and the numerical results for these constants.

IV.1 The $g_{a_{1}NN}$ coupling constant in the framework of hard-wall model

In order to derive the $g_{a_{1}NN}$ axial-vector meson-nucleon coupling constant we use the 5D action for the interaction between the axial-vector, fermion and scalar fields in the bulk of the AdS space. From the generating function the 4D axial-vector current of nucleons will be obtained. Let us write the interaction action in the bulk of AdS space:

[TABLE]

According to holographic principle the 4-dimensional axial-vector current of nucleons is found by taking variation from the generating function $Z$ for the vacuum expectation values of the axial-vector field in dual theory boundary. In our case this principle will be written as:

[TABLE]

where $Z_{QCD}=e^{iS_{int}}$, $\tilde{A}_{\mu}^{0}=\tilde{A}_{\mu}(q,z=0)=A_{\mu}(q)$ is the boundary value of the axial-vector field $\left(A(z=0)=1\right)$ and $J_{\mu}$ is obtained by the variation of $exp\left(iS_{int}\right)$ which will be identified with the nucleon current. In other hand, in the boundary of AdS space, where QCD was defined, the 4D current of axial-vector current is written as follows:

[TABLE]

where, $q=p^{\prime}-p$ is an energy-momentum conservation in the interaction vertex of $a_{1}$ meson and the nucleons. From the equivalence of the right-hand side of (34) and (35) we get the formula for the $g_{a_{1}NN}$ coupling constant as an integral expression over $z$ when two currents, the fermionic current on the boundary and the nucleon current are identified according to AdS/CFT correspondence.

Now, we need an explicit form of $\mathcal{L}_{int}$ interaction Lagrangian in the bulk of AdS space. The interaction Lagrangian is constructed based on the gauge invariance of using the model and include different kinds of interaction terms between the bulk fields  11 ; 21 ; 22 . As was noted above this coupling constant was calculated using only two terms - gauge interaction Lagranian term contained in the covariant derivative of action and Pauli interaction Lagranian terms, within a hard-wall model 11 and within a soft-wall model  20 . In addition this two interaction terms, using a new triple interaction term, which was introduced in  14 similar to the Yukava interaction term we have calculated $a_{1}$ axial-vector meson-nucleon coupling constant within both of soft-wall and hard-wall models. Thus, we use the following three interaction Lagranian terms:

1. a term of a minimal gauge interaction of the axial-vector field with the current of fermions  11 ; 22

[TABLE]

2)a magnetic gauge coupling of spinors with axial-vector field  11 ; 20 ; 22

[TABLE]

where the field stress tensor of the axial-vector field is $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$.

3. a triple interaction term, 14 :

[TABLE]

After substitution the interaction Lagrangian terms (36),(37),(38) in the action for interaction (36), then performing the integrals in momentum space and applying the holography principle this Lagrangian term gives the following contributions to $g_{a_{1}NN}^{h.w.}$ constant represented in terms of integral over the $z$. We carry out calculation for each interaction Lagrangian terms separately and get following formulas for $g_{a_{1}NN}^{h.w.}$ coupling constant within hard-wall model:

[TABLE]

where the $A_{0}$ is the profile function of the axial-vector field, $f_{1,2L,R}$ are the profile function of the nucleons and the superscript indices $n$ and $m$ indicate the number of excited states of the initial and final nucleons respectively.

[TABLE]

[TABLE]

Thus, the $g_{a_{1}NN}$ constant will be sum of three terms:

[TABLE]

We carry out a numerical analysis of these terms separately in Sec. 5.

IV.2 The $g_{a_{1}NN}$ coupling constant within soft-wall model

In the framework of soft-wall model, the 5D action for the interaction between the axial-vector, fermion and scalar fields in the bulk of the AdS space is written as

[TABLE]

After substitution the interaction Lagrangian terms (36), (37), (38) in the interaction Laqrangian in (43), then performing the integrals in momentum space and applying the holography principle the following contributions to $g_{a_{1}NN}$ constant will be represented in terms of integral over the $z$. We carry out separate calculations for each interaction Lagrangian term and get following formulas for $g_{a_{1}NN^{s.w.}}$ coupling constant within soft-wall model:

[TABLE]

where the $A_{0}$ is the profile function of the axial-vector meson expressed as (26), $f_{1,2L,R}^{s}$ are the profile function of the nucleons within soft-wall model of AdS/QCD expressed as (31) and the superscript indices $n$ and $m$ indicate the number of excited states of the initial and final nucleons respectively:

[TABLE]

[TABLE]

Thus, we describe the $g_{a_{1}NN^{s.w.}}$ constant as the sum of three terms:

[TABLE]

We carry out a numerical analysis of these terms separately in Sec. 5.

IV.3 The $g_{a_{1}\Delta\Delta}$ coupling constant in the framework of hard-wall model

Similar to the $g_{a_{1}NN}$ constant in the framework of hard-wall model, we study the $g_{a_{1}\Delta\Delta}$-axial-vector meson-spin 3/2 $\Delta$-baryon coupling constant in the framework of hard-wall AdS/QCD model and predict the numerical values for this coupling constant. We have used interaction Lagrangian terms for this coupling constant similar to ones for the $g_{a_{1}NN}$ constant:

[TABLE]

[TABLE]

[TABLE]

Boundary spin 3/2 baryons is described by the $\Psi_{1,2}^{N}$ and $\Psi_{1,2N}$ Rarita-Schwinger fields in the bulk of AdS space. In the same way by using (33)-(35) and the interaction Lagrangian terms, we get the following contributions for $g_{a_{1}\Delta\Delta}$ constant. We carry out separately calculations for each interaction Lagrangian terms here too:

[TABLE]

where $F_{1,2L,R}$ are the profile function of the $\Delta$ baryons and is given in  7 ; 11 ; 22 :

[TABLE]

and constants $C_{1}^{n}$ are found from the normalization condition:

[TABLE]

Other two contributions are equal to following ones:

[TABLE]

[TABLE]

Thus, we describe the $g_{a_{1}\Delta\Delta}$ constant as the sum of three terms:

[TABLE]

V Numerical analysis

To carry out a numerical analysis for $g_{a_{1NN}}^{hw}$ coupling constant, we calculate the integrals for the expression $g_{a_{1}NN}^{(0)nmhw}$, $g_{a_{1}NN}^{(1)nmhw}$ and $g_{a_{1}NN}^{(2)nmhw}$, corresponding to (39)-(41) and for $g_{a_{1NN}}^{sw}$ coupling constant, we calculate the integrals for the $g_{a_{1}NN}^{(0)nmsw}$, $g_{a_{1}NN}^{(1)nmsw}$ constants and $g_{a_{1}NN}^{(2)nmsw}$, corresponding (44)-(46) expressions. To this end, we have used the Matematica package and compared our results with the experimental and the other results obtained within the soft-wall and hard-wall models of AdS/QCD. We use following fittings for the free parameters $k_{1}$, $g_{Y}$, $m_{q}$ and $\sigma$. The parameter $k_{1}=-0.98$ was fixed from the fitting $g_{\rho NN}$ and $g_{\pi NN}$ coupling constants results within hard-wall model with the experimental data 22 . The constant $g_{Y}$=9.182 was fixed and taken from  26 . The parameters $\sigma=(0.213)^{3}$ and $m_{q}=0.0083$ were fixed and taken from  24 as. All the parameters of values and constants presented here are in GeV units and are same for all coupling constants.

$g_{a_{1}NN}^{phen}=4.7\pm 0.6$ value is given in  27 on the basis of phenomenological estimates. The hard-wall value $g_{a_{1}NN}^{hw}=1.5\sim 4.5$ was taken from  11 , the value $g_{a_{1}NN}^{hw}=-2.93$ was also taken from  11 and the value $g_{a_{1}NN}^{hw}=0.42$ was taken from  11 . The soft-wall value $g_{a_{1}NN}^{sw}=0.14$ was taken from  20 . In order to compare the contribution of triple interaction term with the sum of minimal coupling and magnetic type terms, we present separately the results for $g_{a_{1}NN}^{(0)nm}+g_{a_{1}NN}^{(1)nm}$ and $g_{a_{1}NN}^{(2)nm}$ coupling constants in the framework of hard-wall model. The numerical results for this constant is given in TABLE 1. The numerical results in the framework of soft-wall model is given in the TABLE 2.

Comparison of results shows that for all values of parameters, the results obtained here are close to other results. We also notice that the our result for $g_{a_{1}NN}$ coupling constant is more sensitive to the value of parameter $z_{m}$. Unfortunately, there are no experimental data for the $g_{a_{1}NN}$ coupling constant in the case of excited states of nucleons, so we cannot compare our results with them.

In order to find the value of $g_{a_{1}\Delta\Delta}$ coupling constant, we need to calculate integrals for the $g_{a_{1}\Delta\Delta}^{(0)nm}+g_{a_{1}\Delta\Delta}^{(1)nm}$ and $g_{a_{1}\Delta\Delta}^{(2)nm}$ appearing in equations (51), (54) and (55). We give the numerical results for this constant in TABLE 1.

VI summary

In present letter we calculated the strong coupling constant of $a_{1}$ mesons with nucleons within the hard-wall and soft-wall models AdS/QCD. We found that the predictions of these models for $g_{a_{1}NN}$ coupling constant are close to phenomenological estimate for it. The $g_{a_{1}NN}$ coupling constant is sensitive to the value of parameter $z_{m}$ within hard-wall model of AdS/QCD. Unfortunately, there is no experimental or phenomenological data for the $g_{a_{1}NN}$ coupling constant in the case of excited states of nucleons, so we could not compare our results for this case.

We also calculated the $g_{a_{1}\Delta\Delta}$-axial-vector meson-spin 3/2 $\Delta$-baryon coupling constant in the framework of hard-wall AdS/QCD model and predict the numerical values for this coupling constant.