# Tensor meson couplings in AdS/QCD

Shahin Mamedov 111<EMAIL_ADDRESS>0000-0002-0261-3914 Institute for
Physical Problems, Baku State University, Z.Khalilov 23, Baku, AZ-1148,
Azerbaijan  Institute of Physics, Ministry of Science and Education, H.Javid
33, Baku, AZ 1143, Azerbaijan  Center for Theoretical Physics, Khazar
University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan  Zeynab Hashimli 222
<EMAIL_ADDRESS>0009-0002-1230-8590 Institute of Physics, Ministry of
Science and Education, H.Javid 33, Baku, AZ 1143, Azerbaijan  Center for
Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096,
Azerbaijan  Shahriyar Jafarzade 333<EMAIL_ADDRESS>0000-0002-4353-0760 Institute of Physics, Jan Kochanowski University, ul.
Uniwersytecka 7, 25-406, Kielce, Poland  Institute of Radiation Problems,
Ministry of Science and Education, B.Vahabzade 9, AZ1143, Baku, Azerbaijan
Department of Physics and Chemistry, “Composite Materials” Scientific Research
Center, Azerbaijan State Economic University (UNEC), H. Aliyev 135, AZ1063,
Baku, Azerbaijan

###### Abstract

We study the hadronic and radiative couplings of the $f_{2}(1270)$ meson
within the hard- and soft-wall models of AdS/QCD. The results for the tensor
meson-nucleon-nucleon coupling ($g_{f_{2}NN}$) and tensor meson-photon-vector
meson coupling ($g_{f_{2}\gamma\rho}$) are compared to the ones obtained by
using the dispersion relations and amplitude methods, respectively.
Qualitative agreement with different analyses implies the reliability of the
holographic description of spin-2 meson.

##  I Introduction

Quantum chromodynamics (QCD) describes the strong interaction within the
Standard Model of particle physics. The binding of quarks into nucleons
(protons and neutrons) and other hadrons, such as mesons composed of a quark-
antiquark pair, is accomplished through strong interactions. One of the main
challenges of QCD is the lack of a small dimensionless variable that would
allow for the perturbative calculation of low-energy observables. Lattice QCD
simulations and effective models are alternative approaches to studying low-
energy QCD. Phenomenological bottom-up approaches of holographic QCD are one
of the effective approaches to low energy QCD which presume that the extra-
dimensional fields propagate through a region of anti-de Sitter (AdS) space
inspired by the AdS/CFT correspondence [1].

The holographic description of QCD observables was originally constructed to
test the hadrons, including pseudoscalar, vector, and axial-vector mesons
within the so-called hard-wall model (HW) of AdS/QCD [2]. Three main
experimental measurements, such as pion mass and decay constant and the mass
of $\rho(770)$, were used to predict other quantities of the mesons within the
model.

Later on, the application of holographic QCD is extended to the spin-2 case
within the HW model, in which tensor mesons were introduced similarly to
gravitons [3]. The equation of motion for the profile function of the
$f_{2}(1270)$ meson was solved in this approach, and its decay into two pions
and two photons was evaluated. Higher spin mesons, including $f_{2}(1270)$,
were investigated within the SW model in [4].

Nucleons within the HW model were studied in Ref. [5]. Further investigations
include the coupling between mesons and nucleons, such as the nucleon coupling
with the vector meson in Ref. [6], with the axial-vector meson in Ref. [7] and
with the pseudoscalar meson coupling in Ref.[8]. Similar studies can be found
in Refs. [9, 10, 11, 12] and at finite temperature case in Refs.[13, 14].

The resonance $f_{2}(1270)$ is the spin-$2$ meson with the positive parity (P)
and charge conjugation (C) and the total spin number $J=2$. The lightest
spin-$2$ mesons
$\\{a_{2}(1320),K_{2}^{\star}(1430),f_{2}(1270),f_{2}(1525)\\}$ make up a
multiplet with quantum numbers $J^{PC}=2^{++}$ and has a solid experimental
[15], theoretical [16] and phenomenological [17] foundation compatible with
the quark model. Recent analyses of spin-2 mesons can be found e.g., in Refs
[18, 19, 20]. It has been studied within chiral perturbation theory in [21].
Especially, the mass and two-photon decay of $f_{2}(1270)$ estimates within
holographic QCD fit the Particle Data Group (PDG) [15] very well.
Gravitational form factors of nucleons within AdS/QCD approach are computed in
Refs.[22, 23, 24] and extension to the finite temperature see Ref.[25]. To
what extent can holographic descriptions effectively portray various
couplings? We address this question in this paper.

We first study the tensor meson nucleon coupling constant ($g_{f_{2}NN}$)
within the hard- and soft-wall models of AdS/QCD and compare it to other known
theoretical models, such as Dispersion Relation (DR). Secondly, we estimate
the tensor-vector-vector coupling ($g_{f_{2}\rho\rho}$) and the tensor-vector-
photon coupling ($g_{f_{2}\gamma\rho}$) and compare the latter one to the
results obtained from the CLAS experiment data [26] within amplitude methods.

The organization of the paper is as follows: Holographic formulations of
mesons and nucleons are presented in Sec. II and Sec. III, respectively.
Coupling constants between the tensor meson and nucleons are in Sec. IV. Sec.
V is devoted to tensor-vector-vector and tensor-vector-photon couplings. Our
conclusion is in Sec. VI. An alternative derivation of the tensor meson
profile function is discussed in App.A, and a flavor-invariant form of the
Lagrangian is presented in App.B.

##  II Mesons in Ads/QCD

Background gravity for the AdS/QCD models is the 5D AdS spacetime, and the
metric of it in Poincare coordinates has the following form:

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\Big{(}-dz^{2}+dx^{\mu}dx_{\mu}\Big{)}\quad(\mu=0,1,2,3),$
(1)

where extra $z$ coordinate extends $0<z\leq z_{m}=323\,\text{GeV}^{-1}$ in the
HW model and $0<z<\infty$ in the SW model. The action of the model we are
interested in has four components: free tensor field $h$, vector field $v$,
spinor field $N$, and their interactions:

$\displaystyle S=S_{h}+S_{v}+S_{N}+S_{int}.$ (2)

In the next sections, we shall present the profile functions of these fields
and perform calculations of the couplings within both hard- and soft-wall
models.

###  II.1 Tensor and vector mesons in the HW model

Tensor mesons in AdS/QCD were introduced in an analogy with the graviton field
in the bulk of the AdS spacetime [3], and here we present this solution
briefly. Let us consider the 5D extension of the massive spin-2 field
$h_{MN}(x,z)$ $(M,N=0,1,2,3,5)$. The gauge condition on fifth components of
$h_{MN}$ implies

$\displaystyle h_{MN}=\begin{dcases}0,&\text{if }$M$\,\text{or}\,$N=5$\\\
h_{\mu\nu},&\text{otherwise}\end{dcases}$ (3)

The linear perturbation of gravity around the AdS background is given by the
$h_{\mu\nu}(x,z)$ field:

$\displaystyle
ds^{2}=\dfrac{1}{z^{2}}(\eta_{\mu\nu}+h_{\mu\nu})dx^{\mu}dx^{\nu}-\dfrac{1}{z^{2}}dz^{2}\,,$
(4)

leading to the following action:

$\displaystyle S_{h}^{HW}=-\dfrac{1}{4}\int
d^{5}x\dfrac{1}{z^{3}}\Big{[}(\partial_{z}h_{\mu\nu})(\partial_{z}h^{\mu\nu})+h_{\mu\nu}\Box
h^{\mu\nu}+\text{higher order terms}\Big{]}.$ (5)

The UV boundary value of $h_{\mu\nu}$ corresponds to the wave function of the
graviton. The first mode in the Kaluza-Klein (KK) decomposition of
$h_{\mu\nu}$ will be the lightest tensor meson listed in the introduction,
i.e., the $f_{2}(1270)$ meson. QCD predicts the existence of multiple isospin
singlets spin-2 particles 444 See for instance Ref.[27] for the recent
detailed discussion about spin-2 isoscalars., such as glueballs. Therefore, a
comprehensive formulation of the spin-2 particles should contain various bulk
spin-2 fields. Each spin-2 particle appears as a KK mode
$\Big{(}h_{\mu\nu}(x,z)=\sum_{n}h_{\mu\nu}^{(n)}(x)h^{HW}_{n}(z)\Big{)}$ where
the profile function $h^{HW}_{n}(z)$ satisfy the following equation of motion:

$\displaystyle\partial^{2}_{z}h^{HW}_{n}(z)-\dfrac{3}{z}\partial_{z}h^{HW}_{n}(z)+(m_{n}^{h})^{2}h^{HW}_{n}(z)=0.$
(6)

The generic solution to this equation is expressed in terms of Bessel
functions of the first kind $J_{2}$ and the second kind $Y_{2}$ [3]:

$\displaystyle
h_{n}^{HW}(z)=N_{n}z^{2}[J_{2}(m_{n}^{h}z)+\beta_{n}Y_{2}(m_{n}^{h}z)].$ (7)

UV boundary condition $h^{HW}(0)=0$ implies $\beta_{n}=0$ and the IR boundary
condition $h^{HW\prime}(z_{m})=0$ quantize the tensor meson masses according
to $J_{1}(m_{n}^{h}z_{m})=0$. Thus the profile function for the $f_{2}(1270)$
meson is

$\displaystyle
h^{HW}(z)=3.51\frac{z^{2}}{z_{m}}J_{2}(3.83\frac{z}{z_{m}})\text{ with the
normalization: }\int_{0}^{z_{m}}dz\Big{(}h^{HW}_{n}(z)\Big{)}^{2}/z^{3}=1.$
(8)

The $f_{2}(1270)$ mass is predicted by the IR boundary condition of this
solution and equals to $3.83/z_{m}=1236$ MeV, which is only $3\%$ off of the
observed mass in [15]. However, predictions for the two pion decays are
underestimated compared to the PDG [15].

Vector mesons in the HW AdS/QCD are introduced using a 5D vector field
composed of the left and right gauge fields. The action for the $V_{M}$ vector
field is

$\displaystyle S_{v}^{HW}=-\frac{1}{4g_{5}^{2}}\int
d^{5}x\sqrt{g}V^{MN}V_{MN}=-\dfrac{1}{4g_{5}^{2}}\int
d^{5}x(V^{2}_{\mu\nu}-2V^{2}_{\mu 5})/z\,,$ (9)

where
$V_{\mu\nu}:=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}-i[V_{\mu},V_{\nu}]$,
$V_{\mu}:=V_{\mu}^{a}t^{a}$ and the coupling
$g_{5}=\sqrt{4\pi^{2}N_{f}/N_{c}}$ was obtained in Ref.[2]. The equation of
motion for KK modes
$\Big{(}V_{\mu}(x,z)=\sum_{n}V_{\mu}^{(n)}(x)v^{HW}_{n}(z)\Big{)}$ of the
vector field in the HW model has the following form

$\displaystyle\partial^{2}_{z}v^{HW}_{n}(z)-\dfrac{1}{z}\partial_{z}v^{HW}_{n}(z)+(m^{v}_{n})^{2}v^{HW}_{n}(z)=0\,,$
(10)

and solution to this equation is also expressed in terms of Bessel functions:

$\displaystyle
v^{HW}_{n}(z)=N_{n}z\Big{[}J_{1}(m^{v}_{n}z)+\beta_{n}Y_{1}(m^{v}_{n}z)\Big{]}.$
(11)

One can use boundary conditions $v^{HW\prime}(z_{m})=v^{HW}(0)=0$, which sets
$\beta_{n}=0$ and quantize the masses of the vector mesons according to the IR
condition $J_{0}(m^{V}_{n}z_{m})=0$. Normalization is set by

$\displaystyle\int_{0}^{z_{m}}\dfrac{dz}{z}\Big{(}v^{HW}_{n}(z)\Big{)}^{2}=1\,,$
(12)

and the ground state profile function of $\rho$ meson reads

$\displaystyle v^{HW}(z)\equiv
v^{HW}_{0}(z)=2.72\dfrac{z}{z_{m}}J_{1}(2.4\dfrac{z}{z_{m}})\,,$ (13)

where $z_{m}=(323\text{MeV})^{-1}$ is fixed with experimental mass of
$\rho(770)$ meson.

###  II.2 Tensor and vector meson profile functions in the SW model

In the SW model, the spin-$S$ mesons are described by the rank-$S$ tensor
field $\phi^{\mu_{1}...\mu_{S}}(x,z)$ in the bulk of the 5D AdS space and the
quadratic part of the gauge and coordinate invariant action for this field is
written in the form [4]:

$\displaystyle S_{v/h}^{SW}=\dfrac{1}{2}\int
d^{5}x\,\exp{-\Big{(}\Phi(z)-A(z)(2S-1)\Big{)}}\partial_{N}\phi_{\mu_{1}...\mu_{S}}\partial^{N}\phi^{\mu_{1}...\mu_{S}}.$
(14)

Here $A$ denotes $A(z)=-\log kz$ and the dilaton field $\Phi(z)=k^{2}z^{2}$ is
introduced to regulate the integral over the extra dimension $z$ and to give a
meson spectra agreeing with Regge trajectory. The 5D equation of motion for
the KK modes $\phi_{n}$ of the transverse traceless part of the
$\phi_{\mu_{1}...\mu_{S}}$ field
$\Big{(}\phi_{\mu_{1}...\mu_{S}}(x,z)=\sum_{n}\phi_{n}(x,z)\phi^{(n)}_{\mu_{1}...\mu_{S}}(x)\Big{)}$
can be easily derived from this action by considering
$\Box\phi^{(n)}_{\mu_{1}...\mu_{S}}(x)=m^{2}_{n}\phi^{(n)}_{\mu_{1}...\mu_{S}}(x)$:

$\displaystyle\partial_{z}(e^{-(\Phi(z)-A(z)(2S-1))}\partial_{z}\phi_{n}(z))+m^{2}_{n}e^{-(\Phi(z)-A(z)(2S-1))}\phi_{n}(z)=0.$
(15)

By making substitution
$\phi_{n}(z)=e^{{-(\Phi(z)-A(z)(2S-1))}/2}\tilde{\phi}_{n}(z)$ in the equation
it gives the Schrödinger-like equation form for $\tilde{\phi}$

$\displaystyle-\tilde{\phi}^{\prime\prime}_{n}(z)+\Big{[}k^{4}z^{2}+2(S-1)k^{2}+\dfrac{4S^{2}-1}{4z^{2}}\Big{]}\tilde{\phi}_{n}(z)=E\tilde{\phi}_{n}(z).$
(16)

Solving the Eq. (16) will give the eigenfunctions, which are so-called the
profile functions for the spin-$S$ mesons, and the corresponding eigenvalues
are

$\displaystyle E=4n-2S+2\,,$ (17)

with the mass spectrum

$\displaystyle m_{n,S}^{2}=4k^{2}(n+S).$ (18)

  1. 1.

For the vector meson ($S=1$) the equation of motion (16) gets a form:

$\displaystyle-\tilde{\phi}^{\prime\prime}_{n}(z)+\Big{[}k^{4}z^{2}+\dfrac{3}{4z^{2}}\Big{]}\tilde{\phi}_{n}(z)=m_{n}^{2}\tilde{\phi}_{n}(z).$
(19)

By solving this equation with the substitution
$v_{n}^{SW}(z)=\sqrt{kz}e^{k^{2}z^{2}/2}\tilde{\phi}_{n}(z)$ the profile
function in terms of Laguerre polynomials ($L_{n}^{1}(x)$) reads:

$\displaystyle
v^{SW}_{n}(z)=\sqrt{\dfrac{2n!}{(n+1)!}}k^{2}z^{2}L_{n}^{1}(k^{2}z^{2}).$ (20)

Note that, parameter $k$ is fixed according to the mass of $\rho(770)$ which
leads $k=0.388\,\text{GeV}$.

  2. 2.

For the tensor meson ($S=2$) the equation of motion (16) becomes

$\displaystyle-\tilde{\phi}^{\prime\prime}_{n}(z)+\Big{[}2k^{2}+k^{4}z^{2}+\dfrac{15}{4z^{2}}\Big{]}\tilde{\phi}_{n}(z)=m_{n}^{2}\tilde{\phi}_{n}(z)\,,$
(21)

and the substitution
$h^{SW}_{n}(z)=\sqrt{(kz)^{3}}e^{k^{2}z^{2}/2}\tilde{\phi}_{n}(z)$ leads to
the following profile function:

$\displaystyle
h_{n}^{SW}(z)=(kz)^{4}\sqrt{\frac{2n!}{(n+2)!}}L_{n}^{2}(k^{2}z^{2}),$ (22)

which has the normalization:

$\displaystyle\int_{0}^{\infty}dze^{-k^{2}z^{2}}\,\Big{(}h_{n}^{SW}(z)\Big{)}^{2}/z^{3}=1.$
(23)

In the appendix A, we show that the solution for the tensor meson profile
function in (22) is the same as the one derived in [22] if one approximates
the spin-2 tensor meson as a graviton.

##  III Nucleons in Ads/QCD

###  III.1 Nucleon profile function in the HW model

Two bulk fermion fields, $N_{1}^{HW}$ and $N_{2}^{HW}$ are required in the AdS
space-time to describe the independent left and right chiral components of the
boundary nucleon [5]. The Lagrangian for these spinors has different signs for
the ”5-dimensional mass” $m_{5}$, which is related to the conformal dimension
of the spinor: $m_{5}=\dfrac{5}{2}$ for $N_{1}^{HW}$ and $m_{5}=-\dfrac{5}{2}$
for $N_{2}^{HW}$. In light of the normalizability and chirality analyses,
$N_{1}^{HW}$ describes the left-handed component of the boundary nucleon,
while $N_{2}^{HW}$ describes the right-handed one. The IR boundary condition
eliminates the remaining components of the spinors and gives the mass spectrum
of the nucleon $m_{n}$. The bulk action for the Dirac spinors $N_{1}^{HW}$ and
$N_{2}^{HW}$ is given by [5]:

$\displaystyle S_{N}^{HW}=\int
d^{5}x\sqrt{g}\Bigg{[}\dfrac{i}{2}\overline{N}^{HW}_{1}e^{M}_{A}\Gamma^{A}\nabla_{M}N^{HW}_{1}-\dfrac{i}{2}\Big{(}\nabla^{\dagger}_{M}\overline{N}^{HW}_{1}\Big{)}\,e^{M}_{A}\Gamma^{A}N^{HW}_{1}$
$\displaystyle-m_{5}\overline{N}^{HW}_{1}N^{HW}_{1}$
$\displaystyle+\Big{(}N_{1}^{HW}\rightarrow N_{2}^{HW},m_{5}\rightarrow-
m_{5}\Big{)}\Bigg{]},$ (24)

where $e^{A}_{M}=\dfrac{1}{z}\eta^{A}_{M}$ is the vielbein and the Dirac
matrices $\Gamma^{A}=\Big{(}\gamma^{\mu},-i\gamma^{5}\Big{)}$ satisfy the
Clifford algebra $\Big{\\{}\Gamma^{A},\Gamma^{B}\Big{\\}}=2\eta^{AB}$. The
Lorentz and gauge covariant derivative is:

$\displaystyle\nabla_{M}=\partial_{M}+\dfrac{i}{4}\omega^{AB}_{M}\Gamma_{AB},$
(25)

where the Lorentz generator $\Gamma^{AB}$ is defined as
$\Gamma^{AB}:=\dfrac{1}{2i}\Big{[}\Gamma^{A},\Gamma^{B}\Big{]}$ and the non-
vanishing components of the spin connection $\omega^{AB}_{M}$ equal to
$\omega^{5A}_{M}=-\omega^{A5}_{M}=\dfrac{1}{z}\delta^{A}_{M}$. Extremizing the
action (24) with respect to $\overline{N}_{1}^{HW}$, we get 5D Dirac equation
for $N_{1}^{HW}$:

$\displaystyle(ie^{M}_{A}\Gamma^{A}\nabla_{M}-m_{5})N_{1}^{HW}=0\,,$ (26)

with the boundary condition

$\displaystyle\Big{[}\delta\overline{N}^{HW}_{1}e^{5}_{A}\Gamma^{A}N^{HW}_{1}\Big{]}^{z_{m}}_{\epsilon}=0.$
(27)

Similarly, for the field of $N_{2}^{HW}$. Since the AdS/CFT correspondence
relates the classical solution to the bulk equation of motion with the
corresponding boundary operator, it is useful to decompose the bulk spinor
into left and right components, similar to the 4D chirality projectors, when
evaluated at the UV brane:

$\displaystyle N_{1}^{HW}(x,z)=N_{1L}^{HW}(x,z)+N_{1R}^{HW}(x,z)\,,\qquad
N_{2}^{HW}(x,z)=N_{2L}^{HW}(x,z)+N_{2R}^{HW}(x,z),$ (28)

where $N_{1,2L/R}^{HW}=\dfrac{1\mp i\Gamma^{5}}{2}N_{1,2}^{HW}$ are chiral
spinors. In momentum space, they are written in terms of the left (right) 4D
spinors $u_{L,R}(p)$ satisfying the 4D Dirac equation

$\displaystyle
N_{1}^{HW}(p,z)=F^{HW}_{1L}(p,z){u}_{L}(p)+F_{1R}^{HW}(p,z){u}_{R}(p)\,,\qquad
N_{2}^{HW}(p,z)=F_{2L}^{HW}(p,z){u}_{L}(p)+F_{2R}^{HW}(p,z){u}_{R}(p).$ (29)

We perform KK decomposition for the fermion fields as follows:

$\displaystyle
N_{1,2L/R}^{HW}(p,z)=\sum_{n}N^{n}_{1,2L/R}(p)F^{n\,HW}_{1,2L/R}(p,z).$ (30)

Mode functions for ground state nucleons $F^{HW}_{L,R}(z)\equiv
F^{0\,HW}_{L,R}(z)$ are the solutions of the Dirac equation (26) [5]

$\displaystyle F_{1L}^{HW}(p,z)=c_{1}z^{5/2}J_{2}(m_{n}z)\,,\qquad
F_{1R}^{HW}(p,z)=c_{1}z^{5/2}J_{3}(m_{n}z);$ (31) $\displaystyle
F_{2L}^{HW}(p,z)=-c_{2}z^{5/2}J_{3}(m_{n}z)\,,\qquad
F_{2R}^{HW}(p,z)=c_{2}z^{5/2}J_{2}(m_{n}z),$ (32)

with the normalization constants [6] :

$\displaystyle|c_{1,2}|=\dfrac{\sqrt{2}}{z_{m}J_{2}(m_{n}z)}.$ (33)

###  III.2 Nucleon profile function in the SW model

In the SW model with the negative dilaton field $\Phi$ the action for the 5D
$N_{1}$ spinor fields with $m_{5}=5/2$ (similarly for $N_{2}$ with
$m_{5}=-5/2$ ) is written as [28]:

$\displaystyle S_{N_{1}}^{SW}=\int
d^{4}xdz\sqrt{g}e^{-\Phi(z)}\Big{[}\dfrac{i}{2}\overline{N}_{1}^{SW}(x,z)e^{M}_{A}\Gamma^{A}\nabla_{M}N_{1}^{SW}(x,z)-\dfrac{i}{2}\Big{(}\nabla_{M}N_{1}^{SW}(x,z)\Big{)}^{\dagger}\Gamma^{0}e^{M}_{A}\Gamma^{A}N_{1}^{SW}(x,z)$
$\displaystyle-\overline{N}_{1}^{SW}(x,z)\Big{(}m_{5}+\Phi(z)\Big{)}N_{1}^{SW}(x,z)\Big{]}\,.$
(34)

For removing the dilaton field from the overall exponential, the fermionic
fields are rescaled as $N^{SW}_{1,2}(x,z)\rightarrow
e^{k^{2}z^{2}/2}N^{SW}_{1,2}(x,z)$. We decompose the fermion field in left-
and right-chirality components (28) and KK modes 555Since indices 1 and 2
follow the same analogical structure as the HW model, we shall omit them in
order to avoid repetition.. To obtain the equation of motion, we use the
following substitution:

$\displaystyle F^{n\,{SW}}_{L/R}(z)=z^{2}f^{n\,{SW}}_{L/R}(z).$ (35)

The equation of motion for fermions can be represented as a Schrödinger-type
equation

$\displaystyle\Big{[}-\partial^{2}_{z}+k^{4}z^{2}+2k^{2}\Big{(}m\mp\dfrac{1}{2}\Big{)}+\dfrac{m(m\pm
1)}{z^{2}}\Big{]}f^{n\,{SW}}_{L/R}(z)=m^{2}_{n}f^{n\,{SW}}_{L/R}(z).$ (36)

The solution of the Eq. (36) gives the mode functions for nucleons in the SW
model:

$\displaystyle
f^{n\,{SW}}_{L}(z)=\sqrt{\dfrac{2\Gamma(n+1)}{\Gamma(n+3)}}k^{3}z^{5/2}e^{-k^{2}z^{2}/2}L^{2}_{n}(k^{2}z^{2}),\,\qquad
f^{n\,{SW}}_{R}(z)=\sqrt{\dfrac{2\Gamma(n+1)}{\Gamma(n+2)}}k^{2}z^{3/2}e^{-k^{2}z^{2}/2}L^{1}_{n}(k^{2}z^{2}).$
(37)

Undoing the change of variables, we get the original mode functions

$\displaystyle
F^{n\,{SW}}_{L}(z)=\sqrt{\dfrac{2\Gamma(n+1)}{\Gamma(n+3)}}k^{3}z^{9/2}L^{2}_{n}(k^{2}z^{2}),\,\qquad
F^{n\,{SW}}_{R}(z)=\sqrt{\dfrac{2\Gamma(n+1)}{\Gamma(n+2)}}k^{2}z^{7/2}L^{1}_{n}(k^{2}z^{2})\,.$
(38)

The normalization is set by

$\displaystyle\int_{0}^{\infty}\dfrac{e^{k^{2}z^{2}}}{z^{4}}F^{n\,{SW}}_{L/R}(z)F^{m\,{SW}}_{L/R}(z)=\delta^{mn}\,,$
(39)

and the mass spectrum of the nucleon will be given by

$\displaystyle m^{2}_{n}=4k^{2}(n+2).$ (40)

Note that we will consider the interaction only for the ground states $n=0$
and will use the short-hand notation for the SW profile functions as

$\displaystyle F_{L/R}^{\,SW}(z)\equiv F_{L/R}^{0\,SW}(z)\,.$ (41)

##  IV Tensor-Nucleon couplings

In order to derive the tensor meson-nucleon coupling constant $g_{f_{2}NN}$
within the holographic QCD, we have to write down the interaction Lagrangian
$L^{5D}_{f_{2}NN}$ for the corresponding fields in the bulk of the 5D AdS
space. The interaction Lagrangian between the tensor meson and nucleons in 4D
theory is given in Ref. [29]:

$\displaystyle
L_{f_{2}NN}=-2i\frac{g^{(1)}_{f_{2}NN}}{m_{n}}\overline{N}(x)(\gamma^{\mu}\overleftrightarrow{\partial}^{\nu}+\gamma^{\nu}\overleftrightarrow{\partial}^{\mu})N(x)h_{\mu\nu}(x)+4\frac{g^{(2)}_{f_{2}NN}}{m_{n}^{2}}\partial^{\mu}\overline{N}(x)\partial^{\nu}N(x)h_{\mu\nu}(x)\,,$
(42)

where
$\overleftrightarrow{\partial}=\overleftarrow{\partial}-\overrightarrow{\partial}$
and $M_{N}$ denote the mass of nucleon. It is reasonable to write 4D action in
momentum space using Fourier transforms of the nucleon $N(x)$ and tensor meson
$h_{\mu\nu}(x)$ fields:

$\displaystyle\overline{N}(x)=\int
d^{4}p^{\prime}\overline{N}(p^{\prime})e^{ip^{\prime}x}\,,\qquad N(x)=\int
d^{4}pN(p)e^{-ipx}\,,\qquad h_{\mu\nu}(x)=\int d^{4}qh_{\mu\nu}(q)e^{-iqx}.$
(43)

The action for the interaction of these fields in momentum space has two
terms:

$\displaystyle S^{4D}_{\text{int}}=4\frac{g^{(1)}_{f_{2}NN}}{m_{n}}\int
d^{4}p^{\prime}d^{4}p(p^{\prime}+p)^{\nu}\overline{N}(p^{\prime})\gamma^{\mu}N(p)h_{\mu\nu}(q)+4\frac{g^{(2)}_{f_{2}NN}}{m_{n}^{2}}\int
d^{4}p^{\prime}d^{4}pp^{\prime\mu}p^{\nu}\overline{N}(p^{\prime})N(p)h_{\mu\nu}(q),$
(44)

which are proportional to $(p^{\prime}+p)^{\nu}\gamma^{\mu}$ and
$p^{\prime\mu}p^{\nu}$ in the tensor current of the nucleon. In 5D space-time
(1), the interaction Lagrangian between the bulk tensor $h_{MN}$ and fermionic
fields $N$ is the extension of the Lagrangian (42). To this end, we should
replace partial derivatives with the covariant derivatives (25) of the spinor
field and construct hermitian Lagrangian. Thus, we can write the 5D action

$\displaystyle S_{\text{int}}^{5D}=\int d^{5}x\sqrt{g}\,L^{5D}_{f_{2}NN}\,,$
(45)

based on the following interaction Lagrangian:

$\displaystyle L_{f_{2}NN}^{5D}$
$\displaystyle=2i\overline{N}(x,z)\Big{(}\Gamma^{M}\overleftrightarrow{\nabla}^{N}+\Gamma^{N}\overleftrightarrow{\nabla}^{M}\Big{)}N(x,z)h_{MN}(x,z)$
$\displaystyle+4\Bigg{(}N(x,z)^{\dagger}\Gamma^{0}\nabla^{M}\nabla^{N}N(x,z)+\Big{(}N(x,z)^{\dagger}\Gamma^{0}\nabla^{M}\nabla^{N}N(x,z)\Big{)}^{\dagger}\Bigg{)}h_{MN}(x,z)\,,$
(46)

where
$\overleftrightarrow{\nabla}=\overleftarrow{\nabla}-\overrightarrow{\nabla}$.
Note that the second term with two derivatives is written in such a way as to
ensure the hermiticity of the Lagrangian. One can show that it is a correct 5D
generalization of the corresponding 4D term, and it looks like to the 4D
Lagrangian upon replacing $(\mu,\nu\rightarrow M,N)$

$\displaystyle
L^{5D}_{f_{2}NN}=2i\overline{N}(x,z)\Big{(}\Gamma^{M}\overleftrightarrow{\partial}^{N}+\Gamma^{N}\overleftrightarrow{\partial}^{M}\Big{)}N(x,z)h_{MN}(x,z)+4\partial^{M}\overline{N}(x,z)\partial^{N}N(x,z)h_{MN}(x,z)\,.$
(47)

In momentum space, 5D action in the HW model reads:

$\displaystyle
S^{HW}_{\text{int}}=2\int_{0}^{z_{m}}\dfrac{dz}{z^{4}}\Big{\\{}\Big{(}F^{HW}_{1L}(z)\Big{)}^{2}+\Big{(}F^{HW}_{1R}(z)\Big{)}^{2}\Big{\\}}h^{HW}(z)\int
d^{4}p^{\prime}d^{4}p(p^{\nu\prime}+p^{\nu})\overline{u}(p^{\prime})\gamma^{\mu}u(p)h_{\mu\nu}(q)$
$\displaystyle+2\int_{0}^{z_{m}}\dfrac{dz}{z^{5}}\Big{\\{}\Big{(}F^{HW}_{1L}(z)\Big{)}^{2}-\Big{(}F^{HW}_{1R}(z)\Big{)}^{2}\Big{\\}}h^{HW}(z)\int
d^{4}p^{\prime}d^{4}pp^{\mu\prime}p^{\nu}\overline{u}(p^{\prime})u(p)h_{\mu\nu}(q).$
(48)

Comparing equations (44) and (48), we find the coupling constants
$g^{(1)}_{f_{2}NN}$ and $g^{(2)}_{f_{2}NN}$ for the interaction between the
ground state nucleon and the tensor meson. In the HW model, we find the
following integral representations for these couplings:

$\displaystyle
g^{(1)\,HW}_{f_{2}NN}=\dfrac{m_{n}}{2}\int_{0}^{z_{m}}\dfrac{dz}{z^{4}}\Big{\\{}\Big{(}F^{HW}_{1L}(z)\Big{)}^{2}+\Big{(}F^{HW}_{1R}(z)\Big{)}^{2}\Big{\\}}h^{HW}(z),$
(49) $\displaystyle
g^{(2)\,HW}_{f_{2}NN}=\dfrac{m^{2}_{n}}{2}\int_{0}^{z_{m}}\dfrac{dz}{z^{5}}\Big{\\{}\Big{(}F^{HW}_{1L}(z)\Big{)}^{2}-\Big{(}F^{HW}_{1R}(z)\Big{)}^{2}\Big{\\}}h^{HW}(z).$
(50)

Action in the SW model gets the following form:

$\displaystyle
S^{SW}_{\text{int}}=2\int_{0}^{\infty}dz\dfrac{e^{-k^{2}z^{2}}}{z^{4}}\Big{\\{}\Big{(}F^{SW}_{L}(z)\Big{)}^{2}+\Big{(}F^{SW}_{R}(z)\Big{)}^{2}\Big{\\}}h^{SW}(z)\int
d^{4}p^{\prime}d^{4}p(p+p^{\prime})^{\nu}\overline{u}(p^{\prime})\gamma^{\mu}u(p)h_{\mu\nu}(q)$
$\displaystyle+2\int_{0}^{\infty}dz\dfrac{e^{-k^{2}z^{2}}}{z^{5}}\Big{\\{}\Big{(}F^{SW}_{L}(z)\Big{)}^{2}-\Big{(}F^{SW}_{R}(z)\Big{)}^{2}\Big{\\}}h^{SW}(z)\int
d^{4}p^{\prime}d^{4}pp^{\prime\mu}p^{\nu}\overline{u}(p^{\prime})u(p)h_{\mu\nu}(q).$
(51)

The couplings are as follows:

$\displaystyle
g^{(1)\,SW}_{f_{2}NN}=\dfrac{m_{n}}{2}\int_{0}^{\infty}dz\dfrac{e^{-k^{2}z^{2}}}{z^{4}}\Big{\\{}\Big{(}F^{SW}_{L}(z)\Big{)}^{2}+\Big{(}F^{SW}_{R}(z)\Big{)}^{2}\Big{\\}}h^{SW}(z),$
(52) $\displaystyle
g^{(2)\,SW}_{f_{2}NN}=\dfrac{m^{2}_{n}}{2}\int_{0}^{\infty}dz\dfrac{e^{-k^{2}z^{2}}}{z^{5}}\Big{\\{}\Big{(}F^{SW}_{L}(z)\Big{)}^{2}-\Big{(}F^{SW}_{R}(z)\Big{)}^{2}\Big{\\}}h^{SW}(z).$
(53)

In Table 1, we present the results of our numerical calculations compared to
the analyses performed in various works.

Couplings | Hard Wall | Soft Wall | TMD[30] | DR [31] | DR [32] | DR [33] | DR [34]
---|---|---|---|---|---|---|---
$|g^{(1)}_{f_{2}NN}|$ | $1.06$ | 8.46 | $2.18\pm 0.12$ | $6.44\pm 0.60$ | $3.75$ | $5.16$ | $5.26\pm 2.15$
$|g^{(2)}_{f_{2}NN}|$ | $0.23$ | 0.42 | $\approx 0$ | $0.39\pm 1.10$ | $-$ | $\approx 0$ | $3.15\pm 4.73$
Table 1: Predictions for the couplings compared to the Refs.[30, 31, 32, 33,
34]

Figure 1: Dependence of the couplings on the cutoff value $z_{m}$ within the
HW model

It is interesting to know the sensitivity of the couplings to the value of the
cutoff parameter $z_{m}$. To this end, we plot in Fig.1 the dependencies of
the $g^{(i)}_{f_{2}NN}$ constants on $z_{m}$. These plots are obtained by
interpolating the points (black dots) corresponding to the different cutoff
values. It is seen from Fig.1, in the domain approximately $0<z_{m}<0.24$
$\text{GeV}^{-1}$, the result for the couplings is susceptible to the cutoff
values $z_{m}$. In the domain $0.24\,\text{GeV}^{-1}<z_{m}<\infty$ the
couplings weakly depend on this parameter. In the case of the $z_{m}=0.322$
$\text{GeV}^{-1}$ in which the masses of mesons are fixed to the experiment,
we observe some deviation from other results. While in the case of
$z_{m}=0.205$ $\text{GeV}^{-1}$ in which nucleons are well described, we get
$g^{(1)}_{f_{2}NN}\approx 15$. Therefore, all results obtained by different
groups can be described within the HW model in the domain of
$z_{m}\in[0.205,0.322]\,\text{GeV}^{-1}$. Note that some underestimate for
tensor-nucleon coupling can be understood in the context of Tensor Meson
Dominance (TMD) since the AdS/QCD prediction for
$g^{\text{HW}}_{f_{2}\pi\pi}=0.009$ is two times less than the $SU(3)$
effective model prediction $g^{SU(3)}_{f_{2}\pi\pi}=0.019$ [30]. If we set
$g^{(2)}_{f_{2}NN}=0$ in the following relation (see derivation in the
appendix of Ref.[29]),

$\displaystyle\frac{2}{m_{n}}\Big{(}g^{(1)}_{f_{2}NN}+g^{(2)}_{f_{2}NN}\Big{)}=\frac{g_{f_{2}\pi\pi}}{m_{f_{2}}}$
(54)

we get $g^{(1)}_{f_{2}NN}=1.06$, which is approximately two times less than in
the Ref. [30].

##  V Tensor meson decay constants $g_{f_{2}\rho\rho}$ and
$g_{f_{2}\gamma\rho}$

The tensor meson-vector meson interaction, which includes $f_{2}$ decays to
the vector particles $\rho$ and $\gamma$ ( $f_{2}\rightarrow\rho\rho$,
$f_{2}\rightarrow\gamma\rho$, and $f_{2}\rightarrow\gamma\gamma$) lies in the
bulk action for the vector field in the background of Eq. (9). The action
containing the first order of the $h^{\mu\nu}$ and the second order of
$V_{\mu}$ is written as:

$\displaystyle
S=\dfrac{1}{2g^{2}_{5}}\int\frac{dz}{z}d^{4}xh^{\mu\alpha}\Big{[}\partial_{\mu}V^{(1)}_{\nu}\partial_{\alpha}V^{(2)\nu}-\partial_{\mu}V^{(1)}_{\nu}\partial^{\nu}V^{(2)}_{\alpha}-\partial_{\nu}V^{(1)}_{\mu}\partial_{\alpha}V^{(2)\nu}+\partial_{\nu}V^{(1)}_{\mu}\partial^{\nu}V^{(2)}_{\alpha}-\partial_{z}V^{(1)}_{\mu}\partial_{z}V^{(2)}_{\alpha}\Big{]}.$
(55)

Using KK decomposition and Fourier transformation, we can write the action in
momentum space:

$\displaystyle
S=-\dfrac{1}{2g^{2}_{5}}\int\frac{dz}{z}d^{4}p_{1}d^{4}p_{2}\Big{\\{}$
$\displaystyle
h(q,z)V^{(1)}(p_{1},z)V^{(2)}(p_{2},z)h^{\mu\alpha}(q)\Big{[}p_{1\nu}p_{2}^{\nu}V^{(1)}_{\mu}(p_{1})V^{(2)}_{\alpha}(p_{2})-p_{1\mu}p_{2}^{\nu}V^{(1)}_{\nu}(p_{1})V^{(2)}_{\alpha}(p_{2})$
$\displaystyle-
p_{1\nu}p_{2\alpha}V^{(1)}_{\mu}(p_{1})V^{(2)\nu}(p_{2})+p_{1\mu}p_{2\alpha}V^{(1)}_{\nu}(p_{1})V^{(2)\nu}(p_{2})\Big{]}$
$\displaystyle-h(q,z)\partial_{z}V^{(1)}(p_{1},z)\partial_{z}V^{(2)}(p_{2},z)h^{\mu\alpha}(q)V^{(1)}_{\mu}(p_{1})V^{(2)}_{\alpha}(p_{2})\Big{\\}},$
(56)

where $q=-(p_{1}+p_{2})$ due to $\delta(q+(p_{1}+p_{2}))$-function arising on
$x$ integration. Here $p_{1},p_{2}$ are the four-momenta of the
$\rho^{(1)},\rho^{(2)}$ mesons. As is known from earlier works on tensor meson
couplings [35, 30, 18], the tensor-vector vector meson interaction amplitude
contains five independent couplings due to the tensor structure of the tensor-
meson current [35]:

$\displaystyle\mathcal{A}_{TV_{1}V_{2}}=\frac{g_{f_{2}V_{1}V_{2}}}{m_{f_{2}}^{2}}\tau^{\mu\nu}\Big{\\{}\beta_{0}\epsilon^{(1)}_{\mu}\epsilon^{(2)}_{\nu}+\beta_{1}(\epsilon^{(1)}\cdot
p_{2})\epsilon^{(2)}_{\mu}p_{1\nu}+\beta_{2}(\epsilon^{(2)}\cdot
p_{1})\epsilon^{(1)}_{\mu}p_{2\nu}+\beta_{3}(\epsilon^{(1)}\cdot\epsilon^{(2})p_{1\mu}p_{2\nu}+$
$\displaystyle+\delta(\epsilon^{(1)}\cdot p_{2})(\epsilon^{(2)}\cdot
p_{1})p_{1\mu}p_{2\nu}\Big{\\}}\,,$ (57)

with the polarization tensor $\tau^{\mu\nu}$ for spin-2 field and
$\epsilon_{\mu}^{(1,2)}$-polarization vectors for massive vector fields.

The first scenario, the so-called TMD model in which $\beta_{0}=p_{1}\cdot
p_{2}$, $\beta_{1}=\beta_{2}=-\beta_{3}=-1$, and $\delta=0$ leads to the
following Lagrangian

$\displaystyle
L_{f_{2}\rho\rho}^{\text{TMD}}=g_{f_{2}\rho\rho}^{(1)}h^{\mu\alpha}V_{\mu\nu}V_{\alpha}^{\nu}=g_{f_{2}\rho\rho}^{(1)}h^{\mu\alpha}\Big{(}\partial_{\mu}V^{(1)}_{\nu}\partial_{\alpha}V^{(2)\nu}-\partial_{\nu}V^{(1)}_{\mu}\partial_{\alpha}V^{(2)\nu}-\partial_{\mu}V^{(1)}_{\nu}\partial^{\nu}V^{(2)}_{\alpha}+\partial_{\nu}V^{(1)}_{\mu}\partial^{\nu}V^{(2)}_{\alpha}\Big{)}\,,$
(58)

where the coupling $g_{f_{2}\rho\rho}^{(1)}$ is evaluated as follows

$\displaystyle|g_{f_{2}\rho\rho}^{(1){HW}}|=\frac{4\pi
m_{f_{2}}^{2}}{2g_{5}^{2}\sqrt{5}}\int_{0}^{z_{m}}\dfrac{dz}{z}\Big{[}h^{HW}(z)\Big{(}v^{HW}(z)\Big{)}^{2}\Big{]}=0.31\,\text{GeV}\,,\qquad$
(59) $\displaystyle|g_{f_{2}\rho\rho}^{(1){SW}}|=\frac{4\pi
m_{f_{2}}^{2}}{2g_{5}^{2}\sqrt{5}}\int_{0}^{\infty}\dfrac{dz}{z}e^{-k^{2}z^{2}}\Big{[}h^{SW}(z)\Big{(}v^{SW}(z)\Big{)}^{2}\Big{]}=0.68\,\text{GeV}\,.$
(60)

Here the factor $\frac{4\pi}{\sqrt{5}}$ is inserted to do a compatible
matching with 4-dimensional theory [3].

In the case of the so-called minimal model [18], where all momentum dependent
terms are neglected
$\Big{(}\beta_{0}=m_{f_{2}}^{2},\beta_{1}=\beta_{2}=\beta_{3}=\delta=0\Big{)}$,
the 4D Lagrangian has a simple form

$\displaystyle
L_{f_{2}\rho\rho}^{\text{min}}=g_{f_{2}\rho\rho}^{(0)}h^{\mu\alpha}V_{\mu}V_{\alpha}\,.$
(61)

Numerical estimation of the minimal coupling constant within the HW model
reads from the last term of Eq. (56) :

$\displaystyle|g_{f_{2}\rho\rho}^{(0){HW}}|=\frac{4\pi}{2g_{5}^{2}\sqrt{5}}\int_{0}^{z_{m}}\dfrac{dz}{z}\Big{[}h^{HW}(z)\Big{(}\partial_{z}v^{HW}(z)\Big{)}^{2}\Big{]}=0.04\,\text{GeV}\,,$
(62)

Similarly, within the SW model:

$\displaystyle|g_{f_{2}\rho\rho}^{(0){SW}}|=\frac{4\pi}{2g_{5}^{2}\sqrt{5}}\int_{0}^{\infty}\dfrac{dz}{z}e^{-k^{2}z^{2}}\Big{[}h^{SW}(z)\Big{(}\partial_{z}v^{SW}(z)\Big{)}^{2}\Big{]}=0.08\,\text{GeV}\,.$
(63)

Note that the Lagrangian describing the minimal model (61) can be extended to
the following 5D form:

$\displaystyle
L^{\text{min}-5D}_{f_{2}\rho\rho}=g_{f_{2}\rho\rho}^{(0)\prime}h^{MN}V_{M}V_{N}\,.$
(64)

Numerical results for the coupling of the $g_{f_{2}\rho\rho}^{(0)\prime}$ in
the HW and SW models are $g_{f_{2}\rho\rho}^{(0)\prime\,HW}=0.38\,\text{GeV}$
and $g_{f_{2}\rho\rho}^{(0)\prime\,SW}=0.84\,\text{GeV}$, respectively.

We move on to the next action describing the decay of the tensor meson into a
vector meson and a photon. In this case, the action consists of the
convolution of two field stress tensors $V_{MN}$ and
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$ corresponding to vector meson and
photon fields’ stress tensors in the boundary 6664D analogous of the action
Eq. (66) can be obtained via VMD [36, 37] by applying the following shift in
the Lagrangian (58)  $V_{\mu\nu}\rightarrow
V_{\mu\nu}+\frac{e}{g_{\rho}}\begin{pmatrix}\frac{2}{3}&0&0\\\
0&-\frac{1}{3}&0\\\ 0&0&-\frac{1}{3}\end{pmatrix}F_{\mu\nu}\text{ .}$ (65) The
electromagnetic field tensor is represented by $F_{\mu\nu}$. The electric
coupling constant is denoted by $e$ and is equal to $\sqrt{4\pi\alpha}$. The
photon-vector-meson transition is parameterized by $g_{\rho}$, which is
approximately equal to 5.5. The holographic description of VMD is studied in
[38].:

$\displaystyle S=\int d^{5}x\sqrt{g}F_{MN}V^{MN}.$ (66)

The KK decomposition for the $A_{M}$ field will contain the photon profile
function [3]

$\displaystyle A(p,z):=\gamma(z)=\frac{e}{\sqrt{3\pi^{2}}}\,.$ (67)

In the first order of the $h^{\mu\nu}$, Eq. (66) contains the tensor-vector-
photon interaction terms similar to tensor-vector-vector coupling case and has
the following explicit form:

$\displaystyle S$
$\displaystyle=\int\frac{dz}{z}d^{4}pd^{4}p^{\prime}h(q,z)A(p,z)V(p^{\prime},z)h^{\mu\alpha}(q)\times$
$\displaystyle\times\Big{[}p_{\nu}p^{\prime\nu}A_{\mu}(p)V_{\alpha}(p^{\prime})-p_{\mu}p^{\prime\nu}A_{\nu}(p)V_{\alpha}(p^{\prime})-p_{\nu}p^{\prime}_{\alpha}A_{\mu}(p)V^{\nu}(p^{\prime})+p_{\mu}p^{\prime}_{\alpha}A_{\nu}(p)V^{\nu}(p^{\prime})\Big{]}\,.$
(68)

The 5D action, which describes tensor meson coupling to two photons in the 4D
boundary, is written in the same way:

$\displaystyle S$ $\displaystyle=\int
d^{5}x\sqrt{g}F_{MN}F^{MN}=\int\frac{dz}{z}d^{4}pd^{4}p^{\prime}h(q,z)A(p,z)A(p^{\prime},z)h^{\mu\alpha}(q)\times$
$\displaystyle\times\Big{[}p_{\nu}p^{\prime\nu}A_{\mu}(p)A_{\alpha}(p^{\prime})-p_{\mu}p^{\prime\nu}A_{\nu}(p)A_{\alpha}(p^{\prime})-p_{\nu}p^{\prime}_{\alpha}A_{\mu}(p)A^{\nu}(p^{\prime})+p_{\mu}p^{\prime}_{\alpha}A_{\nu}(p)A^{\nu}(p^{\prime})\Big{]}\,.$
(69)

In an analogous way to the $g_{f_{2}\rho\rho}^{(1)}$ coupling, from the
actions (V) and (V) we write the integral representations for the
$g_{f_{2}\gamma\rho}^{(1)}$ and $g_{f_{2}\gamma\gamma}^{(1)}$

$\displaystyle g_{f_{2}\gamma\rho}^{(1)HW}=\frac{4\pi
m_{f_{2}}^{2}}{\sqrt{5}}\int\dfrac{dz}{z}\Big{[}h^{HW}(z)\gamma(z)v^{HW}(z)\Big{]}=1.16\,\text{GeV}\,,$
(70)

while the holographic model result for the two-photon coupling of the tensor
meson [3] reads

$\displaystyle g_{f_{2}\gamma\gamma}^{(1)HW}=\frac{4\pi
m_{f_{2}}^{2}}{\sqrt{5}}\int\dfrac{dz}{z}\Big{[}h^{HW}(z)\Big{(}\gamma(z)\Big{)}^{2}\Big{]}=0.06\,\text{GeV}\,.$
(71)

Coupling |  $\Gamma_{f_{2}(1270)\rightarrow\pi^{+}\pi^{-}2\pi^{0}}$[18] |  $\Gamma_{f_{2}\rightarrow\gamma\gamma}$[18] |  Chiral model[39] | Our Result (HW)
---|---|---|---|---
$g_{f_{2}\gamma\rho}^{(1)}/m_{f_{2}}$ | $3.32$ | $0.68$ | $2.93$ | $0.90$
Table 2: The HW model estimation for the coupling vs results derived from
Refs. [18, 39]

Considering the following experimental results given in PDG [15]

$\displaystyle\Gamma_{f_{2}(1270)\longrightarrow\rho^{0}\rho^{0}+2\rho^{-}\rho^{+}}\simeq\Gamma_{f_{2}(1270)\longrightarrow\pi^{+}\pi^{-}2\pi^{0}}\approx
19.5\,\text{MeV}\,,\qquad\Gamma_{f_{2}\rightarrow\gamma\gamma}=2.6\pm
0.5\,\text{keV,}$

authors in Ref.[18] extract the coupling $g_{f_{2}\gamma\rho}$ in Table 2
using the following relation derived under the Vector Meson Dominance (VMD)
assumption:

$\displaystyle
g_{f_{2}\rho\rho}=\frac{m_{\rho}}{g_{\rho}}\frac{g_{f_{2}\gamma\rho}}{\sqrt{4\pi\alpha}}\,.$
(72)

The following relation for the couplings of the radiative decays is valid
within the quark model

$\displaystyle
g_{f_{2}\gamma\rho}=\frac{g_{f_{2}\gamma\gamma}}{4\pi\alpha}\frac{1}{\Big{(}\frac{g_{\rho}}{m_{\rho}}+\frac{1}{3}\frac{g_{\omega}}{m_{\omega}}\Big{)}}$
(73)

the decay constants for the vector fields $\rho(770)$ and $\omega(782)$ are
obtained from the experiment

$\displaystyle g_{V}=\sqrt{\frac{3m_{V}\Gamma_{V\rightarrow
e^{-}+e^{+}}}{4\pi\alpha^{2}}}$ (74)

Decay rate $\Gamma_{f_{2}\rightarrow\gamma\gamma}=2.6$ keV implies

$\displaystyle
g_{f_{2}\gamma\gamma}=\sqrt{\frac{320\pi\Gamma_{f_{2}\rightarrow\gamma\gamma}}{m_{f2}}}=0.045$
(75)

which coincides with the HW AdS/QCD model in Ref. [3]. Considering the
relation between the radiative decays of the tensor meson in equation A.46 of
Ref. [33], we obtain $\Gamma_{f_{2}\rightarrow\rho\gamma}=0.6\,\text{MeV}$
which is compatible with the prediction of the chiral model $0.7\pm
1.7\,\text{MeV}$ in Ref. [39].

##  VI Conclusion and future directions

We studied the holographic description of the tensor meson by exploring its
hadronic and radiative coupling constants. The results are consistent with
those obtained from Dispersion Relations and amplitude methods, and the
holographic approach still remains effective in describing the couplings of
tensor mesons. Our results show that the coupling for tensor meson-nucleon
interaction coupling with one derivative term is the dominant one compared to
the two-derivative case. The formalism presented here can be extended to
describe the tensor glueball, which could be tested in the future Panda
experiment [40]. Motivated by the experimental observation of the 4-pion decay
of $f_{2}(1270)$ (the second dominant decay channel after the two-pion
channel), we have computed the coupling ($g^{(i)}_{f_{2}\rho\rho}$) within the
holographic model. Radiative decay channels such as
$f_{2}(1270)\rightarrow\rho(770)\gamma$, which is obtained via the VMD of the
tensor-vector interaction Lagrangian, can be interesting for future experiment
measurements, such as the GlueX experiment [41] at Jefferson Lab, to
effectively test our prediction for the coupling. We predict the decay width
for this channel to be around $0.6\,\text{MeV}$, which is consistent with the
prediction presented in Ref. [39]. One can study the finite temperature
effects [13, 14], the tensor form factors of nucleons given in Ref. [42, 43,
44] and the tensor meson decay constant [45] within our model.

## Acknowledgements

The authors thank Tahmasib Aliev for the useful discussion. S.J. thanks
Francesco Giacosa for earlier collaborations on tensor mesons and Astrid
Hiller Blin for fruitful discussion. S.J. acknowledges the financial support
through the project “Development Accelerator of the Jan Kochanowski University
of Kielce”, co-financed by the European Union under the European Social Fund,
with no. POWR.03.05. 00-00-Z212 / 18 and partial support by the Polish
National Science Centre (NCN) through the OPUS project 2019/33/B/ST2/00613.

##  Appendix A The SW tensor meson profile function revisited

In the SW model, the gravity action in the second-order perturbation
$h_{\mu\nu}$ contains an additional exponential warping factor [22]

$\displaystyle S_{h}^{SW}=-\int
d^{5}x\dfrac{e^{-2\kappa_{1}^{2}z^{2}}}{4z^{3}}(h_{\mu\nu,z}h^{\mu\nu}_{,z}+h_{\mu\nu}\Box
h^{\mu\nu}),$ (76)

where the transverse-traceless gauge conditions $\partial^{\mu}h_{\mu\nu}=0$,
and $h^{\mu}_{\mu}=0$ have been imposed. Solution for the perturbation can be
found in the form $h_{\mu\nu}(x,z)=h_{\mu\nu}(x)h(x,z)$, and the linearized
Einstein equation for the Fourier-transformed profile function $h(p,z)$ takes
the form:

$\displaystyle\Bigg{[}\partial_{z}\big{(}\dfrac{e^{-2\kappa_{1}^{2}z^{2}}}{z^{3}}\partial_{z}\big{)}+\dfrac{e^{-2\kappa_{1}^{2}z^{2}}}{z^{3}}p^{2}\Bigg{]}h(p,z)=0.$
(77)

We get the solution to this equation expressed in terms of the hypergeometric
function:

$\displaystyle
h(z)=N_{1}\big{(}\kappa_{1}z\big{)}^{4}{}_{1}F_{1}\Big{[}2-\dfrac{m^{2}}{8\kappa_{1}^{2}};3;2\kappa_{1}^{2}z^{2}\Big{]}+N_{2}G_{1,2}^{2,0}\left(\begin{array}[]{c}-\frac{m}{8\kappa_{1}^{2}}+1\\\
0,2\\\ \end{array}\middle|-2\kappa_{1}^{2}z^{2}\right),$ (80)

where $p^{2}=m^{2}$ has been taken which implies $m=4\kappa_{1}$ for the mass
of spin-2 tensor meson in the SW model. The second term of the solution is
non-normalizable, so, $N_{2}$ should vanish. To relate this solution with one
in Eq. (21), it is preferred to express the profile function in terms of the
generalized Laguerre polynomials by taking into account the following relation
between the hypergeometric function and Laguerre polynomial:

$\displaystyle{}_{1}F_{1}(-n,\alpha+1,x)=\frac{\alpha!n!}{(n+\alpha)!}L_{n}^{\alpha}(x).$
(81)

For our solution $\alpha=2$ and $n=-\dfrac{m^{2}}{2-8\kappa_{1}^{2}}$. By
taking into account these values the solution (80) will be written in terms of
Laguerre polynomials

$\displaystyle
h_{n}^{SW}(z)=N\big{(}\kappa_{1}z\big{)}^{4}\frac{2n!}{(n+2)!}L_{n}^{2}(2k_{1}^{2}z^{2}).$
(82)

The given solution can be interpreted as the $n$-th KK mode, and the mass
spectrum of the modes can be found from $m^{2}_{n}=8\kappa_{1}^{2}(2+n)$. We
observe that the profile function in (82) coincides with one in (22) with
redefinition $2\kappa_{1}^{2}=\kappa^{2}$.

##  Appendix B The flavor invariant form of the Lagrangian

This appendix shows the flavor invariant form of the Lagrangian studied in
Sec.V. In the case of $N_{f}=2$, the adjoint representation of $U(2)_{f}$ can
be written as

$\displaystyle\overline{2}\otimes 2=3\oplus 1\,.$ (83)

Considering the vector mesons as quark-anti-quark objects, we have triplets
$\Big{(}\rho_{\mu}^{0},\rho_{\mu}^{+},\rho_{\mu}^{-}\Big{)}$ and the singlet
$\Big{(}\omega_{\mu}\Big{)}$ within the following nonet $V_{\mu}$:

$\displaystyle
V_{\mu\,i}^{j}=\overline{q}^{j}\gamma_{\mu}q_{i}=\begin{pmatrix}\overline{u}\gamma_{\mu}u&\overline{u}\gamma_{\mu}d\\\
\overline{d}\gamma_{\mu}u&\overline{d}\gamma_{\mu}d\end{pmatrix}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}\rho^{0}+\omega&\sqrt{2}\rho^{-}\\\
\sqrt{2}\rho^{+}&-\rho^{0}+\omega\end{pmatrix}_{\mu}=\vec{\rho}_{\mu}\vec{\tau}+\omega_{\mu}\mathbb{1}_{2\times
2}\,,$ (84)

where $\vec{\tau}$’s are the three Pauli matrices. We have used the following
substitutions:

$\displaystyle\rho_{\mu}^{0}=\frac{1}{\sqrt{2}}\Big{(}\overline{u}\gamma_{\mu}u-\overline{d}\gamma_{\mu}d\Big{)}\,,\quad\rho_{\mu}^{+}=-i\,\overline{d}\gamma_{\mu}u\,,\quad\rho_{\mu}^{-}=i\,\overline{u}\gamma_{\mu}d\,,\quad\omega_{\mu}^{0}=\frac{1}{\sqrt{2}}\Big{(}\overline{u}\gamma_{\mu}u+\overline{d}\gamma_{\mu}d\Big{)}\,.$
(85)

Thus, we can write $V_{\mu}:=V_{\mu}^{a}t^{a}$ where
$V_{\mu}^{a}=\\{\vec{\rho},\omega\\}$ and
$t^{a}=\\{\vec{\tau},\mathbb{1}_{2\times 2}\\}$. A matrix containing the
vector fields is invariant under the flavor symmetry $U(2)_{f}$ such that

$\displaystyle V_{\mu}\rightarrow UV_{\mu}U^{\dagger}\,,$ (86)

where $U$’s are the elements of the unitary group $U(2)_{f}$ satisfy the
condition

$\displaystyle U^{\dagger}U=\mathbb{1}_{2\times 2}\,.$ (87)

Similarly, for the tensor fields, construct the flavor invariant matrix
$h_{\mu\nu}$

$\displaystyle
h_{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}a_{2}^{0}+f_{2}&\sqrt{2}a_{2}^{-}\\\
\sqrt{2}a_{2}^{+}&-a_{2}^{0}+f_{2}\end{pmatrix}_{\mu\nu}=\vec{a}_{2\,\mu\nu}\vec{\tau}+f_{2\,\mu}\mathbb{1}_{2\times
2}$ (88)

which is invariant under $U(2)_{f}$

$\displaystyle h_{\mu\nu}\rightarrow Uh_{\mu\nu}U^{\dagger}\,.$ (89)

The following lagrangian describes the interaction between the tensor and
vector meson and is invariant under flavor symmetry $U(2)_{f}$

$\displaystyle\mathcal{L}$
$\displaystyle=g^{(0)}\text{Tr}\Big{[}h^{\mu\nu}V_{\mu}V_{\nu}\Big{]}+g^{(1)}\text{Tr}\Big{[}h^{\mu\nu}V_{\mu\alpha}V^{\alpha}_{\nu}\Big{]}=$
(90)
$\displaystyle=\frac{g_{f_{2}\rho\rho}^{(0)}}{\sqrt{2}}f_{2}^{\mu\nu}\Big{(}\rho_{\mu}^{0}\rho_{\nu}^{0}+2\rho_{\mu}^{+}\rho_{\nu}^{-}\Big{)}+\frac{g_{f_{2}\rho\rho}^{(1)}}{\sqrt{2}}f_{2}^{\mu\nu}\Big{(}\rho_{\mu\alpha}^{0}\rho_{\nu}^{0\,\alpha}+2\rho_{\mu\alpha}^{+}\rho_{\nu}^{-\,\alpha}\Big{)}+\cdots\,.$

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