# Study of $B^{0}_{s}\rightarrow\
TT(a_{2}(1320),K^{*}_{2}(1430),f_{2}(1270),f^{{}^{\prime}}_{2}(1525))$ in the
perturbative QCD approach

Jing Dai<EMAIL_ADDRESS>School of Physical Science and Technology,
Southwest University, Chongqing 400715, China  Xian-Qiao Yu<EMAIL_ADDRESS>School of Physical Science and Technology, Southwest University, Chongqing
400715, China

###### Abstract

In this work, we calculate the two-body decays $B^{0}_{s}\rightarrow\ TT$ ( T
denotes tensor mesons, $a_{2}(1320)$, $K^{*}_{2}(1430)$, $f_{2}(1270)$,
$f^{{}^{\prime}}_{2}(1525)$) in the perturbative QCD approach. The predictions
of branching rations, polarization fractions, and direct $CP$ violations are
given in detail. We find that: (a) for the pure annihilation decay, the
longitudinal polarization around $90.0\%$, and the transverse polarizations
are relatively small; (b) the direct $CP$ asymmetry is proportional to the
interference between the tree and penguin contributions. The direct $CP$
asymmetry for most decays are small in this paper; (c) here are only six kinds
of diagrams for $B^{0}_{s}\rightarrow\ TT$, because the tensor mesons cannot
be produced through the $(V\pm A)$ currents or $(S\pm P)$ density, the
factorizable emission diagrams are prohibited, the nonfactorizable and
annihilation contributions are important in these decay modes. The branching
ratios of our calculation for $B^{0}_{s}\rightarrow\ TT$ are at the order of
$10^{-6}$ and $10^{-7}$, which can be tested in the LHCb and Belle II
experiments, hopefully; (d) mixing exists for the $f_{2}(1270)$ and
$f^{{}^{\prime}}_{2}(1525)$, just as the $\eta$ and $\eta^{{}^{\prime}}$
mixing, the branching ratios about the mixing angle $\theta$ are given in this
work. But it is diffferent from $f_{1}(1285)-f_{1}(1420)$, the mixing angle is
very small, so the decay branching ratio only has little change.

##  I Introduction

The model about two-body decay of $B$ mesons has been studied extensively in
the past two decades, both theoretically Liu:2017cwl ; Kim:2013cpa ;
Zou:2015iwa ; Ali:2007ff  and experimentally Munoz:1998sn ; BaBar:2008ozy ;
BaBar:2008lan , and the theoretical framework is well developed. Since there
appeared that charmless hadronic $B$ decays involving a light tensor meson in
the final state, tensor mesons get more attentions. By making use of the
flavor SU(3) symmetry, we study in this paper include nine mesons PDG ;
Wang:2010ni , isovector mesons $a_{2}(1320)$, isodoulet states
$K^{*}_{2}(1430)$ and two isosinglet mesons $f_{2}(1270)$,
$f^{{}^{\prime}}_{2}(1525)$, which forms the first ${1}^{3}P_{2}$ nonet
ParticleDataGroup:2010dbb . According to the latest experimental data of 2022
PDG PDG , the $B^{0}_{s}\rightarrow\
K^{*}_{2}(1430)\overline{\emph{K}}^{*}_{2}(1430)$ decay has already appeared,
which indicates a large of decay modes related to the final state of two
tensor mesons will appear. In Ref. Kim:2013cpa , some researchers have
calculated the decay channel about $B^{0}_{s}\rightarrow\ VT$, at the same
time, they also adopted some other methods to compare. And these calculations
indicated that predictions based on the pQCD that can accommodate experimental
data well. By comparing their predictions with the experimental data, and they
find that ${\cal BR}(B^{0}_{s}\rightarrow\
\Phi(K^{*}_{2}(1430),\overline{\emph{K}}^{*}_{2}(1430)))$ is similar to ${\cal
BR}(B^{0}_{s}\rightarrow\ \Phi(K^{*}(1430),\overline{\emph{K}}^{*}(1430)))$,
but the magnitude is smaller. For $B^{0}_{s}\rightarrow\ VV,TV$, they are
similar but a little different.

In the exploration of two-body decay of $B$ mesons, the perturbative QCD
factorization approach is a method based on factorization Yan:2018si ;
Li:2021qiw ; Rui:2021kbn ; Liang:2022mrz ; Yang:2022jog ; Liu:2021nun ;
Liang:2019eur  to calculate the decay process of $B^{0}_{s}\rightarrow\
M_{1}M_{2}$, in which $M_{i}$ are composed of light non-charmed mesons. In the
perturbative QCD approach, the factorization scale about $1/b$ is used to
separate the perturbative area from the nonperturbative area, and the non-
leptonic decay of the $B$ meson is assumed to be dominated by hard gluon
exchange, so that the hard part of the decay process can be isolated and
calculated directly by the perturbative theory, while the nonperturbative part
is absorbed into the universal hadron wave function. On this basis, the two-
body decay amplitude of $B^{0}_{s}$ meson can be generally described as

${\cal A}={\cal
H}\otimes\phi_{\emph{B}^{0}_{s}}\otimes\phi_{h_{1}}\otimes\phi_{h_{2}}$ (1)

Here the hard decay kernel ${\cal H}$ represents the contributions of Feynman
diagrams, which can be calculated by the perturbative theory. The
nonperturbative inputs $\phi_{\emph{B}^{0}_{s}}$, $\phi_{h_{1}}$ and
$\phi_{h_{2}}$ represent the wave functions of $B^{0}_{s}$ meson, tensor
mesons, respectively, which can be obtained by extracting relevant
experimental data or calculating based on several nonperturbative approaches.

The corresponding theoretical calculations of tensor mesons attract some
researchers. Compared with vector mesons, tensor mesons are more special and
complex. For $B\rightarrow\ PT$, $VT$ Zou:2012td ; Liu:2017cwl , relevant
people have done research about the tensor meson in the final state. But for
$B^{0}_{s}\rightarrow\ TT$, the case where the final states are all tensor
mesons has not yet been studied in the literature, and this article is the
first to do so. For the decays of $B^{0}_{s}\rightarrow\ TT$, the amplitude
can be defined as three invariant helicity components: $A_{0}$, for which the
polarizations of the tensor meson are longitudinal with the respect to their
momenta, and $A_{\Arrowvert}$, $A_{\perp}$ are for transversely polarized
tensor meson Huang:2005if ; Li:2006cva .

The branching rations of $B^{0}_{S}\rightarrow\
f_{2}(f^{{}^{\prime}}_{2})f_{2}(f^{{}^{\prime}}_{2})$ are related to the
mixing angle $\theta$ of $f_{2}(1270)$ and $f^{{}^{\prime}}_{2}(1525)$,
analogous to $\eta$ and $\eta^{{}^{\prime}}$ mixing in the pseudoscalar sector
Feldmann:1998vh ; Xiao:2006mg ; Ali:2007ff ; Escribano:2005qq . From the
experiment that $\pi\pi$ is the dominant decay mode of $f_{2}(1270)$, and
$f^{{}^{\prime}}_{2}(1525)$ decays dominantly into $KK$, we can know that the
physical $f_{2}(1270)$-$f^{{}^{\prime}}_{2}(1525)$ mixing angle is smaller
than the decoupling value: $\theta_{T,ph}-\theta_{dec.}$=
$29.5^{\circ}-35.26^{\circ}=-5.8^{\circ}$ Cheng:2011fk ;
ParticleDataGroup:2010dbb , which indicates $f_{2}(1270)$ is nearly
$f^{u}_{2}+f^{d}_{2}$, while $f^{{}^{\prime}}_{2}(1525)$ is mainly $f^{s}_{2}$
Li:2000zb . Therefore, the mixing angle is very small, and the related
helicity amplitudes are defined by Jiang:2020eml ; Yang:2007zt

$\begin{split}f_{2}(1270)=\frac{1}{\sqrt{2}}(f^{u}_{2}+f^{d}_{2})\cos\theta_{f_{2}}-f^{s}_{2}\sin\theta_{f_{2}},\\\
f^{{}^{\prime}}_{2}(1525)=\frac{1}{\sqrt{2}}(f^{u}_{2}+f^{d}_{2})\cos\theta_{f_{2}}+f^{s}_{2}\sin\theta_{f_{2}},\end{split}$
(2)

where $f^{d}_{2}$ = $d\overline{d}$, $f^{u}_{2}$ = $u\overline{u}$,
$f^{s}_{2}$ = $s\overline{s}$ . Moreover, it is also found that the mixing
angle $\theta_{f_{2}}$ = $7.8^{\circ}$ Li:2000zb  and $(9\pm 1)^{\circ}$
Li:2002mi .

This paper is organized as follows. In Sect.II, the theoretical framework of
the pQCD and the wave functions involved in our calculation for the
$B^{0}_{s}\rightarrow\ TT$ decays are described, and the helicity amplitudes
are collected in Sect.III. The numerical results and discussions are given in
Sect.IV. The main contents of Sect.V are a summary of this work. And the
explicit formulas of all the helicity amplitudes are presented in the
Appendix.

##  II Theoretical Framework

###  II.1 Hamiltonian and kinematics

The relevant weak effective Hamiltonian of the decays $B^{0}_{s}\rightarrow\
TT$ is Buras:1998raa ; Buchalla:1995vs

${\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}\big{\\{}V^{*}_{ub}V_{us}[C_{1}O_{1}+C_{2}O_{2}]-V^{*}_{tb}V_{ts}[\sum^{10}_{i=3}C_{i}O_{i}]\big{\\}},$
(3)

where $V^{*}_{ub}$$V_{us}$ and $V^{*}_{tb}$$V_{ts}$ are Cabibbo-Kobayashi-
Maskawa (CKM) factors, the Fermi coupling constant $G_{F}$=$1.66378\times
10^{-5}{\rm GeV^{-2}}$, and $C_{i}$ is the Wilson coefficient corresponding to
the quark operator, $O_{i}$ are the local four-quark operators, which can be
written as

$\begin{split}\\\
O_{1}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})u_{\beta}\overline{u}_{\beta}\gamma^{\mu}(1-\gamma_{5})X_{\alpha},\\\
O_{2}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})u_{\alpha}\overline{u}_{\beta}\gamma^{\mu}(1-\gamma_{5})X_{\beta},\\\
O_{3}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\alpha}\sum_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1-\gamma_{5})X^{{}^{\prime}}_{\beta},\\\
O_{4}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\beta}\sum_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1-\gamma_{5})X^{{}^{\prime}}_{\alpha},\\\
O_{5}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\alpha}\sum_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1+\gamma_{5})X^{{}^{\prime}}_{\beta},\\\
O_{6}&=\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\beta}\sum_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1+\gamma_{5})X^{{}^{\prime}}_{\alpha},\\\
O_{7}&=\frac{3}{2}\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\alpha}\sum_{X^{{}^{\prime}}}e_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1+\gamma_{5})X^{{}^{\prime}}_{\beta},\\\
O_{8}&=\frac{3}{2}\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\beta}\sum_{X^{{}^{\prime}}}e_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1+\gamma_{5})X^{{}^{\prime}}_{\alpha},\\\
O_{9}&=\frac{3}{2}\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\alpha}\sum_{X^{{}^{\prime}}}e_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1-\gamma_{5})X^{{}^{\prime}}_{\beta},\\\
O_{10}&=\frac{3}{2}\overline{b}_{\alpha}\gamma_{\mu}(1-\gamma_{5})X_{\beta}\sum_{X^{{}^{\prime}}}e_{X^{{}^{\prime}}}\overline{X^{{}^{\prime}}}_{\beta}\gamma^{\mu}(1-\gamma_{5})X^{{}^{\prime}}_{\alpha},\end{split}$
(4)

where $\alpha$ and $\beta$ are color indices, ${X^{{}^{\prime}}}=u,d,s,c$ or
$b$ quarks, and they are the active quarks at the scale $m_{b}$. $O_{1}$ and
$O_{2}$ are current-current operators, $O_{i}(i=3,...,10)$ are penguin
operators, in which $O_{i}(i=7,...,10)$ are the electroweak penguin operators.
The operators $O_{7\gamma}$ and $O_{8g}$ are not listed, because their
contribution is neglected.

In the pQCD approach, the decay amplitude can be factorized into three parts
in detail, the hard scattering kernel, the wave functions of mesons, and the
convolution of the Wilson coefficients. For $B^{0}_{s}\rightarrow\ M_{1}M_{2}$
decay, the decay amplitude can be written as Li:2006cva ; Liu:2016rqu ;
Liu:2012jb ; Li:2005hg

${\cal A}\sim\int dx_{1}dx_{2}dx_{3}b_{1}db_{1}b_{2}db_{2}b_{3}db_{3}\cdot
Tr[C(t)\Phi_{B}(x_{1},b_{1})\Phi_{M_{2}}(x_{2},b_{2})\Phi_{M_{3}}(x_{3},b_{3}){\cal
H}(x_{i},b_{i},t)S_{t}(x_{i})e^{-S(t)}]$ (5)

where $x_{i}$ are the proportions of the momenta for the spectator quark
inside the mesons $B^{0}_{s}$, $T_{1}$ and $T_{2}$, respectively, with the
values ranging from 0 to 1, $b_{i}$ are the conjugate space coordinates of the
transverse momenta $k_{i}$ for the light quarks. $Tr$ denotes the trace over
all Dirac structure and color indices. $C(t)$ is the short distance Wilson
coefficients at the hard scale $t$. $t$ is the largest energy scale of the
hard part $\cal H$. The threshold resummation $S_{t}(x_{i})$ comes from the
large double logarithms Li:2001ay , which can remove the end-point
singularities. The last term $e^{-S(t)}$ is the Sudakov factor, which can
suppress the soft dynamics Li:1997un .

The corresponding physical quantity is represented in the light cone
coordinate system. Assuming that the initial state of the meson $B^{0}_{s}$ is
stationary, the tensor mesons $T_{1}$ and the $T_{2}$ move in the direction of
the light-like vector $v=(0,1,0_{\top})$ and $n=(1,0,0_{\top})$, respectively.
Here we use ${p}_{1}$, ${p_{2}}$ and ${p}_{3}$ to represent the momenta of the
mesons $B^{0}_{s}$, $T_{1}$ and $T_{2}$

$\begin{split}&\emph{p}_{1}=\frac{M_{B^{0}_{s}}}{\sqrt{2}}(1,1,0_{\top}),\\\
&\emph{p}_{2}=\frac{M_{B^{0}_{s}}}{\sqrt{2}}(1-r_{T}^{2},r_{T}^{2},0_{\top}),\\\
&\emph{p}_{3}=\frac{M_{B^{0}_{s}}}{\sqrt{2}}(r_{T}^{2},1-r_{T}^{2},0_{\top}).\end{split}$
(6)

In addition, the momenta of the corresponding light quark to the mesons
$B^{0}_{s}$, $T_{1}$ and $T_{2}$ are

$\begin{split}&k_{1}=(0,\frac{M_{B^{0}_{s}}}{\sqrt{2}}x_{1},k_{1\top}),\\\
&k_{2}=(\frac{M_{B^{0}_{s}}}{\sqrt{2}}(1-r_{T}^{2})x_{2},0,k_{2\top}),\\\
&k_{3}=(0,\frac{M_{B^{0}_{s}}}{\sqrt{2}}(1-r_{T}^{2})x_{3},k_{3\top}).\end{split}$
(7)

where $M_{B^{0}_{s}}$ represents the mass of the meson $B^{0}_{s}$, and
$r_{T}=\dfrac{M_{T}}{M_{B^{0}_{s}}}$, ${M_{T}}$ is the mass of the tensor
meson.

###  II.2 Wave functions

####  II.2.1 B meson

The wave function is decomposed into the following Lorentz structures

$\int\frac{d^{4}z}{(2\pi)^{4}}e^{ik\cdot
z}<0\mid\overline{b}_{\alpha}(0)d_{\beta}(z)\mid B_{s}(P_{1})>\\\
=\frac{i}{\sqrt{2N_{c}}}(\not{p}_{1}+M_{B}){\gamma_{5}}[\phi_{B}({x_{1},b_{1}})+\frac{\not{n}}{\sqrt{2}}\overline{\phi}_{B}(x_{1},b_{1})]$
(8)

where $\phi_{B}({x_{1},b_{1}})$and $\overline{\phi}_{B}({x_{1},b_{1}})$ are
the twist distribution amplitudes, the contribution of
$\overline{\phi}_{B}({x_{1},b_{1}})$ is relatively small, so we neglect it.
Therefore, the meson $B^{0}_{s}$ is considered to be a heavy-light model, the
wave function can be described as Shen:2014wga ; Keum:2000ph ; Beneke:2000wa ;
Lu:2000em

$\Phi_{B}=\frac{i}{\sqrt{2N_{c}}}(\not{p}_{1}+M_{B}){\gamma_{5}}{\phi_{B}({x_{1},b_{1}})},$
(9)

where $N_{c}={3}$ is the number of colors, because we calculate the relevant
parameters in the $b$ space, and the distribution amplitude $\phi_{B}$ can be
expressed as Kurimoto:2001zj ; Li:2003yj

$\phi_{B}(x_{1},b_{1})=\emph{N}_{B}{{x_{1}}^{2}}(1-{x_{1}})^{2}\exp[-\frac{M^{2}_{B}{{x_{1}}^{2}}}{2\omega^{2}_{B}}-\frac{1}{2}(\omega_{B}{b_{1}})^{2}],$
(10)

with the normalization

$\int^{1}_{0}dx\phi_{B}(x,b=0)=\frac{f_{B}}{{2}\sqrt{2N_{c}}},$ (11)

where $N_{B}=91.784$ GeV is the normalization constant, $f_{B}$ is the decay
constant. For $B^{0}_{s}$ meson, we use the shape parameter
$\omega_{B_{s}}=0.50\pm 0.05$ GeV Ali:2007ff .

####  II.2.2 Tensor meson

For the spin-2 polarization tensor $\epsilon^{uv}(\lambda)$ with helicity
$\lambda$, satisfies $\epsilon_{uv}p^{v}_{2}$=0 Cheng:2010hn ; Cheng:2010yd ,
which can be constructed based on the polarization vectors of vector mesons
$\epsilon$, they can be writtten as

$\begin{split}\epsilon_{uv}(\pm 2)&\equiv\epsilon(\pm 1)_{u}\epsilon(\pm
1)_{v},\\\ \epsilon_{uv}(\pm 1)&\equiv\frac{1}{\sqrt{2}}[\epsilon(\pm
1)_{u}\epsilon(0)_{v}+\epsilon(0)_{u}\epsilon(\pm 1)_{v}],\\\
\epsilon_{uv}(0)&\equiv\frac{1}{\sqrt{6}}[\epsilon(+1)_{u}\epsilon(-1)_{v}+\epsilon(-1)_{u}\epsilon(+1)_{v}]+\sqrt{\frac{2}{{3}}}\epsilon(0)_{u}\epsilon(0)_{v},\\\
\end{split}$ (12)

with the tensor meson moving in the plus direction of the $z$ axis, the
polarizations $\epsilon$ can be defined as

$\begin{split}\epsilon_{u}(0)=\frac{1}{m_{T}}(\mid P_{T}\mid,0,0,E_{T}),\\\
\epsilon_{u}(\pm 1)=\frac{1}{\sqrt{2}}(0,\pm 1,i,0),\end{split}$ (13)

where $E_{T}$ is the energy of the tensor meson. In the following calculation,
it is necessary to introduce a new polarization vector $\epsilon_{T}$ for the
considered tensor meson for convenience Wang:2010ni

$\begin{split}\epsilon_{Tu}(h)=\frac{1}{m_{B}}\epsilon_{uv}(h)P^{v}_{B},\\\
\end{split}$ (14)

which satisfies

$\begin{split}\epsilon_{Tu}(\pm 2)&=0,\\\ \epsilon_{Tu}(\pm
1)&=\frac{1}{\sqrt{2}m_{B}}\epsilon(0)\cdot P_{B}\epsilon_{u}(\pm 1),\\\
\epsilon_{Tu}(\pm 0)&=\sqrt{\frac{2}{3}}\frac{1}{m_{B}}\epsilon(0)\cdot
P_{B}\epsilon_{u}(0).\\\ \end{split}$ (15)

It is obvious that the new $\epsilon_{T}$ is similar to the ordinary
polarization vector $\epsilon$, regardless of the dimensionless constants
$\sqrt{\frac{1}{2}}\frac{\mid P_{T}\mid}{m_{T}}$or
$\sqrt{\frac{2}{3}}\frac{\mid P_{T}\mid}{m_{T}}$.

The decay constants of the tensor mesons are defined as Wang:2010ni

$\begin{split}<T\mid j_{uv}(0)\mid 0>=f_{T}m^{2}_{T}\epsilon^{*}_{uv},\\\
<T\mid j_{uv\rho}\mid
0>=-if^{T}_{T}m_{T}(\epsilon^{*}_{u\delta}P_{2v}-\epsilon^{*}_{v\delta}P_{2u}),\end{split}$
(16)

where the currents are given as

$\begin{split}j_{uv}(0)&=\frac{1}{2}[\overline{q}_{1}(0)\gamma_{u}i\overleftrightarrow{D}_{v}q_{2}(0)+\overline{q}_{1}(0)\gamma_{v}i\overleftrightarrow{D}_{u}q_{2}(0)],\\\
j^{+}_{uv\rho}(0)&=\overline{q}_{2}(0)\sigma_{uv}i\overleftrightarrow{D}_{\rho}q_{1}(0),\end{split}$
(17)

with$\overleftrightarrow{D}_{u}=\overrightarrow{D}_{u}-\overleftarrow{D}_{u}$,
$\overrightarrow{D}_{u}=\overrightarrow{\partial}_{u}+ig_{s}A^{a}_{u}\lambda^{a}/{2}$
and
$\overleftarrow{D}_{u}=\overleftarrow{\partial}_{u}-ig_{s}A^{a}_{u}\lambda^{a}/{2}$,
respectively. Here we adopted these decay constants from Ref. Cheng:2010hn
that have been calculated in the QCD sum rules Aliev:1981ju ; Aliev:1982ab ;
Aliev:2009nn , which can be seen from Table 1. We can find that the transverse
decay constants are equal to the longitudinal one for $a_{2}(1320)$,
$f_{2}(1270)$, but it is different from $K^{*}_{2}(1430)$ and
$f^{{}^{\prime}}_{2}(1525)$, their ratio relationship:
$f^{T}_{T}/f_{T}\sim(50\%-65\%)$.

Table 1: Decay constants (in unit of MeV) of tensor mesons | $f_{a_{2}(1320)}=107\pm 6$ | $f_{K^{*}_{2}(1430)}=118\pm 5$ | $f_{f_{2}(1270)}=102\pm 6$ | $f_{f^{{}^{\prime}}_{2}(1525)}=126\pm 4$
---|---|---|---|---
| $f^{T}_{a_{2}(1320)}=105\pm 21$ | $f^{T}_{K^{*}_{2}(1430)}=77\pm 14$ | $f^{T}_{f_{2}(1270)}=117\pm 25$ | $f^{T}_{f^{{}^{\prime}}_{2}(1525)}=65\pm 12$

From Ref. Cheng:2010hn , we can learn about the light cone distribution
amplitudes(LCDAs) of the tensor mesons. The LCDAs up to twist-3 for generic
tensor meson are defined by

$\begin{split}<T(p_{2},\epsilon)\mid{q_{1\alpha}(0)\overline{q}_{2\beta}(z)\mid
0}>=\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0}\emph{d}\emph{x}e^{ixp_{2}\cdot{z}}[m_{T}\not{\epsilon}^{*}_{\bullet
L}\Phi_{T}(x)\\\ +\not{\epsilon}^{*}_{\bullet
L}\not{p_{2}}\Phi^{t}_{T}(x)+m^{2}_{T}\frac{\epsilon_{\bullet}\cdot{v}}{p_{2}\cdot{v}}\Phi^{s}_{T}(x)]_{\alpha\beta},\end{split}$
(18)
$\begin{split}<T(p_{2},\epsilon)\mid{q_{1\alpha}(0)\overline{q}_{2\beta}(z)\mid
0}>=\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0}\emph{d}\emph{x}e^{ixp_{2}\cdot{z}}[m_{T}\not{\epsilon}^{*}_{\bullet
L}\Phi^{v}_{T}(x)\\\ +\not{\epsilon}^{*}_{\bullet
L}\not{p_{2}}\Phi^{T}_{T}(x)+m_{T}i\epsilon_{uv\rho\sigma}\gamma_{5}\gamma^{u}\epsilon^{*v}_{\bullet
T}n^{\rho}v^{\sigma}\Phi^{a}_{T}(x)]_{\alpha\beta}.\end{split}$ (19)

The convention $\epsilon^{0123}$=1 has been adopted. The Eq. (18) is for the
longitudinal polarization$(h=0)$, and Eq. (19) for the transverse
polarizations$(h=\pm{1})$, respectively. Here $n$ is the moving direction of
the tensor meson and $v$ is the opposite direction. The new vector
$\epsilon_{\bullet}$ which plays the same role with the polarization vector
$\epsilon$, which is defined by

$\begin{split}\epsilon_{\bullet
u}=\frac{\epsilon_{uv}v^{v}}{p_{2}\cdot{v}}m_{T}\end{split}$ (20)

with the momenta and polarizations, which can be reexpressed by

$\begin{split}\epsilon_{\bullet
u}=\frac{2m_{T}}{m^{2}_{B}}p^{v}_{B}\epsilon_{uv}\end{split}$ (21)

In Ref. Wang:2010ni ; Cheng:2010hn ; Cheng:2010yd , the amplitudes are given
by

$\begin{split}\Phi_{T}(x)&=\frac{f_{T}}{2\sqrt{2N_{c}}}\Phi_{\parallel}(x),\Phi^{t}_{T}(x)=\frac{f^{T}_{T}}{2\sqrt{2N_{c}}}h^{(t)}_{\parallel}(x),\\\
\Phi^{s}_{T}(x)&=\frac{f^{T}_{T}}{4\sqrt{2N_{c}}}\frac{d}{dx}h^{s}_{\parallel}(x),\Phi^{T}_{T}(x)=\frac{f^{T}_{T}}{2\sqrt{2N_{c}}}\Phi_{\perp}(x),\\\
\Phi^{v}_{T}(x)&=\frac{f_{T}}{2\sqrt{2N_{c}}}g^{v}_{\perp}(x),\Phi^{a}_{T}(x)=\frac{f_{T}}{8\sqrt{2N_{c}}}\frac{d}{dx}g^{a}_{\perp}(x).\\\
\end{split}$ (22)

The twist-2 distribution amplitudes which can be expanded in terms of
Gegenbauer polynomials $C^{3/2}_{n}(2x-1)$, and the asymptotic form is given
by

$\begin{split}\Phi_{\parallel,\perp}(x)=30x(1-x)(2x-1)\end{split}$ (23)

with the normalization conditions

$\begin{split}\int^{1}_{0}\emph{d}\emph{x}(2x-1)\Phi_{\parallel,\perp}(x)=1\end{split}$
(24)

The twist-3 distribution amplitudes are also the asymptotic, which are written
as Cheng:2010hn

$\begin{split}h^{t}_{\parallel}(x)&=\frac{15}{2}(2x-1)(1-6x+6x^{2}),\\\
h^{s}_{\parallel}(x)&=15x(1-x)(2x-1),\\\ g^{a}_{\perp}(x)&=20x(1-x)(2x-1),\\\
g^{v}_{\perp}(x)&=5(2x-1)^{3}.\end{split}$ (25)

##  III Decay amplitudes

In this section, we give the pQCD formulas for all the Feynman diagrams. As
depicted in Fig. 1. The first row shows the annihilation-type diagrams, with
the first two are factorizable and the last two are nonfactorizable. The
second row are nonfactorizable emission diagrams. For the
$B^{0}_{s}\rightarrow\ TT$ decays, both the longitudinal polarization and the
transverse polarization contribute. The symbol F and M represent the
factorizable and nonfactorizable contributions, respectively. The superscripts
LL denotes the amplitude of the $(V-A)(V-A)$ operators, and LR describe the
amplitude of the $(V-A)(V+A)$ operators. The symbol SP is Fierz transformation
of LR. The decay amplitudes of the longitudinal and the transverse
polarization are same in form after simplification, specifically as follows

---
Figure 1: The Feynman diagrams for the $B^{0}_{s}\rightarrow\ TT$ decays

The longitudinal polarization amplitudes of the factorizable annihilation
diagrams are

$\begin{split}{\emph{F}}^{\emph{LL,L}}_{af}=&\frac{16}{3}\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{2}\emph{d}\emph{b}_{2}\emph{b}_{3}\emph{d}\emph{b}_{3}\\\
&\times\\{[2{r}_{2}{r}_{3}{x}_{3}\phi^{s}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})-2{r}_{2}{r}_{3}{x}_{3}({x}_{3}-2)\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+({x}_{3}-1)\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{af}({t}_{e}){h}_{af}(\alpha_{1},\beta,{b}_{2},{b}_{3})\\\
&+[-2{r}_{2}{r}_{3}({x}_{2}-1)\phi^{s}_{T}({x}_{3})\phi^{t}_{T}({x}_{2})+2{r}_{2}{r}_{3}(-{x}_{2}-1)\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+{x}_{2}\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{af}({t}_{f}){h}_{af}(\alpha_{2},\beta,{b}_{3},{b}_{2})\\}\\\
\end{split}$ (26)
$\begin{split}{\emph{F}}^{\emph{SP,L}}_{af}=&-\frac{32}{3}\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{2}\emph{d}\emph{b}_{2}\emph{b}_{3}\emph{d}\emph{b}_{3}\\\
&\times\\{[2{r}_{2}\phi^{s}_{T}({x}_{2})\phi_{T}({x}_{3})+{r}_{3}({x}_{3}-1)\phi^{s}_{T}({x}_{3})\phi_{T}({x}_{2})+{r}_{3}({x}_{3}-1)\phi_{T}({x}_{2})\phi^{t}_{T}({x}_{3})]{E}_{af}({t}_{e}){h}_{af}(\alpha_{1},\beta,{b}_{2},{b}_{3})\\\
&+[2{r}_{3}({x}_{2}-1)\phi_{s}({x}_{3})\phi_{T}({x}_{2})+{r}_{2}{x}_{2}\phi^{s}_{T}({x}_{3})\phi^{t}_{T}({x}_{2})-{r}_{2}{x}_{2}\phi^{s}_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{af}({t}_{f}){h}_{af}(\alpha_{2},\beta,{b}_{3},{b}_{2})\\}\\\
\end{split}$ (27)

The longitudinal polarization amplitudes of the nonfactorizable annihilation
diagrams are given below

$\begin{split}{\emph{M}}^{\emph{LL,L}}_{anf}=&\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[{r}_{2}{r}_{3}((1-{x}_{2}+{x}_{3})\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})+({x}_{2}+{x}_{3}-1)\phi^{t}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})\\\
&+(1-{x}_{2}-{x}_{3})\phi_{T}({x}_{2})\phi_{T}({x}_{3})+({x}_{2}-{x}_{3}+3)\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3}))\\\
&-{x}_{2}\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[-{r}_{2}{r}_{3}((1+{x}_{2}-{x}_{3})\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+({x}_{2}+{x}_{3}-1)\phi^{s}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})\\\
&+(1-{x}_{2}-{x}_{3})\phi^{t}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+({x}_{3}-{x}_{2}-1)\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))\\\
&+({x}_{3}-1)\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{anf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (28)
$\begin{split}{\emph{M}}^{\emph{LR,L}}_{anf}=&\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[{r}_{2}(2-{x}_{2})\phi^{s}_{T}({x}_{2})\phi_{T}({x}_{3})+{r}_{2}(2-{x}_{2})\phi^{t}_{T}({x}_{2})\phi_{T}({x}_{3})\\\
&+{r}_{3}({x}_{3}+1)\phi_{T}({x}_{2})\phi^{s}_{T}({x}_{3})-{r}_{3}({x}_{3}+1)\phi^{t}_{T}({x}_{3})\phi_{T}({x}_{2})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}_{2}x_{2}(\phi^{s}_{T}({x}_{2})\phi_{T}({x}_{3})+\phi^{t}_{T}({x}_{2})\phi_{T}({x}_{3}))+{r}_{3}(1-x_{3})(\phi_{T}({x}_{2})\phi^{s}_{T}({x}_{3})\\\
&-\phi_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))]{E}_{enf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (29)
$\begin{split}{\emph{M}}^{\emph{SP,L}}_{anf}=&-\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[{r}_{2}{r}_{3}((1-{x}_{2}+{x}_{3})\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})-({x}_{2}+{x}_{3}-1)\phi^{t}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})\\\
&+({x}_{2}+{x}_{3}-1)\phi^{s}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})++({x}_{2}-{x}_{3}+3)\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})\\\
&+({x}_{3}-1)\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&-[{r}_{2}{r}_{3}((1+{x}_{2}-{x}_{3})\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+(1-{x}_{2}-{x}_{3})\phi^{s}_{T}({x}_{2})\phi^{t}_{T}({x}_{3})+({x}_{2}+{x}_{3}-1)\phi^{t}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})\\\
&+({x}_{3}-{x}_{2}-1)\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))-{x}_{2}\phi_{T}({x}_{2})\phi_{T}({x}_{3})]{E}_{anf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (30)

The longitudinal polarization amplitudes of the nonfactorizable emission
diagrams are as follows

$\begin{split}{\emph{M}}^{\emph{LL,L}}_{enf}=&\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\phi_{2}({x}_{2})\\\
&\times\\{[({x}_{2}-1)\phi_{T}({x}_{3})+{r}_{3}({x}_{3}-1)\phi^{s}_{T}({x}_{3})-{r}_{3}({x}_{3}-1)\phi^{t}_{T}({x}_{3})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[(x_{2}+x_{3}-2)\phi_{T}({x}_{3})-{r}_{3}(x_{3}-1)\phi^{s}_{T}({x}_{3})-{r}_{3}(x_{3}-1)\phi^{t}_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (31)

$\begin{split}{\emph{M}}^{\emph{LR,L}}_{enf}=&\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}{r}_{2}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[{r}_{3}({x}_{2}-{x}_{3}-1)(\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})-\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))+({x}_{2}+{x}_{3}-1)(\phi^{s}_{T}({x}_{3})\phi^{t}_{T}({x}_{2})-\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3}))\\\
&+({x}_{2}-1)(\phi_{T}({x}_{3})\phi^{s}_{T}({x}_{2})-\phi_{T}({x}_{3})\phi^{t}_{T}({x}_{2}))]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}_{3}(x_{3}-{x}_{2})(\phi^{t}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+\phi^{s}_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))+({x}_{2}+{x}_{3})(\phi^{s}_{T}({x}_{2})\phi^{s}_{T}({x}_{3})+\phi^{t}_{T}({x}_{2})\phi^{t}_{T}({x}_{3}))\\\
&+{x}_{2}(\phi_{T}({x}_{3})\phi^{s}_{T}({x}_{2})-\phi_{T}({x}_{3})\phi^{t}_{T}({x}_{2}))]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (32)
$\begin{split}{\emph{M}}^{\emph{SP,L}}_{enf}=&-\frac{32}{3}\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\phi_{2}({x}_{2})\\\
&\times\\{[({x}_{2}-{x}_{3}-1)\phi_{T}({x}_{3})+{r}_{3}{x}_{3}\phi^{s}_{T}({x}_{3})+{r}_{3}{x}_{3}\phi^{t}_{T}({x}_{3})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[-{r}_{3}{x}_{3}\phi^{s}_{T}({x}_{3})+{r}_{3}{x}_{3}\phi^{t}_{T}({x}_{3})+{x}_{2}\phi_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (33)

The transverse polarization amplitudes of the factorizable annihilation
diagrams are

$\begin{split}{\emph{F}}^{\emph{LL(LR),N}}_{af}=&4\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}{r}_{2}{r}_{3}\int^{1}_{0}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{2}\emph{d}\emph{b}_{2}\emph{b}_{3}\emph{d}\emph{b}_{3}\\\
&\times\\{[({x}_{3}-2)(\phi^{v}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})+\phi^{a}_{T}({x}_{2})\phi^{a}_{T}({x}_{3}))-{x}_{3}\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})\\\
&-{x}_{3}\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})]{E}_{af}({t}_{e}){h}_{af}(\alpha_{1},\beta,{b}_{2},{b}_{3})\\\
&+[({x}_{2}-1)\phi^{v}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})+({x}_{2}-1)\phi^{a}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})\\\
&+({x}_{2}+1)\phi^{a}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})+({x}_{2}+1)\phi^{v}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})]{E}_{af}({t}_{f}){h}_{af}(\alpha_{2},\beta,{b}_{3},{b}_{2})\\}\\\
\end{split}$ (34)

$\begin{split}{\emph{F}}^{\emph{SP,N}}_{af}=&8\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{2}\emph{d}\emph{b}_{2}\emph{b}_{3}\emph{d}\emph{b}_{3}\\\
&\times\\{[{r}_{2}\phi^{a}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+{r}_{2}\phi^{v}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{af}({t}_{e}){h}_{af}(\alpha_{1},\beta,{b}_{2},{b}_{3})\\\
&-[{r}_{3}\phi^{a}_{T}({x}_{3})\phi^{T}_{T}({x}_{2})+{r}_{3}\phi^{v}_{T}({x}_{3})\phi^{T}_{T}({x}_{2})]{E}_{af}({t}_{f}){h}_{af}(\alpha_{2},\beta,{b}_{3},{b}_{2})\\}\\\
\end{split}$ (35)

$\begin{split}{\emph{F}}^{\emph{LL,T}}_{af}=&4\pi\emph{C}_{\emph{F}}f_{B}M^{4}_{B^{0}_{s}}{r}_{2}{r}_{3}\int^{1}_{0}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{2}\emph{d}\emph{b}_{2}\emph{b}_{3}\emph{d}\emph{b}_{3}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[({x}_{3}-2)(\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})+\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3}))-{x}_{3}\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})\\\
&-{x}_{3}\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})]{E}_{af}({t}_{e}){h}_{af}(\alpha_{1},\beta,{b}_{2},{b}_{3})\\\
&+[({x}_{2}-1)\phi^{v}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})+({x}_{2}-1)\phi^{a}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})\\\
&+({x}_{2}+1)\phi^{a}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})+({x}_{2}+1)\phi^{v}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})]{E}_{af}({t}_{f}){h}_{af}(\alpha_{2},\beta,{b}_{3},{b}_{2})\\}\\\
\end{split}$ (36)
$\begin{split}{\emph{F}}^{\emph{SP,N}}_{af}=-{\emph{F}}^{\emph{SP,T}}_{af}\end{split}$
(37)
$\begin{split}{\emph{F}}^{\emph{LL,T}}_{af}=-{\emph{F}}^{\emph{LR,T}}_{af}\end{split}$
(38)

The transverse polarization amplitudes of the nonfactorizable annihilation
diagrams are

$\begin{split}{\emph{M}}^{\emph{LL(SP),N}}_{anf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[-2{r}_{2}{r}_{3}\phi^{a}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})-2{r}_{2}{r}_{3}\phi^{v}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})\\\
&-{r}^{2}_{2}(x_{2}-1)\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+{r}^{2}_{3}{x}_{3}\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}^{2}_{2}x_{2}\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})-{r}^{2}_{3}(x_{3}-1)\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{anf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (39)

$\begin{split}{\emph{M}}^{\emph{LR,N}}_{anf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[{r}_{2}({x}_{2}-2)\phi^{a}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+{r}_{2}({x}_{2}-2)\phi^{v}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})\\\
&-{r}_{3}({x}_{3}+1)\phi^{T}_{T}({x}_{2})\phi^{a}_{T}({x}_{2})+{r}_{3}({x}_{3}+1)\phi^{T}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}_{3}({x}_{3}-1)\phi^{T}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})-{r}_{3}({x}_{3}-1)\phi^{T}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})+{r}_{2}{x}_{2}\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})\\\
&+{r}_{2}{x}_{2}\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})]{E}_{enf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (40)
$\begin{split}{\emph{M}}^{\emph{LR,N}}_{anf}=-{\emph{M}}^{\emph{LR,T}}_{anf}\\\
\end{split}$ (41)

$\begin{split}{\emph{M}}^{\emph{LL,T}}_{anf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[2{r}_{2}{r}_{3}\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})+2{r}_{2}{r}_{3}\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})\\\
&-{r}^{2}_{2}(x_{2}-1)\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})-{r}^{2}_{3}{x}_{3}\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{anf}({t}_{g}){h}_{anf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}^{2}_{2}x_{2}\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+{r}^{2}_{3}(x_{3}-1)\phi^{T}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{anf}({t}_{h}){h}_{anf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (42)
$\begin{split}{\emph{M}}^{\emph{LL,T}}_{anf}=-{\emph{M}}^{\emph{SP,T}}_{anf}\end{split}$
(43)

For $B^{0}_{s}\rightarrow\ TT$ decays, the transverse polarization amplitudes
of the nonfactorizable emission diagrams are follows

$\begin{split}{\emph{M}}^{\emph{LL,N}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}{r}_{2}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[(1-{x}_{2})\phi^{a}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+(1-{x}_{2})\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&-[2{r}_{3}(x_{2}+x_{3})\phi^{a}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})+2{r}_{3}(x_{2}+x_{3})\phi^{v}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})-{x}_{2}(\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})\\\
&-x_{2}\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2}))]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (44)
$\begin{split}{\emph{M}}^{\emph{LR,N}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\phi^{T}_{T}({x}_{2})\\\
&\times\\{[{r}_{3}{x}_{3}\phi^{a}_{T}({x}_{3})-{r}_{3}{x}_{3}\phi^{v}_{T}({x}_{3})-{r}^{2}_{2}({x}_{2}-1)\phi^{T}_{T}({x}_{3})+{x}_{3}{r}^{2}_{3}\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}_{3}{x}_{3}\phi^{a}_{T}({x}_{3})-{r}_{3}{x}_{3}\phi^{v}_{T}({x}_{3})+{r}^{2}_{2}{x}_{2}\phi^{T}_{T}({x}_{3})+{x}_{3}{r}^{2}_{3}\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (45)
$\begin{split}{\emph{M}}^{\emph{SP,N}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}{r}_{2}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[2{r}_{3}({x}_{3}-{x}_{2}+1)\phi^{v}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})-2{r}_{3}({x}_{3}+1-{x}_{2})\phi^{a}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})\\\
&+({x}_{2}-1)\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})-({x}_{2}-1)\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{x}_{2}\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})-{x}_{2}\phi^{v}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (46)
$\begin{split}{\emph{M}}^{\emph{LL,T}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}{r}_{2}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[({x}_{2}-1)\phi^{a}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})+({x}_{2}-1)\phi^{T}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[2{r}_{3}(x_{2}+x_{3})\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})+2{r}_{3}(x_{2}+x_{3})\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})-{x}_{2}\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})\\\
&-x_{2}\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (47)

$\begin{split}{\emph{M}}^{\emph{LR,T}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\phi^{T}_{T}({x}_{2})\\\
&\times\\{[{r}_{3}{x}_{3}\phi^{v}_{T}({x}_{3})-{r}_{3}{x}_{3}\phi^{a}_{T}({x}_{3})-{r}^{2}_{2}({x}_{2}-1)\phi^{T}_{T}({x}_{3})-{x}_{3}{r}^{2}_{3}\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{r}_{3}{x}_{3}\phi^{v}_{T}({x}_{3})-{r}_{3}{x}_{3}\phi^{a}_{T}({x}_{3})+{r}^{2}_{2}{x}_{2}\phi^{T}_{T}({x}_{3})-{x}_{3}{r}^{2}_{3}\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\end{split}$
(48)

$\begin{split}{\emph{M}}^{\emph{SP,T}}_{enf}=&8\sqrt{\frac{2}{3}}\pi\emph{C}_{\emph{F}}M^{4}_{B^{0}_{s}}{r}_{2}\int^{1}_{0}\emph{d}\emph{x}_{1}\emph{d}\emph{x}_{2}\emph{d}\emph{x}_{3}\int^{\frac{1}{\Lambda}}_{0}\emph{b}_{1}\emph{d}\emph{b}_{1}\emph{b}_{2}\emph{d}\emph{b}_{2}\phi_{B_{s}}(\emph{x}_{1},b_{1})\\\
&\times\\{[2{r}_{3}({x}_{2}-{x}_{3}-1)\phi^{v}_{T}({x}_{2})\phi^{a}_{T}({x}_{3})-2{r}_{3}({x}_{3}+1-{x}_{2})\phi^{a}_{T}({x}_{2})\phi^{v}_{T}({x}_{3})\\\
&+({x}_{2}-1)\phi^{T}_{T}({x}_{3})\phi^{a}_{T}({x}_{2})-({x}_{2}-1)\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})]{E}_{enf}({t}_{c}){h}_{enf}(\alpha,\beta_{1},{b}_{1},{b}_{2})\\\
&+[{x}_{2}\phi^{T}_{T}({x}_{3})\phi^{v}_{T}({x}_{2})-{x}_{2}\phi^{a}_{T}({x}_{2})\phi^{T}_{T}({x}_{3})]{E}_{enf}({t}_{d}){h}_{enf}(\alpha,\beta_{2},{b}_{1},{b}_{2})\\}\\\
\end{split}$ (49)

##  IV Numerical results and discussions

In this section, we start our calculation by listing the input parameters. The
decay constant $f_{B}$, the Wolfenstein parameters, the masses of $B$ meson
and tensor mesons, and the corresponding lifetime, which are listed in Table 2
Particle Date Group:2020ac .

Table 2: Various parameters involved in the calculation process Mass of mesons |  $M_{B^{0}_{s}}=5.367$ GeV |  $M_{a_{2}(1320)}=1.317$ GeV |  $M_{K^{*}_{2}(1430)}=1.432$ GeV
---|---|---|---
|  $M_{K^{*\pm}_{2}(1430)}=1.427$ GeV |  $M_{f^{{}^{\prime}}_{2}(1525)}=1.517$ GeV |  $M_{f_{2}(1270)}=1.275$ GeV
|  $m_{b}=4.18$ GeV |  $m_{s}=0.093$ GeV |
Decay constants of mesons |  $f_{B^{0}_{s}}=0.24\pm 0.02$ GeV |  |
Lifetime of meson |  $\tau_{B^{0}_{s}}=1.509$ ps |  |
Wolfenstein parameters | $\emph{A}=0.836\pm 0.015$ | $\lambda=0.22453\pm 0.00044$ |
| $\bar{\eta}=0.355^{+0.012}_{-0.011}$ | $\bar{\rho}=0.122^{+0.018}_{-0.017}$ |

We discuss the numerical calculations in the pQCD approach about the branching
ratios, direct CP violations, and polarization fractions in Tables 3-5. There
are still many uncertainties in our calculation results. Such as Table 3, the
errors are induced by the uncertainties of the shape parameter
$\omega_{B}=(0.50\pm 0.05)$ GeV in the $B^{0}_{s}$ meson distribution
amplitude Ali:2007ff . The second ones are the decay constants of the
$B^{0}_{s}$ meson and the final state tensor mesons, which can be found in
Table 1. The third error arises from $\Lambda_{QCD}$= $(0.25\pm{0.05})$ GeV,
and varies $20\%$ from hard scale $t_{max}=(1.0\pm 0.2)t$ detailed in the
Appendix. Other uncertainties such as the CKM matrix elements V from the
$\bar{\eta}$ and $\bar{\rho}$, angles of the unitary triangle that can be
neglected.

With the amplitudes calculated in Sect.III, the decay width is given as

$\begin{split}\Gamma={\frac{[(1-(r_{2}+r_{3})^{2})(1-(r_{2}-r_{3})^{2})]^{1/2}}{16\pi
m_{B}}}\sum_{i}\mid A_{i}\mid^{2}\end{split}$ (50)

The branching ratio is got through $\cal{BR}$ = $\Gamma\tau_{B^{0}_{s}}$. In
Ref. Bigi:2000yz ; Branco:1999fs , we can learn about direct $CP$ violations,
$A^{dir}_{CP}$ is defined by

$\begin{split}{A^{dir}_{CP}}={\frac{\mid\overline{A}_{\overline{f}}\mid^{2}-\mid
A_{f}\mid^{2}}{\mid\overline{A}_{\overline{f}}\mid^{2}+\mid A_{f}\mid^{2}}}\\\
\end{split}$ (51)

Here the two amplitudes are defined as follows

$\begin{split}A_{f}=<f\mid\cal H\mid B>,\\\
\overline{A}_{\overline{f}}=<\bar{f}\mid\cal H\mid\bar{B}>.\end{split}$ (52)

where the $\overline{B}$ meson has a $b$ quark and $\overline{f}$ is the $CP$
conjugate state of $f$. The results of the polarization fractions $f_{i}$,
which are defined as

$\begin{split}{f_{0,\|,\perp}}={\frac{\mid A_{i}\mid^{2}}{\sum_{i}\mid
A_{i}\mid^{2}}}\end{split}$ (53)

where $A_{i}(i=0,\|,\perp)$ is the amplitude of the longitudinal or transverse
polarization contributions. Based on the helicity amplitudes
$A_{i}(i=0,\|,\perp)$ for longitudinal, parallel, and perpendicular
polarizations, and the three parts amplitudes are as follows

$\begin{split}A_{0}=\xi
m^{2}_{B^{0}_{s}}A_{L},A_{\|}=\xi\sqrt{2}m^{2}_{B^{0}_{s}}A_{N},\\\
A_{\perp}=\xi\sqrt{2(r^{2}-1)}m^{2}_{B^{0}_{s}}r_{2}r_{3}A_{T}.\end{split}$
(54)

here $\xi=\sqrt{\frac{G^{2}_{F}P_{c}}{(16\pi m^{2}_{B}\Gamma)}}$ , and the
ratio $r=\frac{P_{2}\cdot P_{3}}{(m^{2}_{B}r_{2}r_{3})}$.

Table 3: The $CP$-averaged branching ratios of the $B^{0}_{S}\rightarrow T_{1}T_{2}$ decay (in unit of $10^{-6}$), the errors come from the shape parameter, the decay constants, the hard scale and QCD scale. Decay Modes | $B_{0}$ | $B_{\parallel}$ | $B_{\perp}$ | $B_{total}$
---|---|---|---|---
$B^{0}_{S}\rightarrow a^{0}_{2}a^{0}_{2}$ | $0.28^{+0.01+0.08+0.18}_{-0.03-0.02-0.01}$ | $0.03^{+0.00+0.01+0.00}_{-0.00-0.00-0.00}$ | $0.00$ | $0.31^{+0.01+0.09+0.18}_{-0.03-0.02-0.01}$
$B^{0}_{S}\rightarrow a^{+}_{2}a^{-}_{2}$ | $0.42^{+0.01+0.04+0.32}_{-0.03-0.02-0.01}$ | $0.06^{+0.01+0.02+0.00}_{-0.01-0.01-0.02}$ | $0.00$ | $0.48^{+0.02+0.06+0.32}_{-0.04-0.03-0.03}$
$B^{0}_{S}\rightarrow\ K^{*0}_{2}(1430)\overline{\emph{K}}^{*0}_{2}(1430)$ | $2.03^{+1.74+0.91+1.45}_{-1.64-0.65-1.20}$ | $0.22^{+0.17+0.08+0.10}_{-0.10-0.07-0.06}$ | $0.47^{+0.33+0.20+0.24}_{-0.18-0.14-0.20}$ | $2.72^{+2.24+1.18+1.79}_{-1.92-0.86-1.46}$
$B^{0}_{S}\rightarrow\ K^{*+}_{2}(1430){K}^{*-}_{2}(1430)$ | $1.86^{+1.62+0.83+1.34}_{-1.40-0.59-1.11}$ | $0.21^{+0.16+0.08+0.09}_{-0.09-0.06-0.01}$ | $0.19^{+0.13+0.08+0.17}_{-0.08-0.06-0.08}$ | $2.26^{+1.91+0.99+1.60}_{-1.57-0.71-1.20}$
$B^{0}_{S}\rightarrow\ f_{2}f_{2}$ | $0.50^{+0.04+0.08+0.24}_{-0.05-0.04-0.01}$ | $0.04^{+0.00+0.01+0.00}_{-0.00-0.00-0.00}$ | $0.00$ | $0.54^{+0.04+0.09+0.24}_{-0.05-0.04-0.01}$
$B^{0}_{S}\rightarrow\ f_{2}f^{{}^{\prime}}_{2}$ | $1.86^{+1.13+0.76+1.68}_{-0.65-0.57-0.87}$ | $0.16^{+0.14+0.07+0.17}_{-0.07-0.05-0.01}$ | $0.44^{+0.31+0.08+0.42}_{-0.17-0.07-0.30}$ | $2.46^{+1.58+0.91+2.27}_{-0.89-0.69-1.18}$
$B^{0}_{S}\rightarrow\ f^{{}^{\prime}}_{2}f^{{}^{\prime}}_{2}$ | $6.03^{+0.84+1.68+4.51}_{-2.38-1.86-3.49}$ | $0.42^{+0.38+0.14+0.01}_{-0.21-0.12-0.02}$ | $1.23^{+1.23+0.47+0.01}_{-0.65-0.38-0.16}$ | $7.68^{+2.45+2.29+4.53}_{-3.24-2.36-3.67}$

As can be seen in Table 3, on the basis of the data and we would like to make
a few comments:

(1) For $B^{0}_{S}\rightarrow\
a^{0}_{2}a^{0}_{2},a^{+}_{2}a^{-}_{2},f_{2}f_{2}$, they are only the pure
annihilation diagrams, whose branching ratios are at the order of $10^{-7}$.
For $B^{0}_{S}\rightarrow\
f^{{}^{\prime}}_{2}f^{{}^{\prime}}_{2},f_{2}f^{{}^{\prime}}_{2},K^{*0}_{2}\overline{\emph{K}}^{*0}_{2},K^{*+}_{2}{\emph{K}}^{*-}_{2}$,
they have six kinds of diagrams, whose branching ratios are at the order of
$10^{-6}$ and penguin dominated. Under the $SU(3)$ limit, because of the Bose
statistics are satisfied, the meson wave function is antisymmetric when the
momenta fractions of the quark and anti-quark of tensor mesons are exchanged
Cheng:2010hn ; Cheng:2010yd . Due to the commutative antisymmetry of tensor
meson wave function, the nonfactorizable emission diagrams will be enhanced
and provide a greater contribution Cheng:2010hn ; Cheng:2010yd . Take
$B^{0}_{S}\rightarrow\ f^{{}^{\prime}}_{2}f^{{}^{\prime}}_{2}$ for example, if
without nonfactorizable emission contributions, the branching ratio will
decrease $90\%$, and its longitudinal polarization fraction will also reduce
largely, which is different from the $B^{0}_{S}\rightarrow\
f_{1}(1420)f_{1}(1420)$ in Ref.Jiang:2020eml . For $B^{0}_{S}\rightarrow\
f_{1}(1420)f_{1}(1420)$, the annihilation diagrams could contribute a large
imaginary part and plays an important role in calculating the branching
ratios. The reason for the difference may be that, relative to the commutative
antisymmetry of tensor meson wave function, two kinds of nonfactorizable
emission contributions are not offseted but enhanced.

(2) From the Ref. Kim:2013cpa , we find that ${\cal BR}(B^{0}_{S}\rightarrow\
\phi(K^{*-}_{2}(1430),\overline{\emph{K}}^{*0}_{2}(1430)))$ and ${\cal
BR}(B^{0}_{S}\rightarrow\ \phi(K^{*-}(1430)\overline{\emph{K}}^{*0}(1430)))$
are at the same order, but the former is a little small. The author suggested
that only small effects when ${K}^{*0}(1430)$ is substituted for
$K^{*}_{2}(1430)$. We observe the corresponding decays in Ref. PDG , such as
$B^{0}_{S}\rightarrow\ K^{*\pm}_{0}(1430){\emph{K}}^{\pm}$ and
$B^{0}_{S}\rightarrow\ K^{*}_{0}(1430)\overline{\emph{K}}^{0}$, the branching
ratio is also at the order of $10^{-5}$ when vector mesons are replaced by
tensor mesons. In this paper, the branching ratio of $B^{0}_{S}\rightarrow\
K^{*0}(1430)\overline{\emph{K}}^{*0}_{2}(1430)$ or $B^{0}_{S}\rightarrow\
K^{*0}_{2}(1430)\overline{\emph{K}}^{*0}(1430)$ is at the order of $6\times
10^{-6}\sim 9\times 10^{-6}$, and the branching ratio of
$B^{0}_{S}\rightarrow\ K^{*0}_{2}(1430)\overline{\emph{K}}^{*0}_{2}(1430)$ is
at the order of $2.71\times 10^{-6}$, which also supports the view of the
author.

(3) For $B^{0}_{S}\rightarrow\ VT(a_{2},f_{2})$, when a vector meson is
emitted, the factorizable emission contribution of the penguin diagrams will
offset the contribution of the tree annihilation diagrams, and the branching
ratio becomes very small. But the contribution of the annihilation diagrams is
not offset due to the absence of the factorizable emission diagrams in the
decay path of the emission tensor meson. Therefore, $B^{0}_{S}\rightarrow\
TT(a_{2},f_{2})$ is one or two orders of magnitude larger than
$B^{0}_{S}\rightarrow\ VT(a_{2},f_{2})$, which makes it more beneficial to
experimental observation.

(4) For $B^{0}_{S}\rightarrow\
f_{2}(f^{{}^{\prime}}_{2})f_{2}(f^{{}^{\prime}}_{2})$ decays with
$f_{2}(1270)-f^{{}^{\prime}}_{2}(1425)$ mixing, there also adopted the
$\eta-\eta^{{}^{\prime}}$ mixing. In the same way, diagrams about
$f_{2}(1270)-f^{{}^{\prime}}_{2}(1525)$ mixing that we get as Fig. 2 and Fig.
3. In Ref.Jiang:2020eml , the author also plotted the related figures about
the branching ratios of $B^{0}_{S}\rightarrow\ f_{1}f_{1}$ decays dependent on
the free parameter $\theta$. By analyzing two kinds of figures about
$B^{0}_{S}\rightarrow\ f_{1}f_{1}$ and $B^{0}_{S}\rightarrow\
f_{2}(f^{{}^{\prime}}_{2})f_{2}(f^{{}^{\prime}}_{2})$, we can find that when
the $\theta$ is enough large, and its influence on the branching ratio is more
obvious, which can be seen from the line shapes. In addition, there is the
same point. When $\theta$ reaches a certain angle, the branching ratios of
$B^{0}_{S}\rightarrow\ f_{2}(f^{{}^{\prime}}_{2})f_{2}(f^{{}^{\prime}}_{2})$
and $B^{0}_{S}\rightarrow\ f_{1}f_{1}$ will vary an order of magnitude. But
for $f_{2}(1270)-f^{{}^{\prime}}_{2}(1425)$ mixing, it is contrast to
$f_{1}(1285)-f_{1}(1420)$ that the mixing angle is very small. From many
papers, we have known that the mixing $\theta$ is about $5.8^{\circ}\sim
10^{\circ}$ ParticleDataGroup:2008zun ; Li:2000zb ; Cheng:2011fk , and the
related branching ratios are close to the branching ratios of $0^{\circ}$. We
can find the branching ratio of the $B^{0}_{S}\rightarrow\
f^{{}^{\prime}}_{2}(1525)f^{{}^{\prime}}_{2}(1525)$ decay is larger than the
$B^{0}_{S}\rightarrow\ f_{2}(1270)f_{2}(1270)$ decay by one order of
magnitude, which is caused by the former has more Feynman diagrams.

---
Figure 2: (a) The branching ratio of ${B}^{0}_{s}\rightarrow
f^{{}^{\prime}}_{2}(1525)f^{{}^{\prime}}_{2}(1525)$ decay on the mixing angle
$\theta$; (b) The branching ratio of ${B}^{0}_{s}\rightarrow\
f_{2}(1270)f_{2}(1270)$ decay on the mixing angle $\theta$.
---
Figure 3: The branching ratio of ${B}^{0}_{s}\rightarrow
f_{2}(1270)f^{{}^{\prime}}_{2}(1525)$ decay on the mixing angle $\theta$.

For the Table 4,we discuss the predictions for the polarization fraction of
$B^{0}_{S}$ meson:

Table 4: The polarization fraction of the $B^{0}_{S}\rightarrow T_{1}T_{2}$ decay , the errors come from the shape parameter, the decay constants, the hard scale and QCD scale. Decay Modes | $f_{0}$ | $f_{\parallel}$ | $f_{\perp}$
---|---|---|---
$B^{0}_{S}\rightarrow a^{0}_{2}a^{0}_{2}$ | $90.92^{+0.11+0.15+3.58}_{-0.45-0.05-0.53}\%$ | $8.88^{+0.46+0.06+0.56}_{-0.12-0.14-3.56}\%$ | $0.19\%$
$B^{0}_{S}\rightarrow a^{+}_{2}a^{-}_{2}$ | $88.24^{+0.29+0.65+6.83}_{-0.83-1.81-0.52}\%$ | $11.66^{+0.84+1.79+0.54}_{-0.30-0.65-6.82}\%$ | $0.09\%$
$B^{0}_{S}\rightarrow\ K^{*0}_{2}(1430)\overline{\emph{K}}^{*0}_{2}(1430)$ | $75.02^{+1.22+0.43+2.40}_{-25.37-0.55-14.29}\%$ | $8.08^{+7.58+0.14+12.27}_{-0.16-0.30-1.05}\%$ | $16.89^{+18.01+0.42+2.03}_{-1.05-0.13-1.38}\%$
$B^{0}_{S}\rightarrow\ K^{*+}_{2}(1430){K}^{*-}_{2}(1430)$ | $82.50^{+1.01+0.43+0.46}_{-15.80-0.38-11.02}\%$ | $9.16^{+7.70+0.16+9.44}_{-0.41-0.35-1.46}\%$ | $8.34^{+8.10+0.22+0.99}_{-0.60-0.09-1.64}\%$
$B^{0}_{S}\rightarrow\ f_{2}f_{2}$ | $93.7^{+0.01+0.77+2.35}_{-0.07-0.74-0.38}\%$ | $6.78^{+0.08+0.73+0.40}_{-0.02-0.75-2.33}\%$ | $0.14\%$
$B^{0}_{S}\rightarrow\ f_{2}f^{{}^{\prime}}_{2}$ | $75.78^{+1.44+2.22+2.70}_{-1.61-2.78-7.92}\%$ | $6.32^{+0.99+0.18+15.90}_{-0.88-0.30-3.18}\%$ | $17.91^{+0.60+3.07+0.47}_{-0.57-2.47-7.99}\%$
$B^{0}_{S}\rightarrow\ f^{{}^{\prime}}_{2}f^{{}^{\prime}}_{2}$ | $78.56^{+3.78+0.31+7.94}_{-10.88-1.28-15.12}\%$ | $5.44^{+2.45+0.19+4.42}_{-0.77-0.01-1.96}\%$ | $16.99^{+8.43+1.09+10.69}_{-3.02-0.06-5.99}\%$

The pQCD approach has successfully predicted the process of pure annihilation
diagrams about $B^{0}_{S}\rightarrow\ \pi^{+}\pi^{-}$ and $B^{0}\rightarrow\
D^{-}_{S}K^{+}$ theoretically, and corresponding numerical results has been
confirmed by experiments. Based on this, the pQCD approach can also reliably
predict the process dominated by annihilation diagrams Ali:2007ff ; Lu:2002iv
; ParticleDataGroup:2012pjm ; Li:2008ts . For all the decays in this paper,
the main decays are dominated by the longitudinal ones. From the
Ref.Cheng:2009mu ; Zou:2015iwa , we have known that the contributions are
dominated by the longitudinal polarization for the pure annihilation two-body
decays, and the fractions of these decays can even reach nearly $100\%$. It is
generally agree with our predictions of $B^{0}_{S}\rightarrow\ f_{2}f_{2}$,
$B^{0}_{S}\rightarrow\ a^{0}_{2}a^{0}_{2}$ and $B^{0}_{S}\rightarrow\
a^{+}_{2}a^{-}_{2}$. And the longitudinal polarization of the three decays get
around $90\%$, which shows that the transverse polarization cannot be ignored
and can help to make significant contributions in pure annihilation decays.
Here, we take $B^{0}_{S}\rightarrow\ a^{0}_{2}a^{0}_{2}$ and
$B^{0}_{S}\rightarrow\ f_{2}f_{2}$ as an example, from Table 6, we can find
that longitudinal contribution is larger than the transverse contribution, so
the former is leading.

By simple factorial power estimation and quark helicity analysis, the
contribution of longitudinal polarization will play a leading role in the $B$
meson decay Kagan:2004uw ; Korner:1979ci . For $B^{0}_{S}\rightarrow\ TT$, in
the longitudinal polarization part, the contribution of nonfactorizable
emission diagrams is very large, and the cancellation effect with annihilation
diagrams is very weak, while in the transverse part, the contribution of
nonfactorizable emission diagrams is small, which strongly cancels with
annihilation contribution, so the longitudinal polarization is dominant, and
the transverse polarization fraction is about $10\%$-$30\%$.

Table 5: The $CP$-violating asymmetries of the $B^{0}_{S}\rightarrow T_{1}T_{2}$ decay, the errors come from the shape parameter, the decay constants, the hard scale and QCD scale. Decay Modes | ${\cal A}^{dir}_{CP}$ | ${\cal A}^{dir}_{CP}(0)$ | ${\cal A}^{dir}_{CP}(\parallel)$ | ${\cal A}^{dir}_{CP}(\perp)$
---|---|---|---|---
$B^{0}_{S}\rightarrow a^{0}_{2}a^{0}_{2}$ | $(3.30^{+3.34+11.88+13.19}_{-3.75-1.77-38.50})\%$ | $(3.39^{+4.05+15.04+17.03}_{-4.62-2.19-44.62})\%$ | $(2.21^{+0.61+0.13+0.31}_{-0.62-0.15-0.24})\%$ | $(13.80^{+0.88+1.23+1.10}_{-0.90-1.23-1.58})\%$
$B^{0}_{S}\rightarrow f_{2}f_{2}$ | $(-6.12^{+3.06+5.93+9.96}_{-3.00-3.45-33.23})\%$ | $(-6.73^{+3.25+6.34+10.71}_{-4.64-2.35-34.63})\%$ | $(1.80^{+0.48+0.14+0.25}_{-0.46-0.15-0.31})\%$ | $(9.14^{+0.64+0.89+1.87}_{-0.03-0.90-1.59})\%$
$B^{0}_{S}\rightarrow K^{*0}_{2}K^{*0}_{2}$ | $0.00\%$ | … | … | …
$B^{0}_{S}\rightarrow f_{2}f^{{}^{\prime}}_{2}$ | $(1.04^{+0.19+1.81+8.43}_{-0.68-0.35-4.58})\%$ | $(1.04^{+0.31+0.25+3.38}_{-0.55-0.29-7.51})\%$ | $(-71.35^{+2.18+0.10+19.17}_{-1.68-0.11-7.38})\%$ | $(26.72^{+6.84+2.83+18.02}_{-6.20-0.57-4.90})\%$
$B^{0}_{S}\rightarrow f^{{}^{\prime}}_{2}f^{{}^{\prime}}_{2}$ | 0.00% | … | … | …
$B^{0}_{S}\rightarrow a^{+}_{2}a^{-}_{2}$ | $(2.09^{+2.58+11.87+10.92}_{-3.03-1.46-22.78})\%$ | $(2.21^{+2.58+11.87+10.92}_{-3.03-1.46-22.78})\%$ | $(1.06^{+0.30+0.07+61.22}_{-0.29-0.07-0.12})\%$ | $(17.54^{+0.98+1.84+0.81}_{-1.02-1.79-1.84})\%$
$B^{0}_{S}\rightarrow K^{*+}_{2}K^{*-}_{2}$ | ($2.61^{+3.94+0.77+6.18}_{-36.51-0.70-5.13})\%$ | $(3.57^{+3.74+0.99+6.66}_{-53.07-0.99-9.60})\%$ | $(-2.44^{+0.05+0.12+3.51}_{-0.33-0.43-3.87})\%$ | $(-0.91^{+6.62+0.70+2.48}_{-0.84-0.28-4.33})\%$
Table 6: Decay amplitudes (in unit of $10^{-3}~{}\rm GeV^{3}$) of the $B^{0}_{S}\rightarrow T_{1}T_{2}$ modes with three polarizations in the pQCD approach, where only the central values are quoted for clarification. Components | $B^{0}_{S}\rightarrow\ a^{0}_{2}a^{0}_{2}$ | $B^{0}_{S}\rightarrow\ a^{0}_{2}a^{0}_{2}$ | $B^{0}_{S}\rightarrow\ f_{2}f_{2}$ | $B^{0}_{S}\rightarrow\ f_{2}f_{2}$
---|---|---|---|---
classification | Tree diagrams | Penguin diagrams | Tree diagrams | Penguin diagrams
$A_{L}$ | $0.157-0.130i$ | $-0.082-0.224i$ | $-0.290-0.304i$ | $-0.353-0.615i$
$A_{N}$ | $-0.014+0.025i$ | $0.104+0.135i$ | $-0.015+0.030i$ | $0.130+0.148i$
$A_{T}$ | $0.010+0.017i$ | $-7.887\times 10^{-3}+3.704\times 10^{-3}i$ | $0.011+0.020i$ | $-9.728\times 10^{-3}+3.298\times 10^{-3}i$

Direct $CP$ asymmetries of $B^{0}_{s}\rightarrow\ TT$ decays are listed in
Table 5. The magnitude of the direct $CP$ violation is proportional to the
ratio of the penguin and tree contributions ParticleDataGroup:2008zun . For
the $B^{0}_{s}\rightarrow\ VV$ and $B^{0}_{s}\rightarrow\ VT$, when penguin
contributions and tree contributions are at the same level, the direct $CP$
violation appears. For these decays in this paper, because they are dominated
by the penguin contributions, so the direct $CP$ violation is very small.
However, we can find that the direct $CP$ violations of the some special
channels in the transverse polarization are sizable, and the tree
contributions become comparable, which brings relatively large direct $CP$
violation.

For pure annihilation type decay, the $CP$-violating asymmetry is small, which
have been pointed in previous predictions of two-body decays Zou:2015iwa ;
Cheng:2009mu . From Table 5, the $CP$-violating asymmetries about
$B^{0}_{S}\rightarrow a^{0}_{2}a^{0}_{2},f_{2}f_{2}$ and $a^{+}_{2}a^{-}_{2}$,
which also suggest that the $CP$-violating asymmetry of pure annihilation
decay is very small. For $B^{0}_{S}\rightarrow\
K^{*0}_{2}(1430)\overline{\emph{K}}^{*0}_{2}(1430)$,
$f^{{}^{\prime}}_{2}(1525)f^{{}^{\prime}}_{2}(1525)$ (without considering the
mixing angle), in the standard model, there is no contribution of tree
diagrams operator, so the direct $CP$ violation is zero.

##  V Summary

In this work, we predicted the relevant parameters of decays
$B^{0}_{s}\rightarrow\ TT$ in the pQCD approach, where the tensor mesons are
$a_{2}(1320)$, $f_{2}(1270)$, $K^{*}_{2}(1430)$, and
$f^{{}^{\prime}}_{2}(1525)$. We calculate the branching ratios, the
polarization fractions, and the direct $CP$ violations of these decays. Our
calculation results suggest that: (1) the tensor meson cannot be produced
throungh vector or the tensor current, which indicates that the
nonfactorizable emission and annihilation contributions are important.
Especially for the nonfactorizable emission diagram, its contribution will be
enhanced due to the antisymmetry of tensor meson wave function; (2) for only
annihilation type decays, the branching ratio is at the order of $10^{-7}$.
And for $f^{{}^{\prime}}_{2}(1525)$, $K^{*}_{2}(1430)$, they have a sizable
branching ratio with the order of $10^{-6}$, which would be easy to be
verified experimentally; (3) for the polarization fraction, the most of this
paper are dominated by the longitudinal contribution, expecially for the pure
annihilation decay, which can reach around $90\%$; (4) the direct $CP$
violations of these decays are small, the interference of the penguin and tree
contributions will bring the direct $CP$ violation, but the two parts that we
have calculated are not comparable in this paper. Therefore, the direct $CP$
violations of most channels are small, only a few special channels have
relatively large direct $CP$ violation appears. In the end, we hope the
results can be tested by the running LHC-b and Belle II in the near future,
and, of course, we also hope it has positive implications for understanding
the QCD behavior of the tensor mesons.

## acknowledgments

The authors would to thank Hong Yang for some valuable discussions. This work
is supported by the National Natural Science Foundation of China under Grant
No.11047028.

## Appendix : Formulae For The Calculation Used In The Text

In this section, we list the decay amplitude for every considered two-body
decays of $B$ meson.

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
a^{0}_{2}(1320)a^{0}_{2}(1320))&=G_{F}({V^{*}_{ub}V_{us}[(C_{1}+\frac{1}{3}C_{2})F^{LL,h}_{a}+C_{2}M^{LL,h}_{a}]}\\\
&-V^{*}_{tb}V_{ts}[({{2}C_{3}+\frac{2}{3}C_{4}+{2}C_{5}+\frac{2}{3}C_{6}}\\\
&+{\frac{1}{2}C_{7}+\frac{1}{6}C_{8}+\frac{1}{2}C_{9}+\frac{1}{6}C_{10}})F^{LL,h}_{a}\\\
&+({2}C_{4}+\frac{1}{2}C_{10})M^{LL,h}_{a}+({2}C_{6}+\frac{1}{2}C_{8})M^{SP,h}_{a}])\end{split}$
(A.1)

and the decay amplitude of $B^{0}_{S}\rightarrow\ f_{2}(1270)f_{2}(1270)$ is
same to the $B^{0}_{s}\rightarrow a^{0}_{2}(1320)a^{0}_{2}(1320)$ decays.

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
K^{*}_{2}(1430)\overline{\emph{K}}^{*}_{2}(1430))&=-{G_{F}}(V^{*}_{tb}V_{ts}[({\frac{7}{3}C_{3}+\frac{5}{3}C_{4}-\frac{7}{6}C_{9}-\frac{5}{6}C_{10}})F^{LL,h}_{a}\\\
&+2({C_{5}+\frac{1}{3}C_{6}-\frac{1}{2}C_{7}-\frac{1}{6}C_{8}})F^{LR,h}_{a}\\\
&+(-\frac{1}{2}C_{8}-\frac{1}{6}C_{7}+C_{6}+\frac{1}{3}C_{5})F^{SP,h}_{a}\\\
&+(C_{3}-\frac{1}{2}C_{9})M^{LL,h}_{e}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{e}\\\
&+(C_{3}+2C_{4}-\frac{1}{2}C_{9}-C_{10})M^{LL,h}_{a}\\\
&+({2}C_{6}-C_{8})M^{SP,h}_{a}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{a}])\end{split}$ (A.2)

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
f^{{}^{\prime}}_{2}(1525)f^{{}^{\prime}}_{2}(1525))&=-{G_{F}}V^{*}_{tb}V_{ts}[({\frac{4}{3}C_{3}+\frac{4}{3}C_{4}-\frac{2}{3}C_{9}-\frac{2}{3}C_{10}})F^{LL,h}_{a}\\\
&+({C_{5}+\frac{1}{3}C_{6}-\frac{1}{2}C_{7}-\frac{1}{6}C_{8}})F^{LR,h}_{a}\\\
&+({C_{6}+\frac{1}{3}C_{5}-\frac{1}{2}C_{8}-\frac{1}{6}C_{7}})F^{SP,h}_{a}\\\
&+(C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10})M^{LL,h}_{e}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{e}\\\
&+(C_{3}+C_{4}-\frac{1}{2}C_{9}-\frac{1}{2}C_{10})M^{LL,h}_{a}\\\
&+(C_{6}-\frac{1}{2}C_{8})M^{SP,h}_{a}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{a}\\\
&+(C_{6}-\frac{1}{2}C_{8})M^{SP,h}_{e}]\end{split}$ (A.3)

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
f_{2}(1270)f^{{}^{\prime}}_{2}(1525))&=\frac{G_{F}}{{2}}({V^{*}_{ub}V_{us}[C_{2}M^{LL,h}_{e}]}\\\
&-V^{*}_{tb}V_{ts}[({2}C_{4}+\frac{1}{2}C_{10})M^{LL,h}_{e}\\\
&-(\frac{1}{2}C_{8}+{2}C_{6})M^{SP,h}_{e}])\end{split}$ (A.4)

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
a^{+}_{2}(1320)a^{-}_{2}(1320))&=\frac{G_{F}}{\sqrt{2}}({V^{*}_{ub}V_{us}[(C_{1}+\frac{1}{3}C_{2})F^{LL,h}_{a}+C_{2}M^{LL,h}_{a}]}\\\
&-2V^{*}_{tb}V_{ts}[({{2}C_{3}+\frac{2}{3}C_{4}+{2}C_{5}+\frac{2}{3}C_{6}}\\\
&+{\frac{1}{2}C_{7}+\frac{1}{6}C_{8}+\frac{1}{2}C_{9}+\frac{1}{6}C_{10}})F^{LL,h}_{a}\\\
&+({2}C_{4}+\frac{1}{2}C_{10})M^{LL,h}_{a}+({2}C_{6}+\frac{1}{2}C_{8})M^{SP,h}_{a}])\end{split}$
(A.5)

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
K^{*+}_{2}(1430)\overline{\emph{K}}^{*-}_{2}(1430))&=\frac{G_{F}}{\sqrt{2}}({V^{*}_{ub}V_{us}[(C_{1}+\frac{1}{3}C_{2})F^{LL,h}_{a}+C_{2}M^{LL,h}_{a}+C_{1}M^{LL,h}_{e}]}\\\
&-V^{*}_{tb}V_{ts}[({\frac{7}{3}C_{3}+\frac{5}{3}C_{4}+\frac{1}{3}C_{9}-\frac{1}{3}C_{10}})F^{LL,h}_{a}\\\
&+(2{C_{5}+\frac{2}{3}C_{6}+\frac{1}{2}C_{7}+\frac{1}{6}C_{8}})F^{LR,h}_{a}\\\
&+(-\frac{1}{2}C_{8}-\frac{1}{6}C_{7}+C_{6}+\frac{1}{3}C_{5})F^{SP,h}_{a}\\\
&+(C_{3}-\frac{1}{2}C_{9})M^{LL,h}_{e}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{e}\\\
&+(C_{3}+2C_{4}-\frac{1}{2}C_{9}-C_{10})M^{LL,h}_{a}\\\
&+({2}C_{6}-C_{8})M^{SP,h}_{a}\\\
&+(C_{5}-\frac{1}{2}C_{7})M^{LR,h}_{a}])\end{split}$ (A.6)

Based on the mixing scheme, the helicity amplitudes of $B^{0}_{s}\rightarrow
f_{n}f_{n}(f_{s}f_{s})$ and $B^{0}_{s}\rightarrow f_{n}f_{s}$ decays are given
by Jiang:2020eml

$\begin{split}A_{h}(B^{0}_{s}\rightarrow
f_{2}(1270)f^{{}^{\prime}}_{2}(1525))&=\sin(2\theta)[A_{h}(B^{0}_{s}\rightarrow
f_{n}f_{n})-A_{h}(B^{0}_{s}\rightarrow f_{s}f_{s})]\\\
&+\cos(2\theta)A_{h}(B^{0}_{s}\rightarrow f_{n}f_{s}),\end{split}$ (A.7)
$\begin{split}\sqrt{2}A_{h}(B^{0}_{s}\rightarrow(f_{2}(1270)f_{2}(1270))&=2\cos^{2}\theta
A_{h}(B^{0}_{s}\rightarrow f_{n}f_{n})-\sin(2\theta)A_{h}(B^{0}_{s}\rightarrow
f_{n}f_{s})\\\ &+2\sin^{2}\theta A_{h}(B^{0}_{s}\rightarrow
f_{s}f_{s}),\end{split}$ (A.8)
$\begin{split}\sqrt{2}A_{h}(B^{0}_{s}\rightarrow(f^{{}^{\prime}}_{2}(1525)f^{{}^{\prime}}_{2}(1525))&=2\sin^{2}\theta
A_{h}(B^{0}_{s}\rightarrow f_{n}f_{n})+\sin(2\theta)A_{h}(B^{0}_{s}\rightarrow
f_{n}f_{s})\\\ &+2\cos^{2}\theta A_{h}(B^{0}_{s}\rightarrow
f_{s}f_{s}).\end{split}$ (A.9)

In this part, we summarize the functions that appear in the previous sections.
For the factorizable annihilation diagrams that the first two diagrams in the
Fig. 1, whose hard scales $t_{i}$ can be writtten by

$\begin{split}&t_{e}=\mathrm{Max}\\{{\alpha_{1}M_{B_{s}^{0}},\beta
M_{B_{s}^{0}},\frac{1}{b_{2}},\frac{1}{b_{3}}\\}},\\\
&t_{f}=\mathrm{Max}\\{{\alpha_{2}M_{B_{s}^{0}},\beta
M_{B_{s}^{0}},\frac{1}{b_{2}},\frac{1}{b_{3}}\\}},\\\
&h_{af}(\alpha_{i},\beta,b_{2},b_{3})=(\frac{\emph{i}{\pi}}{2})^{2}H^{(1)}_{0}(\beta
M_{B_{s}^{0}}b_{2})\times[\theta(b_{2}-b_{3})H^{(1)}_{0}(\alpha_{i}M_{B_{s}^{0}}b_{2})J_{0}(\alpha_{i}M_{B_{s}^{0}}b_{3})\\\
&+\theta(b_{3}-b_{2})H^{(1)}_{0}(\alpha_{i}M_{B_{s}^{0}}b_{3})J_{0}(\alpha_{i}M_{B_{s}^{0}}b_{2})]S_{t}(x_{3}),\end{split}$
(A.10)

with

$\begin{split}&\alpha_{1}=\sqrt{1-(1-r_{2}^{2})x_{3}},\\\
&\alpha_{2}=\sqrt{[(1-r_{3}^{2})x_{2}+r_{3}^{2}](1-r_{2}^{2})},\\\
&\beta=\sqrt{[(1-r_{2}^{2})(1-x_{3})][r_{3}^{2}+x_{2}(1-r_{3}^{2})]}.\end{split}$
(A.11)

The parameterized expression of the threshold resummation function $S_{t}(x)$
is Li:2001ay :

$\begin{split}S_{t}(x)=[x(1-x)]^{c}\frac{2^{1+2c}\Gamma(\frac{3}{2}+c)}{\sqrt{\pi}\Gamma(1+c)}\end{split}$
(A.2)

where $c=0.04$. The evolution factors $E_{af}(t)$ in the matrix elements are
given by

$\begin{split}&E_{af}(t)=\alpha_{s}(t)\exp[-S_{2}(t)-S_{3}(t)]\end{split}$
(A.2)

The Sudakov exponents can be written as

$\begin{split}&S_{B}(t)=s(x_{1}\frac{M_{B_{s}^{0}}}{\sqrt{2}},b_{1})+\frac{5}{3}\int^{t}_{1/b_{1}}\emph{d}{\bar{\mu}}\frac{\gamma_{q}(\alpha_{s}({\bar{\mu}}))}{\bar{\mu}},\\\
&S_{i}(t)=s(x_{i}\frac{M_{B_{s}^{0}}}{\sqrt{2}},b_{i})+s((1-x_{i})\frac{M_{B_{s}^{0}}}{\sqrt{2}},b_{i})+2\int^{t}_{1/b}\emph{d}{\bar{\mu}}\frac{\gamma_{q}(\alpha_{s}({\bar{\mu}}))}{\bar{\mu}}.\end{split}$
(A.2)

where the Sudakov factors $s(Q,b)$ are derived from the double logarithmic
summation. Its specific expression can be found in the Ref. Keum:2000wi . For
the nonfactorizable contribution, due to the small numerical effect, we drop
the threshold resummation function Li:2002mi . For the nonfactorizable
annihilation diagrams, the scales and the hard functions are Beuzit:2001pt

$\begin{split}&t_{g}=\mathrm{Max}\\{{\alpha
M_{B_{s}^{0}},\sqrt{|\beta_{1}|}M_{B_{s}^{0}},\frac{1}{b_{1}},\frac{1}{b_{2}}\\}},\\\
&t_{h}=\mathrm{Max}\\{{\alpha
M_{B_{s}^{0}},\sqrt{|\beta_{2}|}M_{B_{s}^{0}},\frac{1}{b_{1}},\frac{1}{b_{2}}\\}},\\\
&E_{anf}(t)=\alpha_{s}(t)\exp[-S_{B}(t)-S_{2}(t)-S_{3}(t)]|_{b_{2}=b_{3}},\\\
&h_{anf}(\alpha,\beta_{i},b_{1},b_{2})=\frac{\emph{i}{\pi}}{2}[\theta(b_{1}-b_{2})H^{(1)}_{0}(\alpha
M_{B_{s}^{0}}b_{1})J_{0}(\alpha M_{B_{s}^{0}}b_{2})\\\
&+\theta(b_{2}-b_{1})H^{(1)}_{0}(\alpha M_{B_{s}^{0}}b_{2})J_{0}(\alpha
M_{B_{s}^{0}}b_{1})]\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\begin{cases}\frac{\emph{i}{\pi}}{2}H^{(1)}_{0}(M_{B_{s}^{0}}b_{1}\sqrt{|\beta_{i}|}),&{\beta_{i}<0}\\\
K_{0}(M_{B_{s}^{0}}b_{1}\sqrt{\beta_{i}}),&{\beta_{i}>0}\\\ \end{cases}\\\
\end{split}$ (A.2)

with $i=1,2$

$\begin{split}&\alpha=\sqrt{(1-x_{3})(1-r_{2}^{2})[r_{3}^{2}+x_{2}(1-r_{3}^{2})]},\\\
&\beta_{1}=1-[(1-r_{3}^{2})(1-x_{2})-x_{1}][r_{2}^{2}+x_{3}(1-r_{2}^{2})],\\\
&\beta_{2}=(1-r_{2}^{2})(1-x_{3})[x_{1}-x_{2}(1-r_{3}^{2})-r_{3}^{2}].\end{split}$
(A.2)

For the nonfactorizable emission diagrams, the rest functions are follows

$\begin{split}&t_{c}=\mathrm{Max}\\{M_{B_{s}^{0}}\sqrt{(1-r_{2}^{2})x_{1}x_{3}},M_{B_{s}^{0}}\sqrt{|[(1-r_{3}^{2})(x_{2}-1)+x_{1}][r_{2}^{2}+x_{3}(1-r_{2}^{2})]|},\frac{1}{b_{1}},\frac{1}{b_{2}}\\},\\\
&t_{d}=\mathrm{Max}\\{M_{B_{s}^{0}}\sqrt{(1-r_{2}^{2})x_{1}x_{3}},M_{B_{s}^{0}}\sqrt{|[(r_{3}^{2}-1)x_{2}+x_{1}]x_{3}(1-r_{2}^{2})|},\frac{1}{b_{1}},\frac{1}{b_{2}}\\},\end{split}$
(A.2)

The function $h_{enf}$ can be determined by

$\begin{split}&h_{enf}(\alpha,\beta_{i},b_{1},b_{2})=[\theta(b_{2}-b_{1})I_{0}(\alpha
b_{1})K_{0}(\alpha b_{2})+\theta(b_{1}-b_{2})I_{0}(\alpha b_{2})K_{0}(\alpha
b_{1})]\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\begin{cases}\frac{\emph{i}{\pi}}{2}H^{(1)}_{0}(M_{B_{s}^{0}}b_{2}\sqrt{|\beta^{2}_{i}|}),&{\beta^{2}_{i}<0}\\\
K_{0}(M_{B_{s}^{0}}b_{2}\beta_{i}),&{\beta^{2}_{i}>0}\\\ \end{cases}\\\
\end{split}$ (A.2)

with $i=1,2$

$\begin{split}&E_{enf}(t)=\alpha_{s}(t)\exp[-S_{B}(t)-S_{2}(t)-S_{3}(t)]|_{b_{1}=b_{3}},\\\
&\alpha=M_{B_{s}^{0}}\sqrt{(1-r_{2}^{2})x_{1}x_{3}},\\\
&\beta^{2}_{1}=[(1-r_{3}^{2})(x_{2}-1)+x_{1}][r_{2}^{2}+x_{3}(1-r_{2}^{2})],\\\
&\beta^{2}_{2}=[(r_{3}^{2}-1)x_{2}+x_{1}]x_{3}(1-r_{2}^{2}).\end{split}$ (A.2)

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