# Study of $B_{s}\to\phi(\rho^{0},\omega)$ Decays in Standard Model and Family
Non-universal $Z^{\prime}$ Model

Ying Li<EMAIL_ADDRESS>Yue Sun  Zhi-Tian Zou  Department of Physics,
Yantai University, Yantai 264005,China

###### Abstract

Within QCD factorization approach, we employ the new results of form factors
of $B_{s}\to\phi$ and calculate the branching fractions, CP asymmetries (CPAs)
and polarization fractions of the decay modes $B_{s}\to\phi(\rho^{0},\omega)$
in both the standard model (SM) and the family non-universal $Z^{\prime}$
model. We find that in SM the above observables are relate to two parameters
$\rho_{H}$ and $\phi_{H}$, which characterize the end-point divergence in the
hard spectator-scattering amplitudes. When setting $\rho_{H}=0.5$ and
$\phi_{H}\in[-180,180]^{\circ}$, the theoretical uncertainties are large. By
combining the experimental data, our results can be used to constrain these
two parameters. Supposing $\rho_{H}=0$, we study the effects of $Z^{\prime}$
boson on the concerned observables. With the available branching fraction of
$B_{s}\to\phi\rho^{0}$, the possible ranges of parameter spaces are obtained.
Within the allowed parameter spaces, the branching fraction of
$B_{s}\to\phi\omega$ can be enlarged remarkably. Furthermore, the new
introduced weak phase plays important roles in significantly affecting the
CPAs and polarization fractions, which are important observables for probing
the effects of NP. If these decay modes were measured in the on-going LHC-b
and Belle-II experiments in future, the peculiar deviation from SM could
provide a signal of the family non-universal $Z^{\prime}$ model, which can
also be used to constrain the mass of $Z^{\prime}$ boson in turn.

###### pacs:

13.25.Hw, 12.38.Bx

##  I Introduction

It is well known that the hadronic charmless $B$-meson weak decays play
important roles in testing the flavor dynamics of the standard model (SM) and
searching for possible effects of new physics (NP) beyond of SM. Particularly,
decays dominated by contributions from flavour-changing neutral-currents
(FCNC) provide a sensitive probe for NP because their amplitudes are described
by loop (or penguin) diagrams where new particles may enter. Moreover, the SM
predictions for CP asymmetry (CPA) in several decays are tiny, making them
ideal places to look for effects of NP. In past decades, there are already
many measurements in rare $B$ decays at the B-factories Belle and BABAR, and
the LHC-experiments, such as $B\to\pi\pi$, $B\to\pi K^{*}$, $B\to\phi K^{*}$
and $B_{s}\to\phi\phi$ decays ParticleDataGroup:2022pth .

Although new particles have not been observed directly in the on-going LHC
experiments, several experimental results on rare $b$ decays show poor
agreement with the corresponding SM predictions. For example, the LHCb
collaboration reported deviations in the angular distribution of the $B\to
K^{*}\mu^{+}\mu^{-}$ decay (the “$P_{5}^{\prime}$ anomaly”) and in the
branching fractions of the $B_{s}\to\phi\mu^{+}\mu^{-}$, $B\to
K^{*}\mu^{+}\mu^{-}$, $B\to K\mu^{+}\mu^{-}$ and $B\to K^{(*)}\nu\bar{\nu}$
decays (for recent reviews, see, e.g., Li:2018lxi ; Bifani:2018zmi  and
references therein). Theoretically, in order to explain above deviations, on
the one hand, we need to calculate all possible corrections in SM including
high-order and high-power corrections, and on the other hand we also consider
whether these anomalies are due to contributions of NP. It is found that most
deviations of semi-leptonic $B$ decays are related to the FCNC $b\to
s\mu^{+}\mu^{-}$ transition. If NP is indeed at the origin of the anomalies in
$b\to s\mu^{+}\mu^{-}$ transition, it is natural to expect signals in other
observables induced by other $b\to s$ transitions, possibly with different
realizations though sharing some common features. A natural extension to
explore the possible existence of these signals are the charmless hadronic $B$
decays induced by $b\to sq\bar{q}$ transitions. Based on this strategy, some
decays such as $B_{s}^{0}\to K^{(*)}\overline{K}^{(*)}$ Alguero:2020xca ;
Li:2022mtc  and $B_{s}^{0}\to\phi\phi$ Kapoor:2023txh  have been studied
comprehensively. However, unlike semileptonic decays, $B$ meson hadronic
decays suffer from larger uncertainties arising from the nonperturbative
inputs and corrections from high-order and high-power, so that it is more
difficult to calculate with a high precision. For this reason, we are
encouraged to search for observables of some decay modes that have less
theoretical uncertainties.

Motivated by above, we would like to study the process
$B_{s}^{0}\to\phi\rho^{0}$ and $B_{s}^{0}\to\phi\omega$, which are dominated
by the FCNC $b\to sq\bar{q}$ transition. In SM, the possible feynman diagrams
for the $B_{s}^{0}\to\phi\rho^{0}$ are presented in Figure.1, where Fig. 1(a)
is the electroweak penguin diagram, Fig. 1(b) is the suppressed tree diagram
and Fig. 1(c) is the singlet-annihilation diagram. In particular, the last
diagram called hairpin diagram is viewed as suppressed by Okubo-Zweig-Iizuka
(OZI) rules, because the $\rho^{0}$ meson is produced from at least three
gluons, thus this contribution is neglected. Therefore,
$B_{s}^{0}\to\phi\rho^{0}$ decay is independent of weak annihilation
contributions and only sensitive to hard spectator scattering corrections,
making it an ideal channel for determining the end-point parameters in the
hard spectator scattering (HSS) amplitudes. Furthermore, the interference
between Fig. 1(a) and Fig. 1(b) might lead to possible CPA in this decay.

(a) (b) (c)

Figure 1: The possible feynman diagrams for the $B_{s}^{0}\to\phi\rho^{0}$, in
which (a) is the electroweak penguin diagram, (b) is the suppressed tree
diagram and(c) is the singlet-annihilation diagram.

In order to calculate the two body charmless non-leptonic $B$ decays, several
attractive QCD-inspired approaches, such as QCD factorization (QCDF)
Beneke:1999br ; Beneke:2000ry , perturbative QCD (pQCD) Keum:2000ph ;
Lu:2000em  and soft-collinear effective theory (SCET) Bauer:2000yr ;
Bauer:2001yt , have been proposed in the last decades. In previous studies,
the branching fractions of these decays are shown to be about $10^{-7}$ in SM,
both in QCDF Beneke:2006hg ; Chang:2017brr ; Cheng:2009mu  and in PQCD
Ali:2007ff . In the experimental side, the branching fraction of
$B_{s}\to\phi\rho^{0}$ decay has been measured by the LHCb collaboration
LHCb:2016vqn ,

$\displaystyle{\cal B}(B_{s}\to\phi\rho^{0})=(2.7\pm 0.7_{stat.}\pm
0.2_{syst.})\times 10^{-7},$ (1)

with a significance of about $4\sigma$. However, the polarization fractions
and CPA have not been measured till now. In addition, $B_{s}\to\phi\rho^{0}$
has been suggested as a tool to measure $\gamma$ via the mixing-induced CP
asymmetry Fleischer:1994rs . Since in the era of LHCb and Belle-II these two
processes become interesting objects for tests of isospin-violation and
potential NP Belle-II:2018jsg  we will in the following study their
phenomenology in full detail, in SM and beyond. In the calculation, we will
adopt the QCDF approach, since there is no annihilation contribution in these
decays.

Although most experimental data is consistent with the SM predictions, most of
us believe that SM is just an effective theory of a more fundamental one yet
to be discovered. The presence of a $Z^{\prime}$ boson associated with an
additional $\mathrm{U}(1)^{\prime}$ gauge symmetry is a well-motivated
extension of SM. It should be emphasized that this additional symmetry has not
been invented to solve a particular problem of SM, but rather occurs as a
byproduct in many models like e.g. grand unified theories, various models of
dynamical symmetry breaking and little-Higgs models. An extensive review about
the physics of $Z^{\prime}$ gauge-bosons can be found in Langacker:2008yv .
The most interesting feature is that the family non-universal $Z^{\prime}$
couplings could lead to FCNC in the tree level Buchalla:1995dp ; Nardi:1992nq
. The phenomenological effects of $Z^{\prime}$ in $B$ decays have been studied
extensively Barger:2009qs ; Cheung:2006tm ; Chang:2009wt ; Hua:2010wf ;
Hua:2010we . In this work, we will address the effect of the $Z^{\prime}$ in
the rare decay modes $B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\omega$. Because
these two decays are all penguin dominated process and mediated by $b\to
sq\bar{q}$, they are expected to be sensitive to the effect of the
$Z^{\prime}$.

The layout of this paper is as follows. In Sec.II we will present the
theoretical predictions of $B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\omega$ in
SM based on QCDF. Sec.III is devoted to the contributions of $Z^{\prime}$.
Some discussions are also given in this section. We will summarize this work
at last.

##  II Predictions in SM

In SM, the effective weak Hamiltonian mediating FCNC transition of the type
$b\to sq\bar{q}~{}(q=u,d)$ has the form Buchalla:1995vs :

${\cal
H}_{eff}={\frac{G_{F}}{\sqrt{2}}}\Big{[}\sum_{p=u,c}V_{pb}V^{*}_{ps}\Big{(}C_{1}O_{1}^{p}+C_{2}O_{2}^{p}\Big{)}\\\
-V_{tb}V^{*}_{ts}\sum_{i=3}^{10}C_{i}O_{i}\Big{]},$ (2)

where $G_{F}$ is Fermi coupling constant and $V_{pb}V^{*}_{ps}$ is the product
of the CKM matrix element. $C_{i}$ are the Wilson coefficients with the
renormalization scale $\mu\sim{m_{b}}$. The $O_{i}$ are the local four fermion
operators, and $O_{1,2}^{p}$ are the left-handed current-current operators,
$O_{3,...,6}$ and $O_{7,...,10}$ are the QCD and electroweak penguin
operators, respectively, and they can be expressed as follows:

$\displaystyle O_{1}^{p}=(\bar{s}b)_{V-A}(\bar{q}p)_{V-A},$ $\displaystyle
O_{2}^{p}=(\bar{s}_{\alpha}b_{\beta})_{V-A}(\bar{q}_{\beta}p_{\alpha})_{V-A},$
$\displaystyle
O_{3}=(\bar{s}b)_{V-A}\sum\limits_{q^{\prime}}(\bar{q}q)_{V-A},$
$\displaystyle
O_{4}=(\bar{s}_{\alpha}b_{\beta})_{V-A}\sum\limits_{q}(\bar{q}_{\beta}q_{\alpha})_{V-A},$
$\displaystyle
O_{5}=(\bar{s}b)_{V-A}{\sum\limits_{q^{\prime}}}(\bar{q}q)_{V+A},$
$\displaystyle
O_{6}=(\bar{s}_{\alpha}b_{\beta})_{V-A}{\sum\limits_{q}}(\bar{q}_{\beta}q_{\alpha})_{V+A},$
$\displaystyle
O_{7}=(\bar{s}b)_{V-A}\sum\limits_{q}\frac{3}{2}e_{q}(\bar{q}q)_{V+A},$
$\displaystyle
O_{8}=(\bar{s}_{\alpha}b_{\beta})_{V-A}\sum\limits_{q}\frac{3}{2}e_{q}(\bar{q}_{\beta}q_{\alpha})_{V+A},$
$\displaystyle
O_{9}=(\bar{s}b)_{V-A}\sum\limits_{q}\frac{3}{2}e_{q}(\bar{q}q)_{V-A},$
$\displaystyle
O_{10}=(\bar{s}_{\alpha}b_{\beta})_{V-A}\sum\limits_{q}\frac{3}{2}e_{q}(\bar{q}_{\beta}q_{\alpha})_{V-A},$
(3)

where $\alpha$ and $\beta$ are color indices and $e_{q}$ are charges of the
corresponding quark.

In QCDF approach, the hadronic matrix element of the decay
${\overline{B}}_{s}\to\phi\rho^{0}$ can be written as

$\langle\phi\rho^{0}|O_{i}|\overline{B}_{s}^{0}\rangle=\sum_{j}F_{j}^{B_{s}^{0}\to\phi}\int_{0}^{1}dxT_{ij}^{I}(x)\Phi_{\rho}(x)\\\
+\int_{0}^{1}d\xi\int_{0}^{1}dx\int_{0}^{1}dyT_{i}^{II}(\xi,x,y)\Phi_{B_{s}}(\xi)\Phi_{\phi}(x)\Phi_{\rho}(y),$
(4)

where $T_{ij}^{I}$ and $T_{i}^{II}$ are the perturbative short-distance
interactions and can be calculated perturbatively.
$\Phi_{X}(x)~{}(X=B_{s},\pi,\phi)$ are the universal and non-perturbative
distribution amplitudes, which can be estimated by the non-perturbative
approaches, such as the light cone QCD sum rules, QCD sum rules or lattice
QCD.

Because the initial state heavy meson has spin 0, the two vector mesons must
have the same helicity due to the conservation of angular momentum. Within the
framework of QCDF, the effective Hamiltonian matrix elements are written in
the form

$\displaystyle\left\langle{\phi\rho^{0}}\left|\mathcal{H}_{eff}\right|\overline{B}_{s}^{0}\right\rangle=\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}\lambda_{p}^{(q)}\left\langle{\phi\rho^{0}}\left|\mathcal{T}_{\mathcal{A}}^{h,p}\right|\overline{B}_{s}^{0}\right\rangle$
(5)

where the superscript $h$ denotes the helicity of the final-state meson.
${\mathcal{T}_{\mathcal{A}}^{h,p}}$ describes contributions from naive
factorization, vertex corrections, penguin contractions and spectator
scattering expressed in terms of the flavor operators $a_{i}^{p,h}$.
Specifically,

$\displaystyle\mathcal{T}_{\mathcal{A}}^{p,h}$ $\displaystyle=$ $\displaystyle
a_{1}^{p,h}(M_{1}M_{2})\delta_{pu}(\bar{s}b)_{V-A}\otimes(\bar{u}u)_{V-A}$ (6)
$\displaystyle+$ $\displaystyle
a_{2}^{p,h}(M_{1}M_{2})\delta_{pu}(\bar{u}b)_{V-A}\otimes(\bar{s}u)_{V-A}$
$\displaystyle+$ $\displaystyle
a_{3}^{p,h}(M_{1}M_{2})\sum(\bar{s}b)_{V-A}\otimes(\bar{q}q)_{V-A}$
$\displaystyle+$ $\displaystyle
a_{4}^{p,h}(M_{1}M_{2})\sum(\bar{q}b)_{V-A}\otimes(\bar{s}q)_{V-A}$
$\displaystyle+$ $\displaystyle
a_{5}^{p,h}(M_{1}M_{2})\sum(\bar{s}b)_{V-A}\otimes(\bar{q}q)_{V-A}$
$\displaystyle+$ $\displaystyle
a_{6}^{p,h}(M_{1}M_{2})\sum(-2)(\bar{q}b)_{S-P}\otimes(\bar{s}q)_{S+P}$
$\displaystyle+$ $\displaystyle
a_{7}^{p,h}(M_{1}M_{2})\sum(\bar{s}b)_{V-A}\otimes{\frac{3}{2}}e_{q}(\bar{q}q)_{V+A}$
$\displaystyle+$ $\displaystyle
a_{8}^{p,h}(M_{1}M_{2})\sum(-2)(\bar{q}b)_{S-P}\otimes{\frac{3}{2}}e_{q}(\bar{s}q)_{S+P}$
$\displaystyle+$ $\displaystyle
a_{9}^{p,h}(M_{1}M_{2})\sum(\bar{s}b)_{V-A}\otimes{\frac{3}{2}}e_{q}(\bar{q}q)_{V-A}$
$\displaystyle+$ $\displaystyle
a_{10}^{p,h}(M_{1}M_{2})\sum(\bar{q}b)_{V-A}\otimes{\frac{3}{2}}e_{q}(\bar{s}q)_{V-A}$

where $(\bar{q}_{1}q_{2})_{V\pm
A}\equiv\bar{q}_{1}\gamma_{\mu}(1\pm{\gamma}_{5})q_{2}$ and
$(\bar{q}_{1}q_{2})_{S\pm P}\equiv\bar{q}_{1}(1\pm{\gamma}_{5})q_{2}$, and the
summation is over $q=u,d$. The symbol $\otimes$ indicates that the matrix
elements of the operators in $\mathcal{T}_{\mathcal{A}}$ are to be evaluated
in the factorized form.

The decay constant and form factors is defined by Cheng:2009mu :

$\displaystyle\langle V(p,\varepsilon)|V_{\mu}|0\rangle$ $\displaystyle=$
$\displaystyle f_{V}m_{V}\varepsilon^{*}_{\mu},$ (7) $\displaystyle\langle
V(p,\varepsilon)|V_{\mu}|B(p_{B})\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{m_{B}+m_{V}}\,\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}p_{B}^{\alpha}p^{\beta}V(q^{2}),$
(8) $\displaystyle\langle V(p,\varepsilon)|A_{\mu}|B(p_{B})\rangle$
$\displaystyle=$ $\displaystyle
i\Bigg{\\{}(m_{B}+m_{V})\varepsilon^{*}_{\mu}A_{1}(q^{2})$ (9)
$\displaystyle-$ $\displaystyle\frac{\varepsilon^{*}\cdot
p_{B}}{m_{B}+m_{V}}\,(p_{B}+p)_{\mu}A_{2}(q^{2})$ $\displaystyle-$
$\displaystyle 2m_{V}\frac{\varepsilon^{*}\cdot
p_{B}}{q^{2}}\,q_{\mu}\big{[}A_{3}(q^{2})-A_{0}(q^{2})\big{]}\Bigg{\\}},$

where $q=p-p^{\prime}$, $A_{3}(0)=A_{0}(0)$, and

$\displaystyle
A_{3}(q^{2})=\frac{m_{B}+m_{V}}{2m_{V}}\,A_{1}(q^{2})-\frac{m_{B}-m_{V}}{2m_{V}}\,A_{2}(q^{2}).$
(10)

In this work, the form factors of $B_{s}\to\phi$ we adopted are from
Bharucha:2015bzk  based on the light-cone sum rules.

For a decay $B(p_{B})\to
V_{1}(\varepsilon_{1}^{*},p_{1})V_{2}(\varepsilon_{2}^{*},p_{2})$, its
factorizable matrix elements are thus given as

$\displaystyle X_{h}^{(BV_{1},V_{2})}$ $\displaystyle\equiv$
$\displaystyle\left\langle{V_{2}}\left|J_{\mu}\right|0\right\rangle\left\langle{V_{1}}\left|J^{\mu^{\prime}}\right|B\right\rangle$
(11) $\displaystyle=$ $\displaystyle-
if_{V_{2}}m_{2}\left[(\varepsilon_{1}^{*}\cdot\varepsilon_{2}^{*})(m_{B}+m_{V_{1}})A_{1}^{BV_{1}}(m_{V_{2}}^{2})-(\varepsilon_{1}^{*}\cdot{p_{B}})(\varepsilon_{2}^{*}\cdot{p_{B}})\frac{2A_{2}^{BV_{1}}(m_{V_{2}}^{2})}{m_{B}+m_{V_{1}}}+i\epsilon_{\mu\nu\alpha\beta}\varepsilon_{2}^{*\mu}\varepsilon_{1}^{*\nu}p_{B}^{\alpha}p_{1}^{\beta}\frac{2V^{BV_{1}}(m_{V_{2}}^{2})}{(m_{B}+m_{V_{1}})}\right],$
(12)

where $V_{1}$ takes the spectator quark of the $B$ meson and $V_{2}$ is the
emitted meson. So, the longitudinal $(h=0)$ and transverse $(h=\pm)$
components are obtained as

$\displaystyle X_{0}^{(BV_{1},V_{2})}$ $\displaystyle=$
$\displaystyle\frac{if_{V_{2}}}{2m_{V_{1}}}\left[(m_{B}^{2}-m_{V_{1}}^{2}-m_{V_{2}}^{2})(m_{B}+m_{V_{1}})A_{1}^{BV_{1}}(q^{2})-\frac{4{m_{B}^{2}}p_{c}^{2}}{m_{B}+m_{V_{1}}}A_{2}^{B{V_{1}}}(q^{2})\right],$
(13) $\displaystyle X_{\pm}^{(BV_{1},V_{2})}$ $\displaystyle=$ $\displaystyle-
if_{V_{2}}m_{B}m_{V_{2}}\left[\left(1+\frac{m_{V_{1}}}{m_{B}}\right)A_{1}^{B{V_{1}}}(q^{2})\mp\frac{2p_{c}}{m_{B}+m_{V_{1}}}V^{B{V_{1}}}(q^{2})\right].$
(14)

In QCDF, $a_{i}^{p,h}$ in Eq.(6) are basically the Wilson coefficients in
conjunction with short-distance nonfactorizable corrections including vertex
corrections and hard spectator interactions, and they have the expressions as

$\displaystyle a_{i}^{p,h}(V_{1}V_{2})=\left(C_{i}+\frac{C_{i\pm
1}}{N_{c}}\right)N_{i}^{h}(V_{2})+\frac{C_{i\pm
1}}{N_{c}}\frac{C_{F}\alpha_{s}}{4\pi}\left[V_{i}^{h}(V_{2})+\frac{4\pi^{2}}{N_{c}}H_{i}^{h}(V_{1}V_{2})\right]+P_{i}^{h,p}(V_{2}),$
(15)

where $i=1,...10$. The upper (lower) signs apply when $i$ is odd (even),
$C_{i}$ are the Wilson coefficients, $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ with
$N_{c}=3$. The functions $V_{i}^{h}(V_{2})$ stand for vertex corrections,
$H_{i}^{h}(V_{1}V_{2})$ for hard spectator interactions with a hard gluon
exchange between the emitted meson and the spectator quark and $P_{i}(V_{2})$
for penguin contractions. In addition, the expression of the quantities
$N_{i}^{h}$ reads

$\displaystyle N_{i}^{h}(V_{2})=\Bigg{\\{}\begin{array}[]{rl}0,&i=6,8\\\
1,&{\rm else}.\end{array}$ (18)

Table 1: Summary of input parameters  B meson parameters |  |  |  |  |  |
---|---|---|---|---|---|---
$B$ | $m_{B}({\rm GeV})$ | $\tau_{B}(ps)$ | $f_{B}({\rm MeV})$ | $\lambda_{B}({\rm MeV})$ |  |
$B_{s}$ | $5.366$ | $1.472$ | $230\pm 20$ | $300\pm 100$ |  |
Light vector mensons |  |  |  |  |  |
$V$ | $f_{V}({\rm MeV})$ | $f_{V}^{\perp}({\rm MeV})$ | $a_{1}^{V}$ | $a_{2}^{V}$ | $a_{1}^{\perp,V}$ | $a_{2}^{\perp,V}$
$\rho$ | $216\pm 3$ | $165\pm 9$ | $0$ | $0.15\pm 0.07$ | $0$ | $0.14\pm 0.02$
$\phi$ | $233\pm 5$ | $191\pm 9$ | $0$ | $0.18\pm 0.05$ | $0$ | $0.14\pm 0.02$
$\omega$ | $197\pm 3$ | $148\pm 9$ | $0$ | $0.15\pm 0.07$ | $0$ | $0.14\pm 0.02$
Form factors at $q^{2}=0$ |  |  |  |  |  |
$A_{1}^{B_{s}\phi}$ | $A_{2}^{B_{s}\phi}$ | $V^{B_{s}\phi}$ |  |  |  |
$0.296\pm 0.03$ | $0.25\pm 0.03$ | $0.387\pm 0.03$ |  |  |  |
Quark masses |  |  |  |  |  |
$m_{b}(m_{b})({\rm GeV})$ | $m_{c}(m_{b})({\rm GeV})$ | $m_{c}^{pole}/m_{b}^{pole}$ | $m_{s}(2.1{\rm GeV})({\rm GeV})$ |  |  |
$4.2$ | $0.91$ | $0.3$ | $0.095\pm 0.020$ |  |  |
Wolfenstein parameters |  |  |  |  |  |
$A$ | $\lambda$ | $\bar{\rho}$ | $\bar{\eta}$ | $\gamma$ |  |
$0.825$ | $0.2265$ | $0.1598$ | $0.3499$ | $67.6$ |  |

It is noted that there are end-point divergences when we study power
corrections in QCDF. When we calculate the hard spectator interactions
$H_{i}^{h}(V_{1}V_{2})$ at twist-3 order, soft and collinear divergences arise
from the soft spectator quark Beneke:1999br . Since the treatment of end-point
divergences is model dependent, this subleading power corrections generally
can be studied only in a phenomenological way. In this work, we shall follow
Beneke:1999br ; Beneke:2006hg  to model the end-point divergence in
$H_{i}^{h}(V_{1}V_{2})$ as

$\displaystyle
X_{H}\equiv\int_{0}^{1}\frac{dx}{1-x}=\left(1+\rho_{H}e^{i\phi_{H}}\right)\ln\left(\frac{m_{B}}{\Lambda_{h}}\right),$
(19)

where $\Lambda_{h}=500~{}\rm MeV$ is a typical scale, and $\rho_{H},\phi_{H}$
are the unknown real parameters.

The amplitudes of $\overline{B}_{s}^{0}\to{\phi\rho^{0}}$ and
$\overline{B}_{s}^{0}\to{\phi\omega}$ decays are written as

$\mathcal{A}_{h}(\overline{B}_{s}^{0}\to\phi\rho^{0})=\frac{G_{F}}{2}\sum_{p=u,c}\lambda_{p}^{(s)}\Bigg{[}\delta_{pu}a_{2}^{p,h}\\\
+\frac{3}{2}(a_{7}^{p,h}+a_{9}^{p,h})\Bigg{]}X_{h}^{(\overline{B}_{s}^{0}\phi,\rho)};$
(20)
$\mathcal{A}_{h}(\overline{B}_{s}^{0}\to\phi\omega)=\frac{G_{F}}{2}\sum_{p=u,c}\lambda_{p}^{(s)}\Bigg{[}\delta_{pu}a_{2}^{p,h}\\\
+2(a_{3}^{p,h}+a_{5}^{p,h})+\frac{1}{2}(a_{7}^{p,h}+a_{9}^{p,h})\Bigg{]}X_{h}^{(\overline{B}_{s}^{0}\phi,\omega)}.$
(21)

We note that the transverse amplitudes $\mathcal{A}_{\pm}$ are suppressed by a
factor $m_{2}/m_{B}$ relative to $\mathcal{A}_{0}$, as shown in Eqs.(13) and
(14). In addition, the axial-vector and vector contributions to
$\mathcal{A}_{+}$ are cancelled by each other in the heavy-quark limit, due to
an exact form factor relation Beneke:2000wa . Thus, in quark model
Korner:1979ci  or naive factorization Kramer:1991xw , the hierarchy of
helicity amplitudes

$\displaystyle\mathcal{A}_{0}:\mathcal{A}_{-}:\mathcal{A}_{+}=1:\frac{\Lambda_{\rm
QCD}}{m_{b}}:\left(\frac{\Lambda_{\rm QCD}}{m_{b}}\right)^{2}$ (22)

is therefore expected. This hierarchy can also be explained by the chirality
flip Kagan:2004uw . The transverse amplitudes defined in the transversity
basis are related to the helicity ones via

$\displaystyle\mathcal{A_{\|}}=\frac{\mathcal{A}_{+}+\mathcal{A}_{-}}{\sqrt{2}},\mathcal{A_{\perp}}=\frac{\mathcal{A}_{+}-\mathcal{A}_{-}}{\sqrt{2}}.$
(23)

Figure 2: The possible regions of CP averaged branching fraction, CPA and
longitudinal polarization fraction of decay $B_{s}\to\phi\rho^{0}$ as a
function of the phase $\theta_{H}$, the red regions represent results from the
$\rho=0.5$, and the horizontal (grey) regions are results of $\rho=0$.

Figure 3: The possible regions of CP averaged branching fraction, CPA and
longitudinal polarization fraction of decay $B_{s}\to\phi\omega$ as a function
of the phase $\theta_{H}$, the red regions represent results from the
$\rho=0.5$, and the horizontal (grey) regions are results of $\rho=0$.

With the amplitudes, we then obtain the branching fraction of
$\overline{B}_{s}^{0}\to V_{1}V_{2}$ as

$\displaystyle\mathcal{B}(\overline{B}_{s}^{0}\to
V_{1}V_{2})=\frac{\tau_{B_{s}}\left|p_{c}\right|}{8\pi{m_{B_{s}}}^{2}}\left(\left|\mathcal{A}_{L}\right|^{2}+\left|\mathcal{A}_{\|}\right|^{2}+\left|\mathcal{A}_{\perp}\right|^{2}\right),$
(24)

where $\tau_{B_{s}}$ is the lifetime of the $B_{s}$ meson and
$\left|p_{c}\right|$ is the absolute value of two final-state hadrons’
momentum in the $B_{s}$ rest frame. Then, three polarization fractions are
then defined as

$\displaystyle
f_{\alpha}\equiv\frac{\left|\mathcal{A}_{\alpha}\right|^{2}}{\left|\mathcal{A}_{0}\right|^{2}+\left|\mathcal{A}_{\|}\right|^{2}+\left|\mathcal{A}_{\perp}\right|^{2}}$
(25)

with $\alpha=L,\|,\perp$. In addition, we can also define the direct CPAs as:

$\displaystyle
A_{CP}=\frac{|\mathcal{A}(\overline{B}_{s}\to\overline{V}_{1}\overline{V}_{2})|^{2}-|\mathcal{A}(B_{s}\to
V_{1}V_{2})|^{2}}{|\mathcal{A}(\overline{B}_{s}\to\overline{V}_{1}\overline{V}_{2})|^{2}+|\mathcal{A}(B_{s}\to
V_{1}V_{2})|^{2}};$ (26) $\displaystyle
A_{CP}^{\alpha}=\frac{f_{\alpha}(\overline{B}_{s}\to\overline{V}_{1}\overline{V}_{2})-f_{\alpha}(B_{s}\to
V_{1}V_{2})}{f(_{\alpha}\overline{B}_{s}\to\overline{V}_{1}\overline{V}_{2})+f_{\alpha}(B_{s}\to
V_{1}V_{2})}.$ (27)

Using the input parameters listed in Table. 1, we now calculate the branching
fractions, polarization fractions and direct CPAs. As aforementioned, two of
the most important parameters are $\rho_{H}$ and $\phi_{H}$. We firstly follow
Beneke:2006hg  and adopt the default values $\rho_{H}=0$ and $\phi_{H}=0$.
Three observables of decay $B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\omega$ in
SM are presented as

$\displaystyle\left\\{\begin{array}[]{l}{\cal B}(B_{s}\to\phi\rho^{0})=(4.1\pm
0.5)\times 10^{-7},\\\ {\cal B}(B_{s}\to\phi\omega)=(1.5^{+0.6}_{-0.5})\times
10^{-7};\\\ \end{array}\right.$ (30)
$\displaystyle\left\\{\begin{array}[]{l}f_{L}(B_{s}\to\phi\rho^{0})=(89.5^{+0.9}_{-1.0})\%,\\\
f_{L}(B_{s}\to\phi\omega)=(82.1^{+2.4}_{-2.8})\%;\\\ \end{array}\right.$ (33)
$\displaystyle\left\\{\begin{array}[]{l}A_{CP}(B_{s}\to\phi\rho^{0})=(32.8^{+0.1}_{-0.6})\%,\\\
A_{CP}(B_{s}\to\phi\omega)=(-0.6^{+3.8}_{-2.1})\%;\\\ \end{array}\right.$ (36)

where the errors are from the uncertainties of the nonperturbative parameters.
It can be seen that the longitudinal polarization fractions and the CPAs are
not sensitive to the nonperturbative inputs, such as decay constants, form
factors, and moments in the distribution amplitudes. In addition, we find that
for $B_{s}\to\phi\rho^{0}$ the major uncertainties are from the form factors,
while the decay constants dominate the uncertainties in $B_{s}\to\phi\omega$.
The branching fractions are in agreement with the previous predictions of
Beneke:2006hg , and are larger than those of Cheng:2009mu , because the form
factors of $B_{s}\to\phi$ we used are larger than theirs. Comparing to the
experimental data LHCb:2016vqn  shown in Eq. (1), our results is a bit larger
than the data. The other observables have not been measured till now.

In ref.Chang:2017brr , the authors had fitted $\rho_{H}=0.5$ within the all
$B_{u,d,s}\to VV$ data. Within these ranges, we suppose
$\phi_{H}\in[-180^{\circ},180^{\circ}]$ and obtain the results as

$\displaystyle\left\\{\begin{array}[]{l}{\cal
B}(B_{s}\to\phi\rho^{0})=(4.2^{+0.7+0.6}_{-0.6-0.5})\times 10^{-7},\\\ {\cal
B}(B_{s}\to\phi\omega)=(2.9^{+1.1+0.0}_{-0.9-2.2})\times 10^{-7},\\\
\end{array}\right.$ (39)
$\displaystyle\left\\{\begin{array}[]{l}f_{L}(B_{s}\to\phi\rho^{0})=(86.5^{+1.0+8.0}_{-2.2-5.3})\%,\\\
f_{L}(B_{s}\to\phi\omega)=(75.7^{+2.3+13.4}_{-2.7-16.1})\%,\\\
\end{array}\right.$ (42)
$\displaystyle\left\\{\begin{array}[]{l}A_{CP}(B_{s}\to\phi\rho^{0})=(31.4^{+0.8+15.6}_{-2.8-14.7})\%,\\\
A_{CP}(B_{s}\to\phi\omega)=(-2.3^{+1.6+14.3}_{-0.8-8.9})\%,\\\
\end{array}\right.$ (45)

where the first errors are from the uncertainties of the nonperturbative
parameters, and the second ones come from the phase $\phi_{H}$. From the
results, it is obvious that the strong phase $\phi_{H}$ takes larger
uncertainties, especially for decay $B_{s}\to\phi\omega$. Moreover, when the
$\phi_{H}$ is in the range $[35,160]^{\circ}$, the branching fraction of
$B_{s}\to\phi\rho^{0}$ agrees with the data LHCb:2016vqn . In order to show
the effects of the parameters $(\rho_{H},\phi_{H})$, we set $\rho_{H}=0$ and
$\rho_{H}=0.5$ and plot the variations of the branching fractions, CPAs and
longitudinal polarization fractions of $B_{s}\to\phi\rho^{0}$ and
$B_{s}\to\phi\omega$ decays with the phase $\phi_{H}$ in Fig.2 and Fig.3,
respectively. In fact, if $\rho_{H}$ becomes larger, the uncertainties taken
by $\phi_{H}$ will be even larger. In all figures, only uncertainties of the
form factors (decay constants) in $B_{s}\to\phi\rho^{0}$
($B_{s}\to\phi\omega$) are considered. In future, if these above observables
could be measured with high precision, our theoretical results will be useful
for determining the range of $(\rho_{H},\phi_{H})$ in turn.

##  III Effects of New Physics

Table 2: The Wilson coefficients $C_{i}$ within the SM and with the contribution from $Z^{\prime}$ boson included in NDR scheme at the scale $\mu=m_{b}$ and $\mu_{h}=\sqrt{\Lambda_{h}m_{b}}$ Chang:2009wt . Wilson | $\mu=m_{b}$ | $\mu_{h}=\sqrt{\Lambda_{h}m_{b}}$
---|---|---
coefficients | $C_{i}^{SM}$ | $\Delta C_{i}^{Z^{\prime}}$ | $C_{i}^{SM}$ | $\Delta C_{i}^{Z^{\prime}}$
$C_{1}$ | $1.075$ | $-0.006\xi^{L}$ | $1.166$ | $-0.008\xi^{L}$
$C_{2}$ | $-0.170$ | $-0.009\xi^{L}$ | $-0.336$ | $-0.014\xi^{L}$
$C_{3}$ | $0.013$ | $0.05\xi^{L}-0.01\xi^{R}$ | $0.025$ | $0.11\xi^{L}-0.02\xi^{R}$
$C_{4}$ | $-0.033$ | $-0.13\xi^{L}+0.01\xi^{R}$ | $-0.057$ | $-0.24\xi^{L}+0.02\xi^{R}$
$C_{5}$ | $0.008$ | $0.03\xi^{L}+0.01\xi^{R}$ | $0.011$ | $0.03\xi^{L}+0.02\xi^{R}$
$C_{6}$ | $-0.038$ | $-0.15\xi^{L}+0.01\xi^{R}$ | $-0.076$ | $-0.32\xi^{L}+0.04\xi^{R}$
$C_{7}/{\alpha}_{em}$ | $-0.015$ | $4.18\xi^{L}-473\xi^{R}$ | $-0.034$ | $5.7\xi^{L}-459\xi^{R}$
$C_{8}/{\alpha}_{em}$ | $0.045$ | $1.18\xi^{L}-166\xi^{R}$ | $0.089$ | $3.2\xi^{L}-355\xi^{R}$
$C_{9}/{\alpha}_{em}$ | $-1.119$ | $-561\xi^{L}+4.52\xi^{R}$ | $-1.228$ | $-611\xi^{L}+6.7\xi^{R}$
$C_{10}/{\alpha}_{em}$ | $0.190$ | $118\xi^{L}-0.5\xi^{R}$ | $0.356$ | $207\xi^{L}-1.4\xi^{R}$

Though when $\rho_{H}=0.5$, the SM prediction of branching fraction of
$B_{s}\to\phi\rho^{0}$ can explain the data, we have chances to search for the
effects of NP, because $\rho_{H}$ and $\phi_{H}$ have not completely confirmed
yet. In this section, we will study the effects of an extra $Z^{\prime}$ gauge
boson in these decays $B_{s}\to\phi(\rho^{0},\omega)$. Ignore the mixing
between $Z^{0}$ and $Z^{\prime}$, we write the couplings of the $Z^{\prime}$
to fermions as Langacker:2008yv

$\displaystyle
J_{Z^{\prime}}^{\mu}=g^{\prime}\sum_{i}\bar{\psi}_{i}\gamma^{\mu}[\epsilon_{i}^{\psi_{L}}P_{L}+\epsilon_{i}^{\psi_{R}}P_{R}]\psi_{i},$
(46)

where $i$ is the family index and $\psi$ labels the fermions and
$P_{L,R}=(1\mp\gamma_{5})/2$. In some string and GUT models Chaudhuri:1994cd ;
Cleaver:1998gc ; Cvetic:2001tj ; Cvetic:2002qa ; Kuo:1984gz ; Barger:1987hh ,
the $Z^{\prime}$ couplings are not required to be family universal. When
rotating to the physical basis, FCNCs generally appear at tree level in both
left handed and right handed sectors, explicitly, as

$\displaystyle
B^{L}=V_{\psi_{L}}\epsilon^{\psi_{L}}V_{\psi_{L}}^{\dagger},\;\;\;\;\;B^{R}=V_{\psi_{R}}\epsilon^{\psi_{R}}V_{\psi_{R}}^{\dagger}.$
(47)

For simplicity, we suppose that the right-handed couplings are flavor-diagonal
and $B_{sb}^{R}=0$. As a result, the $Z^{\prime}$ part of the effective
Hamiltonian for $b\to s\bar{q}q\,(q=u,d)$ transition has the form as:

${\cal H}_{eff}^{\rm
Z^{\prime}}=\frac{2G_{F}}{\sqrt{2}}\big{(}\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\big{)}^{2}\,B_{sb}^{L}(\bar{s}b)_{V-A}\\\
\times\sum_{q}\left[B_{qq}^{L}(\bar{q}q)_{V-A}+B_{qq}^{R}(\bar{q}q)_{V+A}\right],$
(48)

where $g_{1}=e/(\sin{\theta_{W}}\cos{\theta_{W}})$ and $M_{Z^{\prime}}$ is the
$Z^{\prime}$ mass. Compared with the effective weak Hamiltonian of SM shown in
Eq.(2), the above Hamiltonian Eq. (48) can be rearranged as

${\cal H}_{eff}^{\rm Z^{\prime}}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}\\\
\times\sum_{q}(\Delta C_{3}O_{3}^{q}+\Delta C_{5}O_{5}^{q}+\Delta
C_{7}O_{7}^{q}+\Delta C_{9}O_{9}^{q})\,,$ (49)

where $O_{i}^{q}(i=3,5,7,9)$ are the effective operators of SM. $\Delta C_{i}$
are the modifications to the corresponding SM Wilson coefficients caused by
$Z^{\prime}$ boson, which are expressed as

$\displaystyle\Delta C_{3}$ $\displaystyle=$
$\displaystyle-\frac{2}{3V_{tb}V_{ts}^{\ast}}\,\big{(}\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\big{)}^{2}\,B_{sb}^{L}\,(B_{uu}^{L}+2B_{dd}^{L})\,,$
$\displaystyle\Delta C_{5}$ $\displaystyle=$
$\displaystyle-\frac{2}{3V_{tb}V_{ts}^{\ast}}\,\big{(}\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\big{)}^{2}\,B_{sb}^{L}\,(B_{uu}^{R}+2B_{dd}^{R})\,,$
$\displaystyle\Delta C_{7}$ $\displaystyle=$
$\displaystyle-\frac{4}{3V_{tb}V_{ts}^{\ast}}\,\big{(}\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\big{)}^{2}\,B_{sb}^{L}\,(B_{uu}^{R}-B_{dd}^{R})\,,$
$\displaystyle\Delta C_{9}$ $\displaystyle=$
$\displaystyle-\frac{4}{3V_{tb}V_{ts}^{\ast}}\,\big{(}\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\big{)}^{2}\,B_{sb}^{L}\,(B_{uu}^{L}-B_{dd}^{L})\,,$
(50)

in terms of the model parameters at the $M_{W}$ scale. It is found that the
$Z^{\prime}$ contributes not only to the EW penguins operators but also to the
QCD penguins ones. In particular, we suppose that new physics is manifest in
the EW penguins by setting $B_{uu}^{L,R}=-2B_{dd}^{L,R}$ as done in
Buras:2003dj . Due to the hermiticity of the effective Hamiltonian, the
diagonal elements of the effective coupling matrices $B_{qq}^{L,R}$ are real.
However, for the off-diagonal one $B_{sb}^{L}$, it could be a complex with a
new weak phase $\phi_{sb}^{L}$. As a result, the resulting $Z^{\prime}$
contributions to the Wilson coefficients are then written as:

$\displaystyle\Delta C_{3,5}\simeq 0,$ $\displaystyle\Delta
C_{9,7}=4\frac{|V_{tb}V_{ts}^{\ast}|}{V_{tb}V_{ts}^{\ast}}\xi_{sb}^{L,R}e^{-i\phi_{sb}^{L}},$
(51)

with

$\displaystyle\xi_{sb}^{L,R}=\left(\frac{g^{\prime}M_{Z}}{g_{1}M_{Z^{\prime}}}\right)^{2}\left|\frac{B_{sb}^{L}B_{dd}^{L,R}}{V_{tb}V_{ts}^{\ast}}\right|.$
(52)

The next step is to constrain the ranges of the new defined parameters
$\xi_{sb}^{L,R}$. In generally, we suppose that both the $U(1)_{Y}$ and
$U^{\prime}(1)$ gauge groups origin from the same grand unified theories, and
$g^{\prime}/g_{1}\sim 1$ is expected. The direct search for $Z^{\prime}$ boson
is one of important physics programs of current and future high-energy
colliders. However, the direct signal of the new ${Z^{\prime}}$ boson have not
been observed in the current experiments such as CMS and ATLAS, implying that
the mass of $Z^{\prime}$ would be larger than the TeV scale. In this work, we
set $M_{Z^{\prime}}\geq 3\rm TeV$ conservatively. Because the family non-
universal $Z^{\prime}$ leads to $\Delta B=2$ and $\Delta S=2$ FCNC, so that
$B_{s}^{0}-\overline{B}_{s}^{0}$ mixing happens at the tree level. Then, the
mass difference $\Delta m_{B_{s}}$ presents the most strong constraint to the
models with $Z^{\prime}$ boson, and
$\left|B_{sb}^{L}\right|\sim\left|V_{tb}V_{ts}^{*}\right|$ is theoretically
required Chang:2009tx ; Li:2012xc ; Barger:2004qc . Meanwhile, in order to
explain CPAs of $B\to K\pi$ and branching fractions of $B\to K\phi$ and $B\to
K^{*}\phi$, the diagonal elements should satisfy
$\left|B_{ss}^{L,R}\right|\sim\left|B_{dd}^{L,R}\right|\sim\mathcal{O}(1)$
Cheung:2006tm ; Barger:2009qs . In addition this parameter region is of
interest for collider detection. Based on the above results obtained, we shall
use

$\displaystyle|\xi|=|\xi_{db}^{L,R}|=|\xi_{sb}^{L,R}|\in(1\sim 2)\times
10^{-3}.$ (53)

We also note that the other SM Wilson coefficients also receive contributions
from the $Z^{\prime}$ boson through renormalization group (RG) evolution.
Given that there is no significant RG running effect between $M_{Z^{\prime}}$
and $M_{W}$ scale, the RG evolution of the modified Wilson coefficients is
exactly the same as the ones in the SM Buchalla:1995vs . The Wilson
coefficients at $m_{b}$ and $\sqrt{\Lambda_{h}m_{b}}$ scale have been
presented in Table. 2.

Figure 4: The CP averaged branching fractions, CPAs and longitudinal
polarization fractions of decay $B_{s}\to\phi\rho^{0}$ as a function of the
new weak phase $\phi_{sb}$, the dotted (red),solid (black) lines represent
results from the $\xi=0.001,0.002$, and the solid lines the horizontal
(green)lines are the center values of SM.

Figure 5: The CP averaged branching fractions, CPAs and longitudinal
polarization fractions of decay $B_{s}\to\phi\omega$ as a function of the new
weak phase $\phi_{sb}$, the dotted (red), solid (black) lines represent
results from the $\xi=0.001,0.002$, and the solid lines the horizontal
(green)lines are the center values of SM.

To illustrate the effect of the $Z^{\prime}$ boson, by setting $\rho_{H}=0$
and $\xi=(1,2)\times 10^{-3}$, one can get the variations of the CP averaged
branching fractions, CPAs and longitudinal polarization fractions of decays
$B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\omega$ as a function of the new weak
phase $\phi_{sb}$, as shown in Fig. 4 and Fig. 5, where the horizontal lines
are the center values predicted in SM. From left panel of Fig. 4, we find that
large $\xi>0.003$, namely a lighter $Z^{\prime}$ boson, is ruled out. When
$\xi=0.001$, $\phi_{sb}\in(-40,80)^{\circ}$ is favored. With this range, the
branching fraction of $B_{s}\to\phi\omega$ would be $(0.8\sim 1.2)\times
10^{-6}$, which is larger than that of SM by one order of magnitude, as shown
in left panel of Fig. 5. As aforementioned, the newly introduced weak phase
$\phi_{sb}$ in the off-diagonal element of $B_{sb}^{L}$ plays a major role in
changing CPA. The CPAs are much sensitive to $\xi$ and $\phi_{sb}$. For
$B_{s}\to\phi\rho^{0}$, if $\xi=0.002$, the range of CPA is $-56\%\sim 38\%$.
If $\xi=0.001$, its CPA could reach $\pm 85\%$ for some special $\phi_{sb}$.
The CPA of $B_{s}\to\phi\omega$ shares the same characteristics with
$B_{s}\to\phi\rho$. These remarkable changes shown in the center panels of
Fig. 4 and Fig. 5 will be important signals in testing the model.

For the decay $B\to\phi\rho^{0}$, the longitudinal polarization fraction
$f_{L}$ is not sensitive to nonperturbative parameters if we adopt
$\rho_{H}=0$ and $\phi_{H}=0$, as shown in the right panel of Fig. 2. When
including the contribution of $Z^{\prime}$, it could decrease to 0.33, when
$\xi=0.001$ and $\phi_{sb}=27^{\circ}$. If $\phi_{sb}=-10^{\circ}$, it could
be enhanced to $0.97$. For the decay $B\to\phi\omega$, if
$\phi_{sb}\in(-40,80)^{\circ}$, the $f_{L}$ is in the range $(0.84,1)$, which
is larger than the prediction of SM. In addition, we also note that when
$\xi=0.002$, the changes is not as remarkable as the case of $\xi=0.001$,
which can be seen in the right panels of of Fig. 4 and Fig. 5. The reason is
that, when $\xi=0.001$, the contribution of $Z^{\prime}$ is comparable with
that of SM, however when $\xi=0.002$, the contribution of $Z^{\prime}$ becomes
larger than that of SM and dominate the amplitude. If we only consider effect
of $Z^{\prime}$, $f_{L}=0.91$ for both decays. In future, these observables
could be used to probe the effect of new physics. If the $Z^{\prime}$ were
detected in the colliders directly, these decays would also be useful to
constrain the couplings.

##  IV Summary

In this work, we calculated the branching fractions, CP asymmetries and
polarization fractions of the decay mode $B_{s}\to\phi\rho^{0}$ and
$B_{s}\to\phi\omega$ within the QCD factorization approach in both the SM and
the family non-universal $Z^{\prime}$ model. This approach is suitable because
these decay modes have no contributions from annihilation diagrams. In SM,
both decays are sensitive to two parameters $\rho_{H}$ and $\phi_{H}$, which
are from end-point singularities in the hard spectator scattering.

Using the latest results of form factors of $B_{s}\to\phi$, we obtained ${\cal
B}(B_{s}\to\phi\rho^{0})=(4.1\pm 0.5)\times 10^{-7}$ with $\rho_{H}=0$, and is
$(4.2^{+0.7+0.6}_{-0.6-0.5})\times 10^{-7}$ with $\rho_{H}=0.5$ and
$\phi_{H}\in[-180,180]^{\circ}$. These results are a bit larger than the
current experimental data. In addition, the longitudinal polarization fraction
$f_{L}$ is $89.4\%$ and $86.3\%$ for different values of $\rho_{H}$. Due to
the interference between tree and penguin contributions, the CPA $A_{CP}$ is
about $30\%$. The future measurements of $f_{L}$ and $A_{CP}$ are helpful to
determine the parameters ($\rho_{H},\phi_{H}$). The decay $B_{s}\to\phi\omega$
was also studied.

Because when we adopt $\rho_{H}=0.5$, the effects of $Z^{\prime}$ will be
buried in the large uncertainties taken from $\phi_{H}$, we only study its
effects in $B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\omega$ by setting
$\rho_{H}=0$. In comparison with experimental data, $\xi>0.003$ (light
$Z^{\prime}$) is ruled out. By setting $\xi=0.001$, the range
$\phi_{sb}\in(-40,80)^{\circ}$ is favoured. The branching fraction of
$B_{s}\to\phi\omega$ may be enlarged by one ordr of magnitude by $Z^{\prime}$
boson within the allowed parameter space. The $f_{L}$ of
$B_{s}\to\phi\rho^{0}$ decreases to 0.33 with $\phi_{sb}=27^{\circ}$, and
reaches to 0.97 with $\phi_{sb}=-10^{\circ}$. Furthermore, as the direct CPA
is concerned, it can reach $-85\%$ with suitable parameter space. The
contribution of $Z^{\prime}$ to $B_{s}\to\phi\omega$ was also calculated. All
above observables could be measured in the on-going LHC-b experiment and
Belle-II, and the future measurements with high precision will provide a plate
to test the non-universal $Z^{\prime}$ model, and can be used to constrain the
mass of the $Z^{\prime}$ boson in turn.

## Acknowledgment

This work was supported in part by the National Science Foundation of China
under the Grant Nos. 12375089, and the Natural Science Foundation of Shandong
province under the Grant No. ZR2022ZD26.

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