# Searching for possible evidences of new physics in $B\to V_{1}V_{2}$

C.Q. Geng1,2,3 and Chia-Wei Liu1,2 1School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China
2International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou,
China
3Chongqing University of Posts & Telecommunications, Chongqing 400065, China

###### Abstract

We explore the decays of $B\to V_{1}V_{2}$ ($V_{1,2}=(\rho,\omega,K^{*},\phi)$
and $B=(B^{0},B^{+},B_{s})$) with transverse polarizations. We explicitly
evaluate the eigenstates of T-odd scalar operators involving spins for the
first time, which offer physical insight among the T violating observables.
Based on the helicity suppression of tree operators for transverse
polarizations in the standard model (SM), we deduce that
$\Delta\phi_{p}=\Delta\phi_{\parallel}-\Delta\phi_{\perp}=0$ with
$\Delta\phi_{\perp,\parallel}$ the weak phases of the transverse amplitudes.
In contrast, the experiments show that $\Delta\phi_{p}(B^{0}\to
K^{*0}\omega)=-0.84\pm 0.54$, which could be a signal of new physics. There is
also a discrepancy between our result in the SM and the experimental data for
the transverse polarized branching ratio in $B^{0}\to K^{*0}\omega$. In
addition, by counting the helicity flips, we argue that
$\sin(\phi_{\parallel}-\phi_{\perp})\approx 0$, which meets well with most of
the experiments except for the ones with $B^{0}\to K^{*0}\omega$, where
$\phi_{\parallel,\perp}$ are the strong phases in the transverse amplitudes.

##  I Introduction

It is known that the final state mesons in the $B$ decays of $B\to VT$ can be
decomposed into three orthogonal states, where
$B=(B^{0},B^{+},B_{s})\,,V=(\rho,\omega,K^{*}(892),\phi)$, and $T$ represent
spin-$n$ ($n\geq 1$) mesons. Although these decays suffer large theoretical
uncertainties, they can be used to explore the helicity structures of the
hadrons in the standard model (SM). In particular, we are allowed to observe T
violating observables originated from the triple momentum correlations
Valencia:1988it ; Chiang:1999qn ; Datta:2003mj ; London:2004ws ; LondonBtoVT ;
Gronau:2011cf ; Kramer:1993yu .

Note that among over a hundred measured parameters in the experiments for the
$B$-meson decays pdg , nearly forty of them with more in the future are
related to the T violating observables. Recently, $B^{0}\to
K^{*0}\rho^{0}/\omega$ have been fully investigated for the first time by the
LHCb Collaboration Krho . Particularly, the precision measurement in $B^{0}\to
K^{*0}\rho^{0}$ is extraordinary, where the uncertainties of the polarized
fractions are less than $4\%$. For instance, the data of LHCb has shown that
pdg ; Krho

$f_{L}(B^{0}\to K^{*0}\rho^{0})=0.173\pm 0.026\,,$ (1)

where $f_{L}$ is the longitudinal polarized fraction. In addition, the LHCb
Collaboration has also updated the experimental data for
$f_{L}(B_{s}\to\phi\phi)$ with the uncertainty is lower than $2\%$ Bsphiphi .
Clearly, these polarization parameters grant us new methods to test the SM and
search for new physics (NP) Kagan:2004uw ; Kagan:2004ia .

On the theoretical side, considerable progresses have been achieved
Suzuki:2001za ; Bartsch:2008ps ; Cheng:2008gxa ; Cheng:2009xz ; Cheng:2009cn ;
Zou:2015iwa ; FQCDBu ; FQCDBs ; SU3 ; Yan:2018fif ; FSI ; Cheng:2001aa ;
Cheng:2009mu ; Lu:2000hj ; Lu:2005be ; Li:2004mp ; Li:2005hg ; Beneke:2005we ;
Beneke:2006hg ; Huang:2005if ; Zhu:2005rt ; Zhu:2005rx ; Ali:2007ff ;
Bauer:1986bm ; Ali:1997nh ; Ali:1998gb ; Ali:2007ff ; Li:2004mp ;
Colangelo:2004rd ; Cheng:2009ms ; Raslan:2020dzh ; Raslan:2020qch ;
Rui:2021kbn ; Liu:2017cwl ; Bell:2020qus ; Lu:2006nza ; Lu:2002ny ;
Cheng:2004ru ; Ball:2006eu ; Wang:2017rmh ; Li:2021qiw ; Jia:2021avh . In the
naive factorization, the amplitudes always contain $\langle
V|(\overline{q}q^{\prime})_{V\pm A}|0\rangle$, implying that $q$ and
$\overline{q}^{\prime}$ have opposite helicities in $V$, and leading to
$f_{L}\approx 1$ Cheng:2001aa ; Bauer:1986bm ; Korner:1979ci . It agrees well
with the data in the decays dominated by tree diagrams. For instance, it is
reported that $f_{L}(B^{0}\to\rho^{+}\rho^{-})=0.990^{+0.021}_{-0.019}$ pdg .
However, the argument is obviously invalid in non-factorizable processes. In
2003, $f_{L}(B^{0}\to\phi K^{*0})$ was measured around $0.5$ by BarBar
BaBar:2003spf  and Belle Chen:2003jfa , which made theorists confront the
penguin annihilation type of diagrams Li:2004mp ; Colangelo:2004rd , or solve
the puzzle with the charm-anticharm rescattering, despite a large theoretical
uncertainty FSI ; Cheng:2004ru .

In the perturbative QCD (PQCD), the decays are calculated through six quark
effective operators Zou:2015iwa ; Li:2021qiw . Although the calculation can be
done systematically with the heavy quark expansion, it suffers from the
uncertainties in the hadron structure and QCD energy scale. On the other hand,
in the improved QCD factrorization (QCDF), it is known that the effective
Wilson coefficients depend on the helicity, which sheds light on the
underlying complexity Cheng:2009cn ; Cheng:2008gxa . Furthermore, in QCDF, one
is forced to add parameters by hand to overcome the endpoint divergence. Both
frameworks have poor precisions in decay rates dominated by penguin operators,
in which the uncertainties could be as large as $50\%$ Zou:2015iwa ;
Cheng:2009cn ; Cheng:2009mu ; FQCDBs ; FQCDBu . However, in PQCD, the
uncertainties in $f_{L}$ are canceled due to the correlations. For instance,
the latest study in PQCD gives that Li:2021qiw

$f_{L}(B^{0}\to
K^{*0}\rho^{0})=0.633^{+0.013}_{-0.012}\,{}^{+0.103}_{-0.099}\,{}^{+0.013}_{-0.015}\,,$
(2)

where the uncertainties are attributed to the shape parameters, Gegenbauer
moments and the QCD energy scale, respectively. The difference between Eqs.
(1) and (2) indicates that the underlying physics is not well understood yet.

On the kinematic aspect, as the vector mesons of $V_{1,2}$ in $B\to
V_{1}V_{2}$ are polarized, the decays provide the possibility for observing
the triple vector product (TVP) asymmetries. In the literature, the decay
amplitudes of $B\to V_{1}V_{2}$ are expanded by the momenta and the spins.
Explicitly, one has that Valencia:1988it ; Datta:2003mj

$A\left({B(p)\to
V_{1}(p_{1},\varepsilon_{1})V_{2}(p_{2},\varepsilon_{2})}\right)=a\varepsilon_{1}\cdot\varepsilon_{2}+b(p\cdot\varepsilon_{1})(p\cdot\varepsilon_{2})+i\frac{c}{m_{B}^{2}}\epsilon_{\mu\nu\rho\sigma}p^{\mu}(p_{1}^{\nu}-p_{2}^{\nu})\varepsilon_{1}^{*\rho}\varepsilon_{2}^{*\sigma}\,,$
(3)

where $A$ stands for the amplitude, $a$, $b$ and $c$ correspond to the $s$,
$d$ and $p$ wave amplitudes, $p_{i}$ and $\varepsilon_{i}$ ($i=1,2$) are the
4-momenta and polarized vectors for the vector mesons of $V_{i}$,
respectively, and $\epsilon_{\mu\nu\rho\sigma}$ is the total antisymmetric
tensor. Without the final state interactions, the T symmetry would demand $a$,
$b$ and $c$ to be real. As the term associated with $c$ in Eq. (3) takes the
TVP form in the rest frame of the $B$ meson, the results based on
$\Im(ac^{*})$ and $\Im(bc^{*})$ are referred to as the “triple product
correlations” in the literature. However, the spins $(\varepsilon_{1,2})$ do
not commute with the TVP. To be explicit, we have spinoperators

$[(\vec{s}_{1}\times\vec{s}_{2})\cdot\hat{p}_{1},\vec{s}_{1}]=i(\vec{s}_{1}\cdot\hat{p}_{1})\vec{s}_{2}-i(\vec{s}_{1}\cdot\vec{s}_{2})\hat{p}_{1}\,,$
(4)

where the subscripts label the particles, while $\vec{s}$ and $\hat{p}$ are
the spin and the norm of the three-momentum operators, respectively. Although
$\Im(ac^{*})$ and $\Im(bc^{*})$ are related to the T violating effects, it is
still questionable to discuss the TVP with Eq. (3), and a reanalysis on the
TVP asymmetry is therefore required. To this end, we expand the states with
T-odd scalar operators ($\hat{T}$), to naturally demonstrate the T violating
observables.

This work is organized as follows. In Sec. II, we construct T-odd observables
for $B\to VT$. In Sec. III, we discuss the numerical results of $B\to
V_{1}V_{2}$. We present the summary in Sec. IV.

##  II T-odd observables

In this section, we study the possible T-odd observables in $B\to VT$. For a
given T-odd operator, $\hat{T}_{i}$, one can define an asymmetry parameter as

$\displaystyle\Delta_{t_{i}}\equiv\frac{\Gamma(\lambda_{t_{i}}>0)-\Gamma(-\lambda_{t_{i}}<0)}{\Gamma(\lambda_{t_{i}}>0)+\Gamma(-\lambda_{t_{i}}<0)}=\frac{|A(\lambda_{t_{i}}>0)|^{2}-|A(-\lambda_{t_{i}}<0)|^{2}}{|A(\lambda_{t_{i}}>0)|^{2}+|A(-\lambda_{t_{i}}<0)|^{2}}$
(5)

where $\lambda_{t_{i}}$ is the eigenvalue of $\hat{T}_{i}$, and
$\Gamma(\lambda_{t_{i}})$ and $A(\lambda_{t_{i}})$ represent the partial decay
rate and amplitude for $B\to\lambda_{t_{i}}$, respectively. Here, we have
chosen $\lambda_{t_{i}}>0$ to avoid the ambiguity. If the relevant effective
Hamiltonian for the decays is invariant under the time reversal
transformation, we have GroupTheory

$|\langle\lambda_{i};{\text{``out''}}|{\cal
H}_{eff}|B\rangle|^{2}-|\langle-\lambda_{i};{\text{``in''}}|{\cal
H}_{eff}|B\rangle|^{2}=0\,,$ (6)

where “out” (“in”) denotes $t\to\infty(-\infty)$. Note that if the final state
interactions (FSIs) are absent, “out” and “in” can be interchanged freely, and
the left side of Eq. (6) is proportional to the numerator in the right side of
Eq. (5), resulting in that $\Delta_{t_{i}}$ is a T-odd observable.

Clearly, to calculate $\Delta_{t_{i}}$, one has to explicitly obtain the
eigenstates of $\hat{T}_{i}$. To do so, it is important to choose a commuting
set of operators to describe the states as emphasized in the previous section.
In the final states, the angular momenta are always constrained by the parent
particles, so it is preferable to describe the final states with $J^{2}$ and
$J_{z}$.

To be compatible with $J^{2}$ and $J_{z}$, we choose $\hat{T}_{i}$ as an
$SO(3)$ rotational scalar, which automatically commutes with the angular
momenta. In this study, we consider three of the most simple $\hat{T}_{i}$,
given as

$\displaystyle\hat{T_{2}}$ $\displaystyle=$
$\displaystyle(\vec{s}_{1}\times\vec{s}_{2})\cdot\hat{p}_{1}\,,~{}~{}~{}~{}~{}~{}~{}\hat{T}_{3}=(\vec{s}_{1}\cdot\hat{p}_{1})\hat{T}_{2}+\hat{T}_{2}(\vec{s}_{1}\cdot\hat{p}_{1})\,,$
$\displaystyle\hat{T}_{4}$ $\displaystyle=$
$\displaystyle(\vec{s}_{1}\cdot\vec{s}_{2})\hat{T}_{2}+\hat{T}_{2}(\vec{s}_{1}\cdot\vec{s}_{2})-(\hat{p}_{1}\cdot\vec{s}_{1})(\hat{p}_{1}\cdot\vec{s}_{2})\hat{T}_{2}-\hat{T}_{2}(\hat{p}_{1}\cdot\vec{s}_{1})(\hat{p}_{1}\cdot\vec{s}_{2})\,,$
(7)

where the subscripts in $\hat{T}_{i}$ denote the numbers of spin operators.
The eigenstates can be directly computed and expressed with the helicity
states as demonstrated in Appendix A. Notably, we find that $\lambda_{t_{i}}$
are discrete, in contrast to Ref. Valencia:1988it , which treats
$\lambda_{t_{i}}$ as continuous variables.

Plugging Eqs. (A), (A) and (A) in Eq. (5), we obtain

$\displaystyle\Delta_{t_{2}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{f_{L}f_{\perp}}\sin(\phi^{B}_{\perp})}{f_{L}+f_{\perp}}\,,$
$\displaystyle\Delta_{t_{3}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{f_{L}f_{0}}\sin(\phi^{B}_{\parallel})}{f_{L}+f_{0}}\,,$
$\displaystyle\Delta_{t_{4}}$ $\displaystyle=$
$\displaystyle\frac{-2\sqrt{f_{\perp}f_{\parallel}}\sin(\phi_{\parallel}^{B}-\phi_{\perp}^{B})}{f_{\perp}+f_{\parallel}}\,,$
(8)

where the superscript of “$B$” denotes the parent $B$-meson, and

$\displaystyle f_{i}=\frac{|A(H_{i})|^{2}}{\sum_{j=0,\pm}|A(H_{j})|^{2}}\,,$
(9)

where the definitions of $H_{i}$ are given in Eq. (48), and
$\phi_{\perp,\parallel}$ correspond to the phases between
$A(H_{\perp,\parallel})$ and $A(H_{0})$.

Since the oscillation between $|\pm\lambda_{i}\rangle$ is possible via the
final state interactions, to obtain a true T violating observable, we have to
subtract the effect of the final state interactions from the charge conjugated
part. Under the parity, $\hat{T_{i}}$ transform according to the number of
spin operators, which are given as

$I_{s}^{\dagger}\hat{T}_{i}I_{s}=(-1)^{i-1}\hat{T}_{i}\,,$ (10)

resulting in

$\hat{T}_{i}I_{s}|\lambda_{t_{i}}\rangle=(-1)^{i-1}\lambda_{t_{i}}I_{s}|\lambda_{t_{i}}\rangle\,,$
(11)

where $I_{s}$ is the parity operator. Accordingly, if CP is conserved, the
decay widths are related to the charge conjugated ones as

$\Gamma(\lambda_{t_{i}})=\overline{\Gamma}\left((-1)^{i-1}\lambda_{t_{i}}\right)\,.$
(12)

Consequently, the T or CP asymmetries are given by

$\displaystyle\Delta^{CP}_{t_{i}}=\Delta_{t_{i}}+(-1)^{i}\,\overline{\Delta}_{t_{i}}\,,$
(13)

where the overline denotes the charge conjugation. Note that strong phases are
not needed to have a non-zero value of the asymmetry in Eq. (13). In terms of
the complex phases, the CP-odd parameters are given as

$\displaystyle\Delta\phi_{\perp}=(\phi_{\perp}^{\overline{B}}-\phi_{\perp}^{B}\pm\pi)/2\,,\,\,\,\,\,\,\Delta\phi_{\parallel}=(\phi_{\parallel}^{\overline{B}}-\phi_{\parallel}^{B})/2\,,\,\,\,\,\,\,\Delta\phi_{p}=\Delta\phi_{\parallel}-\Delta\phi_{\perp}\,,$
(14)

which are related to $\Delta_{t_{2}}^{CP}$, $\Delta_{t_{3}}^{CP}$ and
$\Delta_{t_{4}}^{CP}$, respectively. Here, $\pm\pi$ in $\Delta\phi_{\perp}$
take account the fact that $\Delta_{t_{2}}+\overline{\Delta}_{t_{2}}=0$ if CP
is conserved. The signs are chosen such that
$\pi/2>\Delta\phi_{\perp}>-\pi/2$.

In the literature, $\Delta\phi_{\parallel}$ and $\Delta\phi_{\perp}$ are the
so-called “true” CP violating phases, whereas the “fake” ones are given by

$\phi_{\perp}=\phi_{\perp}^{B}+\Delta\phi_{\perp}\,,\,\,\,\,\,\,\phi_{\parallel}=(\phi_{\parallel}^{\overline{B}}+\phi_{\parallel}^{B})/2\,,\,\,\,\,\,\,\phi_{p}=\phi_{\parallel}-\phi_{\perp}\,,$
(15)

which are originated by the strong interaction. Likewise, we define that

$\Delta^{S}_{t_{i}}=\Delta_{t_{i}}-(-1)^{i}\overline{\Delta}_{t_{i}}\,,$ (16)

which vanish if the final state interaction and CP violation are both absent.
Clearly, non-zero values of $\Delta^{S}_{t_{i}}$ do not necessarily imply CP
or T violation as they can purely arise from the final state interactions.

In the experiments, the information of T-odd observables is often extracted
via the cascade decays with four final state particles, namely $B\to V_{1}(\to
0^{-}0^{-})T_{2}(\to 0^{-}0^{-})$, represented by the triple momentum
correlation of $(\vec{p}_{1}\times\vec{p}_{2})\cdot\vec{p}_{3}$, which is a
P-odd operator, and therefore $\Delta_{t_{3}}$, which is P-even, is not
expected to be observed. On the other hand, if the final states of the cascade
decays involve more than four particles, it is possible to construct P-even
quantities with the form $(\vec{L}_{1,2}\times\vec{p}_{3})\cdot\vec{p}_{4}$
related to $\Delta_{t_{3}}$, where $\vec{L}_{1,2}$ are the orbital angular
momentum operators formed by two particles. The argument is affirmed by the
explicit forms of the angular momentum distributions, where
$\sin(\phi_{\parallel})$ appears only in the final states with more than four
particles LondonBtoVT .

Since $\hat{T}_{4}$ involves interchanging the helicity twice (see Eq. (A)),
one naturally expects that

$\Delta^{S}_{t_{4}}\approx 0,$ (17)

which will be taken as an assumption and checked on its validity throughout
this work. In the decays with $f_{\perp}\approx f_{\parallel}$, such as those
of $B\to V_{1}V_{2}$, it means that

$\sin(\phi_{p})\approx 0\,.$ (18)

The available data for $B\to V_{1}T_{2}$ from the particle data group (PDG)
pdg  are collected in Table 1. The correlations between $\phi_{\parallel}$ and
$\phi_{\perp}$ in the experiments are assumed to be zero, unless explicitly
given by the data.

It is known that the decay parameters, obtained from the angular
distributions, exist two-fold solutions, which are related by Chiang:1999qn

$\displaystyle\phi_{\perp}$ $\displaystyle\rightarrow$
$\displaystyle\pi-\phi_{\perp}\,,$ $\displaystyle\phi_{\parallel}$
$\displaystyle\rightarrow$ $\displaystyle-\phi_{\parallel}\,.$ (19)

These two solutions have very different meanings, since the values of
$\phi_{p}=0$ and $\pi$ correspond to $H_{-}=0$ and $H_{+}=0$, respectively. In
$B^{0}\to K^{*0}(892)\rho^{0}/\omega$, we have adopted a different set of
solutions compared to the original one Krho  with the reason explained in the
next section.

Table 1:  Data from the PDG pdg , where we have adopted a different set of solution compared to the original one for the decays of $B^{0}\to K^{*0}(892)\rho^{0}/\omega$ Krho , related as $\phi_{\perp}\to\pi-\phi_{\perp}$ and $\phi_{\parallel}\to-\phi_{\parallel}$. channel | $\phi_{\perp}$ | $\phi_{p}$ | $\Delta\phi_{\perp}$ | $\Delta\phi_{p}$
---|---|---|---|---
$B^{+}\to\phi K^{*}(892)^{+}$ Aubert:2007ac ; Chen:2005zv  | $2.58\pm 0.17$ | $-0.22\pm 0.16$ | $-0.19\pm 0.21$ | $0.12\pm 0.19$
$B^{0}\to K^{*}(892)^{0}\rho^{0}$ Krho  | $0.79\pm 0.11$ | $-0.018\pm 0.056$ | $0.123\pm 0.092$ | $-0.014\pm 0.043$
$B^{0}\to K^{*}(892)^{0}\omega$ Krho  | $0.7\pm 0.9$ | $0.3\pm 0.9$ | $0.28\pm 0.56$ | $-0.84\pm 0.54$
$B^{0}\to\phi K^{*}(892)^{0}$ pdg  | $2.53\pm 0.09$ | $-0.10\pm 0.14$ | $0.08\pm 0.05$ | $-0.03\pm 0.07$
$B^{0}\to J/\psi K^{*}(892)^{0}$ pdg  | $2.96\pm 0.05$ | $-5.88\pm 0.06$ | - | -
$B^{0}\to\phi K^{*}_{2}(1430)^{0}$ pdg  | $4.5\pm 0.4$ | $-0.5\pm 0.6$ | $-0.2\pm 0.4$ | $-0.7\pm 0.6$
$B^{0}\to\psi(2s)K^{*0}(892)$ xc1  | $2.8\pm 0.3$ | $-5.6\pm 0.5$ | - | -
$B_{s}\to K^{*}(892)^{0}\overline{K}^{*}(892)^{0}$ BsKK  | $2.62\pm 0.69$ | $-0.22\pm 0.77$ | - | -
$B_{s}\to\phi\phi$ Bsphiphi  | $2.56\pm 0.06$ | $-0.26\pm 0.18$ | $0.044\pm 0.062$ | $-0.030\pm 0.084$
$B_{s}\to J/\psi\phi$ pdg  | $2.9\pm 0.4$ | $0.2\pm 0.4$ | - | -
$B_{s}\to\psi(2s)\phi$ pdg  | $3.3\pm 0.4$ | $0.4\pm 0.4$ | - | -
$B_{s}\to J/\psi\phi$ Aaij:2021mus  | $2.41^{+0.44}_{-0.43}$ | $0.70^{+0.43}_{-0.44}$ |  |

While $\phi_{\perp}$ can be in general unbounded, $\phi_{p}$ in Eq. (18) is
consistent with the data except for the decay channels involving
$c\overline{c}$. In these cases, the interacting periods for the final states
are longer, due to the smaller velocity. In the future,
$\Delta_{t_{4}}^{S}\approx 0$ can be a useful benchmark to test the theory. In
fact, our ansatz is well consistent with the theoretical approaches in the
literature Zou:2015iwa ; FQCDBs ; FQCDBu ; SU3 ; Beneke:2006hg ; Wang:2017rmh
.

Likewise, for the $B$ decay with a pair of spin-2 mesons we find that

$\displaystyle\sin(\phi_{\perp 1}-\phi_{\parallel 1})$ $\displaystyle\approx$
$\displaystyle 0\,,$ $\displaystyle\sin(\phi_{\perp 2}-\phi_{0})$
$\displaystyle\approx$ $\displaystyle 0\,.$ (20)

Here, $\phi_{\lambda}$ are the strong complex phases of $A(H_{\lambda})$ and

$\displaystyle
A(H_{\parallel_{i},\perp_{i}})=\frac{1}{\sqrt{2}}\left[A(H_{i})\pm
A(H_{-i})\right]\,,$ (21)

where $i=0,1,2$ denote the helicities. As there has been only one experiment
with a large uncertainty so far BsKK , it is hard to draw any conclusion at
this stage.

##  III $B\to V_{1}V_{2}$ decays

In this section, we would like to explore the consequences of the ansatz in
Eq. (17) with the $SU(3)$ flavor $(SU(3)_{F})$ symmetry. To this end, we
concentrate on the decays of $B\to V_{1}V_{2}$. For simplicity, we do not
explicitly write down the meson masses. We begin our study with the
constituent quark approach, in which the spins of vector mesons are originated
from the quark spins. Accordingly, $H_{+}$ requires the quarks and antiquarks
to have positive helicities, while $H_{-}$ negative ones. Since the decays of
$\overline{b}$ release a great amount of energy, the chirality is related to
the helicity. Explicitly, it means that $H_{\pm}$ contain a pair of
$\overline{q}_{L(R)}$ and $q_{R(L)}$.

The effective Hamiltonian for the charmless $b$ decays is given by
Buchalla:1995vs

$\mathcal{H}_{\mathrm{eff}}=\frac{G_{F}}{\sqrt{2}}\sum_{f=d,s}\left[V_{ub}^{*}V_{uf}(C_{1}O_{1}^{f}+C_{2}O_{2}^{f})-V_{tb}^{*}V_{tf}\left(\sum_{i=3}^{10}C_{i}O_{i}^{f}+C_{7\gamma}O^{f}_{7\gamma}+C_{8G}O_{8G}^{f}\right)\right]\,,$
(22)

with

$\displaystyle
O_{1}^{f}=(\overline{b}_{\alpha}u_{\beta})_{L}(\overline{u}_{\beta}f_{\alpha})_{L}\,,\,\,\,\,O_{2}^{f}=(\overline{b}_{\alpha}u_{\alpha})_{L}(\overline{u}_{\beta}f_{\beta})_{L}\,,$
$\displaystyle
O_{3}^{f}=(\overline{b}_{\alpha}f_{\alpha})_{L}\sum_{q}(\overline{q}_{\beta}q_{\beta})_{L}\,,\,\,\,\,O_{4}^{f}=(\overline{b}_{\alpha}f_{\beta})_{L}\sum_{q}(\overline{q}_{\beta}q_{\alpha})_{L}\,,$
$\displaystyle
O_{5}^{f}=(\overline{b}_{\alpha}f_{\alpha})_{L}\sum_{q}(\overline{q}_{\beta}q_{\beta})_{R}\,,\,\,\,\,O_{6}^{f}=(\overline{b}_{\alpha}f_{\beta})_{L}\sum_{q}(\overline{q}_{\beta}q_{\alpha})_{R}\,,$
$\displaystyle
O_{7}^{f}=\frac{3}{2}(\overline{b}_{\alpha}f_{\alpha})_{L}\sum_{q}e_{q}(\overline{q}_{\beta}q_{\beta})_{R}\,,\,\,\,\,O_{8}^{f}=\frac{3}{2}(\overline{b}_{\alpha}f_{\beta})_{L}\sum_{q}e_{q}(\overline{q}_{\beta}q_{\alpha})_{R}\,,$
$\displaystyle
O_{9}^{f}=\frac{3}{2}(\overline{b}_{\alpha}f_{\alpha})_{L}\sum_{q}e_{q}(\overline{q}_{\beta}q_{\beta})_{L}\,,\,\,\,\,O_{10}^{f}=\frac{3}{2}(\overline{b}_{\alpha}f_{\beta})_{L}\sum_{q}e_{q}(\overline{q}_{\beta}q_{\alpha})_{L}\,,$
$\displaystyle
O^{f}_{7\gamma}=\frac{e}{4\pi^{2}}m_{\mathrm{b}}\bar{b}_{\alpha}\sigma^{\mu\nu}f_{\alpha
L}F_{\mu\nu}\,,\,\,\,\,O_{8G}=\frac{g}{4\pi^{2}}m_{\mathrm{b}}\bar{b}_{\alpha}\sigma^{\mu\nu}T_{\alpha\beta}^{a}f_{\beta
L}G_{\mu\nu}^{a}$ (23)

where $q=(u\,,d\,,s\,,c)$, $G_{F}$ stands for the Fermi constant, $C_{i}$
represent the Wilson coefficients, $\alpha$ and $\beta$ correspond to the
color indexes, $O_{i(j)}$ with $i$=3,4,5,6 ($j$=7,8,9,10) are the so-called
QCD penguin (electroweak pengiun) operators, the subscript of $L(R)$ implies
the current structure of $V-A~{}(V+A)$, $O_{7\gamma}$ and $O_{8G}$ are named
as magnetic penguin operators, $F_{\mu\nu}~{}(G_{\mu\nu}^{a})$ is the photon
(gluon) field strength tensor, $f_{\alpha L}~{}(f_{\alpha}=d_{\alpha}$ or
$s_{\alpha})$ is the left-handed quark, $m_{b}$ is the $b$ quark mass, and
$T^{a}$ are the Gell-Mann matrices. Note that
$\sum_{q}\overline{q}_{\alpha}q_{\beta}$ is an $SU(3)_{F}$ flavor singlet,
which gives the simple flavor structure for the QCD penguin operators. The
operators $O_{1,2,3,4,9,10}$ produce at least one $q_{\alpha L}$ and one
$\overline{q}^{\prime}_{\beta L}$ in final states, which are suppressed by the
helicity flipping in $A(H_{\pm})$. On the other hand, $O_{5,6,7,8}$ can
contribute to either $A(H_{0})$ or $A(H_{+})$. Furthermore, due to the
hierarchy of $C_{7,8}/C_{5,6}\approx\alpha_{em}$ with $\alpha_{em}$ the fine
structure constant, the electroweak operators $O_{7,8}$ can also be safely
neglected Buchalla:1995vs .

Since $f_{\alpha L}$ is left-handed and only the transverse photon operators
are physical, $O_{7\gamma}$ only contributes to $A(H_{+})$ Beneke:2005we . The
leading order amplitudes are factorizable, described by $B\to V_{1}$ and
$\gamma\to V_{2}$, given as Beneke:2005we

$A(P_{7\gamma})=-ie^{i\delta_{1,2}}\frac{\text{G}_{F}}{\sqrt{2}}V_{tb}^{*}V_{tf}m_{b}T_{1}^{V_{1}}f_{V_{2}}a^{\prime}_{V_{1}}a_{V_{2}}{\cal
C}^{\prime}_{7\gamma}\frac{4\alpha_{em}}{3\pi}\frac{m_{B}^{2}}{m_{V_{2}}}\,,$
(24)

where $\delta_{1,2}$ are the strong phases, $T_{1}^{V_{1}}\approx 0.4$ is the
tensor form factor Lu:2002ny , $f_{V_{2}}$ is the $V_{2}$-meson decay
constant, ${\cal C}_{7\gamma}^{\prime}$ represents the effective Wilson
coefficient including spectator-scattering effects Beneke:2001at , and
$a_{V_{1}}^{\prime}$ and $a_{V_{2}}$ correspond to the overlapping factors for
$B\to V_{1}$ and $\gamma\to V_{2}$ with their explicit values, given by
Lu:2006nza

$\displaystyle
a_{\rho^{+},K^{*+},K^{*}0,\overline{K}^{*0},\phi}^{\prime}=1\,,\,\,\,\,a_{\omega}^{\prime}=-a_{\rho^{0}}^{\prime}=\frac{1}{\sqrt{2}}\,,$
$\displaystyle
a_{\rho^{0}}=\frac{3}{2\sqrt{2}}\,,\,\,\,\,a_{\omega}=\frac{1}{2\sqrt{2}}\,,\,\,\,\,a_{\phi}=-\frac{1}{2}\,,$
(25)

respectively.

In short distance contributions, we conclude that $A(H_{+})$ is dominated by
$O_{5,6,7\gamma,8G}$, whereas $A(H_{-})$ is highly suppressed, which agrees
with the analyses of PQCD and QCDF Zou:2015iwa ; Cheng:2009cn ; Beneke:2005we
. It is important to emphasize that the above argument is only valid for short
distance physics. $A(H_{+})$ may potentially oscillate to $A(H_{-})$, via the
complex phases between $A(H_{\perp})$ and $A(H_{\parallel})$, induced by the
final state interactions Suzuki:2001za . However, by demanding Eq. (18), we
find that the oscillation is forbidden and $A(H_{-})$ remains suppressed. This
is the reason that we choose a different set of solutions for $B^{0}\to
K^{*0}\rho^{0}/\omega$ compared to that in Ref. Krho  (see Table 1). The
direct result with $A(H_{-})=0$ is given by

$\delta f\equiv f_{\parallel}-f_{\perp}=0\,.$ (26)

The available experimental data are shown in Table 2. We see that Eq. (26)
meets well with the current measurements. In the following, we use
$A(H_{-})=0$ for a quantitative analysis.

In our numerical calculation, we focus on the branching ratios with positive
helicities and take that $(f_{\phi},f_{\omega},f_{\rho})=(215\pm 5,187\pm
5,216\pm 3)$ MeV Ball:2006eu ; Ball:2004rg  and $|{\cal
C}^{\prime}_{7\gamma}|^{2}=1.65^{+0.018}_{-0.017}$ Beneke:2001at . The
amplitudes from $O_{5,6}$ are extracted from the experimental values with the
$SU(3)_{F}$ symmetry as described by the following. Due to the simple flavor
structure in the penguin operators, there are only few undetermined
parameters. The possible diagrams are illustrated in Fig. 1, where $q$
represents the sum of $(u,d,s)$, and $\otimes(\odot)$ stands for the coupling
of the left (right)-handed current. Figs. 1c and 1d involve a right-handed
quark and antiquark in the final states, so they would contribute to
$A(H_{0})$ only. We are left with Figs. 1a and 1b, representing the penguin
annihilation ($P_{A}$) and flavor singlet penguin annihilation ($P_{S}$),
respectively. The amplitudes of $A[P_{A}(\overline{b}\to\overline{d})]$ and
$A[P_{A}(\overline{b}\to\overline{s})]$ associated with Fig. 1a are related by

$p_{A}\equiv
A[P_{A}(\overline{b}\to\overline{d})]/V_{td}=A[P_{A}(\overline{b}\to\overline{s})]/V_{ts}\,,$
(27)

where we only consider those with positive helicity. Similarly, we have

$p_{S}\equiv
A[P_{S}(\overline{b}\to\overline{d})]/V_{td}=A[P_{S}(\overline{b}\to\overline{s})]/V_{ts}\,.$
(28)

From Eqs. (27) and (28), it is straightforward to see that the amplitudes of
$\overline{b}\to\overline{d}$ are relatively suppressed by
$|V_{td}/V_{ts}|=0.210\pm 0.008$ pdg . With the $SU(3)_{F}$ symmetry, the
decay amplitudes can be parametrized with $p_{A}$ and $p_{S}$, given in the
second columns of Tables III and III, in which $V_{ts}$ $(V_{td})$ has not
been shown explicitly for $\overline{b}\to\overline{s}$
$(\overline{b}\to\overline{d})$.

Note that though the contributions from $O_{8_{G}}$ are not calculated as
$O_{7_{\gamma}}$, still, we can parametrize it with the $SU(3)_{F}$ structure.
We note that in contrast to $O_{7_{\gamma}}$, in which the photon operator
would inevitably break the $SU(3)_{F}$ symmetry, the gluon magnetic operators
in ${\cal H}_{eff}$ share the same $SU(3)_{F}$ structure with the QCD penguin
ones, and therefore the contributions of $O_{8_{G}}$ can be absorbed into
$A(P_{A})$ and $A(P_{S})$, as long as the $SU(3)_{F}$ symmetry is reliable.

(a)

(b)

(c)

(d)

Figure 1:  Topological diagrams for penguin operators $O_{5}$ and $O_{6}$,
where (a), (b) and (c) correspond to the penguin annihilations, while (d) the
penguin induced tree contribution.

If $A(P_{\gamma}),\,A(P_{A})$ and $A(P_{S})$ share the same phases in short
distance contributions, they remain relative real in long distance ones, due
to the same helicity. Hence, there are only 2 real parameters. The results of
the $\chi^{2}$ fitting are given by

$|p_{A}|=(5.5\pm
0.2)10^{-2}\text{G}_{F}\text{GeV}^{3}\,,\,\,\,\,\,p_{S}/p_{A}=0.14\pm
0.04\,,\,\,\,\,\chi^{2}/d.o.f.=18/(15-2)\approx 1.4\,,$ (29)

where $d.o.f.$ stands for degree of freedom. The reproduced branching ratios
with the available experimental data are given in Table II, while our
predictions are shown in Table III, where we have used that

${\cal B}^{+}_{exp}={\cal B}_{exp}-{\cal B}^{0}_{exp}-{\cal
B}^{-}_{exp}=(1-f_{L}){\cal B}_{exp}\,,$ (30)

for the data input pdg  with the superscripts of $0$ and $\pm$ denoting the
helicities, and ${\cal B}^{-}=0$. In our results, the first uncertainties
arise from the experimental input, and the second ones are the estimated
$10\%$ errors caused by the $SU(3)_{F}$ breaking. The upper part of the table
contains the decays with $\overline{b}\to\overline{s}$, which are grouped
according to the isospin group. The lower part corresponds to those with
$\overline{b}\to\overline{d}$, which are suppressed by $|V_{td}/V_{ts}|^{2}$.
We find that $p_{A}$ is about seven times larger than $p_{S}$, which is
consistent with the OZI rule. Most of the reproduced branching ratios are
consistent with the data. However, there is a tension in $B^{0}\to
K^{*0}\omega$, as the theoretical value is almost twice larger than the
experimental one, which could be a signal of NP.

Table 2: Reproduced results in the SM, where the upper (lower) part corresponds to $\overline{b}\to\overline{s}~{}(\overline{b}\to\overline{d})$, and the parametrized decay amplitudes of $O_{5,6}$ are given in the second column, in which the CKM matrix elements of $V_{ts,td}$ have not been shown explicitly for $\overline{b}\to(\overline{s}\,,\overline{d})$. channel | $SU(3)_{F}$ | $10^{6}{\cal B}_{SU(3)}^{+}$ | $10^{6}(1-f_{L}){\cal B}_{exp}$ | $\delta f_{exp}$
---|---|---|---|---
$B^{0}\to K^{*0}\rho^{0}$ | $-p_{A}/\sqrt{2}$ | $4.57\pm 0.31\pm 0.46$ | $3.25\pm 1.09$ | $0.03\pm 0.08$
$B^{0}\to K^{*+}\rho^{-}$ | $p_{A}$ | $5.61\pm 0.49\pm 0.56$ | $6.39\pm 2.10$ |
$B^{+}\to K^{*+}\rho^{0}$ | $p_{A}/\sqrt{2}$ | $1.59\pm 0.19\pm 0.16$ | $1.01\pm 0.60$ |
$B^{+}\to K^{*0}\rho^{+}$ | $p_{A}$ | $6.05\pm 0.53\pm 0.60$ | $4.78\pm 1.07$ |
$B^{0}\to K^{*0}\omega$ | $p_{A}/\sqrt{2}+\sqrt{2}p_{S}$ | $1.14\pm 0.17\pm 0.11$ | $0.62\pm 0.27$ |
$B^{+}\to K^{*+}\omega$ | $p_{A}/\sqrt{2}+\sqrt{2}p_{S}$ | $1.24\pm 0.19\pm 0.12$ | $<4.4$ |
$B^{0}\to\phi K^{*0}$ | $p_{A}+p_{S}$ | $4.98\pm 0.22\pm 0.50$ | $5.03\pm 0.30$ | $0.06\pm 0.04$
$B^{+}\to\phi K^{*+}$ | $p_{A}+p_{S}$ | $5.37\pm 0.24\pm 0.54$ | $5.00\pm 1.12$ | $0.10\pm 0.11$
$B_{s}\to\overline{K}^{*0}K^{*0}$ | $p_{A}$ | $5.51\pm 0.48\pm 0.55$ | $8.44\pm 2.10$ | $0.00\pm 0.24$
$B_{s}\to\phi\phi$ | $2p_{A}+2p_{S}$ | $9.60\pm 0.43\pm 0.96$ | $11.63\pm 0.96$ | $0.038\pm 0.024$
$B^{0}\to\rho^{0}\rho^{0}$ | $p_{A}$ | $0.22\pm 0.01\pm 0.02$ | $0.28\pm 0.10$ |
$B^{0}\to\rho^{-}\rho^{+}$ | $p_{A}$ | $0.25\pm 0.02\pm 0.02$ | $0.28\pm 0.55$ |
$B^{0}\to\overline{K}^{*0}K^{*0}$ | $p_{A}$ | $0.25\pm 0.02\pm 0.02$ | $0.22\pm 0.07$ |
$B^{+}\to\overline{K}^{*0}K^{*+}$ | $p_{A}$ | $0.26\pm 0.02\pm 0.03$ | $0.16^{+0.20}_{-0.15}$ |
$B^{+}\to\omega\rho^{+}$ | $\sqrt{2}p_{A}+\sqrt{2}p_{S}$ | $0.36\pm 0.02\pm 0.04$ | $1.59\pm 0.98$ |
$B_{s}\to\phi\bar{K}^{*0}$ | $p_{A}+p_{S}$ | $0.22\pm 0.01\pm 0.02$ | $0.56\pm 0.24$ | $0.07\pm 0.27$
Table 3: Predictions in the SM along with the parametrized decay amplitudes of $O_{5,6}$, where the upper (lower) part corresponds to $\overline{b}\to\overline{s}~{}(\overline{b}\to\overline{d})$ , while the inequality holds in 95$\%$ confident level.  channel | $SU(3)_{F}$ | $10^{6}{\cal B}_{SU(3)}^{+}$ |
---|---|---|---
$B_{s}\to\rho^{0}\rho^{0}$ | 0 | $0$ |
$B_{s}\to\rho^{-}\rho^{+}$ | 0 | $0$ |
$B_{s}\to K^{*-}K^{*+}$ | $p_{A}$ | $5.51\pm 0.48\pm 0.55$ |
$B_{s}\to\omega\omega$ | $0$ | $0$ |
$B_{s}\to\phi\omega$ | $\sqrt{2}p_{S}$ | $0.34\pm 0.16\pm 0.03$ |
$B_{s}\to\phi\rho^{0}$ | 0 | $0.17\pm 0.0\pm 0.02$ |
$B^{+}\to\phi\rho^{+}$ | $p_{S}$ | $<1.5\times 10^{-3}$ |
$B^{0}\to K^{*-}K^{*+}$ | $0$ | $0$ |
$B^{0}\to\omega\rho^{0}$ | $-p_{A}-p_{S}$ | $0.26\pm 0.01\pm 0.02$ |
$B^{0}\to\omega\omega$ | $p_{A}+2p_{S}$ | $0.05\pm 0.01\pm 0.01$ |
$B^{0}\to\phi\rho^{0}$ | $-p_{S}/\sqrt{2}$ | $<0.7\times 10^{-3}$ |
$B^{0}\to\phi\omega$ | $p_{S}/\sqrt{2}$ | $<0.8\times 10^{-3}$ |
$B^{0}\to\phi\phi$ | 0 | $0$ |
$B_{s}\to K^{*-}\rho^{+}$ | $p_{A}$ | $0.24\pm 0.02\pm 0.02$ |
$B_{s}\to\overline{K}^{*0}\rho^{0}$ | $-p_{A}/\sqrt{2}$ | $0.20\pm 0.01\pm 0.02$ |
$B_{s}\to\omega\bar{K}^{*0}$ | $p_{A}/\sqrt{2}+\sqrt{2}p_{S}$ | $0.05\pm 0.01\pm 0.01$ |

It is interesting to note that the effects of the charm-anticharm ($c\bar{c}$)
rescattering, given as

$\displaystyle B\to\left(\begin{array}[]{ccc}D_{s}^{*}D^{*}\\\ D_{s}DK+X\\\
D_{s}DK+X\\\ \ldots\end{array}\right)\to V_{1}V_{2}\,,$ (35)

have not been considered yet, where “$\ldots$” represents other possible
multiplicities. However, excluding the cases that weak exchanging diagrams are
permitted, in the $\overline{b}\to\overline{s}(\overline{d})$ transition,
there would be at least one left-handed $\overline{s}(\overline{d})$ produced
from the weak vertices, and therefore $A(H_{-})$ is still suppressed. For
instance, $B^{0}\to K^{*0}\omega$ can not receive $A(H_{-})$ from the
$c\bar{c}$ rescattering in the chiral limit. Additionally, the relevant
effective Hamiltonian for the rescattering, given as

${\cal H}_{eff}^{c}\propto
V_{bc}^{*}V_{cs}(\overline{b}c)_{L}(\overline{c}s)_{L}+V_{bc}^{*}V_{cd}(\overline{b}c)_{L}(\overline{c}d)_{L}\,,$
(36)

has the same $SU(3)_{F}$ structure as the QCD penguin operators with the
approximation of

$V_{cd}/V_{cs}\approx V_{td}/V_{cd}\approx-\lambda\,,$ (37)

where $\lambda=V_{us}$ is the parameter in the Wolfenstein parametrization for
the CKM matrix pdg . In all, since the $c\bar{c}$ rescattering approximately
shares the same $SU(3)_{F}$ structure and $A(H_{-})$ remains suppressed, the
effects can be largely absorbed into $A(P_{A})$ and $A(P_{S})$ by
redefinition.

Let us examine CP violation. Since the transverse polarization involves only
one weak phase in $V_{tb}^{*}V_{tf}$, the T violating effects are not expected
to be observed. As a result, we get

$\Delta\phi_{p}=0\,,$ (38)

which is well consistent with those in the literature Zou:2015iwa ; FQCDBs ;
FQCDBu ; SU3 . Notice that there are two different equations concerning
$\phi_{p}$. The first one is $\sin\phi_{p}\approx 0$, argued from the helcitiy
counting, which therefore only constrains the strong phases. The second one is
that $\Delta\phi_{p}=0$, originated from only one weak phase in the transverse
amplitudes. Although their appearances seem similar, the mechanisms behind
them are very different. Consequently, we arrive that

$\phi_{\perp}^{B}=\phi_{\parallel}^{B}\,,\quad\phi_{\perp}^{\overline{B}}=\phi_{\parallel}^{\overline{B}}\pm\pi\,,$
(39)

which are well consistent with the previous theoretical works in Refs.
Zou:2015iwa ; FQCDBs ; FQCDBu ; SU3 ; Beneke:2006hg ; Wang:2017rmh .

In addition, we find that $|A(H_{+})|^{2}=|A(\overline{H}_{-})|^{2}$.
Combining with $A(H_{-})=A(\overline{H}_{+})=0$, we obtain

$A_{CP}^{\perp}=A_{CP}^{\parallel}=-A_{CP}^{dir}\,,$ (40)

where the CP asymmetries are defined by pdg

$\displaystyle A_{CP}^{dir}$ $\displaystyle=$
$\displaystyle\frac{\overline{\Gamma}-\Gamma}{\overline{\Gamma}+\Gamma}\,,$
$\displaystyle A_{CP}^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{\overline{f}_{\lambda}-f_{\lambda}}{\overline{f}_{\lambda}+f_{\lambda}}\,.$
(41)

In Table 4, we show the data related to these CP asymmetries pdg ; Krho ;
Aubert:2007ac .

Table 4: Data for CP violating asymmetries, where $A_{\perp}^{CP}(B^{+}\to\phi K^{*+})$ is obtained in Ref. Aubert:2007ac , while the other ones are taken from the PDG pdg .  channel | $A_{\parallel}^{CP}$ | $A_{\perp}^{CP}$ | $A_{CP}^{dir}$
---|---|---|---
$B^{0}\to K^{*0}\rho^{0}$ | $0.19\pm 0.04$ | $0.05\pm 0.04$ | $-0.06\pm 0.09$
$B^{0}\to K^{*0}\omega$ | $0.3\pm 0.6$ | $0.3\pm 0.9$ | $0.45\pm 0.25$
$B^{0}\to\phi K^{*0}$ | - | $-0.02\pm 0.06$ | $0.00\pm 0.04$
$B^{+}\to\phi K^{*+}$ | - | $-0.22\pm 0.25$ | $-0.01\pm 0.08$

As seen in Table I, the central value of $\Delta\phi_{p}(B^{0}\to
K^{*0}\omega)$ in the data shows a huge discrepancy against Eq. (38), which is
sizable and not expected in the SM. One might look for explanation from the
charm-anticharm rescattering, which induces a weak phase in
$V_{cb}^{*}V_{cs}$. Nevertheless, it would be a futile attempt, since the
relative weak phase between $V_{cb}^{*}V_{cs}$ and $V_{tb}^{*}V_{ts}$ vanishes
in the SM. The another signal could be seen in $B^{0}\to K^{*0}\rho^{0}$.
Although the equality of $A_{\perp}^{CP}=-A^{dir}_{cp}$ holds well, there is a
$2.5~{}\sigma$ deviation between $A_{\parallel}^{CP}$ and $A_{\perp}^{CP}$,
which may arise from the right-handed current Kagan:2004ia ; Kagan:2004uw .
However, the difference is small. Interestingly, in $B^{0}\to K^{*0}\omega$,
inconsistencies exist both in the decay branching ratio and T violating
parameters. If the refined experiments of $B^{0}\to K^{*0}\omega$ in future
confirm the data, it would be a clear signal for NP.

Before ending the section, let us reexamine whether $O_{1,2}$ can be safely
neglected in transverse amplitudes or not. If they played a significant role
in the transverse amplitudes, one would find an enormous inconsistency in the
$\overline{b}\to\overline{d}$ transition in our approach, because of the
hierarchy of $|V_{ud}/V_{td}|\gg|V_{us}/V_{ts}|$, i.e. the tree operators play
much more important roles in $\overline{b}\to\overline{d}$ compared to
$\overline{b}\to\overline{s}$. Nevertheless, no such kind of phenomena has
been found yet, and the systematical hierarchy in Eq. (27) holds as seen in
Table 2. Particularly, the data of
$B^{0}\to\rho^{0}\rho^{0},\overline{K}^{*0}K^{*0}$ fit the theoretical
expectations well. As a result, the helicity flipping via $O_{1,2}^{s}$ is not
capable of explaining the anomaly in ${\cal B}^{+}(B^{0}\to K^{*0}\omega$).
Finally, Eq. (38) is uncontaminated by uncertainties from the experimental
input, and it is very unlikely for the SM to have $\Delta\phi_{p}(B^{0}\to
K^{*0}\omega)\sim 0.8$. However, the uncertainties are still too large at this
stage. Clearly, it is worth a refined experiment in future.

##  IV Summary

The eigenstates of T-odd scalar operators involving spins have been given for
the first time. From these eigenstates, we have explicitly constructed the T
violating observables. In addition, the relations between the TVP asymmetries
and the CP-odd phases have been given consistently for the first time in Eq.
(II). By counting the helicity flip, we have argued that
$\sin(\phi_{p})\approx 0$, which matches the experimental data well. Based on
the helicity suppressions of $O_{1,2}$, we have unwrapped the interference
between tree and penguin operators. Particularly, $O_{1,2}$ could only give
rise to the longitudinal polarization. Our results for the transverse
branching ratios have been represented in Tables 2 and 3. We have found that
there is a tension between our approach and the experimental measurement in
$B^{0}\to K^{*0}\omega$, in which the central value of our result is almost a
factor of two larger than the data.

Since $A(H_{+})$ is dominated by a single CP weak phase, a large CP violating
effect in the transverse amplitudes is not expected. By deducing
$\Delta\phi_{p}=0$, we have shown that the central value of the experimental
data, $\Delta\phi_{p}(B^{0}\to K^{*0}\omega)=-0.84\pm 0.54$ Krho , can not be
explained within the SM. Interestingly, we have demonstrated inconsistencies
in $B^{0}\to K^{*0}\omega$ for both transverse branching ratio and CP
asymmetry. We recommend some dedicated experiments in $B^{0}\to K^{*0}\omega$
to precisely measure these observables in order to claim that NP does exist.

##  Appendix A Eigenstates of T-odd operators

In this Appendix, we calculate the eigenstates of $\hat{T}_{i}$ defined in Eq.
(II) with the following commutation relation, given by spinoperators

$s_{i}L|\vec{p}=0,J_{z}=M\rangle=LJ_{i}|\vec{p}=0,J_{z}=M\rangle\,,$ (42)

where $L$ is an arbitrary Lorentz boost. For the explicit calculation, we take
$\hat{T}_{2}$ as an example. Since $\hat{T}_{2}$ commutes with
$\vec{J}\cdot\hat{p}_{1}$, the meson wave functions can be eigenstates of
$\hat{T_{2}}$ and $\vec{J}\cdot\hat{p}_{1}$ simultaneously. We recall the
canonical states for the two-particle system, given by

$|p\hat{z},m_{s_{1}},m_{s_{2}}\rangle=|p\hat{z},m_{s_{1}}\rangle_{1}\otimes|-p\hat{z},m_{s_{2}}\rangle_{2}\,,$
(43)

where $m_{s_{1,2}}$ correspond to the eigenvalues of $(J_{1,2})_{z}$. We
define the operator

$\hat{C}_{2}=s_{1x}s_{2y}-s_{1y}s_{2x}=\frac{i}{2}\left(s_{1}^{+}s_{2}^{-}-s_{1}^{-}s_{2}^{+}\right)$
(44)

with $s_{i}^{\pm}=s_{ix}\pm is_{iy}$, which coincides with $\hat{T}_{2}$ on
$|p\hat{z}\rangle$ and offers a concrete form for the computation of the
eigenstates. In the system with vector and spin-$n$ mesons in the final
states, with Eq. (42), we find

$\displaystyle|p\hat{z},J_{z}=0,\lambda_{t_{2}}=0\rangle=\frac{1}{\sqrt{2}}\left(|p\hat{z},m_{s_{1}}=1,m_{s_{2}}=-1\rangle+|p\hat{z},m_{s_{1}}=-1,m_{s_{2}}=1\rangle\right)\,,$
$\displaystyle|p\hat{z},J_{z}=0,\lambda_{t_{2}}=\pm\sqrt{n^{2}+n}\rangle=$
$\displaystyle\frac{1}{2}\left(|p\hat{z},m_{s_{1}}=1,m_{s_{2}}=-1\rangle-|p\hat{z},m_{s_{1}}=-1,m_{s_{2}}=1\rangle\mp\sqrt{2}i|p\hat{z},m_{s_{1}}=0,m_{s_{2}}=0\rangle\right)\,,$
(45)

where $\lambda_{t_{2}}$ is the eigenvalue of $\hat{T}_{2}$ and $\hat{C}_{2}$.
Here, we have used $\vec{J}\cdot\hat{p}_{1}$ = $J_{z}$ in the subspace of
$\hat{p}_{1}=\hat{z}$.

The states in Eq. (A) are eigenstates of $\hat{T}_{2}$, $\hat{p}_{1}$ and
$\vec{J}\cdot\hat{p}_{1}$. To relate the states with the decays, we must
construct them in a definite angular momentum to meet the initial condition,
namely $J=J_{z}=0$, imposed by the $B$ meson. To this end, we utilize the
projection operators in $SO(3)$ rotation group, given as GroupTheory ;
Jacob:1959at

$P^{N}_{JM}=\frac{2J+1}{8\pi^{2}}\int d\phi d\cos\theta d\psi
U(\phi,\theta,\psi)D_{J}^{\dagger}(\phi,\theta,\psi)^{N}\,_{M}\,,$ (46)

where $D_{J}$ is the Wigner D-matrix and
$U(\phi,\theta,\psi)=R_{z}(\phi)R_{y}(\theta)R_{z}(\psi)$ with $R_{i}$ the
rotation operators in $\hat{i}$ direction. Here, $P^{N}_{JM}$ projects
$|J,J_{z}=N\rangle$ on $|J,J_{z}=M\rangle$. After the projection, it remains
the eigenstate of $\hat{T}_{2}$ since the scalar operators commute with $R$.

By operating $P^{0}_{00}$ on Eq. (A), we arrive the expected states. To
compare the results with those in the literature, we write down them in terms
of the helicity states

$\displaystyle|\lambda_{t_{2}}=0\rangle$ $\displaystyle=$
$\displaystyle|H_{\parallel}\rangle\,,$
$\displaystyle|\lambda_{t_{2}}=\pm\sqrt{n^{2}+n}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|H_{\perp}\rangle\mp
i|H_{0}\rangle\right)\,,$ (47)

with

$|H_{\perp,\parallel}\rangle=\frac{1}{\sqrt{2}}\left(|H_{+}\rangle\mp|H_{-}\rangle\right)\,,$
(48)

where we have that $J=J_{z}=0$ in the both sides of the equations, and the
subscripts in $H_{0,\pm}$ denote the helicities of the final state mesons.

Notice that $\hat{T}_{2}$ and $\hat{T}_{4}$ are parity odd, whereas
$\hat{T}_{3}$ parity even. All $T_{i}~{}(i=2,3,4)$ are designed as Hermitian
operators. In analogy to $\hat{C}_{2}$ in Eq. (44), we write

$\displaystyle\hat{C}_{3}$ $\displaystyle=$ $\displaystyle
s_{1z}\hat{C}_{2}+\hat{C}_{2}s_{1z}\,,$ $\displaystyle\hat{C}_{4}$
$\displaystyle=$
$\displaystyle\frac{i}{2}\left(s_{1}^{+}s_{1}^{+}s_{2}^{-}s_{2}^{-}-s_{1}^{-}s_{1}^{-}s_{2}^{+}s_{2}^{+}\right)\,,$
(49)

which coincide with $\hat{T}_{3}$ and $\hat{T}_{4}$, respectively, in the
subspace of $\hat{p}_{1}=\hat{z}$. Consequently, the eigenvectors are given by

$\displaystyle|\lambda_{t_{3}}=0\rangle=|H_{\perp}\rangle\,,$
$\displaystyle|\lambda_{t_{3}}=\pm\sqrt{n^{2}+n}\rangle=\frac{1}{\sqrt{2}}\left(|H_{\parallel}\rangle\mp
i|H_{0}\rangle\right)\,,$ (50)

and

$\displaystyle|\lambda_{t_{4}}=0\rangle=|H_{0}\rangle\,,$
$\displaystyle|\lambda_{t_{4}}=\pm(n^{2}+n)\rangle=\frac{1}{\sqrt{2}}\left(|H_{\parallel}\rangle\pm
i|H_{\perp}\rangle\right)\,,$ (51)

respectively, with $J=J_{z}=0$.

Since we only concern spin-0 meson decays, it is sufficient to work on the
subspace with $\vec{J}\cdot\hat{p}_{1}=0$. As a result, one can substitute
$\vec{s}_{1}\cdot\hat{p}_{1}$ by $-\vec{s}_{2}\cdot\hat{p}_{1}$ in the
definition of $\hat{T}$ without alternating the final results.

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