# Scrutinizing new physics of $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$
decay modes

Manas K. Mohapatra<EMAIL_ADDRESS>Department of Physics, IIT
Hyderabad, Kandi - 502285, India

###### Abstract

We inspect the exclusive hadronic decay modes
$B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$, induced by quark level
transition as $b\to d$ $(\Delta S=0)$, in vector like down quark model. As
these decay modes insist highly suppressed followed by the predicted branching
fraction $\mathcal{O}(10^{-9})$ which reflects to scrutinize physics beyond
the standard model. We constrain the new parameter space inferred from
experimental limits on leptonic $B_{d}\to\ell\ell(\ell=e,\mu,\tau)$ and
nonleptonic decay modes $B_{d}\to\eta^{\prime}\pi^{0}$ and
$B_{u}\to\rho^{-}\eta^{\prime}$. We then check the new physics contributions
can have significant impact on the prominent observable so called branching
ratio of $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ processes.

###### pacs:

13.30.-a,14.20.Mr, 14.80.Sv

##  I Introduction

The Standard model (SM) of particle physics, one of the biggest achievements
in twentieth century science, encompasses the beauty of fundamental particles
and their interactions which is ruled by strong, weak and electromagnetic
forces. Despite its spectacular success, it, however, has some important voids
that couldn’t filled out such as matter dominance over antimatter in the
present universe, dark matter and dark energy, hierarchy problem, neutrino
mass etc. The source of matter-antimatter asymmetry is the violation of
combined discrete symmetry of charge conjugation (C) and parity (P) where
Cabbibo–Kobayasi–Maskawa (CKM) matrix is the main cornerstone to account for.
Among various indirect searches, the study of B decay modes provide an insight
to analyze in the SM, and to explore the possible existence of new physics
(NP) beyond it. In one way, it is of interest to study the potential
implications in the sector of flavor changing neutral current (FCNC)
transition at electroweak scale which mainly occurs at loop level.

In this paper, we will perform the decay modes
$B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ induced by $b\to d$ quark level
transition which are suppressed in the SM indicating an ideal place to search
for the new physics effects. We repredict the CP averaged branching ratios
based on the framework of QCD factorisation approach which includes next to
leading order (NLO) contributions where the previous predictions have been
done in Cheng and Chua (2009); Beneke and Neubert (2003); Beneke et al.
(2007). The upper limit of branching fractions of the aforesaid decay modes
are given as Zyla et al. (2020)

$\displaystyle Br(B_{d}^{0}\to\phi\eta)<5\times
10^{-7},Br(B_{d}^{0}\to\phi\eta^{\prime})<5\times 10^{-7},$ $\displaystyle
Br(B_{d}^{0}\to\phi\pi)<1.5\times 10^{-7},Br(B_{d}^{0}\to\phi\omega)<7\times
10^{-7}.$ (1)

where as the SM predictions are of $\mathcal{O}(10^{-9})$ with no CPV (CP
violation) observables. Inspired by these discrepancies, we would like to
probe in the presence of vector like down quark (VLDQ) model where an extra
$SU(2)_{L}$ singlet down type quark has been added to the SM and observe the
significant impact on the branching fractions. As the SM include 3 generations
of quarks but there may be possibility of having a heavier exotic quark in
another generation. Such fermion can appear in $E_{6}$ grand unified theories
and models with large extra dimensions. Because of the addition of this
particle to the SM particle spectrum, it modifies the CKM matrix, of course
not unitary. Thus it implies the FCNC transition at tree level mediated by Z
in the down quark sector. We would like to see such effect in our model on the
above decay modes.

Among $b\to d$ quark level transitions are leptonic decays
$B_{d}\to\ell\ell(\ell=e,\mu,\tau)$. The SM and experimental results are shown
in the TABLE - 1. The deviations, between the predictions and experimental
values, need modification of the branching fractions in the search of NP
scenario.

Table 1: Branching fractions (leptonic) induced by $b\to d$ transition Decay processes |  Predicted Br |  Experimental values/upper limits Zyla et al. (2020)
---|---|---
$B_{d}^{0}\to ee$ |  $(2.23\pm 0.21)\times 10^{-15}$ |  $<0.83\times 10^{-8}$
$B_{d}^{0}\to\mu\mu$ |  $(0.95\pm 0.09)\times 10^{-10}$ |  $(1.1\begin{subarray}{c}+1.4\\\ -1.3\end{subarray})\times 10^{-10}$
$B_{d}^{0}\to\tau\tau$ |  $(2.00\pm 0.19)\times 10^{-8}$ |  $<2.1\times 10^{-3}$
Table 2: CP av. Branching fractions (non leptonic) induced by $b\to d$ transition Decay modes |  Our results |  Previous results Cheng and Chua (2009) |  Experimental value Zyla et al. (2020)
---|---|---|---
$\bar{B}_{d}\to\pi^{0}\eta^{\prime}$ |  $(0.50\pm 0.57\pm 0.03)\times 10^{-6}$ |  $(0.42\begin{subarray}{c}+0.21+0.18\\\ -0.09-0.12\end{subarray})\times 10^{-6}$ |  $(1.2\pm 0.6)\times 10^{-6}$
$B^{-}\to\rho^{-}\eta^{\prime}$ |  $(6.26\pm 1.70\pm 0.34)\times 10^{-6}$ |  $(5.6\begin{subarray}{c}+0.9+0.8\\\ -0.7-0.5\end{subarray})\times 10^{-6}$ |  $(9.7\pm 2.2)\times 10^{-6}$

On the other side, the decay channels having final state meson $\eta^{\prime}$
play a vital role in the non leptonic $B_{d}\to\eta^{\prime}\pi^{0}$ and
$B_{u}\to\rho^{-}\eta^{\prime}$ decay modes where we consider
$\eta-\eta^{\prime}$ mixing effect in the investigation of decay mode having
final state meson $\eta$. Now due to the potential deviations between SM and
experimental results given in TABLE -2, it allows a room to search for physics
beyond the SM. The associated new couplings in the presence of VLDQ model can
be constrained by using the experimental limits on both the leptonic as well
as non leptonic decay modes. Using the allowed parameter space, we scrutinize
the new physics impact on the aforesaid decay processes.

The layout of this work is organized as follows. In section II, we briefly
study the effective Hamiltonian accountable for the quark level transition
$b\to d$ for $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ decay modes. We
include the discussion of the amplitudes for the abovesaid nonleptonic decay
modes along with the numerical results of the so called branching fractions in
the SM with all necessary input parameters. In section III, we then constrain
the new parameter space using the existing experimental limits on branching
fractions of both leptonic $B_{d}\to\ell\ell$ and also non leptonic decay
modes $B_{d}\to\eta^{\prime}\pi^{0}$ and $B_{u}\to\rho^{-}\eta^{\prime}$,
discussed above in the presence of VLDQ model. Next we implement the above
model on the decay modes $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ in the
new physics scenario by using new coupling paramaters. Finally we bring to an
end with brief summary and conclusion of our results in section IV.

##  II Standard Model Predictions

The weak effective Hamiltonian of the decay mode having the quark level
transition $b\to d$, can be written as Buchalla et al. (1996)

$\displaystyle\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}\bigg{[}V_{ub}V_{ud}^{*}(C_{1}(\mu)O_{1}(\mu)+C_{2}(\mu)O_{2}(\mu))-V_{tb}V_{td}^{*}\sum_{i=3}^{10}C_{i}(\mu)O_{i}(\mu)\bigg{]},$
(2)

where the six-dimensional four-quark operators $O_{i}(i=1,...,10)$ given in
the above effective Hamiltonian are specified as below:

$\displaystyle O_{1}$
$\displaystyle=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}).(\bar{u}_{\beta}\gamma_{\mu}Lb_{\alpha}),$
$\displaystyle
O_{6}=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}).(\sum_{q}\bar{q}_{\beta}\gamma_{\mu}Rq_{\alpha}),$
$\displaystyle O_{2}$
$\displaystyle=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}).(\bar{u}_{\beta}\gamma_{\mu}Lb_{\beta}),$
$\displaystyle
O_{7}=\frac{3}{2}(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}).(\sum_{q}e_{q}\bar{q}_{\beta}\gamma_{\mu}Rq_{\beta}),$
$\displaystyle O_{3}$
$\displaystyle=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}).(\sum_{q}\bar{q}_{\beta}\gamma_{\mu}Lq_{\beta}),$
$\displaystyle
O_{8}=\frac{3}{2}(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}).(\sum_{q}e_{q}\bar{q}_{\beta}\gamma_{\mu}Rq_{\alpha}),$
$\displaystyle O_{4}$
$\displaystyle=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}).(\sum_{q}\bar{q}_{\beta}\gamma_{\mu}Lq_{\alpha}),$
$\displaystyle
O_{9}=\frac{3}{2}(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}).(\sum_{q}e_{q}\bar{q}_{\beta}\gamma_{\mu}Lq_{\beta}),$
$\displaystyle O_{5}$
$\displaystyle=(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\alpha}).(\sum_{q}\bar{q}_{\beta}\gamma_{\mu}Rq_{\beta}),$
$\displaystyle
O_{10}=\frac{3}{2}(\bar{d}_{\alpha}\gamma^{\mu}Lu_{\beta}).(\sum_{q}e_{q}\bar{q}_{\beta}\gamma_{\mu}Lq_{\alpha}),$
(3)

where $G_{F}$ is Fermi coupling constant, all
${V_{ab}}^{\prime}s(a,b=u,b,s,t)$ are the CKM matrix elements, $e_{q}$ is the
electromagnetic charge of quark field ‘q’, L (R) is the left (right) handed
projection operator, and $\alpha,\beta$ are the color indices. The quark field
q runs over active flavors i.e., $q\epsilon\\{u,d,c,s,b\\}$ at the scale
$\mu=m_{b}$. In addition to this, the operators belong to $i=1,2$ are current-
current, $i=3,..,6$ are the QCD penguin and $i=7,...,10$ are the EW penguin
operators. The corresponding coupling constants so called Wilson coefficients
$C_{i}^{\prime}s(i=1,...,10)$ are used in the next-to-leading order (NLO) at
the scale of $O(m_{b})$ in order to cancel the $\mu$ dependence of the
amplitude.

## $B_{d}\to\phi\eta^{(^{\prime})}$

The weak decay amplitude in the presence of QCDF approach Beneke and Neubert
(2003) can be written in the form as

$\displaystyle\langle\phi\eta^{(^{\prime})}|O_{\mathit{i}}|B_{d}\rangle=\langle\phi\eta^{(^{\prime})}|O_{\mathit{i}}|B_{d}\rangle_{fact}\big{[}1+\sum
r_{n}\alpha_{s}^{n}+\mathit{O}(\frac{\Lambda_{QCD}}{m_{b}})\big{]},$ (4)

where the factorized matrix element of four-quark operators
$\langle\phi\eta^{(^{\prime})}|O_{\mathit{i}}|B_{d}\rangle_{fact}$ includes
form factors and decay constants. The second term in the paranthesis involve
to higher order contributions which include the QCD effect, more to say, gluon
corrections, and the third term directs to power corrections containing
troublesome end-point divergence. The decay mode $B_{d}\to\phi\eta$ can be
produced from the $B_{d}\to\omega\eta$ process followed by $\omega-\phi$
mixing along with the angle $\delta=3.3^{\circ}$ (Cheng and Chua, 2009;
Benayoun et al., 1999; Kucukarslan and Meissner, 2006; Benayoun et al., 2008;
Qian and Ma, 2008). Now the amplitude for the decay mode $B_{d}\to\omega\eta$
is given by (Cheng and Chua, 2009)

$\displaystyle\mathit{A}(B_{d}\to\omega\eta)\approx\frac{1}{2}\lambda_{p}\bigg{[}A_{\omega\eta_{q}}\big{\\{}\delta_{pu}(\alpha_{2}+\beta_{1})+2\alpha_{3}^{p}+\hat{\alpha}_{4}^{p}\big{\\}}+A_{\eta_{q}\omega}\big{\\{}\delta_{pu}(\alpha_{2}+\beta_{1})+2\alpha_{3}^{p}+\hat{\alpha}_{4}^{p}\big{\\}}\bigg{]},$
(5)

where the parameter $\lambda_{p}$ is the CKM matrix element and is summed over
the quark element $p=u,c$. The required parameters
$\alpha_{i}^{p},\beta_{i}^{p}$ and$\hat{\alpha}_{i}^{p}$ and the factorized
matrix elements are given in the Appendix A. The relation between both the
branching fractions is given by (Cheng and Chua, 2009)

$\displaystyle
Br(B_{d}\to\phi\eta)=Br(B_{d}\to\omega\eta)\times\rm\sin^{2}\delta$ (6)

Similarly, we can proceed for the final states $\omega(\phi)\eta^{\prime}$
where $\eta$ is replaced by $\eta^{\prime}$ in the previous decay mode. For
our study of B decay modes having final state particle $\eta$ and
$\eta^{\prime}$, we consider $\eta-\eta^{\prime}$ mixing effect in our study
of the physical observable of given decay modes. Due to different flavor
states of $\eta^{(^{\prime})}(\eta_{q}^{(^{\prime})},\eta_{s}^{(^{\prime})}$
and $\eta_{c}^{(^{\prime})})$, the corresponding decay constants $f_{q}$ and
$f_{s}$ correlated by a mixing angle $\theta$ which is given by

$\displaystyle\begin{pmatrix}f_{\eta}^{q}&f_{\eta}^{s}\\\
f_{\eta^{\prime}}^{q}&f_{\eta^{\prime}}^{s}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}f_{q}&0\\\
0&f_{s}\end{pmatrix},$ (7)

where the $\eta_{q}-\eta_{s}$ mixing angle $\theta=(39.3\pm 1.0)^{\circ}$
Feldmann et al. (1998) and the mixing with $\eta_{c}$ has been neglected and
the useful parameters $f_{q}$ and $f_{s}$ are given by

$\displaystyle f_{q}=(1.07\pm 0.02)f_{\pi},\hskip 14.22636ptf_{s}=(1.34\pm
0.06)f_{\pi}.$ (8)

## $B_{d}\to\phi\pi^{0}$

Similar to the previous decay mode, the process $B_{d}\to\phi\pi^{-}$ also can
be produced from the chanel $B_{d}\to\omega\pi$ in the presence of
$\omega-\phi$ mixing effect. Now the amplitude for the decay mode
$B^{-}\to\pi^{-}\omega$ is given by(Beneke and Neubert, 2003)

$\displaystyle A(B^{-}\to\pi^{-}\omega)$
$\displaystyle\approx\frac{1}{\sqrt{2}}\lambda_{p}\bigg{[}A_{\pi\omega}\big{\\{}\delta_{pu}(\alpha_{2}+\beta_{2}+2\beta_{S2})+2\alpha_{3}^{p}+\alpha_{4}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}$
(9)
$\displaystyle+\beta_{3}^{p}+\beta_{3,EW}^{p}+2\beta_{S3}^{p}+2\beta_{S3,EW}^{p}\big{\\}}$
$\displaystyle+A_{\omega\pi}\big{\\{}\delta_{pu}(\alpha_{1}+\beta_{2})+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}\big{\\}}\bigg{]},$

where $\lambda_{p}$ is the CKM parameter and the other contributions in the
above amplitude are given in the Appendix A. Now, the mixing relation between
the decay modes $B^{-}\to\omega^{-}\pi$ and $B^{-}\to\phi\pi^{-}$ is given by
(Cheng and Chua, 2009)

$\displaystyle Br(B^{-}\to\phi\pi^{-})_{{\phi-\omega}\hskip
2.84544ptmixing}=Br(B^{-}\to\omega\pi^{-})\times\rm\sin^{2}\delta,$ (10)

and the decay amplitude of $B_{d}\to\phi\pi^{0}$ mode is given by the relation
as (Beneke and Neubert, 2003)

$\displaystyle A(B^{-}\to\pi^{-}\phi)=-\sqrt{2}A(B_{d}\to\phi\pi^{0})$ (11)

For our calculations of above discussed modes
$B_{d}\to\phi(\pi^{0},\eta^{(^{\prime})})$, we consider the $\phi-\omega$
mixing angle $\delta=(3.32\pm 0.09)^{\circ}$ from Ambrosino et al. (2009).

## $B_{d}\to\phi\omega$

From the effective Hamiltonian (2) the matrix element for the four quark
operators is given by

$\displaystyle\big{\langle}V_{1}(\lambda_{1})V_{2}(\lambda_{2})|(\bar{q}_{2}q_{3})_{V-A}(\bar{q}_{1}b)_{V-A}|\bar{B}_{d}\big{\rangle},$
(12)

where $\lambda_{1},\lambda_{2}$ are the helicities of the final state vector
mesons $V_{1}$ and $V_{2}$ respectively. Now the amplitude of the penguin
dominated decay mode $B_{d}\to\phi\omega$ is expressed as Beneke et al. (2007)

$\displaystyle
A(B_{d}\to\phi\omega)=A_{\omega\phi}\big{[}\lambda_{p}(\alpha_{3}^{p}-\frac{1}{2}\alpha_{3,EW}^{p})\big{]}$
(13)

The details of the parameters used in the above process has been provided in
the Appendix B. The helicity amplitudes corresponding to this decay mode are
$A_{0},A_{+}$ and $A_{-}$ and the hierarchy of helicity amplitudes are

$\displaystyle
A_{0}:A_{-}:A_{+}=1:\frac{\Lambda_{QCD}}{m_{b}}:\bigg{(}\frac{\Lambda_{QCD}}{m_{b}}\bigg{)}^{2},$
(14)

where the transverse amplitudes $A_{+}$ and $A_{-}$ are suppressed relative to
the longitudinal one $A_{0}$.
Now in this section, all the discussed non leptonic decay modes include
factorized matrix elements $A_{VP}$ and $A_{VV}$ (P = pseudoscalar meson, V =
vector meson) as well as the higher order corrections such as vertex
corrections, hard spectator interactions, penguin contractions and
annihilation contributions. Now all the discussed amplitudes can be written in
the parameterized form symbolically as

$\displaystyle A(B\to VP)$ $\displaystyle=\lambda_{u}A_{u}+\lambda_{c}A_{c}$
(15)
$\displaystyle=\lambda_{c}A_{c}\big{[}1+rae^{i(\delta_{1}-\gamma)}\big{]},$

where $V=\phi,P=\eta^{(^{\prime})},\pi$. $\lambda_{u,c}$ are the CKM elements
and $A_{u,c}$ are the amplitudes correspond to $u$ and $c$ quark.
$a=|\lambda_{u}/\lambda_{c}|$, $r=|A_{u}/A_{c}|$, $\gamma$ is the weak phase
of CKM element $V_{ub}$ and the relative strong phase is $\delta_{1}$. The
formula for CP averaged branching fraction is given by

$\displaystyle\mathfrak{B}=\frac{1}{2}\left[Br(\mathcal{A}_{B_{d}\to
M_{1}M_{2}})+Br(\mathcal{\bar{A}}_{B_{d}\to M_{1}M_{2}})\right].$ (16)

Now for all the non leptonic decay modes
$B_{d}\to\phi(\eta^{(^{\prime})},\pi)$ can be written as

$\displaystyle\mathfrak{B}=\frac{p_{cm}\tau_{B}}{8\pi
m_{B}^{2}}|\lambda_{c}A_{c}|^{2}\big{\\{}1+r^{2}a^{2}+2ra\rm\cos\delta_{1}\rm\cos\gamma\big{\\}},$
(17)

where the center of mass momentum in $B_{d}$ rest frame is given by

$\displaystyle
p_{cm}=\sqrt{(m_{B_{d}}^{2}-(m_{1}+m_{2})^{2})(m_{B_{d}}^{2}-(m_{1}-m_{2})^{2})}\,,$
(18)

where $m_{1}$ and $m_{2}$ are the masses of final states. Similarly the CP av.
branching fraction for $B_{d}\to\phi\omega$ is given by

$\displaystyle\mathfrak{B}_{(B_{d}\to\phi\omega)}=\frac{\tau_{B_{d}}p_{cm}}{8\pi^{2}m_{B}}(|A_{0}|^{2}+|A_{-}|^{2}+|A_{+}|^{2}).$
(19)

Here CP av. branching fraction correspond to all the indivisual helicity
amplitude can be written similar to the expressions in $B_{d}\to VP$ decay
mode.

Now the CP averaged branching ratio can be calculated for the considered
nonleptonic $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ decay modes. For the
numerical predictions of the CP averaged branching ratio, we use the input
parameters given in S4 scenario of QCDF approach Beneke and Neubert (2003).
The Wilson coefficients in NDR scheme at NLO are taken from de Groot et al.
(2003) at $m_{b}$ scale and the relevant input parameters are given in the
Table 4. Many studies have been done in the Ref. Beneke and Neubert (2003);
Cheng and Chua (2009); Beneke et al. (2007); Zhang and Xiao (2008); Cheng et
al. (2015). We repredict SM values of the CP averaged branching fractions of
$B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$ decay modes which are given in
the TABLE 3 along with the previous results. Here the first theoretical error
correspond to the uncertainties occured due to quak masses, form factor, decay
constants, Gegenbauer moments, the wave function of $B_{d}^{0}$ meson and
$\phi-\omega$ mixing angle where as the parameters due to weak annhilation and
hard spectator interactions are lumped into the second uncertainty. As per the
ref. (Cheng and Chua, 2009) we assign $0.1$ and $20^{\circ}$ uncertainties to
the annihilation parameters $\rho_{A}$ and $\phi_{A}$ respectively.

Table 3: SM predictions of CP av. branching fractions (non leptonic) induced by $b\to d$ transition Decay modes |  Our results |  Previous results Beneke and Neubert (2003); Bao et al. (2008) |  Expt. values Zyla et al. (2020)
---|---|---|---
$B_{d}\to\phi\eta$ |  $(1.18\pm 0.84\pm 0.03)\times 10^{-9}$ |  $0.001\times 10^{-6}$ |  $<5\times 10^{-7}$
$B_{d}\to\phi\eta^{\prime}$ |  $(2.26\pm 1.8\pm 0.09)\times 10^{-9}$ |  $0.003\times 10^{-6}$ |  $<5\times 10^{-7}$
$B_{d}\to\phi\pi^{0}$ |  $(6.91\pm 1.23\pm 0.03)\times 10^{-9}$ |  $0.004\times 10^{-6}$ |  $<1.5\times 10^{-7}$
$B_{d}\to\phi\omega$ |  $(3.16\pm 1.23\pm 0.006)\times 10^{-9}$ |  $0.0017\times 10^{-6}$ |  $<7\times 10^{-7}$
Table 4: Input parameters used in the numerical analysis Running quark masses and coupling constants: |  Ref.
---|---
$G_{F}=1.166\times 10^{-5}$ GeV-2; $\alpha_{em}=1/129$ |  (Zyla et al., 2020)
$\alpha_{s}(M_{Z})=0.1185$; $\tau_{B_{d}}=(1.52\pm 0.004)\times 10^{-12}\hskip 2.84544pts$ |  (Zyla et al., 2020)
$m_{b}(m_{b})=4.2\hskip 2.84544pt\rm GeV$; $m_{c}(m_{b})=0.91\hskip 2.84544pt\rm GeV$; $m_{c}^{\rm pole}/m_{b}^{\rm pole}=0.3$ |  (Cheng and Chua, 2009)
$m_{u}(2\hskip 0.28436pt\rm GeV)=2.15\pm 0.15$ MeV; $m_{d}(2\hskip 0.28436pt\rm GeV)=4.7\pm 0.2$ MeV |  Lü et al. (2019)
$m_{s}(2\hskip 0.28436pt\rm GeV)=93.8\pm 1.3\pm 1.9$ MeV |  Lü et al. (2019)
CKM parameters: |
$V_{ub}=(3.82\pm 0.24)\times 10^{-3}$; $V_{ud}=0.97370\pm 0.00014$; $V_{cb}=(41\pm 1.4)\times 10^{-3}$ |  Zyla et al. (2020)
$V_{cd}=0.221\pm 0.004$; $V_{td}=(8.0\pm 0.3)\times 10^{-3}$; $V_{tb}=1.013\pm 0.030$ |  Zyla et al. (2020)
$\gamma=(72.1\begin{subarray}{c}+4.1\\\ -4.5\end{subarray})^{\circ}$; $\sin 2\beta_{d}=0.699\pm 0.017$ |  Zyla et al. (2020)
Form factors and decay constants: |
$F_{B\to\eta}(0)=0.168\begin{subarray}{c}+0.041\\\ -0.047\end{subarray}$; $F_{B\to\eta^{\prime}}(0)=0.130\begin{subarray}{c}+0.036\\\ -0.032\end{subarray}$; |  Duplancic and Melic (2015)
$F_{B\to\pi}(0)=0.21\pm 0.07$; $A_{B\to\rho}(0)=0.356\pm 0.042$; |  Gubernari et al. (2019); Bharucha et al. (2016)
$A^{0}_{B\to\omega}(0)=0.328\pm 0.048$; $A^{1}_{B\to\omega}(0)=0.243\pm 0.031$; $V_{B\to\omega}(0)=0.304\pm 0.038$; $f_{\omega}=(187\pm 5)$ MeV; $f^{\perp}_{\omega}=(151\pm 9)$ MeV |  Bharucha et al. (2016); Cheng and Chua (2009)
$f_{\eta}^{q}=107$ MeV; $f_{\eta}^{s}=-112$ MeV; $f_{\eta^{\prime}}^{q}=89$ MeV; $f_{\eta^{\prime}}^{s}=137$ MeV |  Cheng and Chua (2009)
$f_{B_{d}}=(190.5\pm 1.3)$ MeV; $f_{\pi}=(130.2\pm 1.4)$ MeV; |  Aoki et al. (2020); Gubernari et al. (2019)
$f_{\rho}=(216\pm 3)$ MeV; $f_{\rho}^{\perp}=(165\pm 9)$ MeV |  Cheng and Chua (2009)
$f_{\phi}=(215\pm 5)$ MeV; $f_{\phi}^{\perp}=(186\pm 9)$ MeV |  Cheng and Chua (2009)
Gegenbauer moments: |
$a_{1}^{\phi}=0$; $a_{2}^{\phi}=0.18\pm 0.08$; $a_{1}^{\perp,\phi}=0$; $a_{2}^{\perp,\phi}=0.14\pm 0.06$ |  Cheng and Chua (2009)
$a_{1}^{\rho}=0$; $a_{2}^{\rho}=0.15\pm 0.07$; $a_{1}^{\perp,\rho}=0$; $a_{2}^{\perp,\rho}=0.14\pm 0.07$ |  (Cheng and Chua, 2009)
$a_{1}^{\pi}=0$; $a_{2}^{\pi}=0.25\pm 0.15$ |  (Cheng and Chua, 2009)
$a_{1}^{\omega}=0$; $a_{2}^{\omega}=0.15\pm 0.07$; $a_{1}^{\perp,\omega}=0$; $a_{2}^{\perp,\omega}=0.14\pm 0.06$; $\lambda_{B}=300\pm 100$ MeV |  (Cheng and Chua, 2009)
Annihilation and hard spectator parameters: |
PP mode: $\rho_{A}=1.1$; $\phi=-50^{\circ}$ |  (Cheng and Chua, 2009)
PV mode: $\rho_{A}=0.87$; $\phi=-30^{\circ}$ |  (Cheng and Chua, 2009)
VP mode: $\rho_{A}=1.07$; $\phi=-70^{\circ}$;$X_{H}=2.4\pm 0.024$ |  (Cheng and Chua, 2009)

##  III New-Physics Contributions

In the standard model, flavor changing neutral current (FCNC) occurs at loop
level and provide a very strong suppression because of the intermediate light
quark contributions. Therefore it would be more challenging to explore the NP
beyond the SM. In this work we include a self-consistent framework where a
minimal extension of the standard model in other words enlarging the matter
sector having an extra iso singlet vector-like down quark represent to this
where Z boson is mediated with FCNC transition at tree level. Now due to the
addition of down type quark, the interaction lagrangian for Z boson in the
weak eigen state basis can be represented as Deshpande et al. (2004)

$\displaystyle
L_{Z}=-\frac{g}{2c_{W}}\big{[}\bar{U}_{L}^{0}\gamma^{\mu}U_{L}^{0}-\bar{D}_{L}^{0}\gamma^{\mu}D_{L}^{0}-2s_{W}^{2}(Q_{u}\bar{U}^{0}\gamma^{\mu}U^{0}+Q_{d}\bar{D}^{0}\gamma^{\mu}D^{0}+Q_{d}\bar{D}^{0^{\prime}}\gamma^{\mu}D^{0^{\prime}}\big{]}Z_{\mu},$
(20)

where $Q_{u,d}$ are the electric charges of up and down type quarks. The up
type quark $U^{0}$ and the down type quark $D^{0}$ are embeded in the SM three
generations of quarks and the additional down type quark is given by
$D^{0^{\prime}}=d^{0^{\prime}}$. Now because of the extension of down type
quark, the down quark matrix and the up quark matrix can be diagonalized by
$4\times 4$ and $3\times 3$ matrix respectively. So the corresponding
interaction Lagrangian mediated by Z, is given as Giri and Mohanta (2003)

$\mathcal{L}_{Z}=\frac{g}{2c_{W}}\big{[}\bar{\mathit{U}}_{L\mathit{i}}\gamma^{\mu}\mathit{U}_{L\mathit{i}}-\bar{D}_{L\alpha}U_{\alpha\beta}\gamma^{\mu}D_{L\alpha}-2s^{2}_{W}J_{em}^{\mu}\big{]}Z_{\mu},$

where $i$ ($\alpha,\beta$) denote the generation indices for up (down)-type
and $L$ indicate for the left chiral particles. Here the focus point is the
second term where the matrix $U_{\alpha\beta}$ is a $4\times 4$ matrix and the
expression is represented as

$\displaystyle
U_{\alpha\beta}=\sum_{\mathit{i}=\mathit{u,c,t}}V_{\alpha\mathit{i}}^{\dagger}V_{\mathit{i}\beta}=\delta_{\alpha\beta}-V_{4\alpha}^{*}V_{4\beta}.$
(21)

And this is the distinctive feature of this model. The corresponding CKM
matrix for the charge current interaction would be
$V=V_{u}^{L\dagger}V_{d}^{L}$ which is a $3\times 4$ pseudo matrix. It is
usually different from the CKM matrix present in the standard model. Since
$U_{\alpha\beta}(\alpha,\beta=b,d,s)\neq 0$, it motivates to study FCNC
mediated by Z boson at tree levelBuras and Lindner (1998); Alok et al. (2016);
Mohapatra (2020); Giri and Mohanta (2004).

Now the non unitrary matrix V arrises due to the addition of the extra 4th
quark to the SM sector. Thus it provides a new signal to scrutinize the
physics beyond the SM. Now we constrain the new parameter space arising due to
both leptonic as well non leptonic modes.

###  III.1  Constraint from leptonic
$B_{d}\to\ell^{+}\ell^{-}(\ell=e,\mu,\tau)$, and non leptonic modes
$B_{d}\to\eta^{\prime}\pi^{0}$ and $B_{u}\to\rho^{-}\eta^{\prime}$:

The leptonic modes $B_{d}\to\ell^{+}\ell^{-}(\ell=e,\mu,\tau)$ are suppressed
in the SM, still it can be investigated in the new physics scenario in the
presence of VLDQ model. The branching fraction of $B_{d}\to\ell^{+}\ell^{-}$
in $Z$ mediated VLDQ model is given by Chen et al. (2010)

$\displaystyle\mathfrak{B}_{(B_{d}\to\ell^{+}\ell^{-})}=\frac{G_{F}^{2}\alpha^{2}m_{B_{d}}m_{\ell}^{2}f_{B_{d}}^{2}\tau_{B_{d}}}{16\pi^{3}}|V_{tb}V_{td}^{*}|^{2}\sqrt{1-4(\frac{m_{\ell}^{2}}{m_{B_{d}}^{2}})}\left|C_{10}^{\rm
tot}\right|^{2},$ (22)

where

$\displaystyle C_{10}^{\rm
tot}=C_{10}-\frac{\pi}{\alpha}\frac{U_{bd}}{V_{tb}V_{td}^{*}}\,.$ (23)

Here the term $U_{bd}$ is the coupling parameter when b quark talks to d quark
in the presence of mediating Z particle and because of quark mixing it may
behave as a complex with weak phase $\phi_{d}$. Now the amplitude of the non
leptonic $B^{-}\to\rho^{-}\eta^{\prime}$ process is given byBeneke and Neubert
(2003),

$\displaystyle-\sqrt{2}\mathcal{A}_{B^{-}\to\rho\eta^{\prime}}$
$\displaystyle=$ $\displaystyle
A_{\rho\eta^{\prime}_{q}}\big{[}\delta_{pu}(\alpha_{2}-\beta_{2}+2\beta_{S2})+2\alpha_{3}^{p}+\alpha_{4}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}$
(24) $\displaystyle+$
$\displaystyle\beta_{3}^{p}+\beta_{3,EW}^{p}+2\beta_{S3}^{p}+2\beta_{S3,EW}^{p}\big{]}$
$\displaystyle+$
$\displaystyle\sqrt{2}A_{\rho\eta^{\prime}_{s}}\big{[}\delta_{pu}\beta_{S2}+\alpha_{3}^{p}-\frac{1}{2}\alpha_{3,EW}^{p}+\beta_{S3}^{p}+\beta_{S3,EW}^{p}\big{]}$
$\displaystyle+$
$\displaystyle\sqrt{2}A_{\rho\eta_{c}}\big{[}\delta_{pc}\alpha_{2}+\alpha_{3}^{p}\big{]}$
$\displaystyle+$ $\displaystyle
A_{\eta^{\prime}_{q}\rho}\big{[}\delta_{pu}(\alpha_{+}\beta_{2})+\alpha_{4}^{p}+\alpha_{4,EW}^{p}+\beta_{3}^{p}+\beta_{3,EW}^{p}\big{]}.$

And the amplitude of the decay mode $B_{d}\to\pi^{0}\eta^{\prime}$ is given as
Beneke and Neubert (2003),

$\displaystyle-2\mathcal{A}_{\bar{B}_{d}\to\pi^{0}\eta^{\prime}}$
$\displaystyle=$ $\displaystyle
A_{\pi\eta_{q}}\big{[}\delta_{pu}(\alpha_{2}-\beta_{1}-2\beta_{S1})+2\alpha_{3}^{p}+\alpha_{4}^{p}+\frac{1}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\alpha_{4,EW}^{p}$
(25) $\displaystyle+$
$\displaystyle\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{3}{2}\beta_{4,EW}^{p}+2\beta_{S3}^{p}-\beta_{S3,EW}^{p}-3\beta_{S4,EW}^{p}\big{]}$
$\displaystyle+$
$\displaystyle\sqrt{2}A_{\pi\eta_{s}}\big{[}-\delta_{pu}\beta_{S1}+\alpha_{3}^{p}-\frac{1}{2}\alpha_{3,EW}^{p}+\beta_{S3}^{p}-\frac{1}{2}\beta_{S3,EW}^{p}-\frac{3}{2}\beta_{S3,EW}^{p}\big{]}$
$\displaystyle+$
$\displaystyle\sqrt{2}A_{\pi\eta_{c}}\big{[}\delta_{pc}\alpha_{2}+\alpha_{3}^{p}\big{]}$
$\displaystyle+$ $\displaystyle
A_{\eta_{q}\pi}\big{[}\delta_{pu}(-\alpha_{2}-\beta_{1})+\alpha_{4}^{p}-\frac{3}{2}\alpha_{3,EW}^{p}-\frac{1}{2}\beta_{4,EW}^{p}+\beta_{3}^{p}-\frac{1}{2}\beta_{3,EW}^{p}-\frac{3}{2}\beta_{4,EW}^{p}\big{]},$

where the above decay mode amplitudes are multiplied by the CKM element
$\lambda_{p}$ and summed over $p=u,c$. The required parameters are given in
the Appendix A. Now the effective Hamiltonian corresponding to new interaction
describing quark lvel transition $b\to d$ can be represented as

$\mathcal{H}_{eff}^{Z}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V_{td}^{*}\big{[}\tilde{C}_{3}O_{3}+\tilde{C}_{7}O_{7}+\tilde{C}_{9}O_{9}\big{]},$

where the new Wilson coefficients at the $M_{Z}$ scale are given as Atwood and
Hiller (2003); Deshpande and Ghosh (2004)

$\displaystyle\tilde{C}_{3}(M_{Z})$ $\displaystyle=$
$\displaystyle\frac{1}{6}\frac{U_{bd}}{V_{tb}V_{td}^{*}},$
$\displaystyle\tilde{C}_{7}(M_{Z})$ $\displaystyle=$
$\displaystyle\frac{2}{3}\frac{U_{bd}}{V_{tb}V_{td}^{*}}\sin^{2}\theta_{W}\,,$
$\displaystyle\tilde{C}_{9}(M_{Z})$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\frac{U_{bd}}{V_{tb}V_{td}^{*}}(1-\sin^{2}\theta_{W})\,$
(26)

and the new Wilson coefficients at $m_{b}$ scale can be found in Mawlong et
al. (2008). From the unitary condition (21), we get

$\displaystyle\lambda_{u}+\lambda_{c}+\lambda_{t}=U_{bd}\,.$ (27)

Now the amplitude in the presence of new physics can be parameterized as,

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\lambda_{u}\mathcal{A}_{u}+\lambda_{c}\mathcal{A}_{c}-U_{bd}\mathcal{A}_{NP}$
(28) $\displaystyle=$
$\displaystyle\lambda_{c}A_{c}\Big{[}1+are^{\mathit{i}(\delta_{1}-\gamma)}-a^{\prime}r^{\prime}e^{\mathit{i}(\delta^{\prime}+\phi_{d})}\Big{]}\,,$

where

$\displaystyle a=|\frac{\lambda_{u}}{\lambda_{c}}|,\hskip
8.53581ptr=|\frac{A_{u}}{A_{c}}|,\hskip
8.53581pta^{\prime}=|\frac{U_{bd}}{\lambda_{c}}|,\hskip
8.53581ptr^{\prime}=|\frac{A_{NP}}{A_{c}}|.$ (29)

Here, $\gamma$, the weak phase arises from the CKM matrix element $V_{ub}$.
$\delta_{1}$ and $\delta^{\prime}$ are the relative strong phase of $A_{u}$
and $A_{NP}$ respectively with $A_{c}$ where the subscript u and c quark
correspond the amplitude involved to up and charm quark. Here the new coupling
parameter $U_{bd}$ may have complex phase $\phi_{d}$. From the amplitude given
in Eq.(28), the CP averaged branching ratio can be written as

$\displaystyle\mathfrak{B}$ $\displaystyle=$
$\displaystyle\frac{\tau_{B_{d}}p_{c}}{8\pi
m_{B_{d}}^{2}}|\xi_{c}A_{c}|^{2}\bigg{[}\mathcal{G}+2ra\cos\delta_{1}\cos\gamma-2r^{\prime}a^{\prime}\cos\delta^{\prime}\cos\phi_{d}$
(30)
$\displaystyle-2rr^{\prime}aa^{\prime}\cos(\delta_{1}-\delta^{\prime})\cos(\gamma+\phi_{d})\bigg{]}\,,$

where $\mathcal{G}=1+(ra)^{2}+(r^{\prime}a^{\prime})^{2}$. Now combining both
the leptonic and non leptonic modes with the given experimental and
theoretical values from TABLE - 1 and 2, the new parameter space
$U_{bd}-\phi_{d}$ within $1\sigma$ limit is represented in FIG.1 . Now the new
parameter ranges are shown below.

$\displaystyle 1.60\times 10^{-6}\leq|U_{bd}|\leq 2.22\times 10^{-4},\hskip
14.22636pt92.70^{\circ}\leq\phi_{d}\leq 360^{\circ},$ $\displaystyle
1.32\times 10^{-5}\leq|U_{bd}|\leq 1.69\times 10^{-4},\hskip
14.22636pt211.55^{\circ}\leq\phi_{d}\leq 360^{\circ}.$ (31)

Figure 1: The allowed region of new coupling parameter space $U_{bd}-\phi_{d}$
arised from the branching fractions of both leptonic
$B_{d}\to\ell^{+}\ell^{-}(\ell=e,\mu,\tau)$, and non leptonic
$B_{d}\to\eta^{\prime}\pi^{0}$ and $B_{u}\to\rho^{-}\eta^{\prime}$ processes.

###  III.2 Impact on the non leptonic modes:

####  $B_{d}\to\phi\eta$:

Using the allowed parameter space, we present the variation of CP averaged
branching ratio $\mathcal{B}$ with the weak phase $\phi_{d}$ by considering
three benchmark entries of the parameter $|U_{bd}|$ as $2\times
10^{-5},4\times 10^{-5}$ and $6\times 10^{-5}$ given in FIG.-2 in the left
panel. The black dotted central line correspond to the SM value where as the
red dot-dashed lines shaded with the yellow color represent its $1\sigma$
uncertainty. From this figure we see that during the variation of the weak
phase for the benchmark value $|U_{bd}|=2\times 10^{-5}$ (blue line), the
observable has significantly deviated from its SM contribution in the region
$0\leq\phi_{d}\leq 240^{\circ}$. Similarly one can also observe that for the
other two benchmark entries (purple and green line), the CP av. branching
fraction has also effective contribution from its standard model prediction.
Additionally, in the presence of the constraint parameters $U_{bd}$ and
$\phi_{d}$ from the two regions (III.1), the observable has constructive
contribution to the standard model value. In the right panel we have shown the
variation of the observable (in the units of $10^{-8}$) in the presence of all
possible entries of the parameter $|U_{bd}|$ and with the weak phase
$\phi_{d}$.

Figure 2: $B_{d}\to\phi\eta$: Variation of CP averaged branching ratio (in the
units of $10^{-8}$) of with ($\mathit{i}$) some benchmark points of $U_{bd}$
as $2\times 10^{-5}$ (Blue), $4\times 10^{-5}$ (Purple) and $6\times 10^{-5}$
(Green) with the new weak phase $\phi_{d}$ where the dashed black line
represents to the SM value with the red dot-dashed line along with the yellow
region denote its $1\sigma$ uncertainty (left panel), and with ($\mathit{ii}$)
all possible values of $U_{bd}$ and $\phi_{d}$ (right panel).

####  $B_{d}\to\phi\eta^{\prime}$:

Similarly in FIG.- 3, we display the impact of the new coupling parameter on
variation of the CP averaged branching ratio with the weak phase $\phi_{d}$
for the decay mode $B_{d}\to\phi\eta^{\prime}$. We study with three different
values of the parameter $|U_{bd}|$ whose entries are $5\times 10^{-5},8\times
10^{-5}$ and $1\times 10^{-4}$, and these entries correspond to the line with
blue, cyan and red color in the given figure (left panel) respectively. Also
the black dotted line and the magenta dot-dashed line (shaded with green
color) denote the SM value and its $1\sigma$ error respectively. Now from this
one can notice clearly in the left panel that the observable in the presence
of the benchmark point $|U_{bd}|$ correspond to blue line has deviated towards
the $1\sigma$ range of SM line in the region $240^{\circ}\leq\phi_{d}\leq
360^{\circ}$ while it has contributions above to the SM in the other region.
Moreover it could be significantly enhanced from the SM value in the presence
of the other two benchmark points while in the constraint regions of the NP
parameters given in eq. (III.1), it has remarkable contributions. Additionally
the right panel shows the impact of all the entries of the new physics
parameter on the CP av. branching ratio (in the units of $10^{-8}$).

Figure 3: $B_{d}\to\phi\eta^{\prime}$: Variation of CP averaged branching
ratio (in the units of $10^{-8}$) with the new weak phase $\phi_{d}$
($\mathit{i}$) in the presence of benchmark points of $U_{bd}$ as $5\times
10^{-5}$ (Blue), $8\times 10^{-5}$ (Cyan) and $1\times 10^{-4}$ (Red) (left
panel) where the dashed black line represent to the SM value along with
$1-\sigma$ error (green region), and with ($\mathit{ii}$) all possible values
of $U_{bd}$ and $\phi_{d}$ (right panel).

####  $B_{d}\to\phi\pi^{0}$:

Here we investigate the CP av. branching ratio of the process
$B_{d}\to\phi\pi^{0}$ with respect to the weak phase $\phi_{d}$. The Z - b - d
coupled parameter $U_{bd}$ has important contributions to the observable in
the presence of NP scenario and is displayed in the left panel of FIG. - 4
with three benchmark inputs. The black dotted line corresponds to the SM
prediction where as the light blue colored region along with red dot-dashed
line denote its $1\sigma$ deviation. With the 3 different inputs of
$|U_{bd}|$, we observe that the observable has significantly deviated from the
SM result. Moreover we get more deviations while increasing the coupling
parameter $|U_{bd}|$. For the ranges of $0^{\circ}\leq\phi_{d}\leq
230^{\circ}$, the CP av. branching ratio could be effectively deviated from
the stadard model value. In addition to this, the right panel having the
variation of the new parameter with all entries along with the phase
$\phi_{d}$, the observable displays all its deviations. However the observable
has significant impact in the regions of the new coupling parameters given in
eq. (III.1).

Figure 4: $B_{d}\to\phi\pi^{0}$: Variation of CP averaged branching ratio (in
the units of $10^{-8}$) with ($\mathit{i}$) three benchmark points of $U_{bd}$
as $5\times 10^{-5}$ (Magenta), $9\times 10^{-5}$ (Green) and $2\times
10^{-4}$ (Cyan) with the new weak phase $\phi_{d}$ (left panel) where the
dashed black line represent to the SM value along with shaded region of
$1-\sigma$ uncertainty, and with ($\mathit{ii}$) all possible values of
$U_{bd}$ and $\phi_{d}$ (right panel).

####  $B_{d}\to\omega\phi$:

Now in the study of $B_{d}\to\omega\phi$ process, the new physics parameter
has contributed effectively to the variation of CP av. branching ratio with
respect to the phase $\phi_{d}$. The corresponding FIG. 5 (left panel)
represents that the new physics in the presence of the benchmark values of
$|U_{bd}|$, the above decay mode has effective contributions from its standard
model value. The region of central black dotted line with dot-dashed grey line
shaded with cyan color provides $1\sigma$ uncertainty to the SM. Taking a
careful observation to the contribution corresponding to the input value of
$|U_{bd}|=4\times 10^{-5}$ $(9\times 10^{-5})$, the observable in the range of
$0^{\circ}\leq\phi_{d}\leq 95^{\circ},255^{\circ}\leq\phi_{d}\leq 360^{\circ}$
($0^{\circ}\leq\phi_{d}\leq 110^{\circ},240^{\circ}\leq\phi_{d}\leq
360^{\circ}$) has less deviations while in the span of
$0^{\circ}\leq\phi_{d}\leq 155^{\circ}$ and $200^{\circ}\leq\phi_{d}\leq
360^{\circ}$ from its $1\sigma$ range, it has deviated more effectively than
the other two contributions. Similar to other decay modes discussed above, we
vary the observable with all the contributions of new physics parameters in
the right panel of the given figure. Furthermore, in the regions of the
sizeable parameters given in eq. (III.1), the physical observable has
significant impact in the presence of the VLDQ model.

Figure 5: $B_{d}\to\phi\omega$: Variation of CP averaged branching ratio (in
the units of $10^{-8}$) with ($\mathit{i}$) some benchmark points of $U_{bd}$
as $4\times 10^{-5}$ (Blue), $9\times 10^{-5}$ (Red) and $2.2\times 10^{-4}$
(Purple) with the new weak phase $\phi_{d}$ (left panel) where the dashed
black line represent to the SM value, with ($\mathit{ii}$) all possible values
of $U_{bd}$ and $\phi_{d}$ (right panel).

##  IV Conclusion

We have scrutinized the decay modes of
$B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$, induced by $b\to d$ quark level
transition beyond the standard model. In the new physics scenario, we have
considered vector-like down quark model where a new quark generation has been
added to the SM and consequently it provides the interaction of $Z$ mediated
FCNC at the tree level. As the leptonic modes
$B_{d}\to\ell\ell(\ell=e,\mu,\tau)$ and the non leptonic modes
$B_{d}\to\pi\eta^{\prime}$ and $B_{u}\to\rho^{-}\eta^{\prime}$ have
descrepancies between the SM and experimental values, we investigated in the
presence of VLDQ model. In the presence of new physics, we constrained the
region of parameter space associated with the interactions“$Z-b-d$” at tree
level. We found that with the sizeable new coupling parameter $U_{bd}$,
considered from both leptonic as well as nonleptonic modes, it provides
significant contributions to $B_{d}\to\phi(\eta^{(^{\prime})},\pi,\omega)$
processes.

###### Acknowledgements.

MKM would like to thank to Department of Science and Technology(DST)- Inspire
Fellowship division, Government of India for financial support through ID No -
IF160303. MKM would like to acknowledge Prof. Anjan Giri for his support and
useful discussions.

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##  Appendix A The parameters used in the nonleptonic $B\to PP,PV,VP$ decay
modes

We need the factorized matrix elements for the decay mode $B\to M_{1}M_{2}$
which are given by

$\displaystyle
A_{M_{1}M_{2}}=i\frac{G_{F}}{\sqrt{2}}\begin{cases}m_{B}^{2}F_{0}^{B\to
M_{1}}(0)f_{M_{2}};&{\rm for}\hskip 5.69046ptM_{1}=M_{2}=\rm Pseudoscalar,\\\
-2m_{V}\epsilon^{*}_{M_{1}}.p_{B}A_{0}^{B\to M_{1}}(0)f_{M_{2}};&{\rm
for}\hskip 5.69046ptM_{1}=\rm Vector,M_{2}=\rm Pseudoscalar,\\\
-2m_{V}\epsilon^{*}_{M_{1}}.p_{B}F_{+}^{B\to M_{1}}(0)f_{M_{2}};&{\rm
for}\hskip 5.69046ptM_{1}=\rm Pseudoscalar,M_{2}=\rm Vector.\end{cases}$ (32)

The form factors $F_{+}$ and $F_{0}$ denote pseudoscalar mesons, $A_{0}$
stands for vector meson where as $f_{P}$ and $F_{V}$ denote the decay constant
for pseudoscalar and vector meson respectively.
The expressions of flavor operators in QCD factorisation process are given as
follows:

$\displaystyle\alpha_{1}(M_{1}M_{2})$ $\displaystyle=$ $\displaystyle
a_{1}(M_{1}M_{2}),$ $\displaystyle\alpha_{2}(M_{1}M_{2})$ $\displaystyle=$
$\displaystyle a_{2}(M_{1}M_{2}),$ $\displaystyle\alpha_{3}^{p}(M_{1}M_{2})$
$\displaystyle=$
$\displaystyle\begin{cases}a_{3}(M_{1}M_{2})-a_{5}(M_{1}M_{2});&{\rm
for}\hskip 5.69046ptM_{1}M_{2}=PP,VP,\\\
a_{3}(M_{1}M_{2})+a_{5}(M_{1}M_{2});&{\rm for}\hskip
5.69046ptM_{1}M_{2}=VV,PV,\end{cases}$
$\displaystyle\alpha_{4}^{p}(M_{1}M_{2})$ $\displaystyle=$
$\displaystyle\begin{cases}a_{4}^{p}(M_{1}M_{2})+r_{\chi}^{M_{2}}a_{6}^{p}(M_{1}M_{2});&{\rm
for}\hskip 5.69046ptM_{1}M_{2}=PP,PV,\\\
a_{4}^{p}(M_{1}M_{2})-r_{\chi}^{M_{2}}a_{6}^{p}(M_{1}M_{2});&{\rm for}\hskip
5.69046ptM_{1}M_{2}=VV,VP,\end{cases}$
$\displaystyle\alpha_{3,EW}^{p}(M_{1}M_{2})$ $\displaystyle=$
$\displaystyle\begin{cases}a_{9}(M_{1}M_{2})-a_{7}(M_{1}M_{2});&{\rm
for}\hskip 5.69046ptM_{1}M_{2}=PP,VP,\\\
a_{9}(M_{1}M_{2})+a_{7}(M_{1}M_{2});&{\rm for}\hskip
5.69046ptM_{1}M_{2}=VV,PV,\end{cases}$
$\displaystyle\alpha_{4,EW}^{p}(M_{1}M_{2})$ $\displaystyle=$
$\displaystyle\begin{cases}a_{10}^{p}(M_{1}M_{2})+r_{\chi}^{M_{2}}a_{8}^{p}(M_{1}M_{2});&{\rm
for}\hskip 5.69046ptM_{1}M_{2}=PP,PV,\\\
a_{10}^{p}(M_{1}M_{2})-r_{\chi}^{M_{2}}a_{8}^{p}(M_{1}M_{2});&{\rm for}\hskip
5.69046ptM_{1}M_{2}=VV,VP,\end{cases}$ (33)

where

$\displaystyle a_{i}^{p}(M_{1}M_{2})=\bigg{(}C_{i}+\frac{C_{i\pm
1}}{N_{c}}\bigg{)}N_{i}(M_{2})+\frac{C_{i\pm
1}}{N_{c}}\frac{C_{F}\alpha_{s}}{4\pi}\bigg{[}V_{i}(M_{2})+\frac{4\pi^{2}}{N_{c}}H_{i}(M_{1}M_{2})\bigg{]}+P_{i}^{p}(M_{2}),$
(34)

and $\hat{\alpha}_{4}^{p}=\alpha_{4}^{p}+\beta_{3}^{p}$ with the superscript
$p=u,c$. The details of the parameter $\beta_{3}^{p}$ is given below. The
quantity $N_{i}(M_{2})$ also reads as

$\displaystyle N_{i}(M_{2})=\begin{cases}0;&i=6,8,\\\ 1;&{\rm
otherwise},\end{cases}$ (35)

and $i$ runs from 1 to 10. The lower and upper sign correspond to even and odd
values of $i$, where as $C_{i}^{\prime}s$ and $C_{F}$ are the Wilson
coefficients and the color factor (with $N_{c}$ = 3) respectively. The
relevant contributions $V_{i}(M_{2})$ and $H_{i}(M_{1}M_{2})$ are vertex
corrections and hard spectator interactions where as the term
$P_{i}^{p}(M_{1}M_{2})$ shows as penguin contractions. The explicit
expressions are given below.

  * •

Vertex corrections:

$\displaystyle
V_{i}(M_{2})=\begin{cases}\int_{0}^{1}dx{\rm\Phi}_{M_{2}}(x)\big{[}12{\rm
ln}\frac{m_{b}}{\mu}-18+g(x)\big{]};&{\rm for}\hskip 5.69046pti=1-4,9,10,\\\
\int_{0}^{1}dx{\rm\Phi}_{M_{2}}(x)\big{[}-12{\rm
ln}\frac{m_{b}}{\mu}+6-g(1-x)\big{]};&{\rm for}\hskip 5.69046pti=5,7,\\\
\int_{0}^{1}dx{\rm\Phi}_{m_{2}}(x)\big{[}-6+h(x)\big{]};&{\rm for}\hskip
5.69046pti=6,8,\end{cases}$

where

$\displaystyle g(x)$ $\displaystyle=$ $\displaystyle
3\bigg{(}\frac{1-2x}{1-x}{\rm ln}x-i\pi\bigg{)}$ $\displaystyle+$
$\displaystyle\bigg{[}2Li_{2}(x)-{\rm ln}^{2}x+\frac{2{\rm ln}x}{1-x}-(3+2\pi
i){\rm ln}x-(x\leftrightarrow 1-x)\bigg{]},$ $\displaystyle h(x)$
$\displaystyle=$ $\displaystyle 2Li_{2}(x)-{\rm ln}^{2}x-(1+2\pi i){\rm
ln}x-(x\leftrightarrow 1-x).$ (36)

The terms so called $\Phi_{P,V}(x)$ and $\Phi_{p,v}(x)$ given in the above
expessions are leading twist and twist-3 light-cone distribution amplitudes,
respectively Beneke and Neubert (2003).

  * •

Hard spectator interactions:

$\displaystyle
H_{i}(M_{1}M_{2})=\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\int_{0}^{1}dx\int_{0}^{1}dy\bigg{[}\frac{\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)}{\bar{x}\bar{y}}+r_{\chi}^{M_{1}}\frac{\Phi_{M_{2}}(x)\Phi_{m_{1}}(y)}{x\bar{y}}\bigg{]}$
(37)

for $i=1-4,9,10,$

$\displaystyle
H_{i}(M_{1}M_{2})=-\frac{B_{M_{1}M_{2}}}{A_{M_{1}M_{2}}}\frac{m_{B}}{\lambda_{B}}\int_{0}^{1}dx\int_{0}^{1}dy\bigg{[}\frac{\Phi_{M_{2}}(x)\Phi_{M_{1}}(y)}{x\bar{y}}+r_{\chi}^{M_{1}}\frac{\Phi_{M_{2}}(x)\Phi_{m_{1}}(y)}{\bar{x}\bar{y}}\bigg{]}$
(38)

for $i=5,7$ and $H_{i}(M_{1}M_{2})=0$ for $i=6,8$ where we consider
$\lambda_{B}$= 300 MeV.

  * •

Penguin contractions:

These terms at the order of $\alpha_{s}$ are given as

$\displaystyle P_{4}^{p}(M_{2})$ $\displaystyle=$
$\displaystyle\frac{C_{F}\alpha_{s}}{4\pi
N_{c}}\bigg{\\{}C_{1}\bigg{[}\frac{4}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{2}{3}-G_{M_{2}}(s_{p})\bigg{]}+C_{3}\bigg{[}\frac{8}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{4}{3}-G_{M_{2}}(0)-G_{M_{2}}(1)\bigg{]}\bigg{\\}}$
(39) $\displaystyle+$ $\displaystyle(C_{4}+C_{6})\bigg{[}\frac{4n_{f}}{3}{\rm
ln}\frac{m_{b}}{\mu}-(n_{f}-2)G_{M_{2}}(0)-G_{M_{2}}(s_{c})-G_{M_{2}}(1)\bigg{]}$
$\displaystyle-$ $\displaystyle 2C_{8g^{\rm
eff}}\int_{0}^{1}\frac{dx}{1-x}\Phi_{M_{2}}(x),$ $\displaystyle
P_{6}^{p}(M_{2}=P)$ $\displaystyle=$ $\displaystyle\frac{C_{F}\alpha_{s}}{4\pi
N_{c}}\bigg{\\{}C_{1}\bigg{[}\frac{4}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{2}{3}-\hat{G}_{M_{2}}(s_{p})\bigg{]}+C_{3}\bigg{[}\frac{8}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{4}{3}-\hat{G}_{M_{2}}(0)-\hat{G}_{M_{2}}(1)\bigg{]}$
$\displaystyle+$ $\displaystyle(C_{4}+C_{6})\bigg{[}\frac{4n_{f}}{3}{\rm
ln}\frac{m_{b}}{\mu}-(n_{f}-2)\hat{G}_{M_{2}}(0)-\hat{G}_{M_{2}}(s_{c})-\hat{G}_{M_{2}}(1)\bigg{]}-2C_{8g}^{\rm
eff}\bigg{\\}},$ $\displaystyle P_{6}^{p}(M_{2}=V)$ $\displaystyle=$
$\displaystyle-\frac{C_{F}\alpha_{s}}{4\pi
N_{c}}\bigg{\\{}C_{1}\bigg{[}\hat{G}_{M_{2}}(s_{p})\bigg{]}+C_{3}\bigg{[}\hat{G}_{M_{2}}(0)-\hat{G}_{M_{2}}(1)\bigg{]}$
$\displaystyle+$
$\displaystyle(C_{4}+C_{6})\bigg{[}(n_{f}-2)\hat{G}_{M_{2}}(0)+\hat{G}_{M_{2}}(s_{c})+\hat{G}_{M_{2}}(1)\bigg{]}\bigg{\\}},$
$\displaystyle P_{8}^{p}(M_{2}=P)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{9\pi
N_{c}}\bigg{\\{}(C_{1}+N_{c}C_{2})\bigg{[}\frac{4}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{2}{3}-\hat{G}_{M_{2}}(s_{p})\bigg{]}-3C_{7\gamma}^{\rm
eff}\bigg{\\}},$ $\displaystyle P_{8}^{p}(M_{2}=V)$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{9\pi
N_{c}}(C_{1}+N_{c}C_{2})\hat{G}_{M_{2}}(s_{p}),$ (40) $\displaystyle
P_{10}^{p}$ $\displaystyle=$ $\displaystyle\frac{\alpha}{9\pi
N_{c}}\bigg{\\{}(C_{1}+N_{c}C_{2})\bigg{[}\frac{4}{3}{\rm
ln}\frac{m_{b}}{\mu}+\frac{2}{3}-G_{M_{2}}(s_{p})\bigg{]}-3C_{7\gamma}^{\rm
eff}\int_{0}^{1}\frac{dx}{1-x}\Phi_{M_{2}}(x)\bigg{\\}},$ (41)

where $n_{f}=5$, $s_{u}=(\frac{m_{u}}{m_{b}})^{2}\approx 0$ and
$s_{c}=(\frac{m_{c}}{m_{b}})^{2}$. The parameters so called $\alpha_{s}$ and
$\alpha$ are strongand EM coupling constants respectively. The functions
$G_{M_{2}}(s)$ and $\hat{G}_{M_{2}}(s)$ are defined in Beneke and Neubert
(2003). In addition to this, the power suppressed weak annihilation
contributions are given by

  * •

Annihilation contribution:

$\displaystyle\beta_{i}^{p}(M_{1}M_{2})=\frac{if_{B}f_{M_{1}}f_{M_{2}}}{A_{M_{1}M_{2}}}b_{i}^{p},$
(42)

where

$\displaystyle b_{1}=\frac{C_{F}}{N_{c}^{2}}C_{1}A_{1}^{i},\hskip
5.69046ptb_{3}=\frac{C_{F}}{N_{c}^{2}}\big{[}C_{3}A_{1}^{i}+C_{5}(A_{3}^{i}+A_{3}^{f})+N_{c}C_{6}A_{3}^{3}\big{]},$
$\displaystyle b_{2}=\frac{C_{F}}{N_{c}^{2}}C_{2}A_{1}^{i},\hskip
5.69046ptb_{4}=\frac{C_{F}}{N_{c}^{2}}\big{[}C_{4}A_{1}^{i}+C_{6}A_{2}^{f}\big{]},$
$\displaystyle
b_{3,EW}^{p}=\frac{C_{F}}{N_{c}^{2}}\big{[}C_{9}A_{1}^{i}+C_{7}(A_{3}^{i}+A_{3}^{f})+N_{c}C_{8}A_{3}^{i}\big{]},$
$\displaystyle
b_{4,EW}^{p}=\frac{C_{F}}{N_{c}^{2}}\big{[}C_{10}A_{1}^{i}+C_{8}(A_{2}^{i}\big{]}.$
(43)

Here the expressions of A are given as:
Case - I ($M_{1}=M_{2}=P$):

$\displaystyle A_{1}^{i}\approx A_{2}^{i}\approx
2\pi\alpha_{s}\big{[}9(X_{A}-4+\frac{\pi^{2}}{3})+r_{\chi}^{M_{1}}r_{\chi}^{M_{2}}X_{A}^{2}\big{]},$
$\displaystyle A_{3}^{i}\approx
6\pi\alpha_{s}(r_{\chi}^{M_{1}}-r_{\chi}^{M^{2}})\big{(}X_{A}^{2}-2X^{A}+\frac{\pi^{2}}{3}\big{)},$
$\displaystyle A_{3}^{f}\approx
6\pi\alpha_{s}(r_{\chi}^{M_{1}}+r_{\chi}^{M^{2}})(2X_{A}^{2}-X_{A}),$
$\displaystyle A_{1}^{f}=A_{2}^{f}=0.$ (44)

Case - II ($M_{1}=V,M_{2}=P$):

$\displaystyle A_{1}^{i}\approx-A_{2}^{i}\approx
6\pi\alpha_{s}\big{[}3(X_{A}-4+\frac{\pi^{2}}{3})+r_{\chi}^{M_{1}}r_{\chi}^{M_{2}}(X_{A}^{2}-2X_{A})\big{]},$
$\displaystyle A_{3}^{i}\approx
6\pi\alpha_{s}\bigg{[}-3r_{\chi}^{M_{1}}\big{(}X_{A}^{2}-2X^{A}+\frac{\pi^{2}}{3}+4\big{)}+r_{\chi}^{M_{2}}\big{(}X_{A}^{2}-2X_{A}+\frac{\pi^{2}}{3}\big{)}\bigg{]},$
$\displaystyle A_{3}^{f}\approx
6\pi\alpha_{s}\big{[}3r_{\chi}^{M_{1}}(2X_{A}-1)(2-X_{A})-r_{\chi}^{M_{2}}(2X_{A}^{1}-X_{A})\big{]},$
$\displaystyle A_{1}^{f}=A_{2}^{f}=0.$ (45)

Case - III ($M_{1}=P,M_{2}=V$):

$\displaystyle A_{1}^{i}\approx-A_{2}^{i}\approx
6\pi\alpha_{s}\big{[}3(X_{A}-4+\frac{\pi^{2}}{3})+r_{\chi}^{M_{2}}r_{\chi}^{M_{1}}(X_{A}^{2}-2X_{A})\big{]},$
$\displaystyle A_{3}^{i}\approx
6\pi\alpha_{s}\bigg{[}-3r_{\chi}^{M_{2}}\big{(}X_{A}^{2}-2X^{A}+\frac{\pi^{2}}{3}+4\big{)}+r_{\chi}^{M_{1}}\big{(}X_{A}^{2}-2X_{A}+\frac{\pi^{2}}{3}\big{)}\bigg{]},$
$\displaystyle
A_{3}^{f}\approx-6\pi\alpha_{s}\big{[}3r_{\chi}^{M_{2}}(2X_{A}-1)(2-X_{A})-r_{\chi}^{M_{1}}(2X_{A}^{1}-X_{A})\big{]},$
$\displaystyle A_{1}^{f}=A_{2}^{f}=0.$ (46)

$A_{n}^{i,f}:$ n = 1, 2 and 3 correspond to the the operator structure
$(V-A)(V-A),(V-A)(V+A)$ and $(S-P)(S+P)$ respectively where as the
superscripts - ($i,f$) denote for the gluon emission from initial and final
states. The chiral factor $r_{\chi}$ is given by

$\displaystyle
r_{\chi}^{P}(\mu)=\frac{2m_{P}^{2}}{m_{b}(\mu)(m_{1}+m_{2})(\mu)},\hskip
5.69046ptr_{\chi}^{V}(\mu)=\frac{2m_{V}}{m_{b}(\mu)}\frac{f_{V}^{\perp}(\mu)}{f_{V}}.$
(47)

The end point divergence that has been used, can be given as

$\displaystyle X_{A}={\rm
ln}\frac{m_{B}}{\Lambda_{QCD}}(1+{\rm\rho_{A}}\exp^{i\phi_{A}}),$ (48)

where ${\rm\rho_{A}}$ and $\phi_{A}$ can be found from Cheng and Chua (2009).

Modes | $\rm\rho_{A}$ | $\rm\phi_{A}$
---|---|---
$B\to PP$ | 1.10 | $-50^{\circ}$
$B\to PV$ | 0.87 | $-30^{\circ}$
$B\to VP$ | 1.07 | $-70^{\circ}$

##  Appendix B The parameters used in the nonleptonic $B\to VV$ decay mode

For the decay process $B\to VV$ the helicity amplitudes depend upon the
factorized matrix elements as Beneke et al. (2007)

$\displaystyle X_{B_{d}\to
V_{1},V_{2}}=\big{\langle}V_{2}|(\bar{q}_{2}q_{3})_{V-A}|0\big{\rangle}\big{\langle}V_{1}|(\bar{q}_{1}b)_{V-A}|\bar{B}_{d}\big{\rangle},$
(49)

where the form factor and the decay constants are defined by

$\displaystyle\big{\langle}V(p,\epsilon^{*})|\bar{q}\gamma_{\mu}q^{\prime}|0\
big\rangle=$ $\displaystyle-if_{V}m_{V}\epsilon_{\mu}^{*},$
$\displaystyle\big{\langle}V(p,\epsilon^{*})\bar{q}\gamma_{\mu}(1-\gamma_{5})b|\bar{B}_{d}(p_{B})\big{\rangle}=$
$\displaystyle-\epsilon_{\mu}^{*}(m_{B}+m_{V})A_{1}^{B_{d}V}(q^{2})+(p_{B}+p)_{\mu}(\epsilon^{*}.p_{B})\frac{A_{2}^{B_{d}V}(q^{2})}{m_{B}+m_{V}}$
(50)
$\displaystyle+q_{\mu}(\epsilon^{*}p_{B})\frac{2m_{V}}{q^{2}}[A_{3}^{B_{d}V}(q^{2})-A_{0}^{B_{d}V}(q^{2})]$
$\displaystyle-i\mathcal{E}_{\mu\nu\alpha\beta}\epsilon^{*\nu}p_{B}^{\alpha}p^{\beta}\frac{2V^{B_{d}V}(q^{2})}{m_{B}+m_{V}},$

where $q=p_{B}-p$. The expressions of the helicity amplitudes are given as

$\displaystyle
X^{0}_{V_{1}V_{2}}=i\frac{G_{F}}{\sqrt{2}}m_{B}^{2}f_{V_{2}}A_{0}^{B\to
V_{1}}(0),\hskip
8.5359ptX_{V_{1}V_{2}}^{\pm}=\frac{G_{F}}{\sqrt{2}}m_{B}m_{2}f_{V_{2}}F_{\pm}^{B\to
V_{1}}(0),$ (51)

where the form factor $F_{\pm}^{B\to V_{1}}$ is defined as

$\displaystyle F_{\pm}^{B\to V_{1}}(q^{2})=(1+\frac{m_{1}}{m_{B}})A_{1}^{B\to
V_{1}}(q^{2})\mp(1-\frac{m_{1}}{m_{B}})V^{B\to V_{1}}(q^{2}).$ (52)

The assembled form of the coefficients $a_{i}$ are given as

$\displaystyle a_{i}^{p,h}(M_{1}M_{2})=\bigg{(}C_{i}+\frac{C_{i\pm
1}}{N_{c}}\bigg{)}N_{i}^{h}(M_{2})+\frac{C_{i\pm
1}}{N_{c}}\frac{C_{F}\alpha_{s}}{4\pi}\bigg{[}V_{i}^{h}(M_{2})+\frac{4\pi^{2}}{N_{c}}H_{i}^{h}(M_{1}M_{2})\bigg{]}+P_{i}^{h,p}(M_{2}),$
(53)

where the upper (lower) sign signifies for $i$ odd (even). The superscript $p$
correspond to penguin contributions where it is ommited for $i=1,2$. The
parameters for $h=0$ corrspond to those given in Appendix A where P is
replaced by V in the final staes PV. And due to the suppressed contributions
from positive helicity so only considering the negative helicity amplitudes
(Beneke et al., 2007), the LO parameter $N_{i}$ is given by

$\displaystyle N_{i}^{-}(M_{2})=\begin{cases}0;&i=\big{\\{}6,8\big{\\}}\\\
1;&{\rm otherwise},\end{cases}$ (54)

The vertex corrections are given as

$\displaystyle
V_{i}^{-}(M_{2})=\begin{cases}\int_{0}^{1}dx{\rm\Phi_{b2}}(x)\big{[}12{\rm
ln}\frac{m_{b}}{\mu}-18+g_{T}(x)\big{]};&{\rm for}\hskip
5.69046pti=\big{\\{}1-4,9,10\big{\\}}\\\
\int_{0}^{1}dx{\rm\Phi_{a2}}_{M_{2}}(x)\big{[}-12{\rm
ln}\frac{m_{b}}{\mu}+6-g_{T}(1-x)\big{]};&{\rm for}\hskip
5.69046pti=\big{\\{}5,7\big{\\}}\\\
\int_{0}^{1}dx{\rm\Phi}_{m_{2}}(x)\big{[}-6+h(x)\big{]};&{\rm for}\hskip
5.69046pti=\big{\\{}6,8\big{\\}}\end{cases}$

The parameter $g_{T}(x)$ is given as

$\displaystyle g_{T}(x)=g(x)+\frac{lnx}{1-x},$ (55)

where $g(x)$ is given in the Appendix A. The hard spectator contributions are
given by

$\displaystyle H_{i}^{-}$
$\displaystyle=-\frac{2f_{B}f_{V_{1}}^{\perp}}{m_{B}m_{b}F_{-}^{B\to
V_{1}}(0)}\frac{m_{b}}{\lambda_{B}}\int_{0}^{1}dxdy\frac{\phi_{1}^{\perp}(x)\phi_{b2}(y)}{\bar{x}^{2}y},\hskip
28.45274pti=\big{\\{}1-4,9,10\big{\\}},$ $\displaystyle H_{i}^{-}$
$\displaystyle=\frac{2f_{B}f_{V_{1}}^{\perp}}{m_{B}m_{b}F_{-}^{B\to
V_{1}}(0)}\frac{m_{b}}{\lambda_{B}}\int_{0}^{1}dxdy\frac{\phi_{1}^{\perp}(x)\phi_{a2}(y)}{\bar{x}^{2}\bar{y}},\hskip
28.45274pti=\big{\\{}5,7\big{\\}},$ $\displaystyle H_{i}^{-}$
$\displaystyle=\frac{2f_{B}f_{V_{1}}}{m_{B}m_{b}F_{-}^{B\to
V_{1}}(0)}\frac{m_{b}m_{1}}{m_{2}^{2}}\frac{m_{b}}{\lambda_{B}}\int_{0}^{1}dxdy\frac{\phi_{a1}(x)\phi_{2}^{\perp}(y)}{y\bar{x}\bar{y}},\hskip
28.45274pti=\big{\\{}6,8\big{\\}},$ (56)

where we use the divergent integral can be founf finite through the defining
parameter as Beneke et al. (2007)

$\displaystyle\int_{0}^{1}dx\frac{\phi_{1}^{\perp}}{\bar{x}^{2}}=\bigg{[}\lim_{u\to
1}\frac{\phi_{1}^{\perp}}{\bar{u}}\bigg{]}X_{H}^{V_{1}}+\int_{0}^{1}\frac{dx}{1-x}\bigg{[}\frac{\phi_{1}^{\perp}(x)}{1-x}-\bigg{(}\lim_{u\to
1}\frac{\phi_{1}^{\perp}(u)}{\bar{u}}\bigg{)}\bigg{]},$ (57)

where the asymptotic distribution amplitudes are given by
$\phi^{\perp}(x)=6x(1-x),\hskip 5.69046pt\\\ \phi_{a}(x)=3(1-x)^{2},\hskip
5.69046pt\phi_{b}(x)=3x^{2}$.