#  Double-Lepton Polarization Asymmetries in
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ Decay in the Fourth-Generation Standard
Model

S. M<EMAIL_ADDRESS>F. Falahati, H. Mehranfar
Physics Department, Shiraz University, Shiraz 71454, Iran

###### Abstract

In this paper, we investigate the effects of the fourth generation of quarks
on the double-lepton polarization asymmetries in the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay. It is shown that these
asymmetries in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay compared with
those of $B\rightarrow K\ell^{+}\ell^{-}$ decay are more sensitive to the
fourth-generation parameters. We conclude that an efficient way to establish
the existence of the fourth generation of quarks could be the study of these
asymmetries in the $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay.

PACS numbers: 12.60.-i, 13.30.-a, 14.20.Mr

##  I Introduction

Although Standard Model (SM) is a successful theory, there is no clear
theoretical argument within this model to restrict the number of generations
to three, and therefore the possibility of a new generation should not be
ruled out. Based on this possibility, a number of theoretical and experimental
investigations have been performed. The measurement of the $Z$ decay widths
restricts the number of light neutrino for $m_{\nu}<m_{Z}/2$ to threeewwg .
However, if a heavy neutrino exits, the possibility of extra generations of
heavy quarks is not excluded from the experiment. Moreover the electro weak
data okun  supports an extra generation of heavy quarks, if the mass
difference between the new up and down-type quarks is not too large.

Many authors who support the existence of fourth-generation studied those
effects in various areas, for instance Higgs and neutrino physics, cosmology
and dark matter Polonsky –Ginzburg . For example, in Ginzburg  it is argued
that the fourth generation of quarks and leptons can be generated in the Higgs
boson production at the Tevatron and the LHC, before being actually detected.
By the detailed study of this process at the Tevatron and LHC, the number of
generations in the SM can be determined. Moreover, the flavor democracy
(Democratic Mass Matrix approach) FD1  favors the existence of the nearly
degenerate fourth SM family, while the fifth SM family is disfavored both by
the mass phenomenology and precision tests of the ${\rm SM}$ FD2 . The main
restrictions on the new SM families come from the experimental data on the
$\rho$ and $S$ parameters FD2 . However, the common mass of the fourth quark
($m_{t^{\prime}}$) lies between 320 $GeV$ and 730 $GeV$ considering the
experimental value of $\rho=1.0002^{+0.0007}_{-0.0004}$ PDG . The last value
is close to upper limit on heavy quark masses, $m_{q}\leq 700$ $GeV$ $\approx
4m_{t}$, which follows from partial-wave unitarity at high energies chanowitz
. It should be noted that with preferable value $a\approx g_{w}$ Flavor
Democracy predicts $m_{t^{\prime}}\approx 8m_{w}\approx 640$ $GeV$.

One of the promising areas in the experimental search for the fourth-
generation, via its indirect loop effects, is the rare B meson decays. Based
on this idea, serious attempts to probe the effects of the fourth-generation
on the rare B meson were made by many researchers. The fourth-generation can
affect physical observables, i.e. branching ratio, CP asymmetry, polarization
asymmetries and forward–backward asymmetries. The study of these physical
observables is a good tool to look for the fourth generation of up type quarks
Hou:2006jy –Turan:2005pf .

Recently, the sensitivity of the double-lepton polarization asymmetries to the
fourth-generation in the transition of $B$ to a pseudo scalar meson
($B\rightarrow K\ell^{+}\ell^{-}$) has been investigated and it is found out
that this observable is sensitive to the fourth-generation parameters
($m_{t^{\prime}}$, $V_{t^{\prime}b}V^{*}_{t^{\prime}s}$)Bashiry:2007tf . In
this work, we investigate the effects of the fourth generation of quarks
$(b^{\prime},t^{\prime})$ on the double-lepton polarizations in the transition
of $B$ to a vector meson ( $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$) and compare
our results with those of $B\rightarrow K\ell^{+}\ell^{-}$ decay presented in
Ref.Bashiry:2007tf . It should be mentioned that both decays occur through
$b\rightarrow s$ transition in which the sequential fourth generation of up
quarks $(t^{\prime})$, like $u,c,t$ quarks, contributes at the loop level.
Hence, this new generation will change only the values of the Wilson
coefficients via virtual exchange of the fourth-generation up quark
$t^{\prime}$ and the full operator set is exactly the same as in SM.

The paper is organized as follows. In Section II, the expressions for the
matrix element and double-lepton polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ in the SM have been presented. The
effect of the fourth generation of quarks on the effective Hamiltonian and the
double-lepton polarization asymmetries have been discussed in Section III. The
sensitivity of these polarizations to the fourth-generation parameters
$(m_{t^{\prime}},r_{sb},\phi_{sb})$ have been numerically analyzed in the
final Section.

##  II The Matrix Element and Double-Lepton Polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ in the SM

In the SM, the relevant effective Hamiltonian for
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay which is described by
$b\rightarrow s\ell^{+}\ell^{-}$ transition at quark level can be written as

${\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i=1}^{10}C_{i}(\mu){\cal
O}_{i}(\mu)\,,$ (1)

where the complete set of the operators ${\cal O}_{i}(\mu)$ and the
corresponding expressions for the Wilson coefficients $C_{i}(\mu)$ are given
in R23 . Using the above effective Hamiltonian, the one-loop matrix elements
of $b\rightarrow s\ell^{+}\ell^{-}$ can be written in terms of the tree-level
matrix elements of the effective operators as:

$\displaystyle{\cal M}(b\rightarrow s\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle<s\ell^{+}\ell^{-}|{\cal H}_{\rm eff}|b>$ (2) $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i}C^{\rm
eff}_{i}(\mu)<s\ell^{+}\ell^{-}|{\cal O}_{i}|b>^{tree}.$ $\displaystyle=$
$\displaystyle-\frac{G_{F}\alpha}{2\pi\sqrt{2}}V_{tb}V^{*}_{ts}\Bigg{[}\tilde{C}_{9}^{\rm
eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b~{}\bar{\ell}\gamma_{\mu}\ell+\tilde{C}_{10}^{\rm
eff}\bar{s}\gamma_{\mu}(1-\gamma_{5})b~{}\bar{\ell}\gamma_{\mu}\gamma_{5}\ell$
$\displaystyle\,\,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2C_{7}^{\rm
eff}\frac{m_{b}}{q^{2}}\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b\,\bar{\ell}\gamma_{\mu}\ell\Bigg{]},$

where $q^{2}=(p_{1}+p_{2})^{2}$ and $p_{1}$ and $p_{2}$ are the final leptons
four–momenta and the effective Wilson coefficients at $\mu$ scale, are given
as R23 ; R24 :

$\displaystyle C_{7}^{\rm eff}$ $\displaystyle=$ $\displaystyle
C_{7}-\frac{1}{3}C_{5}-C_{6}$ $\displaystyle C_{10}^{\rm eff}$
$\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}\tilde{C}^{\rm
eff}_{10}=C_{10}$ $\displaystyle C_{9}^{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\tilde{C}^{\rm
eff}_{9}=C_{9}+\frac{\alpha}{2\pi}Y(s).$ (3)

In Eq.(II), $s=q^{2}/m_{b}^{2}$ and the function $Y(s)$ contains the short-
distance contributions due to the one-loop matrix element of the four quark
operators, $Y_{per}(s)$, as well as the long-distance contributions coming
from the real $c\bar{c}$ intermediate states, i.e., $J/\psi$, $\psi^{\prime}$,
$\cdots$.The latter contributions are taken into account by introducing
Breit–Wigner form of the resonance propagator which leads to the second term
in the following formula (see Eq.II) R26 –R28 . As a result the function
$Y(s)$ can be written as:

$\displaystyle Y(s)$ $\displaystyle=$ $\displaystyle
Y_{per}(s)+\frac{3\pi}{\alpha^{2}}(3C_{1}+C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6})$
$\displaystyle\times$
$\displaystyle\sum_{V_{i}=\psi_{i}}\kappa_{i}\frac{m_{V_{i}}\Gamma(V_{i}\rightarrow\ell^{+}\ell^{-})}{m_{V_{i}}^{2}-sm_{b}^{2}-im_{V_{i}}\Gamma_{V_{i}}},$

where

$\displaystyle Y_{per}(s)$ $\displaystyle=$ $\displaystyle
g(\frac{m_{c}}{m_{b}},s)(3C_{1}+C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6})$ (5)
$\displaystyle-$ $\displaystyle\frac{1}{2}g(1,s)(4C_{3}+4C_{4}+3C_{5}+C_{6})$
$\displaystyle-$
$\displaystyle\frac{1}{2}g(0,s)(C_{3}+3C_{4})+\frac{2}{9}(3C_{3}+C_{4}+3C_{5}+C_{6}).$

The explicit expressions for the $g$ functions can be found in R23  and the
phenomenological parameters $\kappa_{i}$ in Eq.(II) can be determined from

${\cal B}(B\rightarrow K^{\ast}V_{i}\rightarrow
K^{\ast}\ell^{+}\ell^{-})={\cal B}(B\rightarrow K^{\ast}V_{i})\,{\cal
B}(V_{i}\rightarrow\ell^{+}\ell^{-}),$ (6)

where the data for the right hand side is given in R29 . For the lowest
resonances, $J/\psi$ and $\psi^{\prime}$ one can use $\kappa=1.65$ and
$\kappa=2.36$, respectively (see R30 ). In this study, we neglect the long-
distance contributions for simplicity and like Ref.R23 , to have a scheme
independent matrix element, we use the leading order as well as the next-to-
leading order QCD corrections to $C_{9}$ and the leading order QCD corrections
to the other Wilson coefficients.

In order to compute the decay width and other physical observables of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay, we need to sandwich the matrix
elements in Eq.(2) between the final and initial meson states. Therefore, the
hadronic matrix elements for the $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ can be
parameterized in terms of form factors. For the vector meson $\phi$ with
polarization vector $\varepsilon_{\mu}$ the semileptonic form factors of the
V–A current is defined as:

$\displaystyle<\phi(p_{\phi},\epsilon)\mid\bar{s}\gamma_{\mu}(1-\gamma_{5})b\mid
B(p_{B_{s}})>=-\frac{2V(q^{2})}{m_{B_{s}}+m_{\phi}}\epsilon_{\mu\nu\rho\sigma}p_{\phi}^{\rho}q^{\sigma}\epsilon^{*\nu}$
$\displaystyle-i\left[\epsilon_{\mu}^{*}(m_{B_{s}}+m_{\phi})A_{1}(q^{2})-(\epsilon^{*}q)(p_{B_{s}}+p_{\phi})_{\mu}\frac{A_{2}(q^{2})}{m_{B_{s}}+m_{\phi}}\right.$
$\displaystyle-\left.q_{\mu}(\epsilon^{*}q)\frac{2m_{\phi}}{q^{2}}(A_{3}(q^{2})-A_{0}(q^{2}))\right],$
(7)

where $q=p_{B_{s}}-p_{\phi}$, and $A_{3}(q^{2}=0)=A_{0}(q^{2}=0)$ (this
condition ensures that there is no kinematical singularity in the matrix
element at $q^{2}=0$). Also, the form factor $A_{3}(q^{2})$ can be written as
a linear combination of the form factors $A_{1}$ and $A_{2}$ :

$\displaystyle
A_{3}(q^{2})=\frac{1}{2m_{\phi}}\left[(m_{B_{s}}+m_{\phi})A_{1}(q^{2})-(m_{B_{s}}-m_{\phi})A_{2}(q^{2})\right].$
(8)

The other semileptonic form factors coming from the dipole operator
$\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b$ can be defined as:

$\displaystyle\left<\phi(p_{\phi},\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b\right|B(p_{B_{s}})\right>=$
(9) $\displaystyle
4\epsilon_{\mu\nu\rho\sigma}\varepsilon^{\ast\nu}p^{\rho}q^{\sigma}T_{1}(q^{2})+2i\left[\varepsilon_{\mu}^{\ast}(m_{B_{s}}^{2}-m_{\phi}^{2})-(p_{B_{s}}+p_{\phi})_{\mu}(\varepsilon^{\ast}q)\right]T_{2}(q^{2})$
$\displaystyle+2i(\varepsilon^{\ast}q)\left[q_{\mu}-(p_{B_{s}}+p_{\phi})_{\mu}\frac{q^{2}}{m_{B_{s}}^{2}-m_{\phi}^{2}}\right]T_{3}(q^{2})~{}.$

As seen From Eqs. (II-9), we have to compute the form factors to obtain the
physical observables at hadronic level.The form factors are related to the
non-perturbative sector of QCD and can be evaluated only by using non-
perturbative methods. In the present work, we use light cone QCD sum rule
predictions for the form factors in which one-loop radiative corrections to
twist-2 and twist-3 contributions are taken into account. The form factors

$\displaystyle
F(q^{2})\in\\{V(q^{2}),A_{0}(q^{2}),A_{1}(q^{2}),A_{2}(q^{2}),A_{3}(q^{2}),T_{1}(q^{2}),T_{2}(q^{2}),T_{3}(q^{2})\\}~{},$

are fitted to the the following functions R31 ; R32 :

$F(q^{2})=\frac{F(0)}{1-a_{F}\frac{q^{2}}{m_{B_{s}}^{2}}+b_{F}(\frac{q^{2}}{m_{B_{s}}^{2}})^{2}},$
(10)

where the parameters $F(0)$, $a_{F}$ and $b_{F}$ are listed in the Table1.

$\begin{array}[]{|l|ccc|}\hline\cr&F(0)&a_{F}&b_{F}\\\ \hline\cr
A_{0}^{B_{s}\rightarrow\phi}&\phantom{-}0.382&1.77&\phantom{-}0.856\\\
A_{1}^{B_{s}\rightarrow\phi}&\phantom{-}0.296&0.87&-0.061\\\
A_{2}^{B_{s}\rightarrow\phi}&\phantom{-}0.255&1.55&\phantom{-}0.513\\\
V^{B_{s}\rightarrow\phi}&\phantom{-}0.433&1.75&\phantom{-}0.736\\\
T_{1}^{B_{s}\rightarrow\phi}&\phantom{-}0.174&1.82&\phantom{-}0.825\\\
T_{2}^{B_{s}\rightarrow\phi}&\phantom{-}0.174&0.70&-0.315\\\
T_{3}^{B_{s}\rightarrow\phi}&\phantom{-}0.125&1.52&\phantom{-}0.377\\\
\hline\cr\end{array}$ Table 1: The form factors for
$B_{s}\rightarrow\phi\,\ell^{+}\ell^{-}$ in a three–parameter fit R31 .

Using Eqs.(II-9), the matrix element of the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay can be written as follows:

$\displaystyle{\cal
M}(B_{s}\rightarrow\phi\ell^{+}\ell^{-})=\frac{G\alpha}{4\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}$
$\displaystyle\times\Bigg{\\{}\bar{\ell}\gamma^{\mu}(1-\gamma_{5})\ell\,\Big{[}-2B_{0}\epsilon_{\mu\nu\lambda\sigma}\varepsilon^{\ast\nu}p_{\phi}^{\lambda}q^{\sigma}-iB_{1}\varepsilon_{\mu}^{\ast}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+iB_{2}(\varepsilon^{\ast}q)(p_{B_{s}}+p_{\phi})_{\mu}+iB_{3}(\varepsilon^{\ast}q)q_{\mu}\Big{]}$
$\displaystyle~{}~{}~{}~{}+\bar{\ell}\gamma^{\mu}(1+\gamma_{5})\ell\,\Big{[}-2C_{1}\epsilon_{\mu\nu\lambda\sigma}\varepsilon^{\ast\nu}p_{\phi}^{\lambda}q^{\sigma}-iD_{1}\varepsilon_{\mu}^{\ast}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+iD_{2}(\varepsilon^{\ast}q)(p_{B_{s}}+p_{\phi})_{\mu}+iD_{3}(\varepsilon^{\ast}q)q_{\mu}\Big{]}\Bigg{\\}}~{},$

where

$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle(\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10})\frac{V}{m_{B_{s}}+m_{\phi}}+4(m_{B_{s}}+m_{s}){C}^{\rm
eff}_{7}\frac{T_{1}}{q^{2}}~{},$ $\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle(\tilde{C}^{\rm eff}_{9}-\tilde{C}^{\rm
eff}_{10})(m_{B_{s}}+m_{\phi})A_{1}+4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}(m_{B_{s}}^{2}-m_{\phi}^{2})\frac{T_{2}}{q^{2}}~{},$ $\displaystyle
B_{2}$ $\displaystyle=$ $\displaystyle\frac{\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10}}{m_{B_{s}}+m_{\phi}}A_{2}+4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}\frac{1}{q^{2}}\left[T_{2}+\frac{q^{2}}{m_{B_{s}}^{2}-m_{\phi}^{2}}T_{3}\right]~{},$
$\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle 2(\tilde{C}^{\rm
eff}_{9}-\tilde{C}^{\rm
eff}_{10})m_{\phi}\frac{A_{3}-A_{0}}{q^{2}}-4(m_{B_{s}}-m_{s}){C}^{\rm
eff}_{7}\frac{T_{3}}{q^{2}}~{},$ $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle B_{0}(\tilde{C}^{\rm eff}_{10}\rightarrow-\tilde{C}^{\rm
eff}_{10})~{},$ $\displaystyle D_{i}$ $\displaystyle=$ $\displaystyle
B_{i}(\tilde{C}^{\rm eff}_{10}\rightarrow-\tilde{C}^{\rm
eff}_{10})~{},~{}~{}~{}~{}(i=1,~{}2,~{}3).$

From the above equations for the differential decay width, we get the
following result:

$\displaystyle\frac{d\Gamma}{d\hat{s}}(B_{s}\rightarrow\phi\ell^{+}\ell^{-})=\frac{G^{2}\alpha^{2}m_{B_{s}}}{2^{14}\pi^{5}}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\lambda^{1/2}(1,\hat{r},\hat{s})v\Delta(\hat{s})~{},$
(12)

with

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{2}{3\hat{r}_{\phi}\hat{s}}m_{B_{s}}^{2}Re[-12m_{B_{s}}^{2}\hat{m_{l}}^{2}\lambda\hat{s}\\{(B_{3}-D_{2}-D_{3})B_{1}^{*}-(B_{3}+B_{2}-D_{3})D_{1}^{*}\\}$
$\displaystyle+12m_{B_{s}}^{4}\hat{m_{l}}^{2}\lambda\hat{s}(1-\hat{r}_{\phi})(B_{2}-D_{2})(B_{3}^{*}-D_{3}^{*})$
$\displaystyle+48\hat{m_{l}}^{2}\hat{r}_{\phi}\hat{s}(3B_{1}D_{1}^{*}+2m_{B_{s}}^{4}\lambda
B_{0}C_{1}^{*})$
$\displaystyle-16m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}\lambda(\hat{m_{l}}^{2}-\hat{s})\\{|B_{0}|^{2}+|C_{1}|^{2}\\}$
$\displaystyle-6m_{B_{s}}^{4}\hat{m_{l}}^{2}\lambda\hat{s}\\{2(2+2\hat{r}_{\phi}-\hat{s})B_{2}D_{2}^{*}-\hat{s}|(B_{3}-D_{3})|^{2}\\}$
$\displaystyle-4m_{B_{s}}^{2}\lambda\\{\hat{m_{l}}^{2}(2-2\hat{r}_{\phi}+\hat{s})+\hat{s}(1-\hat{r}_{\phi}-\hat{s})\\}(B_{1}B_{2}^{*}+D_{1}D_{2}^{*})$
$\displaystyle+\hat{s}\\{6\hat{r}_{\phi}\hat{s}(3+v^{2})+\lambda(3-v^{2})\\}\\{|B_{1}|^{2}+|D_{1}|^{2}\\}$
$\displaystyle-2m_{B_{s}}^{4}\lambda\\{\hat{m_{l}}^{2}[\lambda-3(1-\hat{r}_{\phi})^{2}]-\lambda\hat{s}\\}\\{|B_{2}|^{2}+|D_{2}|^{2}\\}],$

where $\hat{s}=q^{2}/m_{B_{s}}^{2}$,
$\hat{r}_{\phi}=m_{\phi}^{2}/m_{B_{s}}^{2}$ and
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$,
$\hat{m}_{\ell}=m_{\ell}/m_{B_{s}}$, $v=\sqrt{1-4\hat{m}_{\ell}^{2}/\hat{s}}$
is the final lepton velocity.

Having obtained the matrix element for the
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$, we can now calculate the
double–polarization asymmetries. For this purpose, we define the orthogonal
unit vectors $s^{\pm\mu}_{i}$ in the rest frame of leptons, where i=L,N or T
refer to the longitudinal, normal and transversal polarization directions,
respectively:

$\displaystyle s^{-\mu}_{L}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{L}^{\,-}\right)=\left(0,\frac{\vec{p}_{-}}{\left|\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{L}=\left(0,\vec{e}_{L}^{\,+}\right)=\left(0,\frac{\vec{p}_{+}}{\left|\vec{p}_{+}\right|}\right),$
$\displaystyle s^{-\mu}_{N}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{N}^{\,-}\right)=\left(0,\frac{\vec{p}_{\phi}\times\vec{p}_{-}}{\left|\vec{p}_{\phi}\times\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{N}=\left(0,\vec{e}_{N}^{\,+}\right)=\left(0,\frac{\vec{p}_{\phi}\times\vec{p}_{+}}{\left|\vec{p}_{\phi}\times\vec{p}_{+}\right|}\right),$
$\displaystyle s^{-\mu}_{T}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left(0,\vec{e}_{T}^{\,-}\right)=\left(0,\vec{e}_{N}^{\,-}\times\vec{e}_{L}^{\,-}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s^{+\mu}_{T}=\left(0,\vec{e}_{T}^{\,+}\right)=\left(0,\vec{e}_{N}^{\,+}\times\vec{e}_{L}^{\,+}\right).$

In the above equations $\vec{p}_{\mp}$ and $\vec{p}_{\phi}$ are the
three–momenta of the leptons $\ell^{\mp}$ and $\phi$ meson, respectively. Then
by Lorentz transformation these unit vectors are boosted from the rest frame
of leptons to the center of mass (CM) frame of leptons. Under this
transformation only the longitudinal unit vectors $s^{\pm\mu}_{L}$ change, but
the other two vectors remain unchanged. $s^{\pm\mu}_{L}$ in the CM frame of
leptons are obtained as:

$\displaystyle\left(s^{-\mu}_{L}\right)_{CM}=\left(\frac{\left|\vec{p}_{-}\right|}{m_{\ell}},\frac{E\vec{p}_{-}}{m_{\ell}\left|\vec{p}_{-}\right|}\right),~{}~{}~{}~{}~{}~{}~{}~{}\left(s^{+\mu}_{L}\right)_{CM}=\left(\frac{\left|\vec{p}_{-}\right|}{m_{\ell}},-\frac{E\vec{p}_{-}}{m_{\ell}\left|\vec{p}_{-}\right|}\right).$
(14)

The polarization asymmetries can now be calculated using the spin projector
${1\over 2}(1+\gamma_{5}\\!\\!\not\\!\\!s_{i}^{-})$ for $\ell^{-}$ and the
spin projector ${1\over 2}(1+\gamma_{5}\\!\not\\!\\!s_{i}^{+})$ for
$\ell^{+}$.

Considering the above explanations, we can define the double–lepton
polarization asymmetries as in Fukae :

$\displaystyle
P_{ij}(\hat{s})=\frac{\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},\vec{s}_{j}^{+})}-\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},\vec{s}_{j}^{+})\Bigg{)}}-\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},-\vec{s}_{j}^{+})}-\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},-\vec{s}_{j}^{+})\Bigg{)}}}{\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},\vec{s}_{j}^{+})}+\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},\vec{s}_{j}^{+})\Bigg{)}}+\displaystyle{\Bigg{(}\frac{d\Gamma}{d\hat{s}}(\vec{s}_{i}^{-},-\vec{s}_{j}^{+})}+\displaystyle{\frac{d\Gamma}{d\hat{s}}(-\vec{s}_{i}^{-},-\vec{s}_{j}^{+})\Bigg{)}}}~{},$
(15)

where $i,j=L,~{}N,~{}T$, and the first index $i$ corresponds to lepton while
the second index $j$ corresponds to antilepton, respectively. After doing the
straightforward calculation we obtain the following expressions for
$P_{ij}(\hat{s})$:

$\displaystyle P_{LL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{m_{B_{s}}^{2}}{3\hat{r}_{\phi}\hat{s}\Delta}\,\mbox{\rm
Re}\bigg{\\{}-24m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\hat{s}\lambda\Big{[}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
(16) $\displaystyle+$
$\displaystyle\\!\\!\\!12m_{B_{s}}^{3}\hat{m}_{\ell}\hat{s}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}^{2}\lambda(1+3v^{2})(\left|B_{0}\right|^{2}+\left|C_{1}\right|^{2})+12m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}^{2}\lambda\left|B_{3}-D_{3}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda(4-4\hat{r}_{\phi}-\hat{s})(B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})-32\hat{m}_{\ell}^{2}(\lambda+3\hat{r}_{\phi}\hat{s})B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda[\lambda+3(1-\hat{r}_{\phi})^{2}]B_{2}D_{2}^{\ast}-64m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{r}_{\phi}\hat{s}\lambda
B_{0}C_{1}^{\ast}$ $\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\lambda[\hat{s}-\hat{s}(\hat{r}_{\phi}+\hat{s})-3\hat{m}_{\ell}^{2}(2-2\hat{r}_{\phi}-\hat{s})](B_{1}B_{2}^{\ast}+D_{1}D_{2}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{s}\lambda[\lambda(1+3v^{2})-3(1-\hat{r}_{\phi})^{2}(1-v^{2})](\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})$
$\displaystyle+$
$\displaystyle\\!\\!\\!4[6\hat{m}_{\ell}^{2}(\lambda+6\hat{r}_{\phi}\hat{s})-\hat{s}(\lambda+12\hat{r}_{\phi}\hat{s})](\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})\bigg{\\}}~{},$
$\displaystyle P_{LN}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}}{2\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Im}\bigg{\\{}-4m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})B_{2}D_{2}^{\ast}$
(17) $\displaystyle+$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda
B_{2}B_{3}^{\ast}-2m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{[}B_{3}D_{2}^{\ast}+(B_{2}+D_{2})D_{3}^{\ast}\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}\hat{s}(1+3\hat{r}_{\phi}-\hat{s})\Big{(}B_{1}B_{2}^{\ast}-D_{1}D_{2}^{\ast}\Big{)}-4\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}\hat{s}(1-\hat{r}_{\phi}-\hat{s})(B_{1}+D_{1})(B_{3}^{\ast}-D_{3}^{\ast})$
$\displaystyle+$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})]\Big{(}B_{2}D_{1}^{\ast}+B_{1}D_{2}^{\ast}\Big{)}\bigg{\\}}~{},$
$\displaystyle P_{NL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-P_{LN}~{},$ (18) $\displaystyle P_{LT}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}v}{\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Re}\bigg{\\{}m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\left|B_{2}-D_{2}\right|^{2}$
(19) $\displaystyle-$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}\hat{r}_{\phi}\hat{s}\Big{(}B_{0}B_{1}^{\ast}-C_{1}D_{1}^{\ast}\Big{)}+m_{B_{s}}^{4}\hat{s}\lambda\hat{m}_{\ell}B_{2}B_{3}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{(}B_{2}D_{3}^{\ast}+B_{3}D_{2}^{\ast}-D_{2}D_{3}^{\ast}\Big{)}+\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})\left|B_{1}-D_{1}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!m_{B_{s}}\hat{s}(1-\hat{r}_{\phi}-\hat{s})\Big{[}-m_{B_{s}}\hat{m}_{\ell}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})](B_{1}-D_{1})(B_{2}^{\ast}-D_{2}^{\ast})\bigg{\\}}~{},$
$\displaystyle P_{TL}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{\pi
m_{B_{s}}^{2}v}{\hat{r}_{\phi}\Delta}\sqrt{\frac{\lambda}{\hat{s}}}\,\mbox{\rm
Re}\bigg{\\{}m_{B_{s}}^{4}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\left|B_{2}-D_{2}\right|^{2}$
(20) $\displaystyle+$
$\displaystyle\\!\\!\\!8m_{B_{s}}^{2}\hat{m}_{\ell}\hat{r}_{\phi}\hat{s}\Big{(}B_{0}B_{1}^{\ast}-C_{1}D_{1}^{\ast}\Big{)}+m_{B_{s}}^{4}\hat{s}\lambda\hat{m}_{\ell}B_{2}B_{3}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{m}_{\ell}\hat{s}\lambda\Big{(}B_{2}D_{3}^{\ast}+B_{3}D_{2}^{\ast}-D_{2}D_{3}^{\ast}\Big{)}+\hat{m}_{\ell}(1-\hat{r}_{\phi}-\hat{s})\left|B_{1}-D_{1}\right|^{2}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}\hat{s}(1-\hat{r}_{\phi}-\hat{s})\Big{[}m_{B_{s}}\hat{m}_{\ell}(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{m}_{\ell}[\lambda+(1-\hat{r}_{\phi})(1-\hat{r}_{\phi}-\hat{s})](B_{1}-D_{1})(B_{2}^{\ast}-D_{2}^{\ast})\bigg{\\}}~{},$
$\displaystyle P_{NT}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}v}{3\hat{r}_{\phi}\Delta}\,\mbox{\rm
Im}\bigg{\\{}4\lambda\Big{(}B_{1}D_{1}^{\ast}+m_{B_{s}}^{4}\lambda
B_{2}D_{2}^{\ast}\Big{)}-16m_{B_{s}}^{4}\hat{s}\lambda\hat{r}_{\phi}B_{0}C_{1}^{\ast}$
(21) $\displaystyle-$
$\displaystyle\\!\\!\\!4m_{B_{s}}^{2}\lambda(1-\hat{r}_{\phi}-\hat{s})\Big{(}B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast}\Big{)}\bigg{\\}}~{},$
$\displaystyle P_{TN}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-P_{NT}~{},$ (22) $\displaystyle P_{NN}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}}{3\hat{r}_{\phi}\Delta}\,\mbox{\rm
Re}\bigg{\\{}-24\hat{m}_{\ell}^{2}\hat{r}_{\phi}(\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})+16m_{B_{s}}^{4}\hat{s}\lambda\hat{r}_{\phi}v^{2}B_{0}C_{1}^{\ast}$
(23) $\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda\Big{[}-2B_{1}(B_{2}^{\ast}+B_{3}^{\ast}-D_{3}^{\ast})+2D_{1}(B_{3}^{\ast}-D_{2}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{3}\hat{m}_{\ell}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle+$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda(2+2\hat{r}_{\phi}-\hat{s})(\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})+6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}\lambda\left|B_{3}-D_{3}\right|^{2}$
$\displaystyle+$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\lambda[3(2-2\hat{r}_{\phi}-\hat{s})-v^{2}(2-2\hat{r}_{\phi}+\hat{s})](B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\lambda\Big{[}(3+v^{2})\lambda+3(1-v^{2})(1-\hat{r}_{\phi})^{2}\Big{]}B_{2}D_{2}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2[6\hat{r}_{\phi}\hat{s}(1-v^{2})+\lambda(3-v^{2})]B_{1}D_{1}^{\ast}\bigg{\\}}~{},$
$\displaystyle P_{TT}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{2m_{B_{s}}^{2}}{3\hat{r}_{\phi}\hat{s}\Delta}\,\mbox{\rm
Re}\bigg{\\{}8m_{B_{s}}^{4}\hat{r}_{\phi}\hat{s}\lambda\Big{[}4\hat{m}_{\ell}^{2}(\left|B_{0}\right|^{2}+\left|C_{1}\right|^{2})+2\hat{s}B_{0}C_{1}^{\ast}\Big{]}$
(24) $\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\hat{s}\lambda\Big{[}-2(B_{1}-D_{1})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{3}\hat{m}_{\ell}\hat{s}\lambda(1-\hat{r}_{\phi})\Big{[}2m_{B_{s}}\hat{m}_{\ell}(B_{2}-D_{2})(B_{3}^{\ast}-D_{3}^{\ast})\Big{]}$
$\displaystyle-$
$\displaystyle\\!\\!\\!6m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\hat{s}^{2}\lambda\left|B_{3}-D_{3}\right|^{2}+4m_{B_{s}}^{2}\hat{m}_{\ell}^{2}\lambda(4-4\hat{r}_{\phi}-\hat{s})(B_{1}B_{2}^{\ast}+D_{1}D_{2}^{\ast})$
$\displaystyle+$
$\displaystyle\\!\\!\\!2\hat{s}[6\hat{r}_{\phi}\hat{s}(1-v^{2})+\lambda(1-3v^{2})]B_{1}D_{1}^{\ast}$
$\displaystyle-$
$\displaystyle\\!\\!\\!2m_{B_{s}}^{4}\hat{m}_{\ell}^{2}\lambda[\lambda+3(1-\hat{r}_{\phi})^{2}](\left|B_{2}\right|^{2}+\left|D_{2}\right|^{2})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{2}\hat{s}\lambda[2-2\hat{r}_{\phi}+\hat{s}-3v^{2}(2-2\hat{r}_{\phi}-\hat{s})](B_{1}D_{2}^{\ast}+B_{2}D_{1}^{\ast})$
$\displaystyle-$
$\displaystyle\\!\\!\\!8\hat{m}_{\ell}^{2}(\lambda-3\hat{r}_{\phi}\hat{s})(\left|B_{1}\right|^{2}+\left|D_{1}\right|^{2})$
$\displaystyle-$
$\displaystyle\\!\\!\\!m_{B_{s}}^{4}\hat{s}\lambda\Big{[}(1+3v^{2})\lambda-3(1-v^{2})(1-\hat{r}_{\phi})^{2}\Big{]}B_{2}D_{2}^{\ast}\bigg{\\}}~{}.$

The analytical dependence of the double–lepton polarizations on the fourth
quark mass($m_{t^{\prime}}$) and the product of quark mixing matrix elements
($V_{t^{\prime}b}^{\ast}V_{t^{\prime}s}=r_{sb}e^{i\phi_{sb}}$) are studied in
the next section.

##  III Effects of the fourth-generation

As we mentioned in the introduction, the inclusion of the fourth-generation in
the Standard Model (SM4) does not lead to new operators in the ${\cal H}_{\rm
eff}$ and all Wilson coefficients receive additional terms as
$\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm SM4}_{i}$ either via virtual
exchange of the fourth-generation up-type quark $t^{\prime}$
$(C_{3},...,C_{10})$ or via using the unitarity of CKM matrix $(C_{1},C_{2})$
. Consequently, one can write the new effective Hamiltonian as:

${\cal H}_{\rm
eff}=-\frac{G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\sum_{i=1}^{10}C^{\rm
new}_{i}(\mu){\cal O}_{i}(\mu),$ (25)

where $C^{new}_{i}$ are:

$\displaystyle C^{\rm new}_{i}(\mu)$ $\displaystyle=$ $\displaystyle
C_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm
SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1\ldots 10.$ (26)

In the above equation, $\lambda_{f}=V_{fb}^{\ast}V_{fs}$ and
$\lambda_{t^{\prime}}$ can be parameterized as:

$\lambda_{t^{\prime}}=V_{t^{\prime}b}V_{t^{\prime}s}^{*}=r_{sb}e^{i\phi_{sb}}.$
(27)

Now by using the above effective Hamiltonian, we can reobtain the one-loop
matrix elements of $b\rightarrow s\ell^{+}\ell^{-}$ by replacing $C^{\rm
eff}_{i}(\tilde{C}^{\rm eff}_{i})$ with ${C^{\rm
eff\,new}_{i}}({{\tilde{C}}^{\rm eff\,\rm new}_{i}})$ in Eq.(2), where
${C^{\rm eff\,new}_{i}}$ and ${{\tilde{C}}^{\rm eff\,\rm new}_{i}}$are given
as:

$\displaystyle C^{\rm eff\,\rm new}_{i}(\mu)$ $\displaystyle=$ $\displaystyle
C^{\rm eff}_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}C^{\rm eff\,\,\rm
SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=7,$
$\displaystyle\tilde{C}^{\rm eff\,\rm new}_{i}(\mu)$ $\displaystyle=$
$\displaystyle\tilde{C}^{\rm
eff}_{i}(\mu)+\frac{\lambda_{t^{\prime}}}{\lambda_{t}}\tilde{C}^{\rm
eff\,\,\rm SM4}_{i}(\mu),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=9,10.$ (28)

Here the effective Wilson coefficients ${C}^{\rm eff\,\,\rm SM4}_{i}$ and
$\tilde{C}^{\rm eff\,\,\rm SM4}_{i}$ are defined in the same way as Eqs.(II)
by substituting $C_{i}$ with $C_{i}^{SM4}$. It is worth nothing that the
explicit forms of ${C}^{\rm eff\,\,\rm SM4}_{i}$ and $\tilde{C}^{\rm
eff\,\,\rm SM4}_{i}$ can also be found from the corresponding Wilson
coefficients in SM by replacing $m_{t}\rightarrow m_{t^{\prime}}$ R23 . Based
on the preceding explanations, in order to obtain the matrix element and the
double-lepton polarization asymmetries for
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay in the presence of the fourth-
generation, one should replace $C^{\rm eff}_{i}(\tilde{C}^{\rm eff}_{i})$ with
${C^{\rm eff\,new}_{i}}({{\tilde{C}}^{\rm eff\,\rm new}_{i}})$ in all
equations of the previous section.

The unitary quark mixing matrix is now $4\times 4$ which can be written in
terms of $6$ mixing angles and $3$ CP violating phases. The relevant elements
of this matrix for $b\rightarrow s$ transition satisfy the relation:

$\lambda_{u}+\lambda_{c}+\lambda_{t}+\lambda_{t^{\prime}}=0.$ (29)

Consequently, as required by GIM mechanism, the factor $\lambda_{t}C_{i}^{\rm
new}$ should be modified to $\lambda_{t}C_{i}$ when $m_{t^{\prime}}\rightarrow
m_{t}$ or $\lambda_{t^{\prime}}\rightarrow 0$. We can easily check the
validity of this condition by using Eq.(29):

$\displaystyle\lambda_{t}C_{i}^{\rm
new}=\lambda_{t}C_{i}+\lambda_{t^{\prime}}C_{i}^{\rm SM4}$ $\displaystyle=$
$\displaystyle-(\lambda_{u}+\lambda_{c})C_{i}+\lambda_{t^{\prime}}(C_{i}^{\rm
SM4}-C_{i})$ (30) $\displaystyle=$
$\displaystyle-(\lambda_{u}+\lambda_{c})C_{i}$ $\displaystyle=$
$\displaystyle\lambda_{t}C_{i}.$

The numerical analysis of the dependence of the double–lepton polarizations on
the fourth quark mass ($m_{t^{\prime}}$) and the product of quark mixing
matrix elements ($V_{t^{\prime}b}^{\ast}V_{t^{\prime}s}=r_{sb}e^{i\phi_{sb}}$)
are presented in the next section.

##  IV Results and Discussions

The main input parameters in the calculations are the form factors for which
we have chosen the predictions of light cone QCD sum rule method R31 ; R32 ,
as pointed out in section II. Besides the form factors, we use the other input
parameters as follow:

$\displaystyle
m_{B_{s}}=5.37\,\mbox{GeV}\,,\,m_{b}=4.8\,\mbox{GeV}\,,\,m_{c}=1.5\,\mbox{GeV}\,,\,m_{\tau}=1.77\,\mbox{GeV}\,,\,$
$\displaystyle
m_{\mu}=0.105\,\mbox{GeV},\,m_{\phi}=1.020\,\mbox{GeV}\,\,,\,|V_{tb}V_{ts}^{*}|=0.0385\,,\,\alpha^{-1}=129\,,\,$
$\displaystyle G_{f}=1.166\times
10^{-5}\,{\mbox{GeV}}^{-2}\,,\,\tau_{B_{s}}=1.46\times 10^{-12}\,s\,.$ (31)

In order to present a quantitative analysis of the double-lepton polarization
asymmetries, the values of fourth-generation parameters are needed.
Considering the experimental values of $B\longrightarrow X_{s}\gamma$ and
$B\longrightarrow X_{s}\ell^{+}\ell^{-}$ decays the value of the $r_{sb}$
parameter is restricted to the range $\\{.01-.03\\}$ for
$\phi_{sb}\sim\\{0^{\circ}-360^{\circ}\\}$ and
$m_{t^{\prime}}\sim\\{200-600\\}$ GeVArhrib:2002md ; Zolfagharpour:2007ez2 .
Using the $B_{s}$ mixing parameter $\Delta m_{B_{s}}$, a sharp restriction on
$\phi_{sb}$ has been obtained ($\phi_{sb}\sim 90^{\circ}$)Hou:2006jy .
Therefore in our following numerical analysis, the corresponding values of
above ranges are:
$r_{sb}=\\{.01,~{}.02,~{}.03\\},\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\},m_{t^{\prime}}=175\leq
m_{t^{\prime}}\leq 600$.

It is clear from the expressions of all nine double–lepton polarization
asymmetries that they depend on the momentum transfer $q^{2}$ and the new
parameters $(m_{t^{\prime}}$, $r_{sb}$, $\phi_{sb})$. Consequently, it may be
experimentally difficult to investigate these dependencies at the same time.
One way to deal with this problem is to integrate over $q^{2}$ and study the
averaged double-lepton polarization asymmetries. The average of $P_{ij}$ over
$q^{2}$ is defined as:

$\displaystyle\langle
P_{ij}\rangle=\frac{\displaystyle\int_{4\hat{m}_{\ell}^{2}}^{(1-\sqrt{\hat{r}_{\phi}})^{2}}P_{ij}\frac{d{\cal
B}}{d\hat{s}}d\hat{s}}{\displaystyle\int_{4\hat{m}_{\ell}^{2}}^{(1-\sqrt{\hat{r}_{\phi}})^{2}}\frac{d{\cal
B}}{d\hat{s}}d\hat{s}}~{}.$ (32)

We have used the above formula and depicted the dependency of $\langle
P_{ij}\rangle$ on the fourth-generation parameters in Fig.[1-7]. In the
following, we compare our results for $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$
decay with the results of Ref.Bashiry:2007tf  for $B\rightarrow
K\ell^{+}\ell^{-}$ decay. Since the overall behavior of $\langle
P_{ij}\rangle$ versus $m_{t^{\prime}},r_{sb}$ and $\phi_{sb}$ are almost the
same as that of $B\rightarrow K\ell^{+}\ell^{-}$ decay, we discuss the
differences of these two decays and some aspects which have not been discussed
in Ref.Bashiry:2007tf :

  * •

Figrue(1): Similar to the $B\rightarrow K\mu^{+}\mu^{-}$ decay, $\langle
P_{LL}\rangle$ is not sensitive to the fourth-generation quark parameters,
therefore the $\langle P_{LL}\rangle$ plots for $\mu$ channel have been
omitted. However, for the $\tau$ channel, the maximum deviation from SM is
about $50\%$ which can be seen at $m_{t^{\prime}}\sim 600GeV$. In comparison
with the results of Ref.Bashiry:2007tf , it is understood that the deviation
from SM for $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ is twice that of
$B\rightarrow K\tau^{+}\tau^{-}$ decay. Therefore, the magnitude of $\langle
P_{LL}\rangle$ in $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ compared with that in
$B\rightarrow K\tau^{+}\tau^{-}$decay has more chance to show the existence of
the fourth-generation.

  * •

Figrue(2): The value of $\langle P_{LN}\rangle_{max}$ for $\mu$ channel is
about 0.04 which is four times greater than that for $B\rightarrow K$ decay.
However, for $\tau$ channel such value is at most around $0.3$ which is
approximately equal to the maximum value of $\langle P_{LN}\rangle$ for
$B\rightarrow K$ decay. Furthermore, in $\mu$ and $\tau$ channels by
increasing $r_{sb}$ and keeping the values of $\phi_{sb}$ fixed, the maximum
deviation from SM occurs at smaller values of $m_{t^{\prime}}$. This result
can be interesting since the maximum deviation from SM happens for
$r_{sb}\sim\\{0.02-0.03\\}$ and $m_{t^{\prime}}\sim\\{300-400\\}$GeV.
Therefore, the new generation has a chance to be observed around
$m_{t^{\prime}}\sim\\{300-400\\}$GeV. Our analysis shows that to measure the
effect of the fourth-generation in $\langle P_{LN}\rangle$, the $\tau$ channel
of $B_{s}\rightarrow\phi$ and $B\rightarrow K$ are more important than $\mu$
channel of these decays, knowing that in the $\mu$ channel the
$B_{s}\rightarrow\phi$ decay is more significant than the $B\rightarrow K$
decay.

  * •

Figrue(3): For $\mu$ channel, the magnitude of $\langle P_{LT}\rangle$ in
$B_{s}\rightarrow\phi$ decay changes at most about $80\%$ compared with the SM
prediction, while the maximum change in $B\rightarrow K$ decay reaches up to
$60\%$. For $\tau$ case, unlike $B\rightarrow K$ decay, the magnitude of
$\langle P_{LT}\rangle$ in $B_{s}\rightarrow\phi$ transition exhibits the
strong dependence on the fourth quark mass $(m_{t^{\prime}})$ and the product
of quark mixing matrix elements
$(|V_{t^{\prime}b}V_{t^{\prime}s}^{*}|=r_{sb})$. As seen from Fig.(3) the
maximum deviation from SM in $\tau$ channel is much more than that in $\mu$
channel. Therefore for establishing the fourth generation of quarks the
measurement of $\langle P_{LT}\rangle$ for
$B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ decay is more suitable than such
measurement for $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B\rightarrow
K\mu^{+}\mu^{-}$ decays .

  * •

Figrue(4): It is seen from Eqs.(19) and (20) that contrary to $B\rightarrow K$
decay, $P_{TL}$ is neither symmetric nor anti-symmetric under the exchange of
subscripts L and T which leads to different values for $P_{TL}$ and $P_{LT}$.
For $\mu$ channel, the magnitude of $\langle P_{TL}\rangle$ in
$B_{s}\rightarrow\phi$ decay changes at most about $40\%$ compared with the SM
prediction, while the maximum change in the case of $B\rightarrow K$ decay
reaches up to $60\%$. For $\tau$ case, unlike $B\rightarrow K$ decay, the
magnitude of $\langle P_{TL}\rangle$ in $B_{s}\rightarrow\phi$ transition
changes at most about $60\%$ compared with the SM prediction. Therefore, in
the measurement of $\langle P_{TL}\rangle$, the decays
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$($\ell=\mu,\tau$) and $B\rightarrow
K\mu^{+}\mu^{-}$ have the same significance for finding the new generation of
quarks.

  * •

Figrue(5): By comparing this figure with Fig.(2), one can find out that the
overall behavior of $\langle P_{TN}\rangle$ and $\langle P_{LN}\rangle$ are
the same. Furthermore, the magnitude of $\langle P_{TN}\rangle_{max}$ for
$\mu$ channel is about 0.22 which is four times smaller than that for
$B\rightarrow K$ decay and for $\tau$ channel such value is at most around
$0.0075$ which is approximately ten times smaller than $\langle
P_{TN}\rangle_{max}$ for $B\rightarrow K$ decay. Although the measurement of
$\langle P_{TN}\rangle$ in $B\rightarrow K\tau^{+}\tau^{-}$ decay for finding
the new generation is useful, such measurement in the decays
$B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ and $B\rightarrow K\mu^{+}\mu^{-}$ are
more significant.

  * •

Figrue(6): For both $\mu$ and $\tau$ channels in $B_{s}\rightarrow\phi$ decay,
the values of $\langle P_{NN}\rangle$ show stronger dependence on the fourth-
generation parameters $(m_{t^{\prime}},r_{sb},\phi_{sb})$ in comparison with
those in $B\rightarrow K$ decay. Furthermore, the situation for $\tau$ channel
is even more interesting than $\mu$ channel, since for fixed values of
$\phi_{sb}$ and $r_{sb}$, an increase in $m_{t^{\prime}}$ changes the sign of
$\langle P_{NN}\rangle$. So, for $B_{s}\rightarrow\phi$ decay, the study of
the magnitude and the sign of $\langle P_{NN}\rangle$ for $\tau$ channel and
the magnitude of this asymmetry in $\mu$ channel can serve as good tests for
discovering the new physics beyond the SM. It should also be mentioned that
for both $\mu$ and $\tau$ channels of $B\rightarrow K$ decay in general, and
specially for the $\mu$ channel, the deviation of $\langle P_{NN}\rangle$ from
SM can be a measurable quantity, even though it is less sensitive to the
fourth generation of quarks compered with that of $B\rightarrow\phi$ decay(see
Ref.Bashiry:2007tf ).

  * •

Figrue(7): A comparison between this figure and an analogous figure for
$B\rightarrow K\ell^{+}\ell^{-}$ shows that the values of $\langle
P_{TT}\rangle$ for both $\mu$ and $\tau$ channels in $B_{s}\rightarrow\phi$
decay have considerable dependency on the fourth-generation parameters
$(m_{t^{\prime}},r_{sb},\phi_{sb})$. Therefore, compared with $B\rightarrow
K\ell^{+}\ell^{-}$ decay in Ref.Bashiry:2007tf , the study of the magnitude of
$\langle P_{TT}\rangle$ in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ provides a
better opportunity to see the effect of the new physics beyond the SM.

Finally, let us briefly discuss whether it is possible to measure the lepton
polarization asymmetries in experiments or not. Experimentally, to measure an
asymmetry $\left<P_{ij}\right>$ of the decay with branching ratio $\cal{B}$ at
$n\sigma$ level, the required number of events (i.e., the number of
$B\bar{B}$) is given by the formula

$\displaystyle N=\frac{n^{2}}{{\cal B}s_{1}s_{2}\langle
P_{ij}\rangle^{2}}~{},$

where $s_{1}$ and $s_{2}$ are the efficiencies of the leptons. Typical values
of the efficiencies of the $\tau$–leptons vary from $50\%$ to $90\%$ for their
different decay modesR6016  and the error in $\tau$–lepton polarization is
estimated to be about $(10-15)\%$ R6017 . So, the error in measurement of the
$\tau$–lepton asymmetries is approximately $(20-30)\%$, and the error in
obtaining the number of events is about $50\%$.

Looking at the expression of $N$, it can be understood that in order to detect
the lepton polarization asymmetries in the $\mu$ and $\tau$ channels at
$3\sigma$ level, the minimum number of required events are (for the efficiency
of $\tau$–lepton we take $0.5$):

  * •

for $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$ decay

$\displaystyle N\sim\left\\{\begin{array}[]{ll}10^{6}&(\mbox{\rm
for}\left<P_{LL}\right>)~{},\\\ 10^{7}&(\mbox{\rm
for}\left<P_{NT}\right>,\left<P_{TN}\right>)~{},\\\ 10^{8}&(\mbox{\rm
for}\left<P_{LT}\right>,\left<P_{TL}\right>,\left<P_{NN}\right>,\left<P_{TT}\right>)~{},\\\
10^{9}&(\mbox{\rm for}\left<P_{LN}\right>,\left<P_{NL}\right>)~{},\\\
\end{array}\right.$ (37)

  * •

for $B_{s}\rightarrow\phi\tau^{+}\tau^{-}$ decay

$\displaystyle N\sim\left\\{\begin{array}[]{ll}10^{8}&(\mbox{\rm
for}\left<P_{LT}\right>,\left<P_{TL}\right>,\left<P_{NN}\right>,\left<P_{TT}\right>)~{},\\\
10^{9}&(\mbox{\rm
for}\left<P_{LL}\right>,\left<P_{LN}\right>,\left<P_{NL}\right>)~{},\\\
10^{12}&(\mbox{\rm for}\left<P_{NT}\right>,\left<P_{TN}\right>)~{}.\\\
\end{array}\right.$ (41)

Considering the above values for N and the number of $B\bar{B}$ pairs which
will be produced at LHC($\sim 10^{12}$), one can conclude that except
$\left<P_{NT}\right>$ and $\left<P_{TN}\right>$ for $\tau$ channel, all
double-lepton polarizations can be detected at the LHC.

In summary, in this paper we have presented the analyses of the double-lepton
polarization asymmetries in $B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay using
the SM with the fourth generation of quarks. We found out that these
asymmetries have strong dependency on the fourth-generation parameters which
can be detected at the LHC. We compared
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$decay with $B\rightarrow
K\ell^{+}\ell^{-}$ decay, and showed that the double-lepton polarizations of
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ are more sensitive to the fourth-
generation parameters and therefore by looking at
$B_{s}\rightarrow\phi\ell^{+}\ell^{-}$ decay, one has more chance to
investigate the correctness of the fourth generation of quarks hypothesis in
the near future.

##  V Acknowledgment

The authors would like to thank V. Bashiry for his useful discussions. Support
of Research Council of Shiraz University is gratefully acknowledged.

## References

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Figure 1: The dependence of the $\left<P_{LL}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\tau$ channel.

Figure 2: The dependence of the $\left<P_{LN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.

Figure 3: The dependence of the $\left<P_{LT}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.

Figure 4: The dependence of the $\left<P_{TL}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.

Figure 5: The dependence of the $\left<P_{TN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.

Figure 6: The dependence of the $\left<P_{NN}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.

Figure 7: The dependence of the $\left<P_{TT}\right>$ on the fourth-generation
quark mass $m_{t^{\prime}}$ for three different values of
$\phi_{sb}=\\{60^{\circ},~{}90^{\circ},~{}120^{\circ}\\}$ and
$r_{sb}=\\{0.01,~{}0.02,~{}0.03\\}$ for the $\mu$ and $\tau$ channels.