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\newcommand{\sm}{Standard Model }






\title{\bf The rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ in beyond Standard Model scenarios}
\author{Yusuf Din\c{c}er  \\
        Institute of Theoretical Physics, RWTH Aachen \\
        D-52056 Aachen, Germany \\
        email: dincer@physik.rwth-aachen.de}
\date{}
\maketitle




\begin{abstract}

In this paper, I study the branching ratio of the rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ in the framework of the 
Fourth-Generation Standard Model and the Two-Higgs-Doublet Model.
It is shown that this quantity is a very effective tool for establishing new physics beyond the Standard Model. It is observed that there is a region in the parameter
space in both extensions of the Standard Model for which the branching ratio show considerable departure from the Standard Model.







\end{abstract}



\newpage




\section{Introduction}

The rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ of the neutral $B_s$ meson into a $\nu \bar{\nu}-$pair and a 
radiatively emitted photon $\gamma$ is described in the \sm through the effective Lagrangian
\begin{equation}
{\cal L}_{\rm eff} = \frac{\alpha G_F}{\sqrt{2} \sin^2\Theta_W} V_{tb} V^*_{ts} \, C^{\rm SM} \, 
\sum_{l=e,\mu,\tau} [\bar{\nu}_l \gamma_\mu(1-\gamma_5)\nu_l] \,
[\bar{s} \gamma^\mu (1-\gamma_5) b] \,,
\end{equation}
which is determined by one Wilson coefficient having the Standard Model value
\begin{equation}\label{sminami}
C^{\rm SM} = \frac{x_t}{8}
\left[ \frac{x_t+2}{x_t-1} + \frac{3(x_t-2)}{(x_t-1)^2} \ln x_t \right] \,, \; \qquad  x_t = \left( \frac{m_t}{m_W} \right)^2 \,.
\end{equation}
Here, $m_t$ and $m_W$ are the mass of the top quark and the $W$ boson, respectively, and $\alpha, G_F, \Theta_W$ are the fine structure 
constant, the Fermi constant and the Weinberg angle. The $V_{ij}$'s are the CKM matrix elements. 
The hadronic matrix element which is relevant to this process can be parametrized in terms of two gauge invariant and
independent form factors $f$ and $g$ as follows
\begin{equation}\label{formfactors2}
\left< \gamma \vert \bar{s} \gamma^\mu (1-\gamma_5) b \vert B \right> = \sqrt{4 \pi \alpha} \left[ \epsilon_{\mu \alpha \beta \sigma} 
\varepsilon^*_\alpha p_\beta q_\sigma 
\frac{g(p^2)}{m^2_{B_s}} + i \left( \varepsilon^{*}_\mu (pq) - (\varepsilon^{*} p)q_\mu \right) \frac{f(p^2)}{m^2_{B_s}} \right] 
\,. 
\end{equation}
These form factors $f$ and $g$ have been calculated in QCD by Korchemski et al. \cite{korchemsky} and have the form 
\begin{equation*}\label{formfactors}
f \approx g \approx \frac{f_{B_s} \, m_{B_s} \, R_s}{3 \, E_\gamma} \,,
\end{equation*}
where $R_s$ is a parameter related to the light cone function of the $B_s$ meson, $E_\gamma$ is the photon energy and $f_{B_s}$ is the
decay constant of $B_s$.
The equality of $f$ and $g$ is valid up to corrections of order
$(\Lambda_{\rm QCD} / E_\gamma)^2$.
The normalization of these form factors has an uncertainty estimated to be $20 \%-30 \%$. 
However, it will be possible to reduce this uncertainty if 
the decay $B^+ \to e^+ \nu \gamma$ is detected since it depends on the same form factors $f$ and $g$. 
The photon energy spectrum can be calculated straightforwardly
\begin{equation*}
\frac{d \Gamma}{d x} (B_s \to  \nu \bar{\nu} \gamma) = 
\left| \frac{\alpha G_F}{\sqrt{2} \sin^2\Theta_W} \right|^2 \cdot x (1-x)^3 \cdot \left[ |f(x)|^2 + |g(x)|^2 \right] \,.
\end{equation*}
The same form factor combination $|f(x)|^2+|g(x)|^2$ appears in the photon energy spectrum of the reaction $B^+ \to e^+ \nu \gamma$. In
principle, therefore, a measurement of the branching ratio of $B_s \to \nu \bar{\nu} \gamma$ is a direct probe of the Wilson coefficient 
$C^{\rm SM}$ in the Standard Model. Hence, a
significant deviation from the Standard Model expectation for the Wilson coefficient may be a 
signal for new physics beyond the Standard Model.
Such scenarios include Fourth-Generation Standard Model, Two-Higgs-Doublet Model, MSSM, Technicolor or Left-Right Symmetric Models. 
In \cite{dincer}, the effects of a fourth generation 
top quark $t'$ onto the branching ratio were analysed. 
Since I will study the decay $B_s \to \nu \bar{\nu} \gamma$ in the Fourth-Generation Standard Model and the Two-Higgs-Doublet Model, let me summarize the
current status about possible existence of the fourth generation quarks and leptons and the charged Higgs particle.
%\begin{large}
%\vspace{10cm}
\begin{table}
\begin{tabular}{||c|c|c|c||}
\hline
\hline
{\rm {\bf Up-like \, quark}} & {\rm {\bf Down-like \, quark}} & {\rm
  {\bf Charged \, lepton}} & {\rm {\bf Neutral \, lepton}} \\
\hline 
u & d & e & $\nu_e$ \\
c & s & $\mu$ & $\nu_\mu$ \\
t & b & $\tau$ & $\nu_\tau$ \\
\hline
\hline
U & D & L & N \\
\hline
\hline
\end{tabular}
\caption{The elementary particles of the Standard Model and the fourth generation quarks and leptons}
\label{smparticles}
\end{table}
%\end{large}






The \sm of elementary particles contain three copies of generations differing only in the mass (see Table \ref{smparticles}). 
Until today, there is no
theoretical explanation why there are only three generations with light neutrinos. There are a lot of papers discussing possible
extra generations with new type of quarks and leptons. However, the properties of fourth generation quarks and leptons are restricted by
electroweak precision data.
The precision measurements of the invisible width of Z bosons prove that there exist only three light neutrinos, namely 
$\nu_e, \nu_\mu, \text{ and }\nu_\tau$ and that a fourth generation neutrino $N$ must have mass $m_N \gtrsim M_Z/2$ \cite{collab}. 
Precision measurements of other Z observables when compared with calculations of radiative 
corrections exclude the existence of the fourth generation of fermions in the case when the fourth neutrino N is heavy with mass
$M_N \gtrsim 100 \, {\rm GeV}$ \cite{collab}. However, as was shown, the radiative corrections become small if $m_N \sim m_Z/2$. Thus the
existence of a fourth generation with charged lepton and quarks being heavy with mass $M \gtrsim 100 \, {\rm GeV}$ 
but with 'light' neutrino of mass $M_N \sim 50 \, {\rm GeV}$ is still an open possibility.
The extra generations are excluded by the electroweak precision data, if all fermions are heavier than the $Z$ boson : $m \gtrsim  m_Z$.
From direct signature, one obtains:
$ m_N \gtrsim \frac{M_Z}{2} , m_L \gtrsim M_Z \text{ and } m_D,m_U \gtrsim M_Z $.
The first bound is a consequence of the fact that the channel $Z \to N \bar{N}$ is not observed. The second bound is a consequence of the
fact that the channel $e^+ e^- \to L^+ L^-$ is not observed up to $\sqrt{s} \sim 200 \, {\rm GeV}$. 
No sign of $q \bar{q} \to U \bar{U}, D \bar{D}$ in $p \bar{p}$ collisions at $\sqrt{s} \sim 2 \, {\rm TeV}$ results in the last bound. 
Virtual loop calculations give the following bound
$m_L - m_N \lesssim M_Z$ and $M_U - M_D \lesssim M_Z$.
So, finally, the masses of the fourth generation leptons and quarks are bounded as follows
\begin{equation*}
50 < m_N \lesssim 100 \qquad \text{ and } \qquad  100 \lesssim m_U, m_L, m_D \lesssim 200  \,.
\end{equation*}


The fourth generation quarks and leptons can influence the CKM matrix elements.
As well known, the magnitude of the elements of the CKM matrix are \cite{pdg}
\begin{equation*}
\left(
\begin{array}{ccc}
0.9742 \;\; {\rm to} \;\; 0.97575 &  0.219 \;\; {\rm to} \;\; 0.226    & 0.002 \;\; {\rm to} \;\; 0.005 \\
0.219 \;\; {\rm to} \;\; 0.225    &  0.9734 \;\; {\rm to} \;\; 0.9749  & 0.037 \;\; {\rm to} \;\; 0.043 \\
0.004 \;\; {\rm to} \;\; 0.014    &  0.035 \;\; {\rm to} \;\; 0.043    & 0.9990 \;\; {\rm to} \;\; 0.9993
\end{array}
\right) \,.
\end{equation*}
On page $12$ in \cite{pdg}, it is quoted that these matrix elements change if more than three generations exist:
\begin{equation*}
\left(
\begin{array}{cccc}
0.9722 \;\; {\rm to} \;\; 0.9748    &     0.216 \;\; {\rm to} \;\; 0.223     &  0.002 \;\; {\rm to} \;\; 0.005   &   \ldots   \\
0.209  \;\; {\rm to} \;\; 0.228     &     0.959 \;\; {\rm to}  \;\; 0.976    &  0.037 \;\; {\rm to} \;\; 0.043   &   \ldots   \\
0      \;\; {\rm to} \;\; 0.09      &     0     \;\; {\rm to}  \;\; 0.16     &  0.07  \;\; {\rm to} \;\; 0.993   &   \ldots   \\
\vdots                              &               \vdots                   &            \vdots                 & 
\end{array}
\right) \,.
\end{equation*}






The Standard Model of elementary particles needs a Higgs doublet to produce the mass of the known elementary particles. Although this Higgs particle is not detected yet,
there are a lot of papers discussing the possibility of more than one Higgs particle. In this paper, I only regard the Two-Higgs-Doublet Model. 
Although I will present the Two-Higgs-Doublet model in section \ref{2hdm}, let me mention in this introductory part that this extension of the Standard Model contain the two
free parameters $\tan \beta$ and the charged Higgs mass $m_{H^+}$. Actually, they are not arbitrary. They are restricted to (see section \ref{2hdm})
\begin{equation}\label{region1}
0.7 \le \tan \beta \le 0.51 \left( \frac{m_{H^+}}{{\rm GeV}} \right)  \,,
\end{equation}
and 
\begin{equation}\label{regiontwo}
200 {\rm GeV} \le m_{H^+} \le 1 {\rm TeV} \,.
\end{equation}














The aim of this work is to analyse measurable phenomenological effects of fourth generation quarks and leptons and the charged Higgs particle on the branching ratio of 
the rare radiative decay 
$B_s \to \nu \bar{\nu} \gamma$ of the neutral $B_s$ meson. 
These heavy extra generation quarks and leptons and the charged Higgs particle are not decoupled from low energy processes, giving therefore
potentially important effects on the branching ratio of this decay.





This paper is organized as follows. In section \ref{model}, using the effective theory of weak decays I study this decay $B_s \to \nu \bar{\nu} \gamma$ 
in the Fourth-Generation Standard Model. 
In section \ref{2hdm}, this decay is studied in the Two-Higgs-Doublet Model. 
In section \ref{results}, I present and discuss the results.













\section{$B_s \to \nu \bar{\nu} \gamma$ in the Fourth-Generation Standard Model}\label{model}
The rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ of the neutral $B_s$ meson is described by the effective Hamiltonian
\begin{equation}\label{effhamil}
{\cal H}_{\rm eff} = \frac{4 G_F}{\sqrt{2}} V_{tb} V^*_{ts} \sum^{10}_{i=1} C_i(\mu) {\cal O}_i(\mu) \,,
\end{equation}
where the full set of the operators ${\cal O}_i(\mu)$ and the corresponding expressions for the Wilson coefficients $C_i(\mu)$ in the 
\sm are given in \cite{buras}. 
No new operators
appear and clearly the full operator set is exactly the same as in the Standard Model. It is possible that the new quarks and leptons produce new
local operators, but in this paper, only their contributions to the Wilson coefficients are regarded, 
via virtual exchange of the fourth generation quarks and leptons. Thus the effective Lagrangian has the same form as in Eq. (1) with only one Wilson coefficient.
The Wilson coefficient in the Fourth-Generation Standard Model can be written as a sum of the Standard Model contribution (see Diagram \ref{diagramsm}) and three new
box diagrams caused by virtual exchange of the fourth generation quarks and leptons (see Diagrams \ref{diagramtL}, \ref{diagramUL} and \ref{diagramUl}). 
Introducing the new CKM matrix elements $V_{Ub}, V^*_{Us} \text{ and } V_{L\nu_l}$, the Fourth-Generation Wilson coefficient reads as follows
\begin{equation}\label{wilson}
\begin{split}
C^{\rm SM4} 
& =
C^{\rm SM} + \frac{ V_{Ub} V^{*}_{Us} }{ V_{tb} V^{*}_{ts} } C^{4G}(U,l)  +
\sum_{l=e,\mu,\tau} | V_{L\nu_l} |^2 C^{4G}(t,L) \\
& + 
\sum_{l=e,\mu,\tau}  |V_{L\nu_l}|^2 \frac{ V_{Ub} V^{*}_{Us} }{ V_{tb} V_{ts}^{*} } C^{4G}(U,L) \,.
\end{split}
\end{equation}
The Wilson coefficient $C^{4G}$ was analytically calculated in \cite{inamilin}
\begin{equation}
\begin{split}
C^{4G}(x_j,y_i) 
&=
-\frac{1}{8} \frac{y_i x_j}{y_i - x_j} \left( \frac{y_i-4}{y_i-1} \right)^2 \ln y_i + \frac{1}{8} 
\left[ \frac{x_j}{y_i-x_j} \left( \frac{x_j-4}{x_j-1} \right)^2 \right. \\
&+ 
\left. 1 + 3 \frac{1}{(x_j-1)^2} \right] x_j \ln x_j \\
&+ 
\frac{1}{4} x_j - \frac{3}{8} \left( 1 + 3 \frac{1}{y_i-1} \right)
\frac{x_j}{x_j-1} \,,
\end{split}
\end{equation}
where we have defined $ x_j \equiv m^2_{uj}/m^2_W \text{ and } y_i
\equiv m^2_{Li}/m^2_W $ with $L_i$ denoting the $i-th$ generation lepton 
and $u_j$ denotin the $j-$th generation up-like quark. In the fourth generation model, 
$x_4 = m^2_U/m^2_W \text{ and } y_4 = m^2_L /m^2_W$.
Because the measured charged current couplings $V_{e \nu_e},V_{\mu \nu_\mu} \text{ and } V_{\tau \nu_\tau}$ are nearly equal to unity (universality), one expects
$\sum\limits_{l=e,\mu,\tau} | V_{L \nu_l} |^2$ to be very small. Thus, 
the important correction comes from fourth generation $U-$quark and light neutrino (diagram \ref{diagramUl}),
\begin{equation*}
C^{\rm SM4} = C^{\rm SM} (t,0) + \frac{V_{Ub} V^*_{Us}}{V_{tb} V^*_{ts}} C^{\rm 4G} (U,0) \,.
\end{equation*} 




Hence, the effective Hamiltonian for this rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ simplifies to
\begin{equation}
{\cal H}_{\rm eff} = \frac{\alpha G_F}{\sqrt{2} \sin^2 \Theta_W} \cdot
C^{\rm SM4} [ \bar{s} \gamma_\mu(1-\gamma_5) b] \cdot [ \bar{\nu} \gamma^\mu (1-\gamma_5) \nu ] \,,
\end{equation}
with $C^{\rm SM4}$ as in Eq. (\ref{wilson}).
The corresponding matrix element is given by
\begin{equation}
{\cal M} =  \frac{\alpha G_F}{\sqrt{2} \sin^2 \Theta_W} \cdot C^{\rm SM4} \cdot [ \bar{\nu}(p_2) \gamma_\mu (1-\gamma_5) \nu(p_1) ] \cdot
\left< \gamma(q) \vert \bar{s} \gamma^\mu (1-\gamma_5) b \vert B(q+p) \right> \,,
\end{equation}
where $q^2 = (p_1+p_2)^2$ and $p_1$ and $p_2$ are the final neutrino four momenta. The matrix element 
$\left< \gamma(q) \vert \bar{s} \gamma^\mu (1-\gamma_5) b \vert B(q+p) \right>$ 
can be parametrized in terms of the two gauge invariant and independent form factors $f(p^2)$ and 
$g(p^2)$ as in Eq. \ref{formfactors2}.
%\begin{equation}
%\left< \gamma \vert \bar{s} \gamma^\mu (1-\gamma_5) b \vert B \right> = \sqrt{4 \pi \alpha} \left[ \epsilon_{\mu \alpha \beta \sigma} \varepsilon^*_\alpha p_\beta q_\sigma 
%\frac{g(p^2)}{m^2_B} + i \left( \varepsilon^{*}_\mu (pq) - (\varepsilon^{*} p)q_\mu \right) \frac{f(p^2)}{m^2_B} \right] \,. \nonumber
%\end{equation}
Here, $\varepsilon_\mu$ and $q_\mu$ stand for the polarization vector and momentum of the photon, $p+q$ is the momentum of the $B_s$ meson, 
$g(p^2)$ and $f(p^2)$ correspond
to parity conserved and parity violated form factors for the $B_s \to \nu \bar{\nu} \gamma$ decay. 
The branching ratio of this rare radiative decay can be calculated straightforwardly
\begin{equation}\label{branch}
Br^{\rm SM4} ( B_s \to \sum_{i=e,\mu,\tau} \nu_i \bar{\nu_i} \gamma ) = 3 \, \frac{\alpha (C^{\rm SM4})^2 m^5_B \tau_{B_s}}{256 \pi}    \,     I      \,,
\end{equation}
where the expression $I$ is given as the following integral
\begin{equation*}
I = \frac{1}{m^2_{B_s}}  \int\limits_0^1 dx \, (1-x)^3 \, x \, \left( f^2(x) + g^2(x) \right) \,.
\end{equation*}
Here $x=1-x_\gamma$ where $x_\gamma = 2 E_\gamma/M_{B_s}$ is the normalized photon energy.  
The branching ratio (\ref{branch}) versus $x_\gamma$ is plotted in Fig. \ref{branch1} and \ref{branch2}.  
The form factors $f(p^2)$ and $g(p^2)$ were taken universal form
desired in Ref. \cite{korchemsky} with a modification reflecting $B^*-$dominance
at low $x_\gamma$ \cite{dincersehgal}



\begin{equation*}
f(x_\gamma) = g(x_\gamma) = \frac{1}{3} \frac{f_{B_s}}{\bar{\Lambda}_s} \frac{1}{x_\gamma+\epsilon} \,,  \qquad 
( \epsilon = 2 \, (M_{B^*_s} - M_{B_s})/M_{B_s} ) \,.
\end{equation*}
Thereby, $f_{B_s}$ is the $B_s$ decay constant  and $\bar{\Lambda}_s = m_{B_s} - m_b$ will be taken to have the nominal value $0.5 \, {\rm GeV}$.  
Neclecting the fourth generation contributions, we obtain for the branching ratio in the Standard Model
\begin{equation*}
Br ( B_s \to \sum_{i=e,\mu,\tau} \nu_i \bar{\nu_i} \gamma ) = 6.44 \times 10^{-8}  \,.
\end{equation*}
This is in the same order as values obtained earlier.  
In \cite{lue} the branching ratio
for the decay $B_s \to \nu \bar{\nu} \gamma$ in SM was found to be of the order of $10^{-8}$. In \cite{Aliev2} 
the branching ratio in SM with three generations was calculated to 
$Br ( B_s \to \nu_i \bar{\nu_i} \gamma ) \approx 7.5 \times 10^{-8}$.
These papers used form factors of dipole form based on light cone QCD
sum rules.



We define the ratio
\begin{equation}
R^{\rm SM4} := \frac{Br^{\rm SM4} (B_s \to \nu \bar{\nu} \gamma)}{Br^{\rm SM} (B_s \to \nu \bar{\nu} \gamma)} \,.
\end{equation}





In order to obtain quantitative results we need the value of the fourth generation CKM matrix element $\vert V^*_{t's} V_{t'b} \vert$. Following \cite{Huang}, we will use the 
experimental results of the decays $Br(B \to X_s \gamma)$ and $Br(B \to X_c e \bar{v}_e)$ to determine the fourth generation CKM factor $V^*_{t's} V_{t'b}$. In order to 
reduce the uncertainties arising from $b$ quark mass, we consider the following ratio
\begin{equation*}
R = \frac{Br(B \to X_s \gamma)}{Br(B \to X_c e \bar{v}_e)} \,.
\end{equation*}
In leading logarithmic approximation this ratio can be written as
\begin{equation}\label{r}
R = \frac{ \vert V^*_{ts} V_{tb} \vert^2}{\vert V_{cb} \vert^2} 
\frac{6 \alpha \vert C^{\rm SM4}_7(m_b) \vert^2}{\pi f(\hat{m}_c) \kappa(\hat{m}_c)} \,,
\end{equation}
where the phase factor $f(\hat{m}_c)$ and ${\cal O}(\alpha_s)$ QCD correction factor $\kappa(\hat{m}_c)$ \cite{buras0} of $b \to c l \bar{\nu}$ are given by
\begin{eqnarray}\label{fkappa}
f(\hat{m}_c) &=& 1 - 8 \hat{m}_c^2 + 8 \hat{m}^6_c - \hat{m}^8_c - 24 \hat{m}_c^4 \ln ( \hat{m}^4_c) \,, \\
\kappa( \hat{m}_c) &=& 1 - \frac{2 \alpha_s(m_b)}{3 \pi} \left[ \left( \pi^2 - \frac{31}{4} \right) (1-\hat{m}_c)^2 + \frac{3}{2} \right] \,.
\end{eqnarray}
It is defined $\hat{m}_c = \frac{m^2_{c,pole}}{m^2_{b,pole}}$.
Solving Eq. (\ref{r}) for $V^*_{t's} V_{t'b}$ and taking into account (\ref{wilson}) and (\ref{fkappa}), we get
\begin{equation}
V^*_{t' s} V_{t' b}^\pm = \left[ \pm \sqrt{ \frac{\pi R \vert V_{cb} \vert^2 f( \hat{m}_c ) \kappa( \hat{m}_c )}{6 \alpha \vert V^{*}_{ts} V_{tb} \vert^2} } -
C_7^{\rm SM}(m_b) \right] \frac{V^{*}_{ts} V_{tb}}{C_7^{\rm SM}(m_b)} \,.
\end{equation}



%In \cite{huo}, it was estimated for $V^*_{t's} V_{t'd}$ :
%\begin{equation}
%-1.0 \times 10^{-4} \le V^*_{t'b} V_{t'd} \le 0.5 \times 10^{-4}
%\end{equation}









%\begin{figure}
%  \epsfig{file=plotratio}
%  \caption{The ratio $R^{\rm SM4}$ versus the fourth generation quark $m_U$ in the range $100 < m_U < 200$}
%\label{figureratio}
%\end{figure}




%\begin{figure}
%\subfigure[1.1:]{\epsfig{file=plot3d.eps}}
%\subfigure[1.2:]{\epsfig{file=plotDDy.eps}}\quad
%\subfigure[1.3:]{\epsfig{file=plotDDt.eps}}\quad  
%\caption{Fig.1.1:The function $C^{\rm 4G}(x,y)$ versus $x=(m_U/m_W)^2 \text{ and } y=(m_L/m_W)^2$ ; 
%    The irregularities in the region $x \sim y$  are numerical
%    fluctuations; Fig.1.2:$C^{\rm 4G} (x,y=0)$ versus $x$ for $y \to
%    0$; Fig.1.3: $C^{\rm 4G}(x=m^2_t/m^2_W , y)$ versus $y$ for $x \to 1$ }
%\label{fig:subfigures}
%\end{figure}


%\begin{figure}
%\subfigure[1.4]{\epsfig{file=plots}}
%\caption{$C^{\rm 4G}(x,y)$ versus $x=m^2_U/m^2_W$ for $L=100,150 \text{ and } 200$}
%\end{figure}






\begin{figure}
  \epsfig{file=plot3d}
  \caption{The function $C^{\rm 4G}(x,y)$ versus $x=(m_U/m_W)^2 \text{ and } y=(m_L/m_W)^2$ ; 
    The irregularities in the region $x \sim y$  are numerical fluctuations}
\label{figure3d}
\end{figure}


\begin{figure}
  \epsfig{file=plotDDy}
  \caption{$C^{\rm 4G} (x,y=0)$ versus $x$ for $y \to 0$}
\label{figureDDy}
\end{figure}



\begin{figure}
  \epsfig{file=plotDDt}
  \caption{$C^{\rm 4G}(x=m^2_t/m^2_W , y)$ versus $y$ for $x \to 1$}
\label{figureDDt}
\end{figure}

%\begin{figure}
%  \epsfig{file=plot3dratio}
%  \caption{The ratio $R^{\rm SM4}$ versus $m_U \text{ and } m_L$}
%\label{ratiosm4}
%\end{figure}



\begin{figure}
  \epsfig{file=plots}
  \caption{$C^{\rm 4G}(x,y)$ versus $x=m^2_U/m^2_W$ for $L=100,150 \text{ and } 200$}
\label{figureplots}
\end{figure}


\begin{figure}
\epsfig{file=plotbnu.eps}
\caption{The ratio $R^{\rm SM4}$ versus $m_U$;
the upper curve is for the solution $V^*_{t's} V^{(-)}_{t'b}$; the lower curve is for the solution $V^*_{t's} V^{(+)}_{t'b}$.}
\label{figureplotbnu}
\end{figure}










%The fourth generation Wilson coefficient $C^{\rm 4G}$ versus $x=(m_U/m_W)^2$ and $y=(m_L/m_W)^2$ is plotted in Fig. \ref{figure3d}.
%The Fig. \ref{figureDDy} show $C^{\rm 4G}$ versus $x$ for $y \to 0$. Fig. \ref{figureDDt}
%show $C^{\rm 4G}$ versus $y$ for $x \to 1$. Fig. \ref{figureplots} show $C^{\rm 4G}$ versus $x$ for $m_L = 100,150,200$. In Fig. \ref{figureplotbnu},
%$R^{\rm SM4}$ versus $m_U$ for $m_L=0$ is plotted.




























\section{\bf $B_s \to \nu \bar{\nu} \gamma$ in the Two-Higgs-Doublet Model}\label{2hdm}

In this section, I want to analyse the rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ within the Two-Higgs-Doublet-Model (2HDM). 
As in the case of the Fourth-Generation-Standard-Model, I discuss the effects of the charged Higgs particle of the 2HDM on the
branching ratio of this decay. So, first of all, let me summarize the main ideas of the Two-Higgs-Doublet-Model.






I briefly review some generalities of the 2HDM; for a more complete treatment, I refer the reader to Ref. \cite{grant} and references therein.
The Two-Higgs-Doublet model is essentially the $SU(2)_L \times U(1)_Y$ Standard Model, modified by the extension of the Higgs sector to 
include two scalar isodoublets $\Phi_1 \text{ and } \Phi_2$, which we write in component form as
\begin{equation*}
\Phi_i =
\left(
\begin{array}{c}
\phi^+_i \\
(\phi^0_i+i \xi^0_i) / \sqrt{2}
\end{array}
\right) \,, \qquad (i=1,2) \,.
\end{equation*}
The Yukawa couplings of the scalars and fermions are constrained by the requirement that there be no flavor changing neutral Higgs
interactions. As explained in Ref. \cite{grant}, a necessary and sufficient condition for the elimination of flavor changing neutral
interactions in the Higgs sector is that each quark of a given charge must receive its mass from at most one Higgs field. 
One finds then that there are two possible arrangements for the Yukawa couplings: one may either couple all of the fermions to a 
\textit{single} Higgs doublet (giving what is known as the "type I" model), or one may couple $\Phi_1$ to the right-handed down-type quarks,
and $\Phi_2$ to the right-handed up-type quarks. Here, we will consider the latter possibility. Imposing the discrete symmetry
$\Phi_2 \to - \Phi_2, (u,c,t)_R \to - (u,c,t)_R$ is enough to ensure that the Yukawa couplings have the correct form. The Yukawa
Lagrangian has the form
\begin{equation*}
{\cal L}_{\rm Yukawa} = \sum_{i=1}^3 \bar{l}_{i,L} \Phi_1 e_{i,R} + \sum_{i,j=1}^3 D_{ij} \bar{q}_{i,L} \Phi_1 d_{j,R} +
\sum_{i,j=1}^3  U_{ij} \bar{q}_{i,L} \Phi_2^c u_{j,R} \,,
\end{equation*}
where $l_{i,L} \text{ and } q_{i,L}$ are the left-handed lepton and quark doublets, U and D are the quark mass matrices, and 
$\Phi^c_2= - i \sigma_2 \Phi^*_2$. The Higgs potential, as in the standard model, has its minimum at non-zero values of the fields 
$\Phi_1 \text{ and } \Phi_2$. After applying the requirement that there be no flavor changing neutral interactions, one finds that the vacuum
expectation values (VEVs) can be chosen both real and positive:
\begin{equation*}
\left< \Phi_1 \right> = 
\left( 
\begin{array}{c} 
0 \\ 
v_1/\sqrt{2} 
\end{array} 
\right) \; \text{ and } \;
\left< \Phi_2 \right> =
\left(
\begin{array}{c} 
0 \\
v_2 / \sqrt{2}
\end{array}
\right) \,.
\end{equation*}
After diagonalizing the Higgs mass matrix, one finds that the fields $\phi^+_i$ mix to form a charged Nambu-Goldstone boson $G^+$ and a 
charged physical scalar $H^+$ of mass $m_+$:
\begin{equation*}
G^+ = \phi^+_1 \, \cos \beta    +  \phi^+_2 \,  \sin \beta    \qquad \; \text{ and } \; \qquad 
H^+ = - \phi^+_1 \, \sin \beta  + \phi^+_2 \, \cos \beta \,.
\end{equation*}
 Similarly, the imaginary parts of the neutral components $\xi^0_{1,2}$ mix to form a neutral Nambu-Goldstone boson $G^0$ and a CP-odd 
physical scalar $H^0_3$, again with mixing angle $\beta$:
\begin{equation*}
G^0 = \xi^0_1 \, \cos \beta  + \xi^0_2 \, \sin \beta   \qquad \; \text{ and } \; \qquad H^0_3 = - \xi^0_1 \, \sin \beta  + \xi^0_2 \, \cos \beta  \,.
\end{equation*}
Finally, the scalars $\phi^0_{1,2}$ mix to form a pair of neutral CP-even scalars, now with mixing angle $\alpha$:
\begin{equation*}
H^0_1 = \phi^0_1 \, \cos \alpha  + \phi^0_2 \, \sin \alpha  \qquad \; \text{ and } \; \qquad 
H^0_2 = - \phi^0_1 \, \sin \alpha  + \phi_2^0 \, \cos \alpha  \,.
\end{equation*}
The masses $m_{1,2}$ of $H^0_{1,2}$ obey $m_1 \le m_2$.
The mixing angle is the ratio of the vacuum expectation values of the two Higgs doublets 
\begin{equation*}
\tan \beta = \frac{v_2}{v_1} \,,
\end{equation*}
and it is a free parameter of the model.
The quantity $\tan \beta$ 
plays a central role in the theory because the Yukawa couplings are often proportional to either $\tan \beta$ or $\cot \beta$.


The interaction Lagrangian between the charged Higgs bosons fields and fermions is given by
\begin{equation}\label{lag}
\begin{split}
{\cal L} 
&=
(2 \sqrt{2} G_F)^{1/2} \, \left[ \tan \beta \, \bar{U}_L \, V_{\rm CKM} \, M_D \, D_R \right. \\
&+  \left.
\cot \beta \, \bar{U}_R \, M_U \, V_{\rm CKM} \, D_L + 
\tan \beta \, \bar{N}_L \, M_E \, E_R \right] \, H^+ + h.c. 
\end{split}
\end{equation}
where $h.c.$ means the hermitian conjugate of the first term in the Lagrangian.
Here, $H^+$ represents the charged physical Higgs field. $U_L$ and $D_R$ represent left-handed up and right-handed down quark fields.
$N_L$ and $E_R$ are left-handed neutral and right-handed charged leptons. $M_D, M_U, \text{ and } M_E$ are the mass matrices for the 
down quarks, up quarks, and charged leptons, respectively. $V_{\rm CKM}$ is the Cabibbo-Kobayashi-Maskawa matrix. 


From Eq.(\ref{lag}), it follows that the box diagrams contribution to the process $b \to s \nu \bar{\nu}$ in 2HDM are proportional to
the charged lepton masses; and therefore, they are giving a negligible contribution. Hence, the transition 
$b \to s \nu \bar{\nu}$ can only include extra contribution due to the charged Higgs interactions. The charged Higgs contribution (see Diagram \ref{higgsloop})
modify only the value of the Standard Model Wilson coefficient $C^{\rm SM}$, and it does not induce any new operators (see also \cite{barakat1,barakat2}).
\begin{equation}\label{wilson2hdm}
C^{\rm 2HDM} = - \frac{1}{8} x y \cot^2 \beta \left\{ \frac{1}{y-1} - \frac{\ln y}{(y-1)^2} \right\} \,,
\end{equation}
where $x= m^2_t / m^2_W \text{ and } y = m^2_t / m^2_{H^+}$. As we can obviously see from Eq. (\ref{wilson2hdm}), $C^{\rm 2HDM}$ contains only the two free parameters
of the 2HDM,
namely the mass $m_{H^+}$ of the charged Higgs particle and $\tan \beta$. In Fig. \ref{c2hdm4}, I have plotted 
the 2HDM Wilson coefficient $C^{\rm 2HDM}$ versus
$m_H \text{ and } \tan \beta$ in the allowed ranges (see Eq. (\ref{region1}) and (\ref{regiontwo})).
In the decoupling limit $m_{H^+} \to \infty$, the contribution of the charged Higgs boson in the loop diagram (see Diag. \ref{higgsloop}) should vanish.
Indeed, one can easily check from (\ref{wilson2hdm}) that
\begin{equation*}
\lim_{m_{H^+} \to \infty} C^{\rm 2HDM} = 0  \; \qquad \text{ and } \; \qquad
\lim_{\tan \beta \to \infty} C^{\rm 2HDM} = 0 \,,
\end{equation*}
so that we are only left with the Standard Model Wilson coefficient $C^{\rm SM}$.



We have to replace the Wilson coefficient of the Standard Model $C^{\rm SM}$ in the effective Hamiltonian in Eq. (\ref{effhamil})
through a new Wilson coefficient as follows 
\begin{equation*}
C^{\rm SM} \to C^{\rm SM} + C^{\rm 2HDM} \,,
\end{equation*}
where the 2HDM Wilson coefficient $C^{\rm 2HDM}$ is given in Eq. (\ref{wilson2hdm}).
The constraints on $\tan \beta$  are usually obtained from $B^0- \bar{B}^0$, $K^0 - \bar{K}^0$ mixings, $b \to s \gamma$ decay width,
$R_b = \frac{ \Gamma(z \to b \bar{b}) }{ \Gamma(z \to {\rm hadrons}) }$, and semileptonic $b \to c \bar{\nu}_\tau \tau$ decay which are 
given by \cite{grossman0} as
\begin{equation*}\label{region1}
0.7 \le \tan \beta \le 0.51 \left( \frac{m_{H^+}}{1 \, {\rm GeV}} \right) \,.
\end{equation*}
A lower bound for the charged Higgs mass comes from the virtual Higgs contributions to $b \to s \gamma$ as
\begin{equation*}\label{region2}
m_{H^+} \ge 200 \, {\rm GeV} \,.
\end{equation*}
Furthermore, there are no experimental upper bounds on the mass of the charged Higgs boson, but one generally expects to have
$m_H < 1 \, {\rm TeV}$ in order that perturbation theory remain valid \cite{veltman}. For large $\tan \beta$ the most stringent constraints on 
$\tan \beta$ and $m_{H^+}$ are actually on their ratio, $\tan \beta / m_{H^+}$. The current limits come from the measured branching ratio for the
inclusive decay $B \to X \tau \bar{\nu}$, giving $\tan \beta / m_{H^+} < 0.46 \, {\rm GeV^{-1}}$ \cite{aleph}, and from the upper limit on the
branching ratio for $B \to \tau \bar{\nu}$, giving $\tan \beta / m_{H^+} < 0.38 \, {\rm GeV^{-1}}$ \cite{acciari}.
We also define
\begin{equation}
R^{\rm 2HDM} := \frac{ {\rm Br}^{\rm 2HDM} (B_s \to \nu \bar{\nu} \gamma) }{ {\rm Br}^{\rm SM} (B_s \to \nu \bar{\nu} \gamma) } =
\left[ 1 + \frac{C^{\rm 2HDM}}{C^{\rm SM}} \right]^2 \,.
\end{equation}



\begin{figure}
  \epsfig{file=plot3}
  \caption{The Wilson coefficient $C^{\rm 2HDM}$ of the Two-Higgs-Doublet Model versus $\tan \beta$ for the charged Higgs mass $m_{H^+} = 200,300,400,500$}
\label{c2hdm1}
\end{figure}




\begin{figure}
  \epsfig{file=plot1}
  \caption{The Wilson coefficient $C^{\rm 2HDM}$ of the Two-Higgs-Doublet Model versus the charged Higgs mass $m_{H^+}$ for $\tan \beta = 0.7,1$}
\label{c2hdm2}
\end{figure}


\begin{figure}
  \epsfig{file=plot2}
  \caption{The Wilson coefficient $C^{\rm 2HDM}$ of the Two-Higgs-Doublet Model versus the charged Higgs mass $m_{H^+}$ for $\tan \beta = 10,50,100$}
\label{c2hdm3}
\end{figure}



\begin{figure}
  \epsfig{file=plot3dhiggs}
  \caption{The Wilson coefficient $C^{\rm 2HDM}$ of the Two-Higgs-Doublet Model versus the charged Higgs mass $m_{H^+}$ and  $\tan \beta$}
\label{c2hdm4}
\end{figure}




\begin{figure}
  \epsfig{file=plotratio}
  \caption{The ratio ${\rm Br}^{\rm 2HDM} (B_s \to \nu \bar{\nu} \gamma) / {\rm Br}^{\rm SM} (B_s \to \nu \bar{\nu} \gamma)$ versus the charged Higgs mass
    $m_{H^+}$ and $\tan \beta$}
\label{c2hdm5}
\end{figure}




\begin{figure}
  \epsfig{file=plotRATIO}
  \caption{The ratio $R^{\rm 2HDM}$ versus $\tan \beta$ for the charged Higgs mass $m_{H^+} = 200,300,400,500$}
\label{c2hdm6}
\end{figure}

\begin{figure}
  \epsfig{file=plotRATIO2}
  \caption{The ratio $R^{\rm 2HDM}$ versus the charged Higgs mass $m_{H^+}$ for $\tan \beta=0.7,1,10,50,100$}
\label{c2hdm7}
\end{figure}





\section{Results and Discussion}\label{results}

In order to analyse the ratios $R^{\rm SM4} , R^{\rm 2HDM}$ and the Wilson coefficients $C^{\rm SM4}, C^{\rm 2HDM}$ as a function of the free parameters $m_{t'}$ and 
$(m_{H^+},\tan \beta)$, respectively, I use the following input parameters
\begin{eqnarray*}
\alpha &=& 1/137 \,, e = \sqrt{4 \pi \alpha} \,,      \sin^2 \Theta_W = 0.2319 \,, G_F = 1.16639 \times 10^{-5} \, {\rm GeV^{-2}} \,, \\
m_W &=& 80.22 \, {\rm GeV} \,, m_b = 4.8 \, {\rm GeV} \,, m_t = 176 \, {\rm GeV} \,, m_{B_s} = 5.3 \, {\rm GeV} \,, \\
\vert V^*_{ts} V_{tb} \vert &=& 0.045 \,.
\end{eqnarray*}


The fourth generation Wilson coefficient $C^{\rm 4G}$ versus $x=(m_U/m_W)^2$ and $y=(m_L/m_W)^2$ is plotted in Fig. \ref{figure3d}.
The Fig. \ref{figureDDy} show $C^{\rm 4G}$ versus $x$ for $y \to 0$. Fig. \ref{figureDDt}
show $C^{\rm 4G}$ versus $y$ for $x \to 1$. Fig. \ref{figureplots} show $C^{\rm 4G}$ versus $x$ for $m_L = 100,150,200$. In Fig. \ref{figureplotbnu},
$R^{\rm SM4}$ versus $m_U$ for $m_L=0$ is plotted.





The plot in Fig. \ref{c2hdm5} 
shows $R^{\rm 2HDM}$ versus $200 \, {\rm GeV} \le m_{H^+} \le 500 \, {\rm GeV}$ and $0.7 \le \tan \beta \le 8$. $R^{\rm 2HDM}$ tends to unity in the
decoupling limit $\tan \beta \to \infty$ and $m_{H^+} \to \infty$. In Fig. \ref{c2hdm1}, the Wilson coefficient $C^{\rm 2HDM}$ versus $\tan \beta$ for 
$m_{H^+} = 200,300,400,500$ is plotted. The smaller $\tan \beta$ is, the greater $C^{\rm 2HDM}$ deviates from zero. The greater $\tan \beta$ is, the more $C^{\rm 2HDM}$
tends to zero. In Fig. \ref{c2hdm2} and \ref{c2hdm3}, $C^{\rm 2HDM}$ versus $m_{H^+}$ for $\tan \beta = 0.7,1,10,50,100$ is plotted. Fig. \ref{c2hdm4} shows
$C^{\rm 2HDM}$ as a function of $m_{H^+} \text{ and } \tan \beta$. In Fig. \ref{c2hdm6}, I have plotted $R^{\rm 2HDM}$ versus $\tan \beta$ for $m_{H^+} = 200,300,400,500$.
The smaller $m_{H^+}$ is, the greater $R^{\rm 2HDM}$ deviates from unity. $R^{\rm 2HDM}$ tends to unity for $m_{H^+} \to \infty$.



























\section{Conclusion}\label{conclusion}

In this work, I study the rare radiative decay $B_s \to \nu \bar{\nu} \gamma$ of the neutral $B_s$ meson into a $\nu \bar{\nu}-$pair and a radiatively emitted photon 
$\gamma$
in the framework of the Fourth-Generation Standard Model and the Two-Higgs-Doublet Model. 
I have observed that the ratios $R^{\rm SM4} \text{ and } R^{\rm 2HDM}$ deviate from unity for certain values of the parameters.
The investigation of this decay in other extensions of the Standard Model is left
for future work.






\section{Acknowledgement}

I want to thank Prof. L.M. Sehgal and P. Poulose for helpful discussions.






\newpage






%*******************************


\begin{thebibliography}{1}
\bibitem{korchemsky} G.P. Korchemsky, D. Pirjol and T.M. Yan, Phys. Rev. {\bf D61} (2000) 114510, 
\bibitem{dincer} Y.Din\c{c}er, Phys.Lett. {\bf B505} (2001) 89-93, 
\bibitem{collab} Mark II Collab., G.S. Abrams et al., Phys. Rev. Lett. {\bf 63} (1989) 2173; \\
  L3 Collab., B. Advera et al., Phys. Lett. {\bf B231} (1989) 509; \\
  OPAL Collab., I. Decamp et al., ibid., {\bf 231} (1989) 519; \\
  DELPHI Collab., M.Z. Akrawy et al., ibid., {\bf 231} (1989) 539
\bibitem{pdg} Particle Data Group
\bibitem{buras} A. J. Buras and M. M\"unz, \textsl{Phys. Rev.} {\bf D52} (1995) 186-195, 
\bibitem{inamilin} T. Inami and C.S. Lim, Prog. of Theo. Phys. Vol. 65, No. 1 (1981) 197-314
\bibitem{lue} C.-D. L\"u and Da-Xin Zhang, Phys. Lett. {\bf B381} (1996) 348-352, 
\bibitem{Aliev2} T.M. Aliev, A. \"Ozpineci and M. Savci, \textsl{Phys. Lett.} {\bf B393} (1997) 143-148, 
\bibitem{dincersehgal} Y. Din\c{c}er and L.M. Sehgal, \textit{Phys.Lett.} B521 (2001) 7-14, 
\bibitem{Huang} C.-S. Huang, W.-J. Huo and Y.-L. Wu, \textsl{Mod. Phys.} {\bf A14} (1999) 2453-2462,  \\
\bibitem{grant} A. K. Grant,Phys.Rev. {\bf D51} (1995) 207-217, 
\bibitem{barakat1} T. Barakat, IL Nuovo Cimento 110 A, (1997) 631
\bibitem{barakat2} T. Barakat, J. Phys. G, (1998) xxx. Accepted For Publication
\bibitem{grossman0} Y. Grossman and Z. Ligeti,Phys.Lett. {\bf B332} (1994) 373-380, 
\bibitem{veltman} M. Veltman, Acta Phys. Polon. {\bf B8} (1977) 475; \\
  B.W. Lee, C. Quigg, H.B. Thakcer, Phys. Rev. {\bf D16} (1977) 253;  \\
  M. Veltman, Phys. Lett. {\bf B70} (1977) 253
\bibitem{aleph} ALEPH Collaboration, contriburted to ICHEP, Warsaw, Poland, 25-31, July 1996, Pr. No: PA10-019
\bibitem{acciari} M. Acciari et al., Phys. Lett. {\bf D396} (1997) 327



\bibitem{Goto} T. Goto, Y.Okada and Y. Shimizu, \textsl{Phys. Rev.} {\bf D58} (1998) 094006, 


\bibitem{huo} W.-J. Huo, 

\bibitem{buras0} For revie see A.J. Buras, M.K. Harlander, "Heavy flavours", p58-201, Eds. A.J. Buras, M. Lindner, World Scientific, Singapore; A.Ali, Nucl. Phys. B,
  Proc. Suppl. 39BC, 408-425 (1995); S. Playfer and S. Stone, HEPSY 95-01
\bibitem{ammar} R. Ammar, et al., CLEO Collaboration, Phys. Rev. Lett. {\bf 71}, 674 (1993)
\bibitem{alam} M.S. Alam et al., CLEO Collaboration, Phys. Rev. Lett. {\bf 74}, 2885 (1995)
\bibitem{grinstein} B. Grinstein, M.J. Savage and M.B. Wise, Nucl. Phys. {\bf B319} (1989) 271; M. Misiak, Nucl. Phys. {\bf B393} 23; (E) ibid, {\bf B439} (1995) 461.
\bibitem{buchalla} G. Buchalla and A.J. Buras, Nucl. Phys. {\bf B400}, 225 (1993)
\bibitem{grossman} Y. Grossman, Z. Ligeti, and E. Nardi, preprint WIS-95-49-PH, 
\bibitem{ligeti} Z. Ligeti and M.B. Wise, preprint CALT-68-2029, 
\bibitem{campbell} B.A. Campbell and P.J. O'Donnell, Phys. Rev. {\bf D25}, 1989 (1982); A. Ali, "B decays", p67, Eds. S. Stone, World Scientific, Singapore
\end{thebibliography}



%******************************************



%***************************




%\begin{figure}
%  \epsfig{file=plot3}
%  \caption{$C^{\rm 2HDM}$ versus $\tan \beta$ for $m_H = 200,300,400,500$}
%\label{c2hdm1}
%\end{figure}




%\begin{figure}
%  \epsfig{file=plot1}
%  \caption{$C^{\rm 2HDM}$ versus $m_H$ for $\tan \beta = 0.7,1$}
%\label{c2hdm2}
%\end{figure}


%\begin{figure}
 % \epsfig{file=plot2}
  %\caption{$C^{\rm 2HDM}$ versus $m_H$ for $\tan \beta = 10,50,100$}
%\label{c2hdm3}
%\end{figure}



%\begin{figure}
%  \epsfig{file=plot3dhiggs}
%  \caption{$C^{\rm 2HDM}$ versus $m_H \text{ and } \tan \beta$}
%\label{c2hdm4}
%\end{figure}




%\begin{figure}
%  \epsfig{file=plotratio}
%  \caption{The ratio ${\rm Br}^{\rm 2HDM} (B_s \to \nu \bar{\nu} \gamma) / {\rm Br}^{\rm SM} (B_s \to \nu \bar{\nu} \gamma)$ versus
%    $mH$ and $\tan \beta$}
%\label{c2hdm5}
%\end{figure}




%\begin{figure}
%  \epsfig{file=plotRATIO}
%  \caption{The ratio versus $\tan \beta$ for $m_H = 200,300,400,500$}
%\label{c2hdm6}
%\end{figure}

%\begin{figure}
%  \epsfig{file=plotRATIO2}
%  \caption{The ratio versus $m_H$ for $\tan \beta=0.7,1,10,50,100$}
%\end{figure}





%***************

\begin{figure}\label{figuresm}
  \epsfig{file=diagramsm,height=6cm,width=8cm}
  \caption{The \sm diagram to $b \bar{s} \to \nu \bar{\nu}$ with virtual top quark $t$ and lepton $l$ exchange}
\label{diagramsm}
\end{figure}



\begin{figure}
  \epsfig{file=diagram2,height=6cm,width=8cm}
  \caption{The diagram to $b \bar{s} \to \nu \bar{\nu}$ with \sm top quark $t$ and fourth generation lepton $L$}  
\label{diagramtL}
\end{figure}




\begin{figure}
  \epsfig{file=diagram3, height=6cm, width=8cm}
  \caption{The diagram $b \bar{s} \to \nu \bar{\nu}$ with fourth generation up-like quark $U$ and lepton $L$}
\label{diagramUL}
\end{figure}


\begin{figure}
  \epsfig{file=diagram1, height=6cm, width=8cm}
  \caption{The diagram $b \bar{s} \to \nu \bar{\nu}$ with fourth generation up-like quark $U$ and \sm lepton $l$}
\label{diagramUl}
\end{figure}

\begin{figure}
  \epsfig{file=higgs, height=6cm, width=8cm}
  \caption{The diagram $b \bar{s} \to \nu \bar{\nu}$ with charged Higgs in the Two-Higgs-Doublet Model}
\label{higgsloop}
\end{figure}






\newpage


\begin{figure}
  \epsfig{file=brationunugamma}
  \caption{$\tau_{B_s} \cdot d \Gamma / d x_\gamma$ versus $x_\gamma$} 
\label{branch1}
\end{figure}

\begin{figure}
  \epsfig{file=brationunugamma2}
  \caption{$\tau_{B_s} \cdot d \Gamma / d x_\gamma$ versus $0 \le x_\gamma \le 1$} 
\label{branch2}
\end{figure}





\end{document}

