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\title{{\small {\bf The $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ Rare Decay in 
Two Higgs Doublet Model }}}
\author{T. BARAKAT\\
{\it {\small Civil Engineering Department, Near East University }}\\
{\it {\small Lefko\c{s}a, Mersin- Turkey }}}
\date{}
\begin{document}
\begin{titlepage}
\maketitle
\begin{abstract}
\baselineskip .9cm
The rare $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ decay is investigated in the 
context of type II two-Higgs-doublet model (2HDM). By using the existing 
experimental data of the branching ratio, restrictions on the free parameters 
of the model $m_{H}$, and $tan\beta$ are obtained: 
$0.7\leq tan\beta \leq 0.8$, and 
$500 GeV\leq m_{H} \leq 700GeV$. 
\end{abstract}
\end{titlepage}
\textwidth 16cm
\rightmargin 4.cm
\parskip .5cm
\baselineskip .9cm
\section{ Introduction}
\hspace{0.6cm} The determination of the elements of the Cabibbo-Kobayashi
-Maskawa matrix (CKM) is still an important issue in the flavor 
physics. The precise determination of the CKM parameters will be one of the 
most important progresses to understand the nature, physics of violated 
symmetry.  

In this sense the rare   $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ decay has 
attached a special interest due its sensitivity for the determination of 
CKM parameters, in particular the element $V_{td}$, and considered one 
of the cleanest decays from a theoretical standpoint. Moreover this decay 
occupies a special place, since for this decay the short distance effects 
dominated over the long distance effects. Over the years important 
refinements have been added in the theoretical treatment of 
$K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$,  long-distance contributions to the 
branching ratio were estimated quantitatively and could be shown to be 
essentially negligible as expected, two to three orders of magnitude smaller 
than the short distance contribution at the level of the branching ratio [1]. 
On the other hand, the calculation [2] of next-to-leading QCD correction 
reduced considerably the theoretical uncertainty due to the choice of the 
renormalization scales present in the leading order expression. Since the 
relevant hadronic matrix element of the operator 
$\bar{s}\gamma_{\mu}(1-\gamma_{5})d\bar{\nu}\gamma_{\mu}(1-\gamma_{5})\nu$ 
can be extracted in the leading decay 
$K^{+}\rightarrow \pi^{0} e^{+}\nu$. Conventionally, the 
Br($K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$) is related to the experimental 
well-known quantity Br($K^{+}\rightarrow \pi^{+}e^{+}\nu$)=0.0482, measured 
to ${1\%}$ accuracy. The resulting 
theoretical expression for Br($K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$) is only 
a function of the CKM parameters, the QCD scale $\Lambda_{\bar{M}s}$ and 
the quark masses $m_{t}$ and $m_{c}$. 

Experiments in the K meson system have entered new period. That the branching 
ratio of the flavor-changing neutral current (FCNC) 
process $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ has been recently measured, 
and it has turned out to be 
$Br(K^{+}\rightarrow \pi^{+}\nu\bar{\nu})=(4.2^{+9.7}_{-3.5}).10^{-10}$ [3]. 
The central value seems to be 4-6 times larger than the predictions of the 
Standard Model (SM) Br(0.6-1.5)x$10^{-10}$ [4]. Hence the rare  
$K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ decay is very sensitive 
to a new physics beyond the SM. Therefore, careful investigation 
of this decay can provide useful information about new physics [5]. For this 
reason different new physics scenarios for this decay will become very actual.
 
In this work the decay $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ is investigated 
in the framework of the two Higgs doublet model (2HDM). We estimate the 
constraints of the 2HDM parameters namely, $tan\beta$ and $m_{H}$, using 
the result coming from the measurement of [3]. Subsequently, this paper is 
organized as follows: In Section 2, the relevant effective 
Hamiltonian for the decay $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ in 2HDM is 
presented. Section 3, being devoted to the numerical analysis of our 
results; and finally a brief discussion of the results is given. 
\section{Effective Hamiltonian}
\hspace{0.6cm} In the Standared Model  (SM) the process 
$K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ is described at quark level by the 
$s\rightarrow d\nu\bar{\nu}$ transitions and received contributions from 
$Z^{0}$-penguin and box diagrams. The effective Hamiltonian relevant to  
$s\rightarrow d\nu\bar{\nu}$ transition is described by only one Wilson 
coefficient, and its explicit form is:   
\begin{eqnarray}
H_{eff}&=&\frac{G}{\sqrt{2}}\frac{ \alpha}{2\pi sin^{2}\theta_{w}}V^{*}_{ts}
V_{td}C_{11}^{SM}\bar{s}\gamma_{\mu}(1-\gamma_{5})d\bar{\nu}\gamma_{\mu}
(1-\gamma_{5})\nu,
\end{eqnarray}
where G is the Fermi coupling constant, $\alpha$ is the fine structure 
constant, $V_{tb}V^{*}_{ts}$ are products of Cabibbo-Kabayashi-Maskawa 
matrix elements and $x_{t}=\frac{m_{t}^{2}}{m_{w}^{2}}$. The resulting 
expression of Wilson coefficient $C_{11}$, which
was derived in the context of the SM including $O(\alpha_{s})$ corrections is 
[6,7]  
\begin{eqnarray}
C_{11}^{SM}=\left[X_{0}(x)+\frac{\alpha_{s}}{4\pi}X_{1}(x)\right],
\end{eqnarray} 
with 
\begin{eqnarray}
X_{0}(x)=\eta\frac{x}{8}\left[\frac{x +2}{x-1}+\frac{3x-6}
{(x-1)^{2}}lnx\right],
\end{eqnarray}    
\begin{eqnarray}
X_{1}(x)&=&\frac{4x^{3}-5x^{2}-23x}{3(x-1)^{2}}-
\frac{x^{4}+x^{3}-11x^{2}+x}{(x-1)^{3}}lnx+
\frac{x^{4}-x^{3}-4x^{2}-8x}{2(x-1)^{3}}ln^{2}x \nonumber \\
&+&\frac{x^{3}-4x}{(x-1)^{2}}Li_{2}(1-x)+8x\frac{\partial X_{0}(x)}{\partial x}
lnx_{\mu}.
\end{eqnarray}
Here $Li_{2}(1-x)=\int_{1}^{x}\frac{lnt}{1-t}dt$ and 
$x_{\mu}=\frac{\mu^{2}}{m_{w}^{2}}$ with $\mu=O(m_{t})$.

At $\mu=m_{t}$, the QCD correction for $X_{1}(x)$ term is very small (around 
$\sim 3\%$), and $\eta =0.985$ is the next-to-leading order (NLO) QCD 
correction to the t- exchange calculated in [2]. 

From the theoretical point of view, the transition 
$s\rightarrow d\nu\bar{\nu}$ is a very clean process as pointed out, since it 
is practically free from the scale dependence, and free from any long 
distance effects. In addition, the presence of a single operator 
governing the inclusive $s \rightarrow d \nu\bar{\nu}$ transition is an 
appealing property. The theoretical uncertainty within the SM is only related 
to the value of the Wilson coefficient $C_{11}$ due to the 
uncertainty in the top quark mass. In this work, we have considered possible 
new physics in $s \rightarrow d \nu\bar{\nu}$ only through the value of that 
Wilson coefficient.
 
 In this spirit, the process $s\rightarrow d\nu\bar{\nu}$ in the context of 
the 2HDM  has additional contributions from $Z^{0}$-penguin and box diagrams 
through H boson exchanges. The relevant Feynman diagrams correspond to the 
transition $s\rightarrow d\nu\bar{\nu}$ has been given in [8,9]. The first 
three diagrams describe the effective Hamiltonian in the SM, while the last 
three diagrams represent the 2HDM contributions 
to the $s\rightarrow d\nu\bar{\nu}$ transition, due to the charged Higgs 
boson exchanges. The interaction lagrangian between the 
charged Higgs bosons fields and fermions are then given by:
\begin{eqnarray}
  L=(2\sqrt{2}G_{F})^{1/2}\left[ tan\beta\bar{U_{L}} V_{CKM}M_{D}D_{R}+
ctg\beta\bar{U_{R}}M_{U}V_{CKM}D_{L}+tan\beta\bar{N_{L}}M_{E}E_{R}\right ] 
H^{+}+h.c.
\end{eqnarray}
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.
$M_{D}$, $M_{U}$, and $M_{E}$ are the mass matrices for the down quarks, up 
quarks, and charged leptons respectively. $V_{CKM}$ is the Cabibbo-Kobayashi-
Maskawa matrix. $tan\beta$-is the ratio of the vacuum expectation values of 
the two Higgs doublets in 2HDM, and it is a free parameter of the model. 

 From eq.(5), it follows that the box diagrams contribution to the process 
$s\rightarrow d\nu\bar{\nu}$ in 2HDM are proportional to the charged lepton 
masses; and therefore, they are giving a negligible contribution.
So in this model, the transition $s\rightarrow d\nu\bar{\nu}$ in eq.(1) can 
only include extra contribution due to the charged Higgs interactions. Hence, 
the charged Higgs contribution modify only the value of the Wilson 
coefficient $C_{11}$ (see eq.(1)), and it does not induce any new operators 
(see also [8,9]):
\begin{eqnarray}
C_{11}^{ 2HDM}&=&-\frac{1}{8}xy ctg^{2}\beta \left\{\frac{1}{y-1}-
\frac{lny}{(y-1)^{2}}\right\},
\end{eqnarray}
where $x=\frac{m^{2}_{t}}{m^{2}_{W}}$  and  $y=\frac{m^{2}_{t}}{m^{2}_{H}}$.

As we noted earlier the QCD corrections practically do not change the value of 
$C_{11}$. If so, eq. (2) and eq.(6), are plugged in eq.(1), to obtain a 
modified effective Hamiltonian, which represents 
$s \rightarrow d \nu\bar{\nu}$ decay in 2HDM: 
\begin{eqnarray}
H_{eff}=\frac{G}{\sqrt{2}}\frac{\alpha}{2\pi sin^{2}\theta_{w}}V_{td}V^{*}_{ts}
[X_{tot}]\bar{s}\gamma_{\mu}(1-\gamma_{5})d\bar{\nu}\gamma_{\mu}
(1-\gamma_{5})\nu,  
\end{eqnarray}
where $X_{tot}= C_{11}^{SM}+C_{11}^{2HDM}$.

However, in spite of such theoretical advantages, it would be a very difficult 
task to detect the inclusive $s \rightarrow d \nu\bar{\nu}$  decay 
experimentally, because the final state contains two missing neutrinos and 
many hadrons. Therefore, only the exclusive channels are expected, namely 
$K^{+} \rightarrow \pi^{+} \nu\bar{\nu}$, are well suited to search for and 
constrain for possible "new physics" effects.  

In order to compute $K^{+} \rightarrow \pi^{+}\nu\bar{\nu}$ decay, we need the 
matrix elements of the effective Hamiltonian eq.(7) between the final and 
initial meson states. This problem is related to the non-perturbative sector 
of QCD and can be solved only by using non-perturbative methods. The matrix 
element  $<\pi^{+} \mid H_{eff}\mid K^{+}>$ has been investigated in a 
framework of different approaches, such as chiral perturbation theory [10], 
three point QCD sum rules [11], relativistic quark model by the light front 
formalism [12], effective heavy quark theory [13], and light cone QCD sum 
rules [14,15]. As a result, the hadronic matrix element for the 
$K^{+} \rightarrow \pi^{+} \nu\bar{\nu}$ can be parameterized in terms of form 
factors:
\begin{eqnarray}
<\pi \mid \bar{s}\gamma_{\mu}(1-\gamma_{5})d\mid K>
&=& f_{+}^{\pi^{+}}(q^{2})(p_{K}+p_{\pi})_{\mu}+f_{-}q_{\mu},
\end{eqnarray}   
where  $q_{\mu}=p_{K}-p_{\pi}$, is the momentum transfer. In our calculations
the form factor $f_{-}$ part do not give any contributions since its 
contribution $\sim m_{\nu}$=0.
After performing the mathematics and taking into 
account the number of light neutrinos $N_{\nu}=3$ the differential 
decay width is expressed as:  
\begin{eqnarray}
\frac{d\Gamma(K^{+} \rightarrow \pi^{+} \nu\bar{\nu})}{dq}&=&
\frac{G^{2}\alpha^{2}\eta^{2}}
{2^{8}\pi^{5}sin^{4}\theta_{w}}m_{K^{+}}^{3}\mid V_{tb}V^{*}_{ts}\mid^{2}
\lambda^{3/2}(1,r_{+},s)
\mid X_{tot} \mid^{2} \mid f_{+}^{\pi^{+}}(q^{2})\mid^{2},
\end{eqnarray}
where  $r_{+}=\frac{m^{2}_{\pi^{+}}}{m^{2}_{K^{+}}}$  and   
$s=\frac{q^{2}}{m^{2}_{K^{+}}}$.

Similar calculations for  $K^{+}\rightarrow \pi^{0} e^{0}\bar{\nu}$ 
lead to the following result:
\begin{eqnarray}
\frac{d\Gamma(K^{+} \rightarrow \pi^{0} e^{+}\bar{\nu})}{dq}&=&
\frac{G^{2}}{192\pi^{3}}\mid V_{us}\mid^{2} \lambda^{3/2}(1,r_{-},s)
m_{K^{+}}^{3}\mid f_{+}^{\pi^{0}}(q^{2})\mid^{2},
\end{eqnarray}
where  $r_{-}=\frac{m^{2}_{\pi^{0}}}{m^{2}_{K^{+}}}$, and 
$\lambda (1,r_{\pm},s)=1+r_{\pm}^{2}+s^{2}-2r_{\pm}s-2r_{\pm}-2s$ is the usual 
triangle function.
 In derivation of eq.(10) we neglect the electron mass, and we use the form 
factors $f_{+}^{\pi^{+}}=\sqrt{2}f_{+}^{\pi^{0}}$ which follows from isotopic 
symmetry. Using eq.(9) and eq.(10)
 one can relate the branching ratio of  $K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ 
to the well known measured decay $K^{+} \rightarrow \pi^{0} e^{0}\bar{\nu}$
branching ratio:
\begin{eqnarray}
B(B^{+}\rightarrow K^{+}\nu\bar{\nu})=k \left[\left( \frac{Im \lambda_{t}}
{\lambda^{5}}X_{tot} \right)^{2}+\left(\frac{Re \lambda_{c}}{\lambda}P_{0}
(K^{+})+\frac{Re \lambda_{t}}{\lambda^{5}}X_{tot}\right)^{2}\right],
\end{eqnarray}
where 
$k=r_{K^{+}}\frac{3\alpha^{2}B(K^{+} \rightarrow \pi^{0} e^{+}\nu)}{2\pi^{2} 
sin^{4}\theta_{w}}\lambda^{8}=4.11.~10^{-11}$.

Here $r_{K^{+}}=0.901$ summaries the isospin-breaking corrections which 
come from phase space factors due to the difference of masses of $\pi^{+}$ and 
$\pi^{0}$.

In derivation eq.(11) we have used the wolfenstein parametrization of the CKM 
matrix, in which each element is expanded as a power series in the small 
parameter \\
$\lambda =\mid V_{us}\mid$=0.22, $\lambda_{i}=V^{*}_{is}V_{id}$ and 
$P(K^{+})$ represent the sum of charm contributions to the two diagrams 
including the (NLO) QCD corrections [2]. At $m_{c}=1.3$ GeV, 
$\Lambda_{\bar{M}s}=0.325 $ GeV and at renormalization scale $\mu_{c}=m_{c}$ 
in [16] it is found that $P_{0}(K^{+})=0.4 \pm 0.06$. 

\section{Numerical Analysis}
In the numerical analysis, the following values have been used as input 
parameters:\\
$G_{F}=1.17{~}.10^{-5}~ GeV^{-2}$, $\alpha =1/137$, and $\lambda=0.22$
As we noted early we used the wolfeustein parametrization of CKM matrix 
elements. In this parametrization \\
$Im \lambda_{t}=A^{2}\lambda^{5}\eta$,~~~~
$Re \lambda_{c}=-\lambda (1-\lambda^{2}/2)$, and
$Re \lambda_{t}=-A^{2}\lambda^{5}(1-\rho)$.

The parameter A determines from $b\rightarrow c$ transition and its 
$ A=0.80 \pm 0.075$  [17]. The other two CKM parameters $\rho$ and $\eta$ 
are constrained by the measurements of $\mid V_{ub}/V_{cb} \mid$, 
$x_{d}(B^{0}_{d}- \bar{B^{0}_{d}})$  mixing, and  $\mid \epsilon \mid$ (the CP 
violation parameter in the kaon system). For typical values of the necessary 
input parameters of   $\rho$ and $\eta$ we have adopt the following two sets 
:\\
\begin{eqnarray}
\mbox{set I}: \left\{\begin{array}{l}    
             \rho =0.06 \\              
             \eta =0.35         
       \end{array}
            \right.
~~~~~~~~~~~~~~~~\mbox{set II}: \left\{\begin{array}{l}
             \rho =-0.25 \\
             \eta =0.3
            \end{array}
              \right..
\end{eqnarray}

The free parameters of the 2HDM model which we have used namely 
$tan\beta$ and $m_{H}$ are not arbitrary, but there are some constraints on 
them by using the existing experimental data. These constraints are usually 
obtained from $B^{0}-\bar{B}^{0}$, 
$K^{0}-\bar{K}^{0}$ mixings, $b\rightarrow s\gamma$ decay width, 
$R_{b}=\frac{\Gamma(z\rightarrow b\bar{b})}{\Gamma(z\rightarrow hadrons)}$, 
and semileptonic  $b\rightarrow c\bar{\nu_{\tau}}\tau$ decay which are given 
by [18] as
\begin{equation} 
0.7\leq tan\beta \leq 0.6 (\frac{m_{H}^{+}}{1GeV}),
\end{equation}
where as a lower bound for the charged Higgs mass  $m_{H}\geq 300$ GeV at 
$\mu=5$ scale has been estimated in 2HDM [19]. If these constraints are 
respected, an upper and lower bound for $ctg\beta$ is extracted:
\begin{eqnarray}
0.004\leq ctg\beta \approx 2.
\end{eqnarray}
In Figures 1 and 2, we represent the branching ratio of 
$K^{+} \rightarrow \pi^{+} \nu\bar{\nu}$ as a function of $ctg\beta$ 
for various values of $m_{H}$, and as a function of $m_{H}$ for various values 
of $ctg\beta$. For illustrative purposes we consider three values of 
$ctg\beta$, namely $ctg\beta=$1, 1.5 and 2 and we allow $m_{H}$ to range 
between 300 GeV and 1000 GeV, and then we consider three values of $m_{H}$, 
namely $m_{H}$=300, 500, 1000 GeV and we allow $ctg\beta$ to range between 0 
to 2.  It can be seen that for $ctg\beta=1$, the branching ratio (BR) for 
$K^{+} \rightarrow \pi^{+} \nu\bar{\nu}$ decay increases slowly 
with the increasing of $m_{H}$; whereas, for larger values of $ctg\beta$, 
the BR decreases at all values of $m_{H}$. Furthermore, when the $ctg\beta$ 
is increased the BR rapidly grows up. Therefore, it can be 
concluded that the main contribution to the decay width comes from the charged 
Higgs exchange diagrams (see [8,9]).

The question now is; what kind of restrictions on $tan\beta$ and $m_{H}$
can be obtained if the recent experimental result of 
$Br(K^{+}\rightarrow \pi^{+}\nu\bar{\nu})=(4.2^{+9.7}_{-3.5}).~10^{-10}$ [3] 
is respected that is:
\begin{equation} 
(0.7\leq BR^{exp.} \leq 13.9).~10^{-10},
\end{equation}
and whether or not it coincide with the restrictions given in [18]. For this 
aim, in Figure 3 we present the dependence of $tan\beta$ on  $m_{H}$ using 
both sets of values of $\rho$ and $\eta$. We see that when 
$300 GeV \leq m_{H} \leq 1 TeV$ it gives:
\begin{equation}
0.18\leq tan\beta \leq 0.5\pm 0.2~~~~~~~~  (set I),
\end{equation}
\begin{equation}
0.18\leq tan\beta \leq 0.8\pm 0.3 ~~~~~~~~~(set II). 
\end{equation}
If we use the lowest bound for $tan\beta=0.7$ (see eq.(13)) we  see that the 
set I predictions is ruled out and for set II  we have small room for 
$tan\beta$, namely from eq.(16) and from eq.(17) we have: 
\begin{equation} 
0.7\leq tan\beta \leq 0.8.
\end{equation}
If we increase a little bit the upper bound and if we put a lower 
value for $m_{H}$=500 GeV we can see that in this case
\begin{equation} 
0.7\leq tan\beta \leq 0.9.
\end{equation}

Using these results we can conclude that the mass of the charged Higgs boson 
must be lie in the interval 
\begin{equation} 
500 GeV \leq m_{H} \leq 700 GeV.
\end{equation}

In conclusion, using the experimental result of the branching ratio for 
$K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$ and the CLEO measurements on
$b\rightarrow s\gamma$ [20] we find new restrictions on the free parameters 
$tan\beta$ and $m_{H}$ of the 2HDM model. In summay it is found that the 
contribution of type II two-Higgs-doublet model to the branching ratio is 
exceed at most by $\sim 20\%$ from the standard model ones. 
\pagebreak
\begin{center}
Figure Captions
\end{center}
~~~~\\
Figure 1 : The dependence of the Br($K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$) 
             on  $ctg\beta$ at fixed values of $m_{H}$.\\
Figure 2  : The dependence of the Br($K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$)
             on $m_{H}$ at fixed values of $ctg\beta$.\\
Figure 3 : The dependence of $tan\beta$ on the charged Higgs boson mass 
          $m_{H}$. Curves (A, B), and (C, D) describes upper 
          and lower bound of the experimental values of the 
          Br($K^{+}\rightarrow \pi^{+}\nu\bar{\nu}$) for Set I and 
          Set II values of $\rho$ and $\eta$ respectively.
 
\pagebreak
\clearpage
\begin{thebibliography}{99}
\bibitem{R1} S. Fajfer, Nuovo Cimento 110 A (1997) 397; C. Q. Gang, I. J. Hsu 
             and Y. C. Lin, Phys. Lett. B355 (1995) 569; J.S. Littenberg, 
             Prog. Part. Nucl. Phys. (1989) 1; D. Rein and L. M. Sehgal, 
             Phys. Rev. D39 (1989) 3325; M. Lu and M. B. Wise, Phys. Lett. 
             B324 (1994) 461. 
\bibitem{R2} G. Buchalla and A. J. Buras, Nucl. Phys. B412 (1994) 106.
\bibitem{R3} S. Adler et al., E787 Collaboration, Phys. Rev. Lett. 79 (1997) 
              2204. 
\bibitem{R4} G. Buchalla and A. J. Buras and M. E. Lautenbacher, 
             Rev. Mod. Phys. 68 (1996) 1125.
\bibitem{R5} Y. Nir and M. P. Worah Prep.  (1997).
\bibitem{R6} T. Inami and C. S. Lim, Prog. Theor. Phys. 65 (1981) 287.
\bibitem{R7} G. Buchalla and A. J. Buras, Nucl. Phys. B400 (1993) 225.
\bibitem{R8} T. Barakat, IL Nuovo Cimento 110 A, (1997) 631. 
\bibitem{R9} T. Barakat, J. Phys. G , (1998) xxx. Accepted For Publication.  
\bibitem{R10} R. Casalbuoni et al., Phys. Reports 281 (1997) 145. 
\bibitem{R11} P. Colangelo, F. De Fazio, P. Santorelli, E. Scrimieri, Phys. 
             Rev. D53 (1996) 3672.
\bibitem{R12} W. Jaus and D. Wyler, Phys. Rev. D41 (1991) 3405; D. Melikhov, 
             N. Nikitin and S. Simula,  (1997). 
\bibitem{R13} W. Roberts, Phys. Rev. D54 (1996) 863.
\bibitem{R14} T. M. Aliev, A. $\ddot{O}$zpineci, M. Savci, Phys.Rev. D (1996) 
             4260.
\bibitem{R15} P. Ball and V. M. Braun, Phys. Rev. D55 (1997) 5561. 
\bibitem{R16} G. Buchalla and A. J. Buras, Phys. Rev. D54 (1996) 6782.
\bibitem{R17} A. Ali, Prep. DESY 96-106 (1996). 
\bibitem{R18} A. K. Grant, Phys. Rev. D51, (1995) 207.
\bibitem{R19}T. M. Aliev, G. Hiller and E. Iltan, Nucl. Phys. B515 (1998) 
              321. 
\bibitem{R20} R. Ammar et al., CLEO Collaboration, Phys. Rev. Lett 71 (1993) 
              674. 
\end{thebibliography}
\end{document}










 










































As we are already noted our aim in this work is to study 
$B \rightarrow K^{*}(\rho) \nu\bar{\nu}$ decay in framework of two Higgs 
doublet model. 

The interaction lagrangian of the charged Higgs bosons with fermions are 
given by 
\begin{eqnarray}
  L=(2\sqrt{2}G_{F})^{1/2}\left[ X\bar{U_{L}} V_{CKM}M_{D}D_{R}+
Y\bar{U_{R}}M_{U}V_{CKM}D_{L}\right ] H^{+}.
\end{eqnarray} 
In current literature usually are dissaced two type 2HDM: (a) when all 
fermions get mass via vacuum expectation value of one Higgs boson (Model I). 
In this case XY=-1; (b) when up fermions get mass with one Higgs doublet and 
down fermions second Higgs doublet (Model 2).

From phenomenological point of view Model 2 is more interesting, since its 
Higgs sector coincide with minimal supersymmetric extention of the Standard 
Model.

Note that X and Y are free parameters of 2HDM. Present phenomenological 
constraints on X and Y are disscussed in [ ].

In this work we will negligt the contributions due to the neutral Higgs boson 
exchange since their are small (probortional to the mass of the external 
particles).







machines appears possibility to measure B meson decay channels, for which

the future B-factories. The $B\rightarrow \nu\bar{\nu}\gamma$ 
is one example of such decays. This process represents an interesting model 
for the following reasons. Firstly, from the experimental point of view it 
has a very clear signature, i.e. "missing energy" and isolated photon. 
Secondly, from the theoretical point of view, the Branching ratio is dependent
 quadratically on leptonic decay constant $f_{B}$. Therefore, the 
investigation of this decay give us an alternative way for determining the 
leptonic decay constant $f_{B}$. Besides, an interesting peculiarity of this 
decay is that, the QCD corrections to this decay practically equal zero.
 
The rare flavor changing decay 
$B \rightarrow K^{*}\nu\bar{\nu}$ has attracted new interest in view of the 
planned experiments at the upcoming $KEK$ and SLAC B-factories. In these 
machines appears possibility to measure B meson decay channels, for which SM 
predicts low branching ratio.

Rare B-decays provide us essential information on the flavor structure of the 
Standard Model (SM) at loop level and on the origin of the CP violation. These 
decays are also very "rich laboratory" searching for new physics beyond SM via 
virtual effects of exotic particles.

In this work we study the $B \rightarrow K^{*}\nu\bar{\nu}$ in two Higgs 
doublet model.

In the standard model the process $B \rightarrow K^{*}\nu\bar{\nu}$ is 
described at the quark level by the $b \rightarrow s \nu\bar{\nu}$ transition 
and receives contribution from the Z-penguin and box diagrams, where dominant 
contributions come from intermediate top quarks. The effective Hamiltonian 
for $b \rightarrow s \nu\bar{\nu}$ transition described by only one Wilson 
coefficient, namely $C_{10}^{\nu}$ and its explicit form is   
\begin{eqnarray}
H_{eff}&=&\frac{G \alpha}{2\sqrt{2}\pi}C_{10}^{\nu}
V_{tb}V^{*}_{ts}\bar{s}\gamma_{\mu}(1-\gamma_{5})b\bar{\nu}\gamma_{\mu}
(1-\gamma_{5})\nu,
\end{eqnarray}
where G is the Fermi coupling constant, $\alpha$ is the fine structure 
constant, $V_{tb}V^{*}_{ts}$ element of Cabibbo-Kabayashi-Maskawa matrix. The 
Wilson coefficient $C_{10}^{\nu}$, has the following form, including 
$O(\alpha_{s})$ corrections:
\begin{eqnarray}
C_{10}^{\nu}=\left[X_{0}(t)+\frac{\alpha_{s}}{4\pi}X_{1}(t)\right]
/sin^{2}\theta_{w},
\end{eqnarray} 
where 
\begin{eqnarray}
X_{0}(x)= \frac{x}{8}\left[\frac{x +2}{x-1}+\frac{3x-6}
{(x-1)^{2}}lnx\right],
\end{eqnarray}    
and
\begin{eqnarray}
X_{1}(x)&=&\frac{4x^{3}-5x^{2}-23x}{3(x-1)^{2}}-
\frac{x^{4}+x^{3}-11x^{2}+x}{(x-1)^{3}}lnx+
\frac{x^{4}-x^{3}-4x^{2}-8x}{2(x-1)^{3}}ln^{2}x \nonumber \\
&+&\frac{x^{3}-4x}{(x-1)^{2}}Li_{2}(1-x)+8x\frac{\partial X_{0}(x)}{\partial x}
lnx_{\mu}.
\end{eqnarray}
Here $Li_{2}(1-x)=\int_{1}^{x}\frac{lnt}{1-t}dt$ and 
$x_{\mu}=\frac{\mu^{2}}{m_{w}^{2}}$ with $\mu=O(m_{t})$.

Note that explicit form (3) and (4)  where calculated in [1] and [2,3] 
correspondingly.

At $\mu=m_{t}$ QCD corrections for $X_{1}(t)$ term is very small, around $3\%$ 
and therefore the transition $b\rightarrow s\nu\bar{\nu}$ are 
free from the scale dependence.

This transition is also free from any long distance effects. So, from the 
theoretical point of view the inclusive $b \rightarrow s \nu\bar{\nu}$ is 
 very attractive and theoretical uncertainty is related only to the value of 
the Wilson cofficient  $C_{10}^{\nu}$ (due to the top quark mass uncertainty).

In this respect, the transition  $b \rightarrow s \nu\bar{\nu}$ is represents 
a clean process even in comparison with the $b \rightarrow s \gamma$, where 
long-distance contributions are to be present, although small [4]. Moreover, 
possible new physics, which we are interesting in this work, only modified the 
SM value of the Wilson coefficient $C_{11}$. 

However, in spite of such theoretical advantages, it would be very difficult 
to detect the $b \rightarrow s \nu\bar{\nu}$ inclusive decay in experements 
because the final state contains two missing neutrinos and (many) hadrons.

For this reason only exclusive channels, namely 
$B \rightarrow K^{*}(\rho) \nu\bar{\nu}$, may be studied in experiments. 

The planned experiments in this direction will be held at near future at KEK 
and SLAC B-factories, which may be test the decays, whose Branching ratios as 
low as $10^{-8}$ times the B-meson decay width.

Therefore, above mentioned decays are receives a special attention for 
checking prediction of the SM and looking for possible new physics beyond its.

As we are already noted our aim in this work is to study 
$B \rightarrow K^{*}(\rho) \nu\bar{\nu}$ decay in framework of two Higgs 
doublet model. 

The interaction lagrangian of the charged Higgs bosons with fermions are 
given by 
\begin{eqnarray}
  L=(2\sqrt{2}G_{F})^{1/2}\left[ X\bar{U_{L}} V_{CKM}M_{D}D_{R}+
Y\bar{U_{R}}M_{U}V_{CKM}D_{L}\right ] H^{+}.
\end{eqnarray} 
In current literature usually are dissaced two type 2HDM: (a) when all 
fermions get mass via vacuum expectation value of one Higgs boson (Model I). 
In this case XY=-1; (b) when up fermions get mass with one Higgs doublet and 
down fermions second Higgs doublet (Model 2).

From phenomenological point of view Model 2 is more interesting, since its 
Higgs sector coincide with minimal supersymmetric extention of the Standard 
Model.

Note that X and Y are free parameters of 2HDM. Present phenomenological 
constraints on X and Y are disscussed in [ ].

In this work we will negligt the contributions due to the neutral Higgs boson 
exchange since their are small (probortional to the mass of the external 
particles).

So, the charged Higgs interactions in ( ) can include extra contribution to 
the   $b \rightarrow s \nu\bar{\nu}$ transition in this model. Here we would 
like to note that the charged Higgs contribution modified only value of the 
Wilson coefficient $C_{{10}}^{\nu}$ (see eq.(1)), but not induce new operators:
\begin{eqnarray}
C_{10}^{\nu new}(Mw)&=&-\frac{1}{8}\mid Y \mid^{2}xy \frac{1}{(y-1)^{3}}
\left\{y^{2}-yx+1+(1-y)lny\right\} \nonumber \\
&=&-\frac{1}{8}\mid Y \mid^{2}xy \left\{\frac{1}{y-1}-\frac{lny}{(y-1)^{2}}
\right\}.
\end{eqnarray}
Here $x=\frac{m^{2}_{t}}{m^{2}_{W}}$, $y=\frac{m^{2}_{t}}{m^{2}_{H}}$.

As we noted early the QCD corrections practically does not changed value of 
$C_{10}^{\nu}$ at $m_{b}$: $C_{10}^{\nu}(M_{w})=C_{10}^{\nu}(m_{b})$.

Using equations (1), (2), and () for effective Hamiltonian, which responsible 
for $b \rightarrow s \nu\bar{\nu}$ decay in 2HDM we get
\begin{eqnarray}
H_{eff}=\frac{G\alpha}{2\sqrt{2}\pi}V_{tb}V^{*}_{ts}
[C_{11}^{SM}+C_{11}^{2HDM}]
\bar{s}\gamma_{\mu}(1-\gamma_{5})b\bar{\nu}\gamma_{\mu}
(1-\gamma_{5})\nu.  
\end{eqnarray}
As we noted the exclusive $b \rightarrow s \nu\bar{\nu}$ decay would be very 
difficult to detect in experimants and only exclusive channel 
$B \rightarrow K^{*} \nu\bar{\nu}$ may be studied in experiments.

In order to compute $B \rightarrow K^{*} \nu\bar{\nu}$ decay we need the 
matrix elements of the effective Hamiltonian ( ) between the final and initial 
meson states. This problem is related to the non-perturbative sector of QCD 
and it can be solved only by using non-pertarbative methods. The matrix 
element  $<K^{*} \mid H_{eff}\mid B>$ has been investigated in framework 
of different approaches, such as chiral theory [], three point QCD sum 
rules [], relativistic quark model [], effective heavy quark theory [], 
light cone QCD sum rules []. For the $B \rightarrow K^{*} \nu\bar{\nu}$, the 
hadronic matrix element can be written in terms of five form factors,
\begin{eqnarray}
<K^{*}(p_{2},\epsilon) \mid \bar{s}\gamma_{\mu}(1-\gamma_{5})b\mid B(p_{1})>
&=& -\frac{2V(q^{2})}{m_{B}+m_{K^{*}}}
\epsilon_{\mu\nu\rho\sigma}p_{2}^{\rho}q^{\sigma} \nonumber \\
&-&i \left[ \epsilon_{\mu}^{*}(m_{B}+m_{K^{*}})A_{1}(q^{2})-(\epsilon^{*}q)
(p_{1}+p_{2})_{\mu}\frac{A_{2}(q^{2})}{m_{B}+m_{K^{*}}} \right. \nonumber \\
&-& \left. q_{\mu}(\epsilon^{*}q)\frac{2m_{K^{*}}}{q^{2}}
(A_{3}(q^{2})-A_{0}(q^{2})) \right],
\end{eqnarray}   
with the condition $A_{3}(q^{2}=0)=A_{0}(q^{2}=0)$, and 
\begin{eqnarray} 
A_{3}(q^{2})=\frac{1}{2m_{K^{*}}}\left[(m_{B}+m_{K^{*}})A_{1}(q^{2})-
(m_{B}-m_{K^{*}})A_{2}(q^{2})\right].
\end{eqnarray}
Here $q=p_{1}-p_{2}$. In eq.( ), $\epsilon_{\mu}$, is the polorization 
4-vector of $K^{*}$ meson. Using eqs.(), () and after performing summation 
over  $K^{*}$ meson polarization and taking into account the number of light 
neutrions $N_{\nu}=3$ we have 
\begin{eqnarray}
\frac{d\Gamma(B \rightarrow K^{*} \nu\bar{\nu})}{ds}&=&
\frac{G^{2}\alpha^{2}\mid V_{tb}V^{*}_{ts}\mid^{2}}{2^{10}\pi^{5}} 
\lambda^{1/2}(1,r,s)m_{B}^{5}\mid C_{11} \mid^{2} 
\left\{8\lambda s\frac{V^{2}}{(1+\sqrt{r})^{2}} \right. \nonumber \\
&+& \left. \frac{1}{r}
\left[\lambda^{2}\frac{A_{2}}{(1+\sqrt{r})^{2}}+(1+\sqrt{r})^{2}(\lambda+12rs)
A_{1}^{2}-2\lambda (1-r-s)Re A_{1}A_{2}\right]\right\}.
\end{eqnarray}
In eq.( ) $\lambda (1,r,s)$ is the usual triangle function 
$\lambda (1,r,s)=1+r^{2}+s^{2}-2rs-2r-2s$ and 
 $r=\frac{m^{2}_{k^{*}}}{m^{2}_{B}}$, $s=\frac{q^{2}}{m^{2}_{B}}$.

From eq.( ) we see that decay width  $B \rightarrow K^{*} \nu\bar{\nu}$ 
contain three form factors V, $A_{1}$ and $A_{2}$. For estimating decay 
width we have used the results of the work [], i.e the monopole type 
form factors based on the light cone QCD sum rules ( In [] it was shown 
that predictions of light cone QCD sum rules more reliable than 3 point one). 
The values of the form factors at $q^{2}=0$ are:
\begin{eqnarray}
A_{1}^{B \rightarrow K^{*}}(0)=0.36\pm 0.05,
\end{eqnarray}
\begin{eqnarray}
A_{2}^{B \rightarrow K^{*}}(0)=0.40\pm 0.05,
\end{eqnarray}
\begin{eqnarray}
V^{B \rightarrow K^{*}}(0)=0.55\pm 0.08.
\end{eqnarray}
Note that all errors, which come from the uncetainties of the b-quark mass, 
the Borel parameter, wave functions and radiative corrections are taking into 
account quadrature.

After this theoretical background let us turn our attention to the numerical 
analysis. In numerical analysis we have used following values of inputs 
parameters:
$G_{F}=1.17{~}.10^{-5}~ GeV^{-2}$, $\alpha =1/137$, $m_{b}= 4.8$ GeV, 
$m_{B}= 5.28$ GeV, $\mid V_{tb}V^{*}_{ts}\mid$=0.045.

In figure 1 we presented the dependence of the differetial Branching ratio 
on s for different values $m_{H}$ and $ctg\beta$. 

In figure 2 we depicted the dependence of the Braching ratio on     
$m_{H}$ for fixed values of  $ctg\beta$.

In figure 3 we investigate the ratio of the Branching ratio in 2HDM and SM 
for different values of $m_{H}$ and $ctg\beta$.

For completness we also present the ratio of the Branching ratio in 2HDM, 
to the inclusive decay width, which corresponds to the SM value. It is well 
known that in SM (see [])
\begin{eqnarray}
B(B \rightarrow X_{s} \nu\bar{\nu})&=&
\frac{3\alpha^{2}\mid V_{tb}V^{*}_{ts}\mid^{2}}{(2\pi)^{2}sin^{4}
\theta_{w}}\frac{C_{10}^{2}}{\eta_{0}f(m_{c}/m_{b}}\bar{\eta}B
(B \rightarrow X_{c} l\nu),
\end{eqnarray}  
where the theoretical uncetainties related to the b-quark mass dependence 
disappears. In eq.( ) the factor 3 corresponds to the number of the light 
neutrions. Phase space factor $f(m_{c}/m_{b}) \simeq 0.44$, QCD correction 
factors $\eta_{0} \simeq 0.87 $ and \\
$\bar{\eta}=1+\frac{2\alpha_{s}(m_{b})}{3\pi}
(\frac{25}{4}-\pi^{2})$$\simeq 0.83$
 [] and experimental measurement 
$ B(B \rightarrow X_{c} l\nu)=10.14\%$ leades to the following results 
for inclusive decay rate in SM.
\end{document}












The theoretical and experimental investigations of the rare 
decays has been a subject of continous interest in the existing literature. 
The experimental observation of the inclusive $b\rightarrow s\gamma$
 and exclusive  $B\rightarrow K^{*}\gamma$ [1] decays stimulated the study of
 rare B meson decays on a new footing. These decays take place via 
flavor-changing neutral currents (FCNC) $
b\rightarrow s$ transitions which are absent in the Standard Model (SM) at 
tree level and appear only at the loop level. Thus the study of these decays 
can provide sensitive tests for investigating the 
structure of SM at the loop level, and they are represents the promising 
objects for establishing "new physics" beyond the standard model [2].    
 Moreover, the investigation of these rare 
decays permits precise determination of the fundamental parameters in the SM,
 which are poorly known at present, such as the Cabibbo- Kobayashi-Maskawa 
matrix elements [3], leptonic decay constants $f_{B_{s}}$,
$f_{B_{d}}$. 

Currently, the main interest on rare B-meson decays is focused on decays for 
which the SM predicts relatively large branching ratio and can be potentially
 measurable in the future B-factories. The $B\rightarrow \nu\bar{\nu}\gamma$ 
is one example of such decays. This process represents an interesting model 
for the following reasons. Firstly, from the experimental point of view it 
has a very clear signature, i.e. "missing energy" and isolated photon. 
Secondly, from the theoretical point of view, the Branching ratio is dependent
 quadratically on leptonic decay constant $f_{B}$. Therefore, the 
investigation of this decay give us an alternative way for determining the 
leptonic decay constant $f_{B}$. Besides, an interesting peculiarity of this 
decay is that, the QCD corrections to this decay practically equal zero.
 
As is well known, the SM predicts the Branching ratio of the above mentioned 
decay of the order $\sim 10^{-8}\div 10^{-9}$ and it will be quite measurable 
in future B-factories.  In general, the FCNC is very 
sensitive to extensions of the SM and provides a unique source of constraints
 on some new physics scenarios which predict large enhancement of 
$B\rightarrow \nu\bar{\nu}\gamma$ decay mode. 
(for review see [4] and references therein).

In the present work, we study the process 
$B_{s}(B_{d})\rightarrow \nu\bar{\nu}\gamma$ in the framework of the 
Two-Higgs doublet model. In Section 2, the relevant effective Hamiltonian 
for the $B_{s}\rightarrow \nu\bar{\nu}\gamma$
 is presented. In Section 3, using the light- cone QCD sum rules results 
for estimating form factors, we calculate the dependence of the  
Branching ratio on the photon energy. Then the Branching ratio of 
$B_{s}(B_{d})\rightarrow \nu\bar{\nu}\gamma$ is estimated and a brief 
discussion of the results is given. 
     
\section{Effective Hamiltonian.}
\hspace{0.6cm} Firstly, we consider the quark level process
$b\rightarrow q\nu\bar{\nu}$ (q=s,d). The relevant Feynman diagrams are
displayed in Fig. 1. The first three diagrams describe the SM and the last 
three represent the 2HDM contributions 
to the $b\rightarrow s\nu\bar{\nu}$ decay due to the charged $H^{-}$ boson
 exchange. 
In the calculations we will use the
so-called model II which appears in Two Higgs doublet or minimal
 Supersymmetric version of SM [5].\\

\vspace*{7.0cm}

 In this model the interaction lagrangian of
fermions with the charged Higgs fields is:
\begin{eqnarray}
  L=(2\sqrt{2}G_{F})^{1/2}\left[ tg\beta\bar{U_{L}} V_{CKM}M_{D}D_{R}+
ctg\beta\bar{U_{R}}M_{U}V_{CKM}D_{L}+tg\beta\bar{N_{L}}M_{E}E_{R}\right ] H^{+}
+h.c.
\end{eqnarray}
Here $H^{+}$ is the charged physical Higgs field. $U_{L}$, $D_{R}$ represents 
left handed up and right handed down quark fields. $N_{L}$, $E_{R}$  are 
left handed neutral and right handed charged leptons.
$M_{D}$, $M_{U}$ and $M_{E}$ are the mass matrices for the down, up quarks
and charged leptons respectively. $V_{CKM}$ is the Cabibbo-Kobayashi-Maskawa
 matrix.
 From eq.(1) it follows that the box diagrams contribution to the process 
$B\rightarrow s \nu\bar{\nu}$ in 2HDM is
proportional to the charged lepton mass and therefore gives a negligible
 contribution.

The SM contributions to  $b\rightarrow s\nu\bar{\nu}$  decay are calculated 
in [6] and in this work the 2HDM contributions are taken into account (
see also [7]). 
The resulting effective Hamiltonian is
\begin{eqnarray}
H_{eff}&=&\frac{G\alpha}{2\sqrt{2}\pi sin^{2}\theta_{w}}
V_{tb}V^{*}_{ts}\bar{s}\gamma_{\mu}(1-\gamma_{5})b\bar{\nu}\gamma_{\mu}
(1-\gamma_{5})\nu  \nonumber \\
&*&\frac{x}{8}\left\{\frac{x +2}{x-1}+\frac{3(x-2)}
{(x-1)^{2}}lnx- ctg^{2}\beta y \left[\frac{1}{y-1}-\frac{lny}{(y-1)^{2}}
\right]\right\},
\end{eqnarray}

where $x=\frac{m^{2}_{t}}{m^{2}_{W}}$,
$y=\frac{m^{2}_{t}}{m^{2}_{H}}$ and $sin^{2}\theta_{w}$ is the Weinberg angle.
$tg\beta$-is the ratio of the vacuum expectation values of the two Higgs
 doublets in 2HDM, and it is a free parameter of the model. The constraints on
 the $tg\beta$ are usually obtained from $B^{0}-\bar{B}^{0}$, 
$K^{0}-\bar{K}^{0}$ mixings, $b\rightarrow s\gamma$ decay width, semileptonic
 $b\rightarrow c\bar{\nu_{\tau}}\tau$ decay and given by [8] as
\begin{equation} 
0.7\leq tg\beta \leq 0.6 (\frac{m_{H}^{+}}{1GeV}).
\end{equation}

The next step is, starting from this effective Hamiltonian to calculate the 
$B_{s(d)}\rightarrow \nu\bar{\nu}\gamma$ decay at the hadronic level. Noting 
that the pure leptonic decay $B\rightarrow \nu\bar{\nu}$ is forbidden by the
helicity conservation. The process
 $b\rightarrow  s \nu\bar{\nu}\gamma$ is described by the diagrams in which the
 photon is attached at any charged lines in Fig.1. In this case no 
helicity suppression any longer exists. However, it is not necessary to 
calculate the contributions of all diagrams. 
Indeed, when the photon is radiated from internal
 charged lines, the contributions of such diagrams are suppressed by factor
$(\frac{m^{2}_{b}}{m^{2}_{W}(m^{2}_{H})})$  (see also [9]). 
The reason for that is, these operators are now of
dimension-8 instead of dimension-6. So, it is enough to consider the
contributions only of the diagrams, where the photon is emitted from initial 
b or s quarks.

In order to calculate the process $B_{s}\rightarrow \nu\bar{\nu}\gamma$
we need the following matrix element
\begin{eqnarray}
<\gamma \mid \bar{s}\gamma_{\mu}(1-\gamma_{5})b\mid B>.
\end{eqnarray}

 This matrix element can be parameterized in the following way
\begin{eqnarray}
<\gamma(q) \mid \bar{s}\gamma_{\mu}(1-\gamma_{5})b\mid B>&=&
\frac{\sqrt{4\pi\alpha}}{m_{B}^{2}}
\left[\epsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p^{\rho}q^{\sigma}f(p^{2})+
i(\epsilon_{\mu}(pq)-(\epsilon^{*}p)q_{\mu})g(p^{2})\right],
\end{eqnarray}
where $\alpha$ is the fine structure constant, $\epsilon_{\mu}$ and q are the 
polarization vector and four momenta of the photon, p is the transfer 
momentum, and $f(p^{2})$, $g(p^{2})$ are the parity conserving and the parity 
violation form factors. These form factors were calculated in the light cone 
QCD sum rules [9] and the results are:
\begin{eqnarray}
f(p^{2})=\frac{f(0)}{(1-p^{2}/m^{2}_{1})^{2}},
\end{eqnarray}
\begin{eqnarray}
g(p^{2})=\frac{g(0)}{(1-p^{2}/m^{2}_{2})^{2}},
\end{eqnarray}
where $f(0)=1$ GeV, $m_{1}=5,6$ GeV, $g(0)=0.8$ GeV, $m_{2}=6.5$ GeV.

After standard calculations, and using the above values it is not difficult 
to show that the decay width for $B\rightarrow \nu\bar{\nu}\gamma$ is 
\begin{eqnarray}
\frac{d\Gamma}{dE_{\gamma}}&=&
\frac{G^{2}\alpha^{3}\mid V_{tb}V_{ts}\mid^{2}}{(2\pi)^{4}m_{B}^{3}sin^{4}
\theta_{w}} E_{\gamma}^{3}
\left (m_{B}^{2}-2m_{B}E_{\gamma} \right ) 
\left [ \mid f\mid^{2} +\mid g\mid^{2} \right] \nonumber \\
&*&\left\{\frac{x}{8}\left[\frac{x +2}{x-1}+\frac{3(x-2)}
{(x-1)^{2}}lnx- ctg^{2}\beta y \left(\frac{1}{y-1}-\frac{lny}{(y-1)^{2}}
\right)\right]\right\}^{2}.
\end{eqnarray}
In the numerical calculations we have used the following values as input
 parameters:
$G_{F}=1.17{~}.10^{-5}~ GeV^{-2}$, $\alpha =1/137$, $m_{b}= 4.7$ GeV, 
$m_{B}= 5.28$ GeV, $\mid V_{tb}V^{*}_{ts}\mid$=0.04 and 
$\tau(B_{d})=1.56{~}.10^{-12} sec^{-1}$ . 
$\mid V_{tb}V^{*}_{td}\mid \simeq 0.01$
 and 
$\tau(B_{s})=1.34{~}.10^{-12} sec^{-1}$ [10].

In Fig.2. we present the dependence of the
 Branching ratio of $B_{s}\rightarrow \nu\bar{\nu}\gamma$ on $m_{H}$ for
 different values of $ctg\beta$. From this figure we see that for large values
of $ctg\beta$, (for example $ctg\beta=50$) 2HDM contribution to the Branching 
ratio exceeds to the order of two to three the SM one.

However, using the experimental restrictions on $tg\beta$ it is clear from 
this that $ctg\beta$ can not be a larger one, and the maximum possible value of
 $ctg\beta \simeq 1.5$. Using this value of $ctg\beta$ one can see that the
 2HDM contribution in practice is smaller than SM.  

In fig.3. we present the dependence of the Branching ratio on the photon energy
 $E_{\gamma}$, for $m_{H}$=100 GeV in (a), $m_{H}$=300 GeV in (b) at 
$tg\beta$=1.

We see that the spectrum is slightly asymmetric as a result of a balance 
between a highly asymmetric resonance type behavior given by the 
non-perturbative contributions and perturbative photon emission.

Finally, note that the results for  $B_{d}\rightarrow \nu\bar{\nu}\gamma$
can be easily obtained from  $B_{s}\rightarrow \nu\bar{\nu}\gamma$
 replacing $V_{tb}V^{*}_{ts}$ by $V_{tb}V^{*}_{td}$. Its obvious that the 
Branching ratio of $B_{d}\rightarrow \nu\bar{\nu}\gamma$
 is one order smaller than  $B_{s}\rightarrow \nu\bar{\nu}\gamma$ ones.

{\bf Acknowledgment}
The author thanks  T. M. Aliev for many useful discussions and 
 encouragement during this work.
\clearpage
\begin{center}
Figure Captions
\end{center}
~~~~\\
Figure 1. : The relevant Feynman diagrams, responsible for
    $b \rightarrow  s\nu \bar {\nu}$ decay. \\
Figure 2. : The dependence of the Branching ratio on the charged Higgs 
           boson mass at different values of ctg$\beta$.\\
Figure 3. : The dependence of the differential branching ratio on the photon
            energy $E_{\gamma}$ (GeV).     
\pagebreak
\begin{thebibliography}{99}
\bibitem{R1} M. S. Alam et. al. Phys. Rev. Lett. 74 (1995)
             2885; R. Ammar et. al. Phys. Rev. Lett. 71 (1993) 674. 
\bibitem{R2} J. L. Hewett, in : Proc. of the $21^{th}$ Annual SLAC Summer 
             Institute, ed. L. De Porcel and C. Dunwoodi SLAC- PUB. 6521.  
\bibitem{R3} Z. ligeti and M. Wise Phys. Rev. D53 (1996) 4937.   
\bibitem{R4} Y. Grossman, Z. Ligeti, and E. Nandi, Nucl. Phys. B465 (1996) 369.
\bibitem{R5} J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, The Higgs 
             Hunter's Guide (Addison-Wesley Pub. Comp. 1990); C. Alright, J. 
             Smith and S. H. Tye Phys. Rev. D15 (1977) 1958.   
\bibitem{R6} B. Grinstein, M. J. Savage and M. B. Wise, Nucl. Phys. B319 
             (1989) 271.
\bibitem{R7} Y. Okada, Y. Shimizu and M. Tanaka KEK Preprint 97-3 (1997).  
\bibitem{R8} Cai-Dian L$\ddot{u}$, Da-Xin Zhang Phys.Lett. B381 (1996) 348.
\bibitem{R9} T. M. Aliev, M. Savci, A. $\ddot{O}$zpineci, Phys. Lett. B393 
              (1997) 369. 
\bibitem{R10} Particle Data Group, Phys. Rev. D54 (1996).
\end{thebibliography}
\end{document}


