Instructions for Producing a Camera-Ready Manuscript %using Latex for Publication in Conference %Proceedings\footnote{\uppercase{T}his work is supported by etc, etc.}
C.~E. JIM and O. SPINAS\footnote{\uppercase{W}ork partially %supported by grant 2-4570.5 of the \uppercase{S}wiss %\uppercase{N}ational \uppercase{S}cience \uppercase{F}oundation.}
We present a systematic analysis of the decay \Bsg\ at the leading log within the framework of Supersymmetry without $R$-parity. We point out some new contributions in the form of bilinear-trilinear combination of $R$-parity violating (RPV) couplings that are enhanced by large $\tan \beta $. We also improve by a few orders of magnitude, bounds on several combinations of RPV parameters.
vaidya_pascos04.tex
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%\title{Instructions for Producing a Camera-Ready Manuscript
%using Latex for Publication in Conference 
%Proceedings\footnote{\uppercase{T}his work is supported by etc, etc.}}

\title{\Bsg\ in Supersymmetry without $R$-parity}

%\author{C.~E. JIM and O. SPINAS\footnote{\uppercase{W}ork partially
%supported by grant 2-4570.5 of the \uppercase{S}wiss 
%\uppercase{N}ational \uppercase{S}cience \uppercase{F}oundation.}}

\author{R.~D. VAIDYA\footnote{\uppercase{S}peaker at the conference.}~ and~ O.C.W. Kong}
\address{Department of Physics, \\
National Central University, \\ 
Chung-Li 32054 TAIWAN} 

\maketitle

\abstracts{
We present a systematic analysis of the decay \Bsg\ at the 
leading log within the 
framework of Supersymmetry without $R$-parity. We point out some new contributions in
the form of bilinear-trilinear combination of $R$-parity violating (RPV) couplings 
that are
enhanced by large $\tan \beta $. We also improve by a few orders of magnitude, bounds on 
several combinations of RPV parameters.}

\section{Introduction}
The large number of talks on supersymmetry (SUSY) in this conference provides 
ample proof of the
inadequacy of standard model (SM) as a complete theory, and the appeal of 
SUSY as a most 
popular candidate for the physics beyond SM. In our opinion, the minimal supersymmetry standard model
with conserved $R$-parity, lacks the much needed solution to neutrino mass problem which is naturally
addressed in models with $R$-parity violation (RPV). However, the large number of 
{\it a priori} arbitrary RPV couplings must be constrained from phenomenology in all possible ways.
In this talk we shall discuss the influence of RPV on the decay channel
\Bsg\ . Being loop mediated rare decay, it is sensitive to physics beyond SM. It has already been
well measured by CLEO, BELLE, ALEPH and BABAR and hence can be used to put upper bounds on RPV
couplings. The experimental world average\cite{world_average} is
$Br\left[\bar{B} \rightarrow X_s + \g \;(E_{\g} > 1.6 GeV) \right]_{\ssc \mathrm{SM}} 
= (3.57 \pm 0.30) \times 10^{-4}$. Within $1 \sigma$ this matches very well with the
SM prediction\cite{bur-mis-02} of
%$Br\left[\bar{B} \rightarrow X_s + \g \;(E_{\g} > 1.6 GeV) \right]_{\ssc \mathrm{SM}} 
$ (3.57 \pm 0.30) \times 10^{-4}$.
%\footnote{recently \cite{neubert-new} it has 
%been pointed out that singnificant perturbative uncertainties
%associated with the parameter $\Delta = m_b - 2E_0$ have been ignored in the previous
%works. The new
%estimate now stands at $Br\left[B \rightarrow X_s + \g \;
%(E_{\g} > 1.8 GeV) \right]_{\ssc \mathrm{SM}} = (3.44 \pm 0.53 \pm 0.35)\times 10^{-4}$.
%With the larger theoretical uncertainties in the SM prediction, 
%the case for new physics furhter strengthens.}.  
The good agreement between SM prediction and the
experimental number at $1 \s$ can be used to constrain the large number of {\it a priori}
arbitrary parameters of SUSY without $R$-parity.

There have been few studies on the process within the general framework of 
$R$-parity violation\cite{carlos,besmer}. 
Ref.\cite{carlos},  fails to consider the
additional 18 four-quark operators which, in fact, give the dominant contribution
in most of the cases. 
Ref.\cite{besmer} has considered a complete operator basis. However, we find their
formula for Wilson coefficient (WC) incomplete, and they do not report on the possibility
of a few orders of magnitude improvement on the bounds for certain combinations of
RPV couplings, as we present here\cite{tri-tri}. In fact, the particular type of 
contributions --- namely, the one from a combination of
a bilinear and a trilinear $R$-parity violating (RPV) parameters, we focused 
on \cite{bi-tri}, has not been studied in any detail before. Here we shall briefly report
the results. For the analytical details we refer the readers to\cite{tri-tri,bi-tri}.

We adopt an optimal phenomenological parametrization of  the full model 
Lagrangian -- the single single-{\it vev} parametrization.
It is essentially about choosing a basis
for Higgs and lepton superfields in which all the ``sneutrino" {\it vev} vanish. The formalism
is discussed at length in\cite{otto-gssm}.
%and its efficient application for the case of quark dipole-moment and
%$\m \rightarrow e \gamma$ see the references in\cite{tri-tri}}. 
\section{Formalism} The partonic transition \bsg\ is described by the magnetic penguin diagram.
Under the effective Hamiltonian approach, the corresponding WC 
of the standard $\mathcal{Q}_7$ operator has many RPV contributions at
the scale $M_{\!\ssc W}$. For example, we separate the contributions from
different type of diagrams as 
$C_7 = C^{\ssc W}_7 \,+ \,C^{\ssc \tilde g}_7 \, + \,C^{\ssc \chi^-}_7 \, 
+\, C^{\ssc \chi^{\mbox{\tiny 0}}}_7 \, +\, C^{\ssc \phi^-}_7 \,
+\, C^{\phi^{\mbox{\tiny 0}}}_7$ corresponding to W-boson, gluino, chargino, neutralino,
colorless charged-scalar and colorless neutral-scalar loops (for details please see\cite{tri-tri}). 
Apart from the 8 SM operators with additional contributions, 
we actually   have  to consider many more operators with admissible nonzero  WC 
at  $M_{\!\ssc W}$ resulting from the RPV couplings. These are the
chirality-flip counterparts $\cqt_7$ and $\cqt_8$ 
of the standard (chromo)magnetic 
penguins $\cq_7$ and $\cq_8$, and a whole list of 18 new relevant four-quark 
operators. For the lack of space, we list 8 important operators below.
\bea
\cqt_{9-11}
\!\! & =& \!\!
\left({\bar s}_{L\a}\,\g^{\m} \, b_{L\b}\right)
\, \left({\bar q}_{R\b}\,\g_{\m}\,q_{R\a}\right)\,,~q = d,s,b;  \nonumber\\
%{\widetilde {\cal Q}_{3,4}}
%\!\! & =& \!\!
%\left({\bar s}_{R\a}\,\g^{\m}b_{R\a,\b}\right)
%\,\sum_{i=u,c,d,s,b}\,\left({\bar q}_{Ri\b}\,\g^{\m}\,q_{Ri\b,\a}\right)\,;  \\
%{\widetilde {\cal Q}_{5,6}}
%\!\! & =& \!\!
% \left({\bar s}_{R\a}\,\g^{\m}b_{R\a,\b}\right)
%\,\sum_{i=u,c,d,s,b}\,\left({\bar q}_{Li\b}\,\g^{\m}\,q_{Li\b,\a}\right)\,;\\
\cqt_{9-13} 
\!\! & =& \!\!
 \left({\bar s}_{R\a}\,\g^{\m}\,b_{R\b}\right)
\, \left({\bar q}_{L\b}\, \g_{\m}\,q_{L\a}\right)\,, ~q = d,s,b,u,c;  
\eea
%\\
%{\widetilde {\cal Q}}_{1,2} & =&  \left({\bar s}_{R\a}\,\g^{\m}\,b_{R\a,\b}\right)
%\,\left({\bar c}_{R\b}\,\g_{\m}\,c_{R\b,\a}\right)\;, \\ 
%{\widetilde {\cal Q}}_{14,15} & =&  \left({\bar s}_{R\a}\,\g^{\m}\,b_{R\a,\b}\right)
%\,\left({\bar u}_{R\b}\,\g_{\m}\,u_{R\b,\a}\right)\;, \\ 
%{\widetilde {\cal Q}}_{16,17} & =&  \left({\bar s}_{R\a}\,\g^{\m}\,b_{R\a,\b}\right)
%\,\left({\bar d}_{R\b}\,\g_{\m}\,d_{R\b,\a}\right)\;.
%\eea
and six more operators from $\l''$ couplings\cite{tri-tri}. 
The interplay among the full set of 28 operators is what 
makes the analysis complicated. The effect of the QCD corrections proved to be 
very significant even for the RPV parts.


%We skip here the details involved in the evaluation of the various
%effective Wilson coefficients for the decay rate of \bsg\ and give only the
After the QCD running  of WC from 
scale $M_W$ to $m_b$, dictated by $28 \times 28$  anomalous dimension matrix, 
the effective WC
are given as (at leading log order)\cite{tri-tri}  :
\begin{eqnarray}
\label{c_mb}
C_7^{\mathrm{eff}}(m_b)&=& 
-0.351 \; C_{2}^{\mathrm{eff}}(M_{\!\ssc W})
+0.665 \;  C_{7}^{\mathrm{eff}}(M_{\!\ssc W})
+0.093 \; C_{8}^{\mathrm{eff}} (M_{\!\ssc W})\nonumber\\
&&-0.198 \; C_{9}^{\mathrm{eff}}(M_{\!\ssc W}) 
-0.198 \;  C_{10}^{\mathrm{eff}} (M_{\!\ssc W})
-0.178 \;  C_{11}^{\mathrm{eff}}(M_{\!\ssc W})\;, \nonumber \\
[0.5cm]
{\wtl C}_7^{\mathrm{eff}}(m_b)&=&
0.381 \;  {\wtl C}_{1}^{\mathrm{eff}}(M_{\!\ssc W})
+0.665  \; {\wtl C}_{7}^{\mathrm{eff}}(M_{\!\ssc W})
+0.093  \; {\wtl C}_{8}^{\mathrm{eff}}(M_{\!\ssc W})\nonumber\\
&&-0.198  \; {\wtl C}_{9}^{\mathrm{eff}}(M_{\!\ssc W}) 
-0.198 \;  {\wtl C}_{10}^{\mathrm{eff}}(M_{\!\ssc W})
-0.178  \; {\wtl C}_{11}^{\mathrm{eff}}(M_{\!\ssc W})\nonumber\\
&&+0.510 \;  {\wtl C}_{12}^{\mathrm{eff}}(M_{\!\ssc W})
+0.510 \; {\wtl C}_{13}^{\mathrm{eff}}(M_{\!\ssc W})
+0.381 \; {\wtl C}_{14}^{\mathrm{eff}}(M_{\!\ssc W})\nonumber\\
&&-0.213 \; {\wtl C}_{16}^{\mathrm{eff}}(M_{\!\ssc W})\;.
\end{eqnarray}
The branching fraction for $Br (b \rightarrow s + \g )$  is expressed through the 
semi-leptonic decay $b \rightarrow u|c e{\bar \nu}$ 
(so that the large
bottom mass dependence $( \sim m^5_b)$ and uncertainties in CKM elements
cancel out)
%\be
%Br (b \rightarrow s + \g) = \frac{\Gamma (b \rightarrow s + \gamma )}
%{\Gamma (b \rightarrow u|c \,e\,{\bar \nu_e})} \;
%Br_{\mathrm{exp}} (b \rightarrow u|c \,e\,{\bar \nu_e})\;,
%\ee
with $Br_{\mathrm{exp}} (b \rightarrow u|c\,e\,{\bar \nu_e}) = 10.5 \%$
and $\Gamma (b \rightarrow s \g) \propto (\,|C^{\mathrm{eff}}_7 (\m_b)|^2 + 
|\widetilde{C}^{\mathrm{eff}}_7(\m_b)|^2 )$.
%\be
%\Gamma (b \rightarrow s \g)  =  \f{\a \, m^5_b}{64 \pi^4} \,
%\left(\,|C^{\mathrm{eff}}_7 (\m_b)|^2 + 
%|\widetilde{C}^{\mathrm{eff}}_7(\m_b)|^2 \right) \;.
%\ee
Note that we have also to include RPV contributions to the semi-leptonic rate
for consistency.
\section{Results: Impact of bilinear-trilinear combination of parameters}
{\it Analytical Appraisal.--}We implement our (1-loop) calculations using mass eigenstate 
expressions, hence free from the commonly adopted mass-insertion 
approximation. While a trilinear RPV parameter gives a vertex, a bilinear 
parameter now contributes only through mass mixing matrix elements characterizing 
the effective couplings of the mass eigenstate running inside the loop. The $\mu_i$'s
are involved in fermion, as well as scalar mixings. There are also the
corresponding soft bilinear  $B_i$ parameters involved only in scalar 
mixings\cite{otto-gssm}. Combinations of $\mu_i$'s and  $B_i$'s with the trilinear
$\lambda^{\!\prime}_{ijk}$ parameters are our major focus.
%\footnote{It should be noted
%that there are three kinds of bilinear parameters, $\m_i,B_i$ and 
%${\tilde m}^2_{{\ssc L}_{0i}}$ related by the tadpole equation 
%constraints\cite{otto-gssm}.
%Without loss of generality, we choose $\m_i$ and $B_i$ to be independent.}. 

There are two kinds of $B_i$-$\l'$ combinations that contribute to \bsg\ at 1-loop: 
(a) $B^*_i \l^{'}_{ij2}$, and (b) $B_i \l^{'*}_{ij3}$. These involve quark-slepton loop diagrams.
Case (a) leads to the $b_{\!\ssc L} \rightarrow s_{\!\ssc R}$ transition (where SM and MSSM 
contribution is extremely suppressed) whereas case (b) leads to SM-like $b_{\!\ssc R} 
\rightarrow s_{\!\ssc L}$ transition. For the purpose of illustration, we will assume a 
degenerate slepton spectrum and take the sleptonic index $i=3$ as a representative. 
%The $j=3$ [$2$] contributions for case (a) [(b)] with both sneutrino-Higgs mixings
%and charged-slepton-Higgs mixings are easy to appreciate. 
For the $j$ values,
the charged loop contributions are still possible by invoking CKM mixings. 
Consider the contribution of case (a) with $|B_3^* \l'_{332}|$ to the 
${\wtl C}_7$, for instance. Through the extraction of the bilinear mass mixing
effect under a perturbative diagonalization of the mass matrices\cite{otto-gssm}, 
we obtain\cite{bi-tri},
\bea
{\wtl C}^{\ssc \phi^-}_7  & \approx &
 \frac{ - |V^{tb}_{\!\mbox{\tiny CKM}}|^2 |B_3^* \l'_{332}| }
{M^2_s}
\left\{
y_b   \tan_{\b}
 \left[ 	F_{12}   (x_t)	
\right] 
+ \! \f{y_t  m_t}{m_b} 
\left[ 	F_{34} (x_t)
\right] 		\right\}\nonumber\\
{\wtl C}^{\phi^{\mbox{\tiny 0}}}_7  &\approx &
\f{-2q_d \, y_b \, |B_3^*  \l'_{332}| \tan_{\b}}{M^2_s M_{\!\scriptscriptstyle S}^2}
%\left[\f{1}{M_{\!\scriptscriptstyle S_5}^2}
F_1 (x_b) 
%\left(
%{ m_{\!\scriptscriptstyle b}^2 \over 
%M_{\!\scriptscriptstyle S}^2}
%\right)
%+ \f{1}{M_{\!\scriptscriptstyle S_{10}}^2}
%F_1 \!\! \left(
%{ m_{\!\scriptscriptstyle b}^2 \over 
%M_{\!\scriptscriptstyle S_{10}}^2}
%\right)\right]
\eea
%\left[\l'_{332}B^*_3 \tan \b\right]
%\left[{\tilde m}^2_{L_{33}} |m_3|^2 + M^2_Z \cos 2\b (-\frac{1}{2} + \sin^2 \theta_W)
%\right](-F_2 (x_{ns}) -F_1 (x_{ns}))
%\ee
%\end{widetext}
for the  charged and neutral colorless scalar loop, respectively. Here $x_t$ stands for
$({m}_{\!\scriptscriptstyle t}^2 / M_{\!\scriptscriptstyle \tilde{\ell}}^2)
$ with an obvious replacement for $x_b$. Here $F_{12(34)}(x_t) = q_u F_{1(3)}(x_t) + F_{2(4)}(x_t)$
where $F_i$ are the well known loop functions\cite{tri-tri}.
In the above equations, proportionality to $\tan\!{\b}$ shows the 
importance of these contributions in the large $\tan\!{\b}$ limit. The $M_s^2$,
$M_{\!\scriptscriptstyle \tilde{\ell}}^2$, $M_{\!\scriptscriptstyle S}^2$,
are all scalar (slepton/Higgs) mass parameters.
The term proportional to $y_t$ above has chirality flip inside the loop. Thinking in terms of
the electroweak states, it is easy to appreciate that the loop diagram giving a corresponding
term for ${\wtl C}^{\phi^{\mbox{\tiny 0}}}_7$ ({\it cf.} involving
$\widetilde{\mathcal{ N}}_{\!\scriptscriptstyle nm3}^{\!\scriptscriptstyle L}\,
 \widetilde{\mathcal{ N}}_{\!\scriptscriptstyle nm2}^{\!\scriptscriptstyle R^*}$)
requires a Majorana-like scalar mass insertion, which has to arrive from other RPV 
couplings\cite{otto-gssm}. 
In the limit of perfect mass degeneracy between the scalar and 
pseudoscalar part (with no mixing) of multiplet, it vanishes. Dropping this smaller
contribution, together  with the difference among the Inami-Lim loop functions and the
fact that the charged loop has more places to attach the photon (with also larger charge
values) adding up, we expect the ${\wtl C}^{\ssc \phi^-}_7$ to be larger than
${\wtl C}^{\phi^{\mbox{\tiny 0}}}_7$. We corroborate these features in our numerical 
study.

{\it Numerical Results.--} We take non-vanishing values
for relevant combinations of a bilinear and a trilinear RPV parameters one at a time,
and stick to real values only. Our model choice for parameters is 
(with all mass dimensions
given in GeV):  squark masses 300, down-type Higgs mass 300, $\m_{\ssc 0} = -300 $ 
sleptons mass 150 and gaugino mass $M_2 = 200$ (with $M_1 = 0.5 M_2$ and 
$M_3 = 3.5M_2$), $\tan\!\b = 37$ and $A$ parameter 300. 
We impose the experimental number to obtain bounds
for each combination of RPV parameters independently (given in Table I). 
%While tabulating the
%bounds we list also the  various Wilson coefficients that play a role in the contribution. 
Consider, for instance, the case (b)
combination $|B_{\ssc 3}\l^{'*}_{\ssc 323}|$.  We obtain a bound of $5.0 \cdot 10^{-5}$, when
normalized by a factor of $\m^2_{\ssc 0}$. Since this is a $b_{\!\ssc R} \rightarrow s_{\!\ssc L}$
transition, the RPV contribution interferes with the SM as well as the MSSM contribution.
Over and above
the loop contributions there are contributions coming from four-quark 
operator with $C_{11}$ ($\propto y_b$) which is stronger than 
the other two four-quark quark coefficients ${\wtl C}_{10,13} \propto y_s$.
Since the neutral scalar loop contribution is proportional
to the loop function $F_1$ (which is of order .01), it is suppressed compared to 
current-current contributions. Also here the charged scalar contribution comes only 
with chirality flip inside the loop and has a CKM suppression. So the current-current is 
dominant. It has a more subtle role to play when one writes the regularization
scheme-independent $C^{\mathrm{eff}}_7 = C_7 -C_{11}$ at scale $M_{\ssc W}$(see\cite{tri-tri}).
Due to dominant and negative
sign chargino contribution (because $A_t\, \m_{\ssc 0} < 0$), 
the positive sign $C_{11}$ interferes
constructively with $C_7$ and enhances the rate.These features can be verified from
Fig.1 of Ref.\cite{bi-tri}. We have done the similar analytical and numerical exercise
for all possible combinations of bilinear and trilinear couplings and quote the
relevant bounds obtained for the first time in Table\ref{bounds}.

{\it Conclusions. ---} We have systematically studied the influence of the 
combination of bilinear-trilinear RPV parameters on the decay \bsg\ analytically as 
well as numerically. 
%Such a study has not been attempted before.
%We demonstrate the plausible dominance of the RPV contributions over conventional SM and MSSM contributions in some parameter space regions. 
These contributions are enhanced 
by large $\tan\!\b$. We also demonstrate the importance of QCD corrections and
 obtain strong bounds on several combinations of RPV parameters for the first time. 
Numerical study
has also been performed on combinations of trilinear parameters\cite{tri-tri}. 
We quote here a few
exciting bounds under a similar sparticle spectrum. 
%with bounds on
%most of the combinations of the kind $|\l'_{i3j}\cdot\l^{'*}_{i2j}|$ about order of magnitude
%stronger than the existing bounds. 
For instance $|\l'_{i33}\cdot\l^{'*}_{i23}|$ for $i=2,3$
should be less than $1.6\cdot10^{-3}$ to be compared with rescaled existing bound of
$2\cdot 10^{-2}$. 
\begin{table}[t]
\tbl{Our Bounds for the absolute value of products of RPV couplings.}
{\small
\vspace{0.3cm}
\label{bounds}
\begin{tabular}{llllll}\hline
{\bf Product } & {\bf Bound} & {\bf Product} & {\bf bound} & {\bf Product} & {\bf bound}
 \\\hline  
$\f{B_i \cdot \l'_{i23}}{\m^2_{\ssc 0}}$& $5.0\cdot 10^{-5}$&
$\f{B_i \cdot \l'_{i12}}{\m^2_{\ssc 0}}$& $4.5\cdot 10^{-2}$&
$\f{\m_i \cdot \l'_{i23}}{\m_{\ssc 0}}$& $2.2\cdot 10^{-3}$\\
%% 1
$\f{B_i \cdot \l'_{i32}}{\m^2_{\ssc 0}}$& $7.4\cdot 10^{-3}$&
$\f{B_i \cdot \l'_{i22}}{\m^2_{\ssc 0}}$& $6.5\cdot 10^{-2}$&
$\f{\m_i \cdot \l'_{i32}}{\m_{\ssc 0}}$& $1.0\cdot 10^{-2}$\\
%% 2
$\f{B_i \cdot \l'_{i33}}{\m^2_{\ssc 0}}$& $2.3\cdot 10^{-3}$&
$\f{B_i \cdot \l'_{i13}}{\m^2_{\ssc 0}}$& $8.0\cdot 10^{-2}$&
$\f{\m_i \cdot \l'_{i33}}{\m_{\ssc 0}}$& $8.0\cdot 10^{-2}$\\\hline
%% 3
\end{tabular}}
\end{table} 
\section*{Acknowledgment}
We would like to thank 
National Science Council of Taiwan and Physics department (NCU) for the 
support under post-doc grant number 
NSC 92-2811-M-008-012. 
%%%%%%%%%%%%%%%%%%% 
\bibliographystyle{plain}
\begin{thebibliography}{99}
%
%\bibitem{paper1} 
%K. Hagiwara, A.D. Martin, D. Nomura and T. Teubner, 
%{\it Phys. Rev.}{\bf D69}, 093003 (2004); hep-ph/0312250.   
\bibitem{world_average} C. Jessop, SLAC-PUB-9610.
\bibitem{bur-mis-02} A. Buras \atal\ {\it Nucl. Phys.}{\bf B631} 219(2002).
\bibitem{carlos} B. de Carlos and P.L. White {\it Phys. Rev.}{\bf D55} 4222 (1997).
\bibitem{besmer} Th. Besmer and A. Steffen {\it Phys. Rev.} {\bf D63} 055007 (2001). 
%Also see
%E.Chun \atal\ \prr{D62}{076006}{00}.
\bibitem{tri-tri}O.C.W. ~Kong and R. Vaidya hep-ph/0403148.
\bibitem{bi-tri}O.C.W. ~kong and R. Vaidya hep-ph/0408115.
%\bibitem{ru12}
%M. Bisset \atal\,
%{\it Phys. Lett.} {\bf B430}, 274 (1998);
%\bibitem{ru2}
%M. Bisset, O.C.W. Kong, C. Macesanu, and L.H. Orr,
%Phys. Rev. {\bf D62},  {\it 035001} (2000).
\bibitem{otto-gssm}O.C.W.~Kong,  {\it Int.J.Mod.Phy.} A{\bf 19}, 1863  (2004).
%\bibitem{otto-keum-kingman} For EDM applications see Y.Y.Keum \atal\ 
%\prll{86}{393}{01}; \prr{D63}{113012}{01}. For $\m \rightarrow e 
%\gamma$ see K. Cheung \atal\ \prr{D64}{095007}{01}.
\end{thebibliography}
\end{document}

\section{Guidelines}
\subsection{Producing the Hard Copy}\label{subsec:prod}
The hard copy may be printed using the advice given in the file
{\em procs-readme9x6$\_$2e.pdf}, which is repeated in this section. 
Total there are seven files given.\footnote{You can obtain these files 
from our WWW pages at:

{\sf http://www.wspc.com.sg/others/style\_files/proceedings/proceedings\_style\_files.shtml}}

\begin{enumerate}
\item {\em procs-readme9x6$\_$2e.pdf} --- the preliminary guide.

\item {\em procs-instruction9x6$\_$2e.pdf} --- general instructions 
for authors.

\item{\em procs-fig1.eps} --- the figure/image file.

\item{\em rotating$\_$pr.sty} --- sty file for landscape figures and tables.

\item {\em ws-procs9x6.cls} --- the class file that provides the higher
level latex commands for the proceedings. Don't change these parameters.

\item {\em ws-procs9x6.tex} --- the main text. 

\item {\em ws-procs9x6.pdf} --- sample pages of the above text.
\end{enumerate}

You can delete our sample text and replace it with your own 
contribution to the volume, however we recommend that you keep an 
initial version of the file for reference.  Strip off any mail 
headers and then latex the tex file.  The command for latexing is 
{\sf latex ws-procs9x6}, do this twice to sort out 
the cross-referencing.

If you wish to use some other form of word-processor, some 
guidelines are given in Sec.~\ref{subsec:wpp} below.  These 
files will work with standard latex2e. If there is an
abbreviation defined in the new definitions at the top of the 
file {\em ws-procs9x6.tex} that conflicts with one of your own 
macros, then delete the appropriate command and revert to
longhand. Failing that, please consult your local texpert to
check for other conflicting macros that may be unique to your
computer system.  Page numbers are included at the top of the
page for your guidance. Do not worry about the final pagination
of the volume which will be done after you submit the paper.

\subsection{Using Other Word-Processing Packages}\label{subsec:wpp}
If you want to use some other form of word-processor to
construct your output, and you are using the final hard copy version
of these files as guidelines; then please follow the style given
here:- 

\begin{romanlist}
\item The title will be in 10~pt boldface and in uppercase; the leading or 
$\backslash$baselineskip is on 13~pt. 

\item The authors' names will be in 9~pt roman and in uppercase. 

\item The addresses will be in 9~pt and $\backslash$baselineskip  
is on 11~pt. 

\item The abstract text will be in 8~pt and $\backslash$baselineskip is 
on 10~pt.  It should be indented on both sides by 0.15~inch from 
the main body of the text.

\item All three section heads are in 10~pt:- 

\noindent
1st section heading is in boldface number run on to boldface
title and to set in uppercase and lowercase.
\eject

\noindent
2nd section heading is in boldface number run on to bold-italic
title and to set in uppercase and lowercase.

\noindent
3rd section heading is roman number run on to italic title and
to set in initial cap only.

\item Figure captions is to set in 8~pt and $\backslash$baselineskip 
is on 10~pt.

\item Table caption, table column heads and table body text 
is to set in 8~pt and $\backslash$baselineskip is on 10~pt. 

\item Footnote text is to set in 8~pt and $\backslash$baselineskip 
is on 10~pt.

\item Reference text is to set in 9~pt and $\backslash$baselineskip 
is on 11~pt.

\end{romanlist}

The proceedings trim size will be 9 by 6 inches; however you
should submit your ms copy on standard A4 paper.  The text area 
excluding page numbers should be 7 inches (textheight) by 
4.5 inches (textwidth).  Paragraphs should have a first line indented 
by about 0.25~inch except where the paragraph is preceded by a heading.

It is also important to reproduce the spacing of the text and
headings as shown here. Text should be slightly more than
single-spaced; use a baselineskip (which is the average 
distance from the base of one line of text to the base of 
an adjacent line) of 13~pt and 10~pt for footnotes, table captions 
and figure captions.

\subsection{Headings, Text and Equations}
Please preserve the style of the headings, text font and line 
spacing in order to provide a uniform style for the 
proceedings volume. 

Equations should be centered and numbered consecutively, as in
Eq.~(\ref{eq:murn}), and the {\em eqnarray} environment may be
used to split equations into several lines, for example in
Eq.~(\ref{eq:sp}), or to align several equations.  An alternative
method is given in Eq.~(\ref{eq:spa}) for long sets of equations 
where only one referencing equation number is wanted.

In latex, it is simplest to give the equation a label, as in
Eq.~(\ref{eq:murn}) where we have used 
{\em $\backslash$label\{eq:murn\}} to identify the equation.
You can then use the reference {\em $\backslash$ref\{eq:murn\}}
when citing the equation in the text which will avoid 
the need to manually renumber equations due to later changes. (Look at
the source file for some examples of this.)

The same method can be used for referring to sections 
and subsections.

\begin{sidewaystable}	
\tbl{Sample statistics for A-share premia for Shanghai companies that 
issued A- and B-shares before April 1994 (sample period: April 1, 
1994--October 31, 1998). \label{table1}}
{\begin{tabular}{@{}ccccccccc@{}}
\hline
\multicolumn{9}{c}{}\\[-2ex]
Year &Taiwan &Dow Jones &Nikkei &South &London's &Hong Seng &Thailand's
&Singapore's\\
{} &Index &Index &Index &Korea's &FTSE &Index &Set Index &Strait Times\\
{} &{} &{} &{} &Kospi &Index &{} &{} &{}\\
{} &{} &{} &{} &Index &{} &{} &{} &{}\\[0.25ex]
\hline
\multicolumn{9}{c}{}\\[-2ex]
1988 &5,119.11 &\phantom{0}2,168.57 &30,159.00 &907.20 &1,455.30
&\phantom{0}2,687.44 &386.73 &1,038.62\\ 
1989 &9,624.18 &\phantom{0}2,753.20 &38,915.87 &909.72 &1,916.60
&\phantom{0}2,836.57 &879.19 &1,481.33\\
1990 &4,530.16 &\phantom{0}2,633.70 &23,848.71 &696.11 &1,673.40
&\phantom{0}3,243.30 &612.86 &1,154.48\\
1991 &4,600.67 &\phantom{0}3,168.83 &22,983.77 &610.92 &1,891.30
&\phantom{0}4,297.33 &711.36 &1,490.70\\
1992 &3,377.06 &\phantom{0}3,301.11 &16,924.95 &678.44 &2,185.20
&\phantom{0}5,512.39 &893.42 &1,524.40\\
1993 &6,070.56 &\phantom{0}3,754.09 &17,417.24 &866.18 &2,559.50
&11,888.39 &1,565.12\phantom{0.} &2,425.68\\
1994 &7,124.66 &\phantom{0}3,834.44 &19,723.06 &1,027.37\phantom{0.}
&3,065.50 &\phantom{0}8,191.04 &1,360.09\phantom{0.} &2,239.56\\
1995 &5,173.73 &\phantom{0}5,117.12 &19,868.15 &882.94 &3,689.30
&10,073.39 &1,280.81\phantom{0.} &2,266.54\\
1996 &6,933.94 &\phantom{0}6,448.27 &19,361.35 &651.22 &4,118.50
&13,451.45 &831.57 &2,216.79\\
1997 &8,187.27 &\phantom{0}7,905.25 &15,258.74 &376.31 &5,135.50
&10,722.76 &372.69 &1,529.84\\
1998 &6,418.43 &\phantom{0}9,181.43 &13,842.17 &562.46 &5,882.60
&10,048.58 &355.81 &1,392.73\\
1999 &8,448.84 &11,497.12 &18,934.34 &1,028.07\phantom{0.} &6,930.20
&16,962.10 &481.92 &2,479.58\\
2000 &5,544.18 &10,971.14 &14,539.60 &514.48 &6,438.40 &14,895.34
&271.84 &1,976.54\\ 
1 &9,744.89 &10,940.53 &19,539.70 &943.88 &6,268.50 &15,532.34 &477.57
&2,230.28\\
2 &9,435.94 &10,128.31 &19,959.52 &828.38 &6,232.60 &17,169.44 &374.32
&2,120.50\\
3 &9,854.95 &10,921.92 &20,337.32 &860.94 &6,540.20 &17,406.54 &400.32
&2,132.59\\
4 &8,777.35 &10,733.91 &17,973.70 &725.39 &6,327.40 &15,519.30 &390.40
&2,164.11\\
5 &8,939.52 &10,522.33 &16,332.45 &731.88 &6,359.30 &14,713.86 &323.29
&1,795.13\\
6 &8,265.09 &10,447.89 &17,411.05 &821.22 &6,312.70 &16,155.78 &325.69
&2,037.97\\
7 &8,114.92 &10,521.98 &15,727.49 &705.97 &6,365.30 &16,840.98 &284.67
&2,051.21\\
%8 &7,616.98 &11,215.10 &16,861.26 &688.62 &6,672.70 &17,097.51 &307.83
%&2,147.77\\[0.5ex]
\hline
\end{tabular}}
\begin{tabnote}
The state-budget funds as a dominant financial source of investment 
of SOEs, regardless of investment decisions being made by governments 
or enterprises. The retained profits and non-bank debts are also used by
SOEs to finance their investment and operation.
\end{tabnote}
\end{sidewaystable}

\subsection{Tables}
The tables are designed to have a uniform style throughout the 
paper. It does not matter how you choose to place
the inner lines of the table, but we would prefer the border
lines to be of the style shown in Table~\ref{table1}.  
The top and bottom horizontal lines should be single 
(using {\em $\backslash$hline}), and there should be single 
vertical lines on the perimeter, (using 
{\em  $\backslash$begin\{tabular\}\{$|...|$\}}).  For the inner
lines of the table, it looks better if they are kept to a
minimum. We've chosen a more complicated example purely as an 
illustration of what is possible.

An imaging experiment with OH$^+$ in CRYRING led to discovery of a
very small kinetic energy release when the electron energy was
sufficient for the ${\rm O}(^3P) + {\rm H}(n=2)$ dissociation limit
(see Table~\ref{table2}) to become energetically allowed.

\looseness1 The caption heading for a table should be placed at the
top of the table.

\begin{table}[ph]
\tbl{First five normalized natural frequencies.}
{\footnotesize
\begin{tabular}{@{}crrrr@{}}
\hline
{} &{} &{} &{} &{}\\[-1.5ex]
{} & $A=0.56$ & $B=0.69$ & $C=0.75$ & $D=0.100$\\[1ex]
\hline
{} &{} &{} &{} &{}\\[-1.5ex]
$AB_1$ &14.0640 &18.5620 &22.0817 &18.90732\\[1ex]
$AC_2$ &61.6728 &44.7844 &44.5884 &60.17496\\[1ex]
$AD_3$ &88.1380 &118.1564 &101.2240 &120.72693\\[1ex]
$DB_4$ &199.8594 &173.1269 &194.4907 &188.75258\\[1ex] 
$DA_5$ &246.7889 &255.9483 &284.6633 &262.24264\\[1ex]
\hline
\end{tabular}\label{table2} }
\vspace*{-13pt}
\end{table}

\subsection{Figures/Illustrations/Images}
Authors are advised to prepare their figures in black and white. 
Please prepare the figures in high resolution (300 dpi) for half-tone 
illustrations or images. Half-tone pictures must be sharp enough 
for reproduction, otherwise they will be rejected.

Colour images are allowed only when they are stated in the 
publishing agreement. The colour images must be prepared in CMYK 
(Cyan, Magenta, Yellow and Black). RGB colour images are not acceptable 
for colour\break
separation.

It is best to embed the figures in the text where they are first
cited, e.g. see Figure~\ref{inter}. Please ensure that all labels in the
figures are legible irregardless of whether they are drawn
electronically or manually.

If you wish to `embed' a postscript figure in the file, then remove
the \% mark from the declaration of the postscript figure 
{\em epsfbox} within the figure description and change the filename 
to an appropriate one. Also remove the comment \% mark from the 
{\em epsfxsize} command and specify the required width of the figure.
System will automatically enlarge or reduce the figure based on the
{\em x-size} provided with {\em epsfxsize} command.  You may need to
play around with this as different computer systems appear to use
different commands. If you like to have an empty box size to the image
you can just fill in the {\em x, y size} against the command 
{\em figurebox}, which has three arguments. The third argument is 
for actual figure name.

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
\centerline{\epsfxsize=4.1in\epsfbox{procs-fig1.eps}}   
\caption{First 3 normalized frequencies versus release location for
clamped simply supported beam with internal slide
release. \label{inter}}
\end{figure}

Next adjust the scaling of the figure until it is correctly 
positioned, and remove the declarations of the lines and any 
anomalous spacing.

If you prefer to submit glossy prints of figures, then it is very 
important to leave sufficient blank spaces in your manuscript 
to accommodate your figures. Send the hard copy of the figures 
on separate pages with clear instructions on where to match 
them to the repsective blank spaces in the final hard copy 
text. Please ensure that each figure is correctly
scaled (ensure legibility) to fit the space available.

The caption heading for a figure should be placed below
that figure.

\subsection{Limitations on the Placement of Tables, 
Equations and Figures}\label{sec:plac}
Very large figures and tables should be placed on a page by themselves. One
can use the instruction {\em $\backslash$begin\{figure\}$[$h$]$} or
{\em $\backslash$begin\{table\}$[$h$]$}
to position these, and they will appear on a separate page devoted to
figures and tables. We would recommend making any necessary
adjustments to the layout of the figures and tables
only in the final draft. It is also simplest to sort out line and
page breaks in the last stages.

\subsection{Acknowledgments, Appendices and the Bibliography}
If you wish to acknowledge funding bodies etc., 
the acknowledgments may be placed in a separate section at the end of the
text, before the Appendices. This should not be numbered so use
{\em $\backslash$section$\ast$\{Acknowledgments\}}.

It is preferable not to have Appendices in a brief article, but if
more than one Appendix is necessary then set headings as Appendix~A,
Appendix~B, etc. It is to type as 
{\em $\backslash$section$\ast$\{Appendix~A\}}.

\subsubsection{Footnotes and the citation}
Footnotes are denoted by a character superscript
in the text,\footnote{Just like this one.} and references
are denoted by a number superscript.
We have used {\em $\backslash$bibitem} to produce the bibliography.
Citations in the text use the labels defined in the bibitem declaration,
for example, the first paper by Jarlskog\cite{ja} is cited using the command
{\em $\backslash$cite\{ja\}}.

If you use square brackets for citation e.g. [2] please note that  
the citation should appear before the punctuation mark, 
e.g. [2], in the body text. 

\subsection{Final Manuscript}
The final hard copy that you submit must be absolutely clean and unfolded.
It will be printed directly without any further editing. Use a printer
that has a good resolution printout (600 dpi or higher). There should
not be any corrections on the printed pages, nor should adhesive
tape cover any lettering. Photocopies are not acceptable.

Your manuscript will not be reduced or enlarged when filmed so please
ensure that indices and other small pieces of text are legible.

\section{Sample Mathematical Text}

To begin with let us write down the covariant Dirac equation in a
curved spacetime, for a massless spinor field $\Psi $, which is given
by
\begin{equation} 
[i\gamma^\mu(x)\partial_\mu+i\gamma^\mu(x)\Gamma_\mu(x)]\Psi(x)=0\,,
\label{eq:murn}
\end{equation}
where $\gamma^\mu(x)$ are the generalized Dirac matrices and are given
in terms of the standard flat space Dirac matrices $\gamma^{(a)}$ as
\begin{equation} 
\gamma^\mu(x)=e_{(a)}^\mu(x)\gamma^{(a)}\,,
\label{eq:sp}
\end{equation}
where $e_{(a)}^\mu(x)$ are tetrad components defined by
\begin{equation} 
e_{(a)}^\mu e_{(b)}^\nu\eta^{(a)(b)}=g^{\mu\nu}\,.
\label{eq:spa}
\end{equation}
Here and in what follows Greek indices are connected with tensor world
indices (coordinate basis system) and Latin indices denote Lorentz
indices which are connected with a local Minkowski coordinate system
(tetrads).

We can bring down powers of $q(t)$ as follows:
\begin{eqnarray} 
e^{\int dt q(t)} &=&1+\int_0^a dt'q(t')
+\int_0^a dt'\int_0^{t'}dt''q(t')q(t'')\nonumber\\[5pt]
&&+\,\int_0^a dt'\int_0^{t'}dt'''\int _0^{t''}dt'''q(t')q(t'')q(t''')\,.
\end{eqnarray}
The leading ``1'' will be assumed not to contribute anything. This is
equivalent to setting $\langle kn\cdots\rangle=0$ when there are
no $q$'s. Also the order of the $J^A(t)$'s has to be preserved after
making contractions of $q^A(t)$ with $kn$'s.

\begin{theorem} \label{theos}
Let $V$ be a closed complex analytic subvariety of a complex
hyperbolic space form of finite volume$.$ Then the Gauss mapping for
$V$ is non-degenerate unless $V$ is totally geodesic$.$
\end{theorem}

Let $V$ be a $k$-dimensional complex submanifold of ${\mathbb P}^n$
and $\gamma: V\rightarrow {\mathbb G}(k,n)$ be the Gauss mapping. At a
point $x\in V$, Lemma~\ref{ste} let $\hat{T}_x(V)$ be the $(k +
1)$-dimensional affine tangent space of $V$ at $x$ so that the tangent
space of $V$.

\begin{lemma}  \label{ste}
There exists a point $(a;b)\in \Delta^m\cdot {\mathbb C}^l$ and an
open neighborhood $W$ of $(a;b)$ such that $F(a;b)\in \partial 
{\mathbb B}^n$. A closed complex analytic subvariety of 
a complex torus has degenerate Gauss mapping if and only 
if it is invariant under the translation by a complex subtorus.
\end{lemma}

Let $F$ be a symmetric and reflexive Borel relation on the standard
Borel space $X$. $F$ is {\it locally finite} if for all $x\in X,
F(x)=\{y\in X: yFx\}$ is finite.

\section*{Acknowledgments}
This is where one acknowledge funding bodies etc.  Note that section
numbers are not required for Acknowledgments, Appendix or References.

\appendix

\section{First Appendix}

\renewcommand{\theequation}{A.\arabic{equation}}

Appendices should be used only when absolutely necessary. They
should come before the References. If there is more than one
appendix, number them alphabetically. Number displayed equations
occurring in the Appendix in this way, e.g.~(\ref{appen1}), (A.2),
etc.
\begin{equation}
\mu(n, t) = \frac{\sum^\infty_{i=1} 1(d_i < t, 
N(d_i) = n)}{\int^t_{\sigma=0} 1(N(\sigma) = n)d\sigma}\,. \label{appen1}
\end{equation}

\section{Second Appendix}

\renewcommand{\theequation}{B.\arabic{equation}}

References in the text are to be numbered consecutively in Arabic
numerals, in the order of first appearance. They are to be typed in
superscripts after punctuation marks, e.g.

(1) $ \qquad $ ``$\ldots$ in the statement.\cite{xyan}''

(2) $ \qquad $ ``$\ldots$ have proven\cite{bu}\cdash\cite{xyan} 
that this equation $\ldots$''

\noindent
This is done using the command: ``$\backslash$cite\{name\}''.

When the reference forms part of the sentence it should not 
be superscripts, e.g.

(1) $ \qquad $  ``One can deduce from Ref.~\refcite{ppz} that $\ldots$''

(2) $ \qquad $  ``See Refs.~\refcite{ja}--\refcite{ma}, \refcite{bu} 
and \refcite{xyan} for more details.'' 

(3) $ \qquad $  ``We refer the readers to Ref.~\refcite{xyan}.''

\noindent
This is done using the command: ``Ref.$\sim$$\backslash$refcite\{name\}''.

\vspace*{8pt}
\noindent
(Alternatively you may opt to use the default square bracket [\ ] 
citation throughout.)

\section{Footnotes}
Footnotes should be numbered sequentially in superscript lowercase
roman letters.\footnote{Footnotes should be typeset in 8~pt Times
roman at the bottom of the page.}

\section{Standard Abbreviations}
\begin{alphlist}[(d)]
\item Do not abbreviate the first word of any sentence: 

$ \qquad $ ``Figure~2 shows us $\ldots.$''

\item Some abbreviation:

$ \qquad $ ` figure ' $ \, $ = $ \, $  ` Fig. '

$ \qquad $ ` figures ' $ \, $ = $ \, $ ` Figs. '

$ \qquad $ ` equation ' $ \, $ = $ \, $ ` Eq. '

$ \qquad $ ` equations ' $ \, $ = $ \, $ ` Eqs. '

$ \qquad $ ` Section~5 ' $ \, $ = $ \, $ ` Sec.~5 '

$ \qquad $ ` Sections~5 and 6 ' $ \, $ = $ \, $ ` Secs.~5 and 6 '

$ \qquad $ ` for example ' = ` e.g. '

\noindent
Note that the first letter is capitalized. There is also a dot. 

\item When it is not appropriate, DO NOT abbreviate. Hence the word `Table'
is not abbreviated. We also do not write `Eq. of motion'.

\item Depends on authors' preference, sometimes ` Eq. ' and ` Eqs. '
are not used at all because it is understood that it is an equation.
For example,  

\begin{center}
We can see a summation and an integration in (\ref{appen1}).
\end{center}
\end{alphlist}

\section{Single Quotation and Double Quotations}
Use double quotation when possible so that there are fewer mix-ups with
differentiation and `prime'. For quotation within a 
quotation you may use the single-quotation.

Open-quotation ` is located at the top left-hand corner of the keyboard.
Close-quotation ' is near the ENTER-key of the keyboard.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%Use this if your figures are put in a subdirectory having the same
%name as the main latex file, ie: 
%
%      ws-procs9x6/procs-fig1.eps      
%      ws-procs9x6/procs-fig2.eps      
%      ws-procs9x6/procs-fig3.eps      
%      ws-procs9x6/procs-fig4.eps      
%      etc.
%
%\begin{figure}[htbp] %ORIGINAL SIZE: width=1.4TRUEIN; height=1.5TRUEIN
%\figurebox{}{}{procf1} %100 percent
%\caption{Labeled tree {\it T}.}
%\end{figure}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{0}
\bibitem{ja} M. Barranco and J. R. Buchler, {\it Phys. Rev.}
{\bf C34}, 1729 (1980).

\bibitem{ppz} G. Pang and H. Zhao, {\it J. Phys. A: Math. Gen.}  
{\bf 25}, L527 (1992).

\bibitem{ma} H. M\"uller and B. D. Serot, {\it Phys. Rev.} {\bf C52}, 
2072 (1995).

\bibitem{bu} V. Baran, M. Colonna, M. Di Toro and A. B. Larionov, 
{\it Nucl. Phys.} {\bf A632}, 287 (1998).

\bibitem{bd} V. Baran, M. Colonna, M. Di Toro and V. Greco, 
{\it Phys. Rev. Lett.} {\bf 86}, 4492 (2001).

\bibitem{xyan} X. Yan, {\it Phys. Rev.} {\bf E61}, 4745 (2000).

\end{thebibliography}

\end{document}

hep-ph0411336.extracted_bib
\begin{thebibliography}{99}
%
%\bibitem{paper1} 
%K. Hagiwara, A.D. Martin, D. Nomura and T. Teubner, 
%{\it Phys. Rev.}{\bf D69}, 093003 (2004); hep-ph/0312250.   
\bibitem{world_average} C. Jessop, SLAC-PUB-9610.
\bibitem{bur-mis-02} A. Buras \atal\ {\it Nucl. Phys.}{\bf B631} 219(2002).
\bibitem{carlos} B. de Carlos and P.L. White {\it Phys. Rev.}{\bf D55} 4222 (1997).
\bibitem{besmer} Th. Besmer and A. Steffen {\it Phys. Rev.} {\bf D63} 055007 (2001). 
%Also see
%E.Chun \atal\ \prr{D62}{076006}{00}.
\bibitem{tri-tri}O.C.W. ~Kong and R. Vaidya hep-ph/0403148.
\bibitem{bi-tri}O.C.W. ~kong and R. Vaidya hep-ph/0408115.
%\bibitem{ru12}
%M. Bisset \atal\,
%{\it Phys. Lett.} {\bf B430}, 274 (1998);
%\bibitem{ru2}
%M. Bisset, O.C.W. Kong, C. Macesanu, and L.H. Orr,
%Phys. Rev. {\bf D62},  {\it 035001} (2000).
\bibitem{otto-gssm}O.C.W.~Kong,  {\it Int.J.Mod.Phy.} A{\bf 19}, 1863  (2004).
%\bibitem{otto-keum-kingman} For EDM applications see Y.Y.Keum \atal\ 
%\prll{86}{393}{01}; \prr{D63}{113012}{01}. For $\m \rightarrow e 
%\gamma$ see K. Cheung \atal\ \prr{D64}{095007}{01}.
\end{thebibliography}