%From: y.rodriguezgarcia@lancaster.ac.uk


\documentclass[12pt]{iopart}
\usepackage{iopams}
\begin{document}
\title[Spontaneous CP violation for the neutral B systems]{Spontaneous CP violation
for the neutral B systems in the left-right symmetric model}	

\author{Yeinzon Rodr\'{\i}guez\dag\footnote[2]{On leave of absence from the Centro
de Investigaciones of the Universidad Antonio Nari\~no, Cll 58A {\#} 37-94, Bogot\'a
D.C., Colombia.} and Carlos Quimbay\S\footnote[4]{Associate researcher of the Centro
Internacional de F\'{\i}sica, Ciudad Universitaria, Bogot\'a D.C., Colombia.}}
\address{\dag\ Department of Physics, Lancaster University, Lancaster LA1 4YB, UK}
\address{\S\ Departamento de F\'{\i}sica, Universidad Nacional de Colombia,
Ciudad Universitaria, Bogot\'a D.C., Colombia}

\eads{\mailto{y.rodriguezgarcia@lancaster.ac.uk},
\mailto{carloqui@ciencias.unal.edu.co}}

\begin{abstract}
We investigate the effect of the spontaneous breakdown of $CP$ for the
$B_\mathrm{d,s}^0-\overline{B}_\mathrm{d,s}^0$ systems, in the Left-Right
Symmetric Model, by means of the calculation of the dilepton asymmetry
$a_\mathrm{ll}=\left[ N\left( l^{+}l^{+}\right)-N\left( l^{-}l^{-}\right) \right]
/[ N\left( l^{+}l^{+}\right)$
$+N\left( l^{-}l^{-}\right) ] $. We use the
fact that the right scale must be very large $( \upsilon _\mathrm{R}\sim 10^7\textbf{ }
GeV) $ in order to have a phenomenologically consistent model. We find
that the prediction for the dilepton asymmetry is the same both in the Left-Right
Symmetric Model and in the Standard Model. We compare these predictions to the
present experimental bounds.

\vspace{5mm}
{\it Keywords}: $CP$ Violation; Left-Right Models; B Mesons; Dilepton Asymmetry
\end{abstract}

\submitto{\jpg}
\pacs{11.30.Er, 12.60.Fr, 13.20.Fc, 13.20.He}

\maketitle

\section{Introduction}
$CP$ violation is one of the most intriguing puzzles in particle physics and a
lot of work has been devoted to the search of its origin.
The most accepted way to generate $CP$ violation in the Standard Model $%
\left( SM\right) $ is through the Cabibbo-Kobayashi-Maskawa $\left(
CKM\right) $ phase. \cite{cabibbo} This way has demonstrated to be
very accurate to describe the phenomenology of the neutral kaon system. \cite
{wakaizumi,buchalla,rosner} However, there are other ways which can give account
of the $CP$ violation effects on the neutral kaon system while exhibiting
new interesting features. One of these ways consists on searching for a
natural origin of the $CP$ violating phase, for example, through complex
vacuum expectation values. This can be achieved in the Left-Right Symmetric
Model $\left( LRSM\right) $ \cite{leftright,deshpande,barenboim,yeinzon,kiers}
where there is one \textit{genuine} complex vacuum expectation value due to the
presence of a scalar bidoublet, and which is responsible for the quark sector $CP$
violating phase. There is also another \textit{genuine} complex vacuum expectation
value due to the presence of two scalar triplets which, together with the scalar
bidoublet vacuum expectation value, are the responsibles for the $CP$ violation in
the lepton sector. The main advantages of this model is that it permits to explain
both parity and $CP$ as spontaneously broken symmetries, and identify the hypercharge
with the quantum number $B-L$ giving it thus a physical meaning. Another property of
this model is that it gives an explanation of the smallness of the neutrino masses,
through the see-saw mechanism, \cite{seesaw} as well as a framework for the study of
the $CP$ violation in the lepton sector, which is not present in the $SM$ and that
is source of lots of present experimental and theoretical works. \cite{cplepton}

As the $SM$ explains succesfully the effects of $CP$ violation on the
neutral kaon system, we expect that it can explain these effects on the $B$
systems too. The dilepton asymmetry $a_\mathrm{ll}=\left[ N\left( l^{+}l^{+}\right)
-N\left( l^{-}l^{-}\right) \right] /\left[ N\left( l^{+}l^{+}\right)
+N\left( l^{-}l^{-}\right) \right] $ relates directly to the parameter of
indirect $CP$ violation $\overline{\varepsilon }_\mathrm{B}$. The most recent
experimental results show that $Re\overline{\varepsilon }_\mathrm{B}$, for the
$B_\mathrm{d}$ system, lies in the range $1.2\pm 2.9\pm 3.6\times 10^{-3}$ \cite{babar}
which has a central value different to zero but with a too long width. Then,
it is possible to have many different models which explain correctly the effects
of $CP$ violation on the neutral kaon system, giving a value for $Re
\overline{\varepsilon }_\mathrm{B}$ compatible with the experimental width but,
in principle, different to the $SM$ one. Many of these models are also able to give
some insight in the understanding of the $CP$ violation in the lepton sector. Among
these models of new physics is the $LRSM$.

The main goal of this article is to study the predictions of the $LRSM$ about
the parameter $Re\overline{\varepsilon }_\mathrm{B}$ for both the $B_\mathrm{d}$ and
$B_\mathrm{s}$ systems, by means of the calculation of the dilepton asymmetry
$a_\mathrm{ll}$, and to compare them to the predictions of the $SM$ and to the
experimental results. As we will see, the new physics sector of the $LRSM$ decouples
from the quark sector and, therefore, the new physics effects involving interactions
with quarks are negligible. So, both the $SM$ and the $LRSM$ predict the same value
for the parameter $Re\overline{\varepsilon}_\mathrm{B}$ in the $B_\mathrm{d,s}$
systems.

This paper is organised as follow: in the section 2 we study the phenomenology of
the $CP$ violation for the $B$ systems. In the section 3, we describe the $LRSM$,
while in the section 4 we calculate the parameter $Re\overline{\varepsilon }_%
\mathrm{B}$ for the $B_\mathrm{d,s}$ systems in the frames of both the $LRSM$
and the $SM$. We compare their predictions to the experimental data. Finally, in
the section 5, we conclude.

\section{Phenomenology of $CP$ Violation for $B$ Systems}

There are two kinds of $B$ mesons: $B_\mathrm{L}$ and $B_\mathrm{H}$, which we can
write as linear combinations of $CP$ eigenstates and that at the same time are linear
combinations of the flavour eigenstates $B^0$ and $\overline{B}^0$:

\begin{eqnarray}
&\mid &B_\mathrm{L}\rangle =\left( \mid B_1\rangle +\overline{\varepsilon }_%
\mathrm{B}\mid
B_2\rangle \right) /\sqrt{1+\left| \overline{\varepsilon }_\mathrm{B}\right| ^2}, \\
&\mid &B_\mathrm{H}\rangle =\left( \mid B_2\rangle +\overline{\varepsilon }_%
\mathrm{B}\mid
B_1\rangle \right) /\sqrt{1+\left| \overline{\varepsilon }_\mathrm{B}\right| ^2},
\end{eqnarray}
where $\mid B_1\rangle $ and $\mid B_2\rangle $ are the $CP$ eigenstates,
even and odd respectively:

\begin{eqnarray}
&\mid &B_1\rangle =\left( \mid B^0\rangle +\mid \overline{B}^0\rangle
\right) /\sqrt{2}, \\
&\mid &B_2\rangle =\left( \mid B^0\rangle -\mid \overline{B}^0\rangle
\right) /\sqrt{2}.
\end{eqnarray}
We can see that $\mid B_\mathrm{L}\rangle $ is almost the $CP$-even state $\mid
B_1\rangle $ with a tiny admixture of the $CP$-odd state $\mid B_2\rangle $,
and that $\mid B_\mathrm{H}\rangle $ is almost the $CP$-odd state $\mid B_2\rangle $
with a tiny admixture of the $CP$-even state $\mid B_1\rangle $. The
parameter $\overline{\varepsilon }_\mathrm{B}$ represents the measure of the $CP$
violation in the ``state mixing''. The two states $\mid B_\mathrm{L}\rangle $ and
$\mid B_\mathrm{H}\rangle $ have their own proper mass and lifetime. \cite{pdg}

One of the products in a $e^+e^-$ collision is a pair $B^0-\overline{B}^0$.
Working in the basis of flavour eigenstates, we can determine the evolution
of this pair by: \cite{wakaizumi}

\begin{eqnarray}
&\mid &B_\mathrm{phys}^0\left( t\right) \rangle =f_{+}\left( t\right) \mid
B^0\rangle +\frac{1-\overline{\varepsilon }_\mathrm{B}}{1+\overline{\varepsilon }_%
\mathrm{B}}%
f_{-}\left( t\right) \mid \overline{B}^0\rangle , \\
&\mid &\overline{B}_\mathrm{phys}^0\left( t\right) \rangle =\frac{1+\overline{%
\varepsilon }_\mathrm{B}}{1-\overline{\varepsilon }_\mathrm{B}}f_{-}\left( t\right)
\mid B^0\rangle +f_{+}\left( t\right) \mid \overline{B}^0\rangle ,
\end{eqnarray}
with

\begin{equation}
f_{\pm }\left( t\right) =\frac 12\left\{ \exp \left[ -\rmi\left( M_\mathrm{L}-\frac
\rmi2\Gamma _\mathrm{L}\right) t\right] \pm \exp \left[ -\rmi\left( M_\mathrm{H}-%
\frac \rmi2\Gamma _\mathrm{H}\right) t\right] \right\} ,
\end{equation}
where $M_\mathrm{L\left( H\right) }$ and $\Gamma _\mathrm{L\left( H\right) }$ are the
mass and the inverse of the lifetime of $B_\mathrm{L\left( H\right) }$.

For an event where both $B$ mesons undergo semileptonic decay, we can define
a charge asymmetry of such events as: \cite{cleo}

\begin{equation}
a_\mathrm{ll}=\frac{N\left( l^{+}l^{+}\right) -N\left( l^{-}l^{-}\right) }{N\left(
l^{+}l^{+}\right) +N\left( l^{-}l^{-}\right) },
\end{equation}
where $N\left( l^{\pm }l^{\pm }\right) $ indicates the number of pairs $%
l^{\pm }l^{\pm }$ produced in the decay.
The numbers of leptonic pairs $N\left( l^{+}l^{+}\right) $ and $N\left(
l^{-}l^{-}\right) $ are related to $\overline{\varepsilon }_\mathrm{B}$ by:

\begin{eqnarray}
N\left( l^{\pm }l^{\pm }\right) &=&\int_0^\infty \left| \left\langle l^{\pm
}l^{\pm }\nu _\mathrm{l}\nu _\mathrm{l}\left( \overline{\nu }_\mathrm{l}\overline%
{\nu }_\mathrm{l}\right) X^{\mp
}X^{\mp }\left| H\right| B_\mathrm{phys}^0\overline{B}_\mathrm{phys}^0\right\rangle
\right| ^2\rmd t \\
&=&\int_0^\infty \left| \left\langle l^{+}l^{+}\nu _\mathrm{l}\nu _\mathrm{l}X^{-}X^%
{-}\left| H\right| B^0B^0\right\rangle f_{+}\left( t\right) f_{-}\left( t\right)
\left( \frac{1\pm \overline{\varepsilon }_\mathrm{B}}{1\mp \overline{\varepsilon }_%
\mathrm{B}}\right) \right| ^2\rmd t.  \nonumber
\end{eqnarray}
Thus:

\begin{equation}
a_\mathrm{ll}=\frac{\left| 1+\overline{\varepsilon }_\mathrm{B}\right| ^4-\left| 1%
-\overline{\varepsilon }_\mathrm{B}\right| ^4}{\left| 1+\overline{\varepsilon }_%
\mathrm{B}\right|^4+\left| 1-\overline{\varepsilon }_\mathrm{B}\right| ^4}\approx 4Re%
\overline{\varepsilon }_\mathrm{B}.
\end{equation}

According to the last result, a dilepton asymmetry is directly related to a
nonvanishing value of $\overline{\varepsilon }_\mathrm{B}$ which measure the amount
of indirect $CP$ violation. This result is model independent and, therefore,
it entitles us to calculate the dilepton asymmetry $a_\mathrm{ll}$ via the calculation
of the parameter $\overline{\varepsilon }_\mathrm{B}$ for any kind of particle model.

\section{Description of the $LRSM$}

The $LRSM$ is based on the group of symmetries $SU\left( 2\right) _\mathrm{L}\otimes
SU\left( 2\right) _\mathrm{R}\otimes U\left( 1\right) _\mathrm{B-L}\otimes C\otimes P$
which assure both the parity and $CP$ conservation. According to this, it is possible
to assign the leptons and quarks the following quantum numbers:

\begin{eqnarray}
Q_\mathrm{L} &=&\left( 
\begin{tabular}{c}
$u_\mathrm{L}$ \\ 
$d_\mathrm{L}$%
\end{tabular}
\right) \equiv \left( 2,1,\frac 13\right) ,\hspace{7mm}Q_\mathrm{R}=\left( 
\begin{tabular}{c}
$u_\mathrm{R}$ \\ 
$d_\mathrm{R}$%
\end{tabular}
\right) \equiv \left( 1,2,\frac 13\right) ,  \nonumber \\
\psi _\mathrm{L} &=&\left( 
\begin{tabular}{c}
$\nu _\mathrm{L}$ \\ 
$e_L$%
\end{tabular}
\right) \equiv \left( 2,1,-1\right) ,\hspace{9mm}\psi _\mathrm{R}=\left( 
\begin{tabular}{c}
$\nu _\mathrm{R}$ \\ 
$e_\mathrm{R}$%
\end{tabular}
\right) \equiv \left( 1,2,-1\right) ,
\end{eqnarray}
where the $U\left( 1\right) $ generator corresponds to the $B-L$ quantum
number of the multiplet. \cite{leftright, deshpande, barenboim, yeinzon, kiers}
The gauge bosons consist of two triplets:

\begin{equation}
\mathbf{W}_{\mu \mathrm{L}}=\left(  
\begin{array}{c}
W_\mu ^{+} \\  
Z_\mu ^0 \\  
W_\mu ^{-}
\end{array}
\right) _\mathrm{L}\equiv \left( 3,1,0\right) ,\hspace{3mm}\mathbf{W}_%
{\mu \mathrm{R}}=\left(  
\begin{array}{c}
W_\mu ^{+} \\  
Z_\mu ^0 \\  
W_\mu ^{-}
\end{array}
\right) _\mathrm{R}\equiv \left( 1,3,0\right) ,
\end{equation}
and one singlet:  
\begin{equation}
\mathbf{B}_\mu =B_\mu ^0\equiv \left( 1,1,0\right) .
\end{equation}

Due to the existence of a discrete parity symmetry, the model must be
invariant under the transformations: $\psi _\mathrm{L}\longleftrightarrow \psi
_\mathrm{R}$, $Q_\mathrm{L}\longleftrightarrow Q_\mathrm{R}$, and
$W_\mathrm{L}\longleftrightarrow W_\mathrm{R}$.

To break the symmetry, and give mass to bosons and fermions, it is necessary
to introduce a bidoublet $\Phi $ and two scalar triplets $\Delta _\mathrm{L}$ and $%
\Delta _\mathrm{R}$ which can be written in a convenient matrix representation $%
2\times 2$: \cite{deshpande,barenboim,yeinzon,kiers} 

\begin{equation}
\Phi =\left( 
\begin{tabular}{cc}
$\phi _1^0$ & $\phi _1^{+}$ \\ 
$\phi _2^{-}$ & $\phi _2^0$%
\end{tabular}
\right) \equiv \left( 2,2,0\right),
\end{equation}

\begin{equation}
\Delta _\mathrm{L}=\left(  
\begin{array}{cc}
\frac{\delta _\mathrm{L}^{+}}{\sqrt{2}} & \delta _\mathrm{L}^{++} \\  
\delta _\mathrm{L}^0 & \frac{-\delta _\mathrm{L}^{+}}{\sqrt{2}}
\end{array}
\right) \equiv \left( 3,1,2\right)\qquad\Delta _\mathrm{R}=\left(  
\begin{array}{cc}
\frac{\delta _\mathrm{R}^{+}}{\sqrt{2}} & \delta _\mathrm{R}^{++} \\  
\delta _\mathrm{R}^0 & \frac{-\delta _\mathrm{R}^{+}}{\sqrt{2}}
\end{array}
\right) \equiv \left( 1,3,2\right).
\end{equation}
We need to introduce these elements so that the new vectorial
physical bosons $W_\mathrm{R}$ and $Z^{\prime} $ get heavy masses compatible with
the experimental bounds.

With these scalar elements, the most general scalar potential $\left(
V\right) $ and Yukawa lagrangian for quarks $\left( \mathcal{L}_{\mathrm{Y}%
}^\mathrm{q}\right) $ and leptons $\left( \mathcal{L}_{\mathrm{Y}}^%
\mathrm{l}\right) $, which are invariants under the manifiest discrete
left-right symmetry $\Phi \longleftrightarrow \Phi ^{\dagger }$ and $\Delta
_\mathrm{L}\longleftrightarrow \Delta _\mathrm{R}$, can be written as:

\begin{equation}
V=V_\Phi +V_\Delta +V_{\Phi \Delta },  \label{potescalar}
\end{equation}
where: \cite{deshpande,barenboim,yeinzon,kiers}

\begin{eqnarray*}
\mathbf{V}_\Phi &=&-\mu _1^2\Tr\left( \Phi ^{\dagger }\Phi \right) -\mu
_2^2\left[ \Tr\left( \widetilde{\Phi }\Phi ^{\dagger }\right) +\Tr\left(  
\widetilde{\Phi }^{\dagger }\Phi \right) \right] \\
&&+\lambda _1\left[ \Tr\left( \Phi \Phi ^{\dagger }\right) \right] ^2+\lambda
_2\left\{ \left[ \Tr\left( \widetilde{\Phi }\Phi ^{\dagger }\right) \right]
^2+\left[ \Tr\left( \widetilde{\Phi }^{\dagger }\Phi \right) \right]
^2\right\} \\
&&+\lambda _3\left[ \Tr\left( \widetilde{\Phi }\Phi ^{\dagger }\right)
\Tr\left( \widetilde{\Phi }^{\dagger }\Phi \right) \right] \\
&&+\lambda _4\left\{ \Tr\left( \Phi ^{\dagger }\Phi \right) \left[ \Tr\left(  
\widetilde{\Phi }\Phi ^{\dagger }\right) +\Tr\left( \widetilde{\Phi }%
^{\dagger }\Phi \right) \right] \right\} ,
\end{eqnarray*}

\begin{eqnarray*}
\mathbf{V}_\Delta &=&-\mu _3^2\left[ \Tr\left( \Delta _\mathrm{L}\Delta
_\mathrm{L}^{\dagger}\right) +\Tr\left( \Delta _\mathrm{R}\Delta _\mathrm{R}%
^{\dagger }\right) \right] \\
&&+\rho _1\left\{ \left[ \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}^%
{\dagger }\right)
\right] ^2+\left[ \Tr\left( \Delta _\mathrm{R}\Delta _\mathrm{R}^{\dagger }\right)
\right]^2\right\} \\
&&+\rho _2\left[ \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}\right) \Tr\left(
\Delta _\mathrm{L}^{\dagger }\Delta _\mathrm{L}^{\dagger }\right) +\Tr\left(
\Delta _\mathrm{R}\Delta _\mathrm{R}\right)
\Tr\left( \Delta _\mathrm{R}^{\dagger }\Delta _\mathrm{R}^{\dagger }\right) \right] \\
&&+\rho _3\left[ \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}^{\dagger }\right)
\Tr\left( \Delta _\mathrm{R}\Delta _\mathrm{R}^{\dagger }\right) \right] \\
&&+\rho _4\left[ \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}\right) \Tr\left(
\Delta _\mathrm{R}^{\dagger }\Delta _\mathrm{R}^{\dagger }\right) +\Tr\left(
\Delta _\mathrm{L}^{\dagger}\Delta _\mathrm{L}^{\dagger }\right) \Tr\left(
\Delta _\mathrm{R}\Delta _\mathrm{R}\right) \right] ,
\end{eqnarray*}

\begin{eqnarray}
\mathbf{V}_{\Phi \Delta } &=&\alpha _1\left\{ \Tr\left( \Phi ^{\dagger }\Phi
\right) \left[ \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}^{\dagger }\right)
+\Tr\left( \Delta _\mathrm{R}\Delta _\mathrm{R}^{\dagger }\right) \right] \right%
\}  \nonumber \\
&&+\alpha _2\{\Tr\left( \widetilde{\Phi }^{\dagger }\Phi \right) \Tr\left(
\Delta _\mathrm{R}\Delta _\mathrm{R}^{\dagger }\right) +\Tr\left( \widetilde{\Phi }\Phi
^{\dagger }\right) \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}^{\dagger }\right%
)  \nonumber \\
&&+\Tr\left( \widetilde{\Phi }\Phi ^{\dagger }\right) \Tr\left( \Delta
_\mathrm{R}\Delta _\mathrm{R}^{\dagger }\right) +\Tr\left( \widetilde{\Phi }^{\dagger
}\Phi \right) \Tr\left( \Delta _\mathrm{L}\Delta _\mathrm{L}^{\dagger }\right)%
\}  \nonumber \\
&&+\alpha _3\left[ \Tr\left( \Phi \Phi ^{\dagger }\Delta _\mathrm{L}\Delta _\mathrm{L}%
^{\dagger}\right) +\Tr\left( \Phi ^{\dagger }\Phi \Delta _\mathrm{R}\Delta _\mathrm{R}%
^{\dagger }\right) \right]  \nonumber \\
&&+\beta _1\left[ \Tr\left( \Phi \Delta _\mathrm{R}\Phi ^{\dagger }\Delta _\mathrm{L}%
^{\dagger}\right) +\Tr\left( \Phi ^{\dagger }\Delta _\mathrm{L}\Phi \Delta _\mathrm{R}%
^{\dagger }\right) \right]  \nonumber \\
&&+\beta _2\left[ \Tr\left( \widetilde{\Phi }\Delta _\mathrm{R}\Phi ^{\dagger }\Delta
_\mathrm{L}^{\dagger }\right) +\Tr\left( \widetilde{\Phi }^{\dagger }\Delta _\mathrm{L}%
\Phi\Delta _\mathrm{R}^{\dagger }\right) \right]  \nonumber \\
&&+\beta _3\left[ \Tr\left( \Phi \Delta _\mathrm{R}\widetilde{\Phi }^{\dagger }\Delta
_\mathrm{L}^{\dagger }\right) +\Tr\left( \Phi ^{\dagger }\Delta _\mathrm{L}\widetilde%
{\Phi }\Delta _\mathrm{R}^{\dagger }\right) \right] ,
\end{eqnarray}

and
\begin{equation}
-\mathcal{L}_{\mathrm{Y}}^\mathrm{q}=\sum_{i,j=1}^3\left( h_{ij}^\mathrm{q}\overline{Q}%
_\mathrm{L}^i\Phi Q_\mathrm{R}^j+\widetilde{h}_{ij}^\mathrm{q}\overline{Q}_\mathrm{L}^i%
\widetilde{\Phi }Q_\mathrm{R}^j\right) +h.c.,
\label{yukawaq}
\end{equation}
\begin{equation}
\fl -\mathcal{L}_{\mathrm{Y}}^\mathrm{l}=\sum_{i,j=1}^3\left(h_{ij}^\mathrm{l}\overline%
{\psi}_\mathrm{L}^i\Phi \psi_\mathrm{R}^j+\widetilde{h}_{ij}^\mathrm{l}\overline{\psi}%
_\mathrm{L}^i\widetilde{\Phi }\psi_\mathrm{R}^j\right)+if_{ij}\left[ {\psi_\mathrm{L}^i}%
^\mathrm{T}C\tau _2\Delta _\mathrm{L}\psi_\mathrm{L}^j+\left(
\mathrm{L}\leftrightarrow \mathrm{R}\right) \right] +h.c.,  \label{leptonyukawa}
\end{equation}
where $\widetilde{\Phi }=\tau _2\Phi ^{*}\tau _2$, $h$, $\widetilde{h}
$, and $f$ are the Yukawa coupling matrices, and $C$ is the Dirac's charge
conjugation matrix.

As a consequence of the discrete left-right symmetry all the terms in the
potential are self-conjugate. We have chosen real coupling constants to avoid
explicit $CP$ violation.

The pattern of symmetry breaking, for the scalar bidoublet, is
achieved by:

\begin{equation}
\left\langle \Phi
\right\rangle =\frac 1{\sqrt{2}}\left( 
\begin{tabular}{cc}
$k_1e^{i\alpha }$ & $0$ \\ 
$0$ & $k_2$%
\end{tabular}
\right) , \label{vevs}
\end{equation}
and, for the scalar triplets, by:

\begin{equation}
\left\langle \Delta _\mathrm{L}\right\rangle  =\frac 1{\sqrt{2}}\left(  
\begin{array}{cc}
0 & 0 \\  
\upsilon _\mathrm{L} & 0
\end{array}
\right) \hspace{1cm} \left\langle \Delta _\mathrm{R}\right\rangle =
\frac 1{\sqrt{2}}\left(  
\begin{array}{cc}
0 & 0 \\  
\upsilon _\mathrm{R}e^{i\theta } & 0
\end{array}
\right),  
\end{equation}
where $k_1$, $k_2$, $\upsilon_\mathrm{L}$, $\upsilon_\mathrm{R}$, $\alpha $,
and $\theta $ are real numbers. There are some constraints on the values that
the vacuum expectation values $k_1,k_2,\upsilon _\mathrm{L},$ and $\upsilon
_\mathrm{R}$ may take: $\upsilon _\mathrm{L}$ must be much smaller than $k_1$
and $k_2\footnote{$\upsilon _\mathrm{L}\simeq k^2/\upsilon _\mathrm{R}$ where
$k^2=k_1^2+k_2^2\simeq \left(246\textbf{ }GeV\right) ^2.$}$ to keep the well
known experimental condition $M_\mathrm{W_L}^2/M_\mathrm{Z_L}^2\simeq \cos ^2%
\theta _\mathrm{W}$. \cite{deshpande} In addition it has been showed that,
without a fine tuning of the coupling constants, the right scale of the model
must be very large in order to avoid flavour changing neutral currents and
ensure the correct order of magnitude for the masses of the left-handed
neutrinos.\footnote{This assures that the right-weak bosons $W_\mathrm{R}^{+},
W_\mathrm{R}^{-},$ and $Z_\mathrm{R}^0$ get really heavy masses compatible with
the experimental bounds.} \cite{yeinzon} In fact, the right scale must be of the
order $\upsilon _\mathrm{R}\sim 10^7$ $GeV$. This is a very important issue because
most of the predictions at low energy of the $LRSM$ will be equal to those of the
$SM$.

A direct consequence of imposing $CP$ as a spontaneously broken symmetry,
together with the manifest left-right discrete symmetry $\Phi
\longleftrightarrow \Phi ^{\dagger }$, is that the Yukawa coupling matrices $%
h$ and $\widetilde{h}$ must be real and symmetric. This leads to a
relationship between the left and right $CKM$ matrices:

\begin{equation}
K_\mathrm{L}=K_\mathrm{R}^{*}.
\end{equation}

The only complex parameter in the quark mass matrices is the complex phase in
$\left\langle \Phi \right\rangle $. To break $CP$ spontaneously, we have to
search for a complex vev of the Higgs bosons. The vev $\left\langle \Phi
\right\rangle $ of the expression (\ref{vevs}) breaks the
$U\left( 1\right) _\mathrm{L-R}$ symmetry and is the only source of $CP$
violation in the quark sector. \cite{yeinzon}

\section{Calculation of $\overline{\varepsilon }_\mathrm{B}$ for the
$B_\mathrm{d}$ and $B_\mathrm{s}$ systems}

We can calculate $\overline{\varepsilon }_\mathrm{B}$, by solving the
Schr\"{o}dinger's equation for the $B^0-\overline{B}^0$ system: \cite
{wakaizumi,buchalla,rosner}

\begin{equation}
\rmi\frac{\rmd\Phi \left( t\right) }{\rmd t}=H\Phi \left( t\right) ,
\end{equation}
where 
\begin{equation}
H=\left( 
\begin{array}{cc}
M_{11}-\rmi\Gamma _{11}/2 & M_{12}-\rmi\Gamma _{12}/2 \\ 
M_{12}^{*}-\rmi\Gamma _{12}^{*}/2 & M_{11}-\rmi\Gamma _{11}/2
\end{array}
\right) ,
\end{equation}
and 
\begin{equation}
\Phi \left( t\right) =a_\mathrm{B}\left( t\right) \mid B^0\rangle
+a_{\overline{\mathrm{B}}}\mid \overline{B}^0\rangle,
\end{equation}
being $\mid B^0\rangle$ and $\mid \overline{B}^0\rangle$ the mutually
orthogonal vectors $\left( 
\begin{array}{c}
1 \\ 
0
\end{array}
\right) $ and $\left( 
\begin{array}{c}
0 \\ 
1
\end{array}
\right) $. 

The solution of this equation gives the following result:

\begin{equation}
Re\overline{\varepsilon }\approx \frac{M_{12}^\mathrm{R}\Gamma_{12}^\mathrm{I}%
-M_{12}^\mathrm{I}\Gamma _{12}^\mathrm{R}}{4\left( M_{12}^\mathrm{R}\right) ^2+
\left( \Gamma_{12}^\mathrm{R}\right) ^2},
\end{equation}
where the superscript indicates the real part $\left( R\right) $ or the
imaginary part $\left( I\right) $.
Thus, to calculate the parameter $Re\overline{\varepsilon }_\mathrm{B}$, we
have to calculate the matrix elements $M_{12}$ and $\Gamma _{12}$ through
the so called ``box diagram'' showed in the Figure1.
\cite{wakaizumi,buchalla,rosner,hagelin,tesis}

In the figure \ref{caja}, $W$ corresponds to the charged vectorial bosons of
the respective model. Aditionally, we have included the contribution of the
scalar and pseudoscalar singly charged bosons.

To make the calculations we have used the central values for the elements
in the $CKM$ matrix as well as the central value for the $CKM$ complex phase
$\delta$.\footnote{$\delta$ is function of $\alpha$ in the context of the
$LRSM$, and matches its experimental value when $\alpha\approx 0$. \cite{%
spontaneous}} \cite{pdg,tesis,ciuchini} The result of the calculation in the
$SM$ shows the following central values, for the $B_\mathrm{d,s}^0-\overline{B}%
_\mathrm{d,s}^0$ systems, which we can compare to the experimental results:
\cite{babar}
\begin{eqnarray}
\fl Re\overline{\varepsilon }_\mathrm{B_d} &\approx &1.1\times 10^{-3},\hspace{1cm}
\left( Re\overline{\varepsilon }_\mathrm{B_d}\right)
_\mathrm{ex.}=1.2\pm 2.9\pm 3.6\times 10^{-3}, \\
\fl Re\overline{\varepsilon }_\mathrm{B_s} &\approx &-1.4\times 10^{-5},\hspace{5mm}
\textit{There are no experimental results for } {}
Re\overline{\varepsilon }_\mathrm{B_s}. 
\end{eqnarray}

What we can see is that the $SM$ prediction for the central value of the parameter
$Re\overline{\varepsilon }_\mathrm{B_d}$ lies in the middle of the experimental
width. Future improvements on the measurement of the dilepton asymmetry will
reduce this width and let us constrain even more the possible candidate models
of new physics. At the present there is no measurement of the dilepton assymetry
in the $B_\mathrm{s}$ system, which is very unpleasant because it would give us more
confidence about the range of validity of our models of $CP$ violation.  

According to the reference \cite{yeinzon} there are four explored cases in the
$LRSM$ corresponding to the combinations of maximum and no $CP$ violation
both in the quark and in the lepton sectors. Following this reference, the cases
$I$ and $II$ corresponding to a maximal $CP$ violation in the quark sector
$(\alpha=\pi/2)$ are ruled out because they present flavour changing neutral
currents at a level which is inconsistent with the current phenomenology.
In contrast, the cases $III$ and $IV$ corresponding to no $CP$ violation in the
quark sector $(\alpha=0)$ do not present any phenomenological inconsistency and
are appropiated to calculate the matrix elements $M_{12}$ and $\Gamma _{12}%
$.\footnote{We have used an universal value for the dimensionless parameters
of the scalar potential equal to $0.7$.} To avoid an explicit origin for the $CP$
violation in the quark sector, we have to adjust $\alpha $ to be small enough so
as not to change the main features and results found, and to lead to the correct
experimental value for the $CKM$ phase of the $SM$. Effectively, to obtain the
correct value for the $CKM$ phase of the $SM$, we need a value close to zero for
the spontaneous $CP$ phase $\alpha $. \cite{spontaneous}

The case $III$ corresponds to no $CP$ violation in the quark sector
$(\alpha=0)$, and maximum $CP$ violation in the lepton sector $(\theta=\pi/2)$
and since the right scale of the model is very large $\left( \upsilon _\mathrm{R}%
\sim10^7\textbf{ }GeV\right)$ there is no any signal of new physics except for the
presence of four additional scalar bosons at the electroweak scale. From these
four bosons two are neutral, one is singly charged, and the other is doubly
charged. \cite{yeinzon} For the present calculation we only need singly charged
scalar bosons interacting  with the quark sector, i.e., linear combinations of
the singly charged fields $\phi _1^{\pm}$ and $\phi _2^{\pm}$ present in the
scalar bidoublets.\footnote{See equation (\ref{yukawaq}).} However, as we can see
in \cite{yeinzon}, the new singly charged scalar boson is a linear combination of
the singly charged fields $\delta _\mathrm{L}^{\pm}$ and $\delta _\mathrm{R}^{\pm}$
in the scalar triplets and, therefore, there is no new physics contribution to the
matrix elements $M_{12}$ and $\Gamma _{12}$.

The case $IV$, corresponding to no $CP$ violation both in the quark sector
$(\alpha=0)$ and in the lepton sector $(\theta=0)$, is the easiest of all the
cases because it reduces to the $SM$ in the limit in which $\upsilon _\mathrm{R}$
goes to the infinity and $\alpha\approx 0$. \cite{deshpande, yeinzon} Therefore,
the predictions of both the $LRSM$ and the $SM$ on the parameter
$Re\overline{\varepsilon }_\mathrm{B}$ are equal.

\section{Conclusions}
The study of the $CP$ violation in the $B$ mesons systems is a very strong
source of developments in particle physics. In particular, the improvements in the
measurement of the dilepton asymmetry will let us understand more about the origin
of the $CP$ violation and the possible models of new physics we
can propose to explain it. Given the present experimental uncertainty in the value
of $Re\overline{\varepsilon }_\mathrm{B_d}$ we have shown that the $SM$ prediction
agrees with the experiment and its order of magnitude is the same as in the kaon
system: $10^{-3}$. We have also shown that the $SM$ predicts a value for
$Re\overline{\varepsilon }_\mathrm{B_s}$ two orders of magnitude below the
prediction for $Re\overline{\varepsilon }_\mathrm{B_d}$ and with opposite sign.

Lots of models can be proposed to explain the origin of the $CP$ violation phenomenon.
Among these models is the $LRSM$ which give us a natural explanation for the
parity and $CP$ violation as well as a physical meaning for the hypercharge
quantum number. Additionally, this model explains the smallness of the
neutrino masses; gives us a framework to study the $CP$ violation in the
lepton sector, which has not been observed yet but that we will surely be able
to observe in the foreseeable future; and exhibits new scalar bosons at the
electroweak scale which are the focus of most current and future experimental work.
\cite{exp} However, since the right scale of the model is very large
$\left( \upsilon _\mathrm{R}\sim10^7\textbf{ }GeV\right)$, the new physics sector
decouples from the quark sector and, therefore, most of the predictions at low energy
of the $LRSM$ will be equal to those of the $SM$, in particular those corresponding
to the parameters $Re\overline{\varepsilon }_\mathrm{B_{d,s}}$.

\ack
This work was supported by Universidad Nacional de Colombia through the contract
DIB 803696. We are specially indebted to Yeong G. Kim and David H. Lyth for their
comments on the manuscript. Y.R. wants to acknowledge Lancaster University for
its financial help and the Colombian agencies Fundaci\'{o}n Mazda para el Arte
y la Ciencia, COLCIENCIAS, and COLFUTURO for their postgraduate scholarships. 

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\Figures
\begin{figure}
\caption{\label{caja}Generic box diagram to calculate the non diagonal matrix
elements $M_{12}$ and $\Gamma _{12}$ for the $B^0-\overline{B}^0$ system.
$q_1$ is the quark $b$, $q_2$ is the quark $d$ for
$B_\mathrm{d}$ or $s$ for $B_\mathrm{s}$ system, and $Q$ runs over the quarks
$u$, $c$, and $t$.}
\end{figure}

\end{document}

