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\title{Minimal 3-3-1 model, lepton mixing and muonium-antimuonium conversion}
\author{ A. Gusso$^{(a)}$\footnote{electronic address:
gusso@ift.unesp.br}, C. A de S. Pires$^{(b)}$\footnote{electronic
address: cpires@fisica.ufpb.br} and P. S. Rodrigues da
Silva$^{(a)}$\footnote{electronic address: fedel@ift.unesp.br}}
\address{\it (a) Instituto de  F\'{\i}sica
Te\'{o}rica, Universidade Estadual Paulista, Rua Pamplona 145,
01405-900 S\~{a}o Paulo - SP, Brazil.\\
\it (b) Departamento de F\'{\i}sica, Universidade Federal da
Para\'{\i}ba, Caixa Postal 5008, 58051-970, Jo\~ao Pessoa - PB,
Brazil.}
\date{\today}
\maketitle
%%%%%%%%%

\begin{abstract}
We investigate constraints imposed by lepton mixing and
muonium-antimuonium conversion in a minimal version of $3-3-1$
model. Demanding agreement with recent results on neutrino
oscillation we show that minimal $3-3-1$ has to be extended in
order to correctly accommodate its mixing pattern. On the other
hand muonium-antimuonium conversion puts a lower bound on the
vector bilepton mass in a specific configuration of this model,
threatening its self-consistency. We then assume that neutrino
physics can be handled separately by the required extension, and
include the scalar bileptons contribution in the
muonium-antimuonium conversion analysis in order to see if this
can soften the vector bilepton bound. We find that effects of
scalars are highly suppressed and do not represent any
improvement on this bound in the minimal version of $3-3-1$.
\end{abstract}

%\pacs{PACS numbers: 12.60.cn; 12.90.+b}
%\maketitle
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%%%%
\section{introduction}
The $3-3-1$ model~\cite{viceframp} is an alternative to the
Standard Model (SM) of electro-weak interactions, possessing very
interesting features concerning its leptonic physics. The
presence of bileptons (bosons carrying double lepton number) could
naturally lead to rare lepton decays that violate family number
but not total lepton number. We must stress, however, that the
presence of such bileptons in $3-3-1$ is not a source of lepton
number violation (which, in contrast, happens when neutrino
oscillation is considered). These peculiar features make such
bileptons very interesting phenomenologically.

Another interesting characteristic of the $3-3-1$ model is that
its framework can naturally provide neutrino masses~\cite{joshi}
at tree level. Notice that when the model was suggested there was
no evidence of neutrino mass, and the model should circumvent this
masslessness of neutrinos by adding a discrete symmetry or a fine
tuning~\cite{footpisano}. However, the most recent experimental
results, including SNO~\cite{SNO} and K2K~\cite{K2K} data, are
corroborating the hypothesis of neutrino
oscillation~\cite{BahcallConcha}, which implies the existence of
at least two massive neutrinos. As a consequence the minimal 3-3-1
model becomes a more attractive alternative to the SM.

However, in the minimal version of the $3-3-1$ model the neutrino
mass and the charged lepton mass are not
independent~\cite{footpisano}, instead, both receive contribution
from a scalar sextet. From the phenomenological point of view, the
updated neutrino oscillation results are responsible for the main
constraint on the possible mixing in the leptonic
sector~\cite{ConchaNir}. Since the charged lepton mass matrix is
necessarily non-diagonal and non-hermitian in this minimal
$3-3-1$ model, we must check if the neutrino mixing suggested by
the experiment admits the texture the model imposes to neutrino
and charged lepton mass matrices.

On the other hand, the interactions of minimal $3-3-1$ model,
besides leading to rare lepton decays, leave room for other
peculiar processes not present in SM. One such process is the
conversion of a bound state called muonium ($\mu^+ e^-$) to the
antimuonium ($\mu^- e^+$). In what concerns muonium-antimuonium
($M-\overline{M}$) conversion, new experimental results were
reported~\cite{muoexp}, and the non-observation of this
transition has led to an upper bound on the coupling describing
this conversion through its effective Hamiltonian. In
Ref.~\cite{muoexp}, it was posed that such a bound implies a
lower bound on the doubly charged vector bilepton in the minimal
$3-3-1$, $M_{U^{++}}\gtrsim 850$ GeV, which is in clear
contradiction to the predicted upper bound required by
self-consistency of the model, namely, $M_{U^{++}}\lesssim 600$
GeV~\cite{mu++limit}\footnote{It is appropriate to note that
extensions adding a Higgs octet of $SU(3)_L$, as observed in
Ref.~\cite{mu++limit}, would be capable of augmenting this upper
limit to $M_{U^{++}}\lesssim 800$, but here we are going to stick
with the minimal version only.}. Facing this situation, the
authors in Ref.~\cite{muoexp} suggested the ruling out of the
minimal version of $3-3-1$.

Afterwards, in Ref.~\cite{remark}, a qualitative analysis was done
claiming that the doubly charged vector bilepton mass can still
escape from this upper bound. The author remarked that the
previous analysis of $M-\overline{M}$ conversion only considered
the $U^{++}$, and no mixing in the leptonic sector. The argument
lies on the fact that the scalars and charged lepton mixing could
help bringing down the upper bound on $M_{U^{++}}$ to values
around 600 GeV, keeping the model on safe grounds.

The purpose of this work is twofold: to find out if the minimal
version of the model accommodates the expected neutrino mass and
mixing and, taking into account the suggestion in
Ref.~\cite{remark}, to determine whether $U^{++}$ can be relieved
from the constraint established by $M-\overline{M}$ conversion in
Ref.~\cite{muoexp}.

We first introduce the necessary ingredients of the model in
Sec.~\ref{sec1}, obtaining the pattern of neutrino and charged
lepton mass matrix. We also examine the distinct possibilities for
the lepton mixing matrices based on results of neutrino
oscillation, verifying that minimal $3-3-1$ needs to be extended
if agreement with neutrino data is demanded. In Sec.~\ref{sec2},
we explore the proposed way out for the model concerning the
constraint from $M-\overline{M}$ conversion. Considering the fact
that neutrino physics can be implemented without any further
consequences to $M-\overline{M}$ conversion, and taking charged
lepton mass diagonal, we analyze the contribution of the remaining
spectrum in order to establish the true role of scalars in
lowering the bound on $M_{U^{++}}$. Finally, in Sec.~\ref{sec3},
we present our conclusions.

\section{Lepton masses and mixing in the minimal 3-3-1 model}
\label{sec1}

In this section, we are going to present the ingredients
necessary to obtain the mass matrix for neutrinos and charged
leptons in the minimal $3-3-1$ model. The gauge group under
consideration is $SU_c(3)\otimes SU_L(3)\otimes U_X(1)$, which
spontaneously breaks to the usual $SU_c(3)\otimes SU_L(2)\otimes
U_Y(1)\rightarrow SU_c(3)\otimes U_{EM}(1)$. In its minimal
version~\cite{footpisano}, the scalar content which interacts
with the lepton sector is composed by a triplet, $ \eta =( \eta^0
, \eta^-_1 , \eta^+_2)^T \sim({\bf 1},{\bf 3}, 0)$  and a sextet
$S$,
%
\bea
S=\left(
\begin{array}{ccc}
 \sigma^0_1 & h_1^{-} & s_2^{+} \\
 h_1^{-} & H_1^{--} & \sigma^0_2 \\
 h_2^{+} & \sigma^0_2 & H_2^{++}
\end{array}
\right)\sim({\bf 1},{\bf 6},0)\,,
 \label{sextet}
 \eea
%
where we have included their assigned transformation properties
under the gauge group $SU_c(3)\otimes SU_L(3)\otimes U_X(1)$.

The leptons are arranged in the fundamental representation of the
gauge group $SU_L(3)$, and for each family we have,
%
\bea \Psi_{a_L} =\left(
\begin{array}{c}
  \nu_a \\
  e_a \\
  e_a^c
\end{array}\right)_L\sim({\bf 1},{\bf 3},0)\,,
\label{leptons} \eea
%
where $a=1,\,2,\,3$ labels the different families, and the
superscript $c$ means the charge conjugated field.

The Yukawa interactions that generate the lepton masses are given
by,
%
\be {\cal L}^Y_l=\frac{1}{2}G_{ab}\overline{ (\Psi_{aL})^c}
S^*\Psi_{b_L}+ \frac{1}{2}F_{ab}\epsilon^{i j k}
\overline{(\Psi_{iaL})^c} \Psi_{jbL} \eta^*_k + \mbox{H.c.}
\label{yukawa} \ee
%
When the neutral scalars develop their respective vacuum
expectation values, $<\sigma^0_1>= v_{\sigma_1}$,
$<\sigma^0_2>=v_{\sigma_2}$  and $<\eta^0>= v_{\eta}$, neutrinos
and charged leptons acquire the following mass matrices,
%
\bea &&M_\nu= \left (
\begin{array}{lcr}
G_{11} & G_{12} & G_{13} \\
G_{21} & G_{22} & G_{23} \\
G_{31} & G_{32} & G_{33}
\end{array}
\right ) v_{\sigma_1}\, \label{lepmass1} \eea
%
and \bea &&M_l= \left (
\begin{array}{lcr}
\,\,\,\,\,\,\,\,\,G_{11}v_{\sigma_2} & (G_{12}v_{\sigma_2}+F_{12}v_\eta )&
(G_{13}v_{\sigma_2}+F_{13}v_\eta ) \\
(G_{21}v_{\sigma_2}-F_{12}v_\eta ) & G_{22}v_{\sigma_2} &
(G_{23}v_{\sigma_2}+F_{23}v_\eta ) \\
(G_{31}v_{\sigma_2}-F_{13}v_\eta ) & (G_{32}v_{\sigma_2}-F_{23}v_\eta ) &
G_{33}v_{\sigma_2}\,\,\,\,\,\,\,\,\,
\end{array}
\right ), \label{lepmass2} \eea
%
respectively.

The question we are going to pursue next is whether it is
possible to have some configuration where these interdependent
matrices assume a realistic form. In other words, if there exist
values of Yukawa couplings, $G_{ab}$ and $F_{ab}$, for which the
expected neutrino mixing as well as the charged lepton mass
spectrum can come about naturally.

We start by noticing that lepton mass matrices in
Eqs.~(\ref{lepmass1}) and (\ref{lepmass2}), can be diagonalized as
follows,
%
\be M_l^D = V_{eL}^\dag M_lV_{eR}\,,\,\,\,\,\,\, M_\nu^D =
V_{\nu}^\dag M_\nu V_{\nu}\,, \label{diagmass} \ee
%
where we have used the notation, $M^D\equiv Diag(m_1,m_2,m_3)$,
with $m_i$ being the physical lepton masses. The matrices $V$
transform the lepton fields in the interaction eigenstates into
mass eigenstates and, in principle, they are different for
left-handed and right-handed fields. These diagonalization
matrices combine themselves in the charged current of the model
which, after the $3-3-1$ breaking to the $SU(3)_c \otimes
U(1)_{EM}$, is given by:
%
\be {\cal L}^{CC}_{l}= -\frac{g}{\sqrt{2}}\left\{\overline{ e_L}
\gamma^\mu O^W\nu_LW^-_\mu + \overline{(e_R)^c}O^V\gamma^\mu
\nu_L V^+_\mu + \overline{(e_R)^c}O^U \gamma^\mu
e_LU^{++}_\mu\right\} + \mbox{H.c.}\,, \label{cc} \ee
%
where we are omitting family indices and $O^W=V^T_{eL} V_{\nu} $,
$O^V=V^T_{eR} V_{\nu}$ and $O^U=V^T_{eR} V_{eL}$ are the mixing
matrices which can be extracted from the
phenomenology~\footnote{The mixing matrices appearing in this
work are of the CKM type and we will be considering the simplest
case of a zero CP violating phase, since this phase is irrelevant
throughout our analysis. This means that these matrices are real,
which allows us to work with the transpose of the matrices
involved, instead of hermitian conjugate.}.

The most recent experimental results from neutrino
oscillation~\cite{SNO,K2K} suggest the so called maximal mixing
matrix~\cite{maximal},
%
\bea O^W=V^T_{eL} V_{\nu L}= \left (
\begin{array}{lcr}
c & s & 0 \\
\frac{-s}{\sqrt{2}} & \frac{c}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
 \frac{s}{\sqrt{2}}& -\frac{c}{\sqrt{2}}  & \frac{1}{\sqrt{2}}
\end{array}
\right )\,, \label{maxmix} \eea
%
where we have denoted $c\equiv\cos{\theta}$ and
$s\equiv\sin{\theta}$.

According to Eq.~(\ref{lepmass2}), the charged lepton mass matrix
is necessarily non-diagonal, compelling us to analyze only two
cases. The first one is the case where neutrino mass matrix is
diagonal, and the mixing in the charged current is due only to
the charged lepton sector. The second possibility is when both,
neutrino and charged lepton mass matrices, are non-diagonal and
the mixing is the product $O^W=V^T_{eL} V_{\nu}$.

\subsection{First Case}

Let us begin our analysis by the case where neutrinos are in a
diagonal mass basis. In this case, $O^W=V^T_{eL}$, $M_\nu =
Diag(G_{11}v_{\sigma_1},G_{22}v_{\sigma_1},G_{33}v_{\sigma_1})$
and $M_l$ takes the form
%
\bea M_l= \left (
\begin{array}{lcr}
G_{11}v_{\sigma_2} & F_{12}v_\eta & F_{13}v_\eta  \\
-F_{12}v_\eta  & G_{22}v_{\sigma_2} & F_{23}v_\eta  \\
-F_{13}v_\eta  & -F_{23}v_\eta  & G_{33}v_{\sigma_2}
\end{array}
\right )\,. \label{charglepmass} \eea
%
This texture is very peculiar since the non-diagonal elements are
anti-symmetric and, as far as we know, there is no approach
dealing with such a texture in literature. Now let us check if we
are able to obtain the mixing in Eq.~(\ref{maxmix}) departing from
this texture. It is suitable to remark that when $V_\nu = I$ then
$O^W=V^T_{eL}$ and $O^V=V^T_{eR}$, meaning that $V^T_{eL}$ is
given solely by the maximal mixing matrix, Eq.~(\ref{maxmix}),
and $O^V$ can have the pattern of a CKM matrix, leading to:
%
\bea V_{eR}= \left (
\begin{array}{ccccc}
c_{13} c_{12} &\,\,&  -s_{12} c_{23}-s_{23} s_{13} c_{12}&\,\,&
s_{23} s_{12} -s_{13}
c_{23} c_{12} \\
s_{12} c_{13} &\,\,& c_{23} c_{12}-s_{23} s_{13} s_{12} &\,\,&
-s_{23} c_{12}-s_{13}
s_{12} c_{23}  \\
s_{13} &\,\,& s_{23} c_{13} &\,\,& c_{23} c_{13}
\end{array}
\right )\,, \label{VeR} \eea
%
where we have used the short form $c_{ij}\equiv\cos{\theta_{ij}}$
and $s_{ij}\equiv\sin{\theta_{ij}}$.

Having determined the matrices $V_{eL}$ and $V_{eR}$ the texture
of the charged lepton mass matrix, Eq.~(\ref{charglepmass}), is
connected to these through $M_l=V_{eL} M^D_l V^T_{eR}$,
%
\bea V_{eL} M^D_l V^T_{eR} &=& \left(
\begin{array}{lcr}
M_{11} &M_{12} & M_{13} \\
M_{21} & M_{22} & M_{23}  \\
M_{31} & M_{32} & M_{33}
\end{array}
\right)\,, \label{clmtexture} \eea
%
where $M^D_l = Diag(m_e , m_\mu , m_\tau)$, and we have defined,
%
\begin{eqnarray} &&M_{12} = cc_{13}s_{12}m_e+\frac{s}{\sqrt{2}}
[(s_{12}s_{13}s_{23}-c_{12}c_{23})m_\mu
-(c_{12}s_{23}+s_{12}s_{13}c_{23})m_\tau]\,,\nonumber \\
&&M_{21}=sc_{12}c_{13}m_e
-\frac{c}{\sqrt{2}}[(s_{12}c_{23}+c_{12}s_{13}s_{23})m_\mu
+(s_{12}s_{23}-c_{12}s_{13}c_{23})m_\tau]\,,\nonumber \\
&&M_{13}=cs_{13}m_e+\frac{s}{\sqrt{2}}[-c_{13}s_{23}m_\mu+
c_{13}c_{23}m_\tau]\,, \nonumber \\
&&M_{31}=\frac{1}{\sqrt{2}}[-(s_{12}c_{23}+c_{12}s_{13}s_{23})m_\mu
+(s_{12}s_{23}-c_{12}s_{13}c_{23})m_\tau]\,,\nonumber \\
&&M_{23}=ss_{13}m_e+\frac{c}{\sqrt{2}}[c_{13}s_{23}m_\mu
-c_{13}c_{23}m_\tau]\,,
\nonumber \\
&&M_{32}=\frac{1}{\sqrt{2}}[(c_{12}c_{23}-s_{12}s_{13}s_{23})m_\mu
-(s_{12}s_{13}c_{23}+c_{12}s_{23})m_\tau ]\,,\nonumber \\
&&M_{11}=cc_{12}c_{13}m_e +\frac{s}{\sqrt{2}}[(s_{12}c_{23}
+c_{12}s_{13}s_{23})m_\mu +(s_{12}s_{23}
-c_{12}s_{13}c_{23})m_\tau]\,,\nonumber \\
&&M_{22}=ss_{12}c_{13}m_e+\frac{c}{\sqrt{2}}[(c_{12}c_{23}-
s_{12}s_{13}s_{23})m_\mu+(c_{12}s_{23}+s_{12}s_{13}c_{23})m_\tau]\,,\nonumber \\
&&M_{33}=
\frac{1}{\sqrt{2}}[c_{13}s_{23}m_\mu+c_{13}c_{23}m_\tau]\,.
\label{elements} \end{eqnarray}
%

According to Eqs.~(\ref{lepmass1}) and (\ref{lepmass2}), once we
impose the neutrino mass matrix is diagonal, the texture of the
charged lepton mass matrix automatically emerges and it is such
that the non-diagonal elements are anti-symmetric as displayed in
Eq.~(\ref{charglepmass}). This implies that the non-diagonal
elements in Eq.~(\ref{elements}) above have to be anti-symmetric,
$M_{ij}=-M_{ji}$ for $i\neq j$. These conditions can be
translated into the following set of non-linear equations for the
mixing angles:
%
\bea &&\sqrt{2}[cs_{12}c_{13} +sc_{12}c_{13}] m_e + [
s(s_{12}s_{13}s_{23}-c_{12}c_{23}) -
c(s_{12}c_{23}+c_{12}s_{13}s_{23})] m_\mu \nonumber \\
&& + [c(c_{12}s_{13}c_{23}-s_{12}s_{23})-s( c_{12}s_{23}
+ s_{12}s_{13}c_{23})]m_\tau=0\,,\nonumber \\
&&\sqrt{2}cs_{13}m_e -[sc_{13}s_{23} +
s_{12}c_{23}+c_{12}s_{13}s_{23}]m_\mu
+[sc_{13}c_{23}+s_{12}s_{23}-c_{12}s_{13}c_{23}]m_\tau=0\,,
\nonumber \\
&&\sqrt{2}ss_{13}m_e +[ cc_{13}s_{23}+
c_{12}c_{23}-s_{12}s_{13}s_{23}]m_\mu  -
[cc_{13}c_{23}+c_{12}s_{23}+s_{12}s_{13}c_{23}]m_\tau=0\,.
\label{nonlineq} \eea
%

The first naive attempt we could try in order to see if this
system admits a solution is to assume that the coefficients of
each mass term has to vanish. In the sequence we are going to
show that no solution can be found in this way and, in this case,
we have to allow for any combination of the mass coefficients to
cancel each other, which will need some numerical computation.

We start by making the coefficients of $m_e$, $m_\mu$ and
$m_\tau$ in Eq.~(\ref{nonlineq}), to vanish identically. This
leads automatically to the following constraints,
%
\be cs_{12}=-sc_{12}\,, \label{cs12} \ee
%
and
%
\be s_{13}=0\,. \label{s13} \ee
%

For simplicity, we are going to assume that
$c_{13}=+1$\footnote{It can be shown that $c_{13}=-1$ would lead
to the same conclusions, although to different charged lepton
mass elements in Eq.~(\ref{elements}).}. The constraint in
Eq.~(\ref{s13}) allows some simplification in Eq.~(\ref{nonlineq})
and we can extract further constraints from it, namely,
%
\bea sc_{23}&=&-s_{12}s_{23}\,, \nonumber \\
ss_{23}&=&-s_{12}c_{23}\,, \nonumber \\
cc_{23}&=&-c_{12}s_{23}\,, \nonumber \\
cs_{23}&=&-c_{12}c_{23}\,. \label{false} \eea
%
Taking into account the constraint Eq.~(\ref{cs12}) together with
these equations, it is straightforward to see that we only have a
solution for this system if $s=0$ or $c=0$, which is not the case
since the small angle MSW solution is already ruled out. Then we
have shown that our naive analytical approach does not allow us
to obtain a conclusion about the existence of a solution.

We jump then to the numerical calculations. The numerical
analysis was done using the package for solving systems of
non-linear equations described in Ref.~\cite{NLS}. We fixed the
lepton masses as, $m_e=0.51$~MeV, $m_\mu =105.66$~MeV and $m_\tau
=1777.0$~MeV, and assumed values for $s$ and $c$ in the range:
$0.439941< s\simeq c <0.686221$~\cite{BahcallConcha}. No solution
for the system Eq.~(\ref{nonlineq}) was found in this range. This
analysis is sufficient to conclude that lepton mixing in minimal
$3-3-1$ model does not allow for the pattern of charged lepton
masses given by Eq.~(\ref{charglepmass}). Hence, we should look
for another scenario where neutrinos and charged lepton mass
matrices are both in a non-diagonal basis, which we will do next.

\subsection{Second Case}

Another possible   scenario is the one  where both neutrino and
charged lepton mass matrices are non-diagonal. In this situation,
the mixing in Eq.~(\ref{maxmix}) is generated partially by the
diagonalization of neutrino mass matrix and partially by the
diagonalization of charged lepton mass matrix. As observed in
Ref.~\cite{desai}, requiring that the mixing angle of the
atmospheric anomaly comes from the neutrino sector and the
maximal mixing angle for the solar neutrino comes from the
charged lepton sector we would need fine tuning. We can, instead,
assume the reverse, that maximal mixing between $\nu_\mu$ and
$\nu_\tau$ comes from the charged lepton sector allowing the
mixing in the $\nu_e$ to $\nu_\mu$ oscillation to come from the
neutrino sector. In this way it is possible to dissociate the
contributions of both sectors as follows,
%
\bea O^W=V^T_{eL} V_{\nu}= \left (
\begin{array}{ccc}
c & s & 0 \\
\frac{-s}{\sqrt{2}} & \frac{c}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
 \frac{s}{\sqrt{2}}& -\frac{c}{\sqrt{2}}  & \frac{1}{\sqrt{2}}
\end{array}
\right )=\left (
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
 0& -\frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}}
\end{array}
\right )\times \left (
\begin{array}{ccc}
 c &  s  & 0 \\
 -s &  c  & 0 \\
 0 &  0  & 1
\end{array}
\right )\,, \label{division} \eea
%
where,
%
\bea V^T_{eL} =\left (
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
 0& -\frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}}
\end{array}
\right ),
\label{clepton}
\eea
%
and,
%
\bea
 V_{\nu}= \left (
\begin{array}{ccc}
c & s & 0 \\
-s & c & 0 \\
 0& 0  & 1
\end{array}
\right )\,. \label{neutpart} \eea
%

From the Lagrangian term containing the singly charged bilepton,
Eq.~(\ref{cc}), we have that $O^V=V^T_{eR} V_{\nu}$, with the
diagonalization matrix, $V_\nu$, given above. We can parameterize
$O^V$ according to the usual CKM notation,
%
\bea O^V= \left (
\begin{array}{ccccc}
c_{12} c_{13} &\,\,& s_{12} c_{13} &\,\,& s_{13}  \\
-s_{12} c_{23}- c_{12} s_{13} s_{23} &\,\,& c_{12} c_{23}-s_{12}
s_{13} s_{23} &\,\,&
c_{13} s_{23} \\
s_{12} s_{23} -c_{12}s_{13}c_{23}&\,\,& -c_{12} s_{23}-s_{12}
s_{13} c_{23} &\,\,& c_{13} c_{23}
\end{array}
\right ), \label{VeRa} \eea
%
and dissociate it as a product of three different rotation
matrices,
%
\bea O^V= \left (
\begin{array}{ccc}
1 & 0 & 0  \\
0 &  c_{23} & s_{23}  \\
0 & -s_{23} & c_{23}
\end{array}
\right )\times \left (
\begin{array}{ccc}
c_{13} & 0 & s_{13}  \\
0 &  1 & 0  \\
-s_{13} & 0 & c_{13}
\end{array}
\right ) \times \left (
\begin{array}{ccc}
c_{12} & s_{12} & 0  \\
-s_{12} &  c_{12} & 0 \\
0 & 0 & 1
\end{array}
\right ),
\label{VeR1}
\eea
%

Looking at the last matrix at the right hand side of
Eq.~(\ref{VeR1}), and once we identify $c_{12}=c$ and $s_{12}=s$,
we can associate it with the neutrino matrix,
Eq.~(\ref{neutpart}). The appropriate form of $V_{eR}$ is then
automatically attributed as follows:
%
\bea V^T_{eR}= \left (
\begin{array}{ccc}
1 & 0 & 0  \\
0 &  c_{23} & s_{23}  \\
0 & -s_{23} & c_{23}
\end{array}
\right )\times \left (
\begin{array}{ccc}
c_{13} & 0 & s_{13}  \\
0 &  1 & 0  \\
-s_{13} & 0 & c_{13}
\end{array}
\right ) =\left (
\begin{array}{ccc}
c_{13} & 0 & s_{13}  \\
-s_{23}s_{13} &  c_{23} & s_{23}c_{13}  \\
-c_{23}s_{13} & -s_{23} & c_{23}c_{13}
\end{array}
\right )\,. \label{VeR2} \eea
%
With $V_{eR}$ given above, Eq.~(\ref{VeR2}), and $V_\nu$ given by
Eq.~(\ref{neutpart}), we are able to obtain the texture of the
neutrino and charged lepton mass matrices. For the neutrinos we
have:
%
\bea V_\nu M_\nu^D V^T_\nu = \left (
\begin{array}{ccccc}
m_1 c^2 + m_2 s^2 &\,\,& (m_2-m_1) cs &\,\,& 0  \\
(m_2-m_1) cs &\,\,&  m_1 s^2 + m_2 c^2 &\,\,& 0  \\
0 &\,\,& 0 &\,\,& m_3
\end{array}
\right )\,. \label{neumass2} \eea
%
When this is confronted with $M_\nu$ in Eq.~(\ref{lepmass1}) we
conclude that $G_{13}=G_{23}=0$. As a consequence, we are left
with a peculiar texture for the charged lepton mass matrix,
%
\bea
M_l=
\left (
\begin{array}{ccc}
G_{11}v_{\sigma_2} & (G_{12}v_{\sigma_2}+F_{12}v_\eta )&
F_{13}v_\eta  \\
(G_{12}v_{\sigma_2}-F_{12}v_\eta ) & G_{22}v_{\sigma_2} &
F_{23}v_\eta  \\
-F_{13}v_\eta  & -F_{23}v_\eta  & G_{33}v_{\sigma_2}
\end{array}
\right )\,. \label{lepmassa2} \eea
%
We must check now if these assignments are plausible by verifying
if the diagonalization matrices so obtained drive us to such a
texture. We then compute $V_{eL}M^D_l V^T_{eR}$ where $V_{eL}$ is
given by the transpose of Eq.~(\ref{clepton}), and $V_{eR}^T$ by
Eq.~(\ref{VeR2}),
%
\bea M_l&&=V_{eL}M^D_l V^T_{eR}= \left (
\begin{array}{ccc}
1 & 0 & 0  \\
0 &  \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}  \\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{array}
\right )\times \left (
\begin{array}{ccc}
m_e & 0 & 0  \\
0 &  m_\mu & 0  \\
0 & 0 & m_\tau
\end{array}
\right ) \times \left (
\begin{array}{ccc}
c_{13} & 0 & s_{13}  \\
-s_{23}s_{13} &  c_{23} & s_{23}c_{13}  \\
-c_{23}s_{13} & -s_{23} & c_{23}c_{13}
\end{array}
\right )= \nonumber \\ && \nonumber \\ && \left (
\begin{array}{ccccc}
c_{13}m_e &\,\,& 0 &\,\,& s_{13}m_e  \\
\frac{1}{\sqrt{2}}(-s_{23}s_{13}m_\mu + c_{23}s_{13}m_\tau)&\,\,&
\frac{1}{\sqrt{2}}( c_{23}m_\mu +s_{23}m_\tau)&\,\,&
\frac{1}{\sqrt{2}}(s_{23}c_{13}m_\mu
-c_{23}c_{13}m_\tau)  \\
\frac{1}{\sqrt{2}}(-s_{23}s_{13}m_\mu-c_{23}s_{13}m_\tau) &\,\,&
\frac{1}{\sqrt{2}}(c_{23}m_\mu-s_{23}m_\tau) &\,\,&
\frac{1}{\sqrt{2}}(s_{23}c_{13}m_\mu+c_{23}c_{13}m_\tau)
\end{array}
\right )\,.
\label{massleptext} \eea
%

Let us look at the consequences of this result. We first observe
that by the form of Eq.~(\ref{lepmassa2}), we obtain from
Eq.~(\ref{massleptext}) the following relations,
%
\bea && -s_{23}s_{13}m_\mu + c_{23}s_{13}m_\tau = 0\,,\nonumber \\
&& s_{13}m_e = \frac{1}{\sqrt{2}}(s_{13}s_{23}m_\mu +
s_{13}c_{23}m_\tau )\,,\nonumber \\
&& c_{23}m_\mu-s_{23}m_\tau = -s_{23}c_{13}m_\mu +
c_{23}c_{13}m_\tau\,. \label{eqnond} \eea
%
We can analyze two possibilities, namely, $s_{13}=0$ and
$s_{13}\neq 0$. For the case $s_{13}=0$ (assuming $c_{13}=+1$),
these equations have a solution for $c_{23}=-s_{23}$. However,
the fact that elements $G_{ab}$ are present in both mass
matrices, of neutrinos and charged leptons, leads to tight
constraints on their values. For instance, by comparing the matrix
in Eq.~(\ref{lepmass1}) with Eq.~(\ref{neumass2}), we can extract
%
\be G_{11}v_{\sigma_1}=m_1 c^2 + m_2 s^2\,,\label{g11} \ee
%
and
%
\be G_{33}v_{\sigma_1}=m_3\,.\label{g33} \ee
%
According to Eqs.~(\ref{lepmass2}) and (\ref{massleptext}), after
substituting the obtained value for the mixing angles, $s_{13}=0
\Rightarrow c_{13}=1$ and $c_{23}=-s_{23}=1/\sqrt{2}$, $G_{11}$
and $G_{33}$ are also constrained to the charged lepton masses,
%
\be G_{11}v_{\sigma_2}=m_e\,,\label{g11m} \ee
%
and
%
\be G_{33}v_{\sigma_2}=\frac{1}{2}(m_\tau - m_\mu)\simeq
\frac{1}{2}m_\tau\,. \label{g33m} \ee
%
But this is not possible because $m_1$, $m_2$ and $m_3$ are of the
order of eV, while $m_e$ is of the order of MeV and $m_\tau$ is
of the order of GeV. As we divide Eq.~(\ref{g11}) and
Eq.~(\ref{g33}) by Eq.~(\ref{g11m}) and Eq.~(\ref{g33m}),
respectively, we obtain that the ratio
$v_{\sigma_1}/v_{\sigma_2}$ is about $10^{-6}$ in one case and
$10^{-9}$ in the other, which is an absurd. This shows that such a
solution is not viable.

Suppose now that $s_{13}$ is not null. In this situation the
second equation in Eq.~(\ref{eqnond}) becomes,
%
\be m_e = \frac{1}{\sqrt{2}}(s_{23}m_\mu + c_{23}m_\tau )\,.
\label{eqme} \ee
%
Since the magnitude of the charged lepton masses are very
different, there is no value for the mixing angle in
Eq.~(\ref{eqme}) that makes the right hand side of this equation
to be equal to the electron mass. For instance, if
$s_{23}\,(c_{23})$ is small, $c_{23}\,(s_{23})$ is close to one
and we would obtain $m_e\sim m_\tau\,(m_\mu)$, an obviously false
result. This analysis is general enough and we do not even need to
appeal to numerical computation to conclude that the minimal 3-3-1
model requires some extension in order to accommodate the leptonic
mixture.

However, this is not the end of the story. Once we can blame a
suitable extension to account for the neutrino mixing, there is
no reason to not assume that charged lepton mass matrix is
diagonal from the beginning. Assuming this, we could impose that
the sextet alone is responsible for the charged lepton masses,
which can be accomplished if we eliminate the anti-symmetric
triplet $\eta$ term in the Yukawa Lagrangian, Eq.~(\ref{yukawa}).
This is naturally done by resorting to some discrete symmetry,
$\eta \rightarrow -\eta$. Then the charged lepton mass take the
form $M_l = diag(G_{11}, G_{22}, G_{33})v_{\sigma_2}$. The
texture of the neutrino mass will depend only on the extension of
the model and, in this case, it is important to reconsider the
implications of $M-\overline{M}$ conversion. This is the subject
considered next.

\section{Muonium-Antimuonium Conversion}
\label{sec2}

The bound state formed by $\mu^+ e^-$, the muonium, was first
introduced by Pontecorvo~\cite{pontecorvo}, who pointed out that
the transition to antimuonium, $\mu^- e^+$, would be suitable for
testing the particle-antiparticle mixing idea. In 1961, Feinberg
and Weinberg~\cite{FW}, in the spirit of Weak interactions,
proposed an effective Hamiltonian of the type $(V-A)$,
%
\be {\cal H}_{M\overline{M}}=\frac{G_{M\overline{M}}}{\sqrt{2}}
\bar{\mu}\gamma_\alpha (1-\gamma_5)e\bar{\mu}\gamma^\alpha
(1-\gamma_5)e + \mbox{H.c.}\,. \label{Heff} \ee
%
The experimental search for $M-\overline{M}$ conversion, has put
an upper limit on the effective coupling, $G_{M\overline{M}}$, in
terms of Fermi constant, $G_F$. The latest
experiment~\cite{muoexp} provides, $G_{M\overline{M}}\leq
3.0\times 10^{-5}$. Since the $M-\overline{M}$ conversion requires
physics beyond SM, it is a suitable tool to test the minimal
$3-3-1$ model, where such conversion would naturally occur
through scalars and vector bileptons exchange. Next we situate
the recent discussion concerning the role of $M-\overline{M}$ in
putting bounds on the vector bilepton mass.

It is the doubly charged vector bilepton $U^{++}$ which has been
the focus on the $M-\overline{M}$ conversion. In
Ref.~\cite{muoexp} its contribution was evaluated taking $O^U=
I$, resulting in the constraint $M_{U^{++}}\geq 850$ GeV. It was
claimed in Ref.~\cite{remark} that such a lower bound appears
because of the restricted conditions assumed on the model, and
when a more thorough inspection is taken into account this bound
can be released to values around 600 GeV, escaping then from
contradiction with the existing upper bound $M_{U^{++}}\lesssim
600$ GeV~\cite{mu++limit}.

According to the view presented in Ref.~\cite{remark}, there are
two ways of evading this contradiction. The first one is to
consider that $O^U$ is not the unity and, instead, $O^U_{ee}
O^U_{\mu \mu} \lesssim 0.7$. If this is not the case, there could
be other contributions from the scalar bileptons and from flavor
changing neutral scalar current(FCNSC). These scalars are
constituents of the sextet and the triplet in the Yukawa
Lagrangian Eq.~(\ref{yukawa}). The scalar bileptons are
$H^{--}_1$ and $H^{++}_2$ that make part of the sextet,
Eq.~(\ref{sextet}). The FCNSC appears because of the charged
lepton mass matrix generated from the interactions in
Eq.~(\ref{yukawa}) is $M_l = G v_{\sigma_2} + F v_\eta $. Of
course the matrices that diagonalize $M_l$ do not diagonalize
individually $G$ and $F$ and as consequence flavor mixing is
still present in the Yukawa interaction which implies FCNSC in
$M-\overline{M}$ conversion. As argued in Ref.~\cite{remark},
when these contributions are taken together the constraint
$M_{U^{++}}$ can be relaxed.


From the results obtained in the preceding section, we saw that
neutrino physics requires an extension of the minimal model. In
this sense, the simplest and most natural assumption we can make
is to take the charged lepton mass matrix diagonal and the only
possibility of relaxing the $M_{U^{++}}$ lower bound lies on
considering the scalar bileptons contribution to $M-\overline{M}$
conversion. In this case, the matrix in Eq.~(\ref{lepmass1}) is
diagonal and the elements $F_{ab}$ in Eq.~(\ref{lepmass2}) are
null. This implies that neutral scalars do not contribute, since
they only appear in the lepton interaction with the triplet $\eta$
(but now $F_{ab}=0$), and through non-diagonal elements in the
interaction with the sextet $S$, which are null in this situation
(since $G_{ab}$ is diagonal). For this scenario we have that
$O^W= V_{\nu}$, $O^V=I$ and $O^U=I$. Then we can compute the
remaining three contributions namely, those coming from the
doubly charged gauge boson, $U^{++}$ and doubly charged scalars,
$H_1^{--}$ and $H_2^{++}$ (see Figure~\ref{fig1}). The doubly
charged vector contribution comes from the third  term in
Eq.~(\ref{cc}),
%
%
\be {\cal L}_{U^{\pm\pm}} = -\frac{g}{\sqrt{2}}
\overline{(l_{aR})^c} \gamma^\mu l_{aL}U^{++}_\mu +
\mbox{H.c.}\,, \label{bilepton} \ee
%
and the doubly charged scalars contribution, derived from the
Yukawa interaction in Eq.~\ref{yukawa}, is given by ,
%
\be {\cal L}_{H^{\pm\pm}} = \frac{1}{2}
\overline{(l_{aL})^c}G_{ab} l_{bL}H_1^{++} + \frac{1}{2}
\overline{(l_{aR})}G_{ab} (l_{bR})^cH_2^{--} + \mbox{H.c.}\,,
\label{Yuk} \ee
%
where $l_{a,b}$ are the charged leptons with $a$ and $b$ running
over family labels.

These terms allow us to write the following low energy effective
Hamiltonian for the $M-\overline{M}$ transition,
%
\bea {\cal H}_{M\overline{M}} &=& \frac{g^2}{2
M_U^2}\overline{\mu_L}\gamma^\mu\mu_R^c\overline{e_R^c}\gamma_\mu
e_L
-\frac{G_{\mu\mu}G_{ee}}{4M_{H_1}^2}\overline{\mu_L}\mu_L^c\overline{e_L^c}e_L
-\frac{G_{\mu\mu}G_{ee}}{4M_{H_2}^2}\overline{\mu_R}\mu_R^c\overline{e_R^c}
e_R + \mbox{H.c.}\,.\label{hamiltef} \eea
%
We can use the Fierz reordering and some manipulation on the
scalar involving the conjugated fields (transposition or
hermitian conjugation), to obtain this Hamiltonian in the same
form as Eq.~(\ref{Heff}),
%
\bea {\cal H}_{M\overline{M}} &=& -\frac{g^2}{8
M_U^2}\bar{\mu}\gamma^\mu (1-\gamma_5)e \bar{\mu}\gamma^\mu
(1+\gamma_5)e + \frac{G_{\mu\mu}G_{ee}}{32
M_{H_1}^2}\bar{\mu}\gamma^\mu (1+\gamma_5)e \bar{\mu}\gamma^\mu
(1+\gamma_5)e \nonumber \\
&& + \frac{G_{\mu\mu}G_{ee}}{32 M_{H_2}^2}\bar{\mu}\gamma^\mu
(1-\gamma_5)e \bar{\mu}\gamma^\mu (1-\gamma_5)e +
\mbox{H.c.}\,.\label{hamiltonian} \eea
%
These terms amount to distinct contributions to the
$M-\overline{M}$ conversion coupling, $G_{M\overline{M}}$.
Moreover, it was observed that on the presence of a magnetic
field, their diverse Lorentz structure, weather $(V\pm A)^2$ or
$(V+A)\times(V-A)$, can show up on the probability
transition~\cite{sasaki}. This is a powerful tool to distinguish
among the different kinds of particles mediating the conversion,
but it is irrelevant for the present discussion. What matters
here is to clarify the importance of the scalar fields which
could soften the bounds imposed by the $M-\overline{M}$ conversion
over the doubly charged vector bilepton mass in the minimal
$3-3-1$ model.

In the case of diagonal charged lepton mass matrix, the values for
the couplings $G_{aa}$ depends on $v_{\sigma_2}$. An optimistic
assumption for $v_{\sigma_2}$ is to take it of the order of 1~GeV,
leading to
%
\be G_{ee}\simeq 5\times 10^{-4}\,,\,\,\,\,\,\,\,
G_{\mu\mu}\simeq 0.1\,,\,\,\,\,\,\,\, G_{\tau\tau}\simeq 1.8\,.
\label{coupl} \ee
%
It is reasonable to assume that the scalar bileptons have masses
at the same order of magnitude, about few hundreds of GeV's,
$\simeq 100$ GeV. Considering this and taking the values for the
scalar couplings as in Eq.~(\ref{coupl}), the bilepton coupling
around $g\sim 0.1$, and $m_{U^{++}} \simeq 600$ GeV, as required
in Ref.~\cite{mu++limit}, it is clear from Eq.~(\ref{hamiltonian})
that the scalar contributions to the effective coupling,
$G_{M\overline{M}}$, are strongly suppressed compared to the
vector bilepton. We can firmly say that scalars play no role in
modifying the bound imposed by $M-\overline{M}$ conversion, which
jeopardizes the minimal version of $3-3-1$ model. So, even taking
into account the improvements in order to drive the appropriate
neutrino mass matrix texture, it would still be necessary to
insert some intricate and less appealing mechanism to
simultaneously relax the bilepton bound due to $M-\overline{M}$
conversion.

It is possible though that the conflict raised by
$M-\overline{M}$ conversion does not really exist. This would be
so if, as remarked in Ref.~\cite{remark}, the true value for the
$U^{++}$ mass upper bound were around $3.5$~TeV~\cite{jain}
instead of that one in Ref.~\cite{mu++limit}. Nevertheless, it is
not clear to us if this discrepancy between these values are due
to the different assumptions over the energy scale where these
masses were computed, or if the particle content considered in
their computations is in disagreement. Anyway, our result shows
that if the conflict exists, it requires subtle changes in the
structure of minimal $3-3-1$ model.

%
\begin{figure}
\centerline{ \epsfxsize=0.10\textwidth
\begin{picture}(100,140)(-45,30)
%
\put(-58,50){\makebox(0,0)[br]{$e^-$}}
\put(20,50){\makebox(60,0)[br]{$e^+$}}
\put(50,95){\makebox(50,0)[br]{$H_1^{--}\,,\,\,H_2^{--}\,,\,\,U_\mu^{--}$}}
\put(20,140){\makebox(60,130)[br]{$\mu^-$}}
\put(-58,140){\makebox(0,0)[br]{$\mu^+$}}
\DashArrowLine(0,60)(0,135)7 \ArrowLine(-85,60)(0,60)
\ArrowLine(85,60)(0,60) \ArrowLine(0,135)(-85,135)
\ArrowLine(0,135)(85,135)
%
\end{picture}}
\caption{Doubly charged scalar and bilepton contribution to
$M-\overline{M}$ conversion through t-channel exchange.}
\label{fig1}
\end{figure}
%
\section{Concluding Remarks}
\label{sec3}

We have investigated the possibility of conciliating the mixing
in leptonic sector, suggested by the neutrino oscillation
hypothesis, with the minimal version of $3-3-1$ model. Due to the
very particular form of the anti-symmetric Yukawa coupling with
the triplet, we have shown that none of the alternatives for the
mass matrix textures of charged leptons and neutrinos could
generate the expected pattern for the maximal mixing matrix,
Eq.~(\ref{maxmix}). This lead us to conclude that the minimal
$3-3-1$ model is incapable of explaining neutrino oscillation and
masses in the way it is formulated, calling for an extension of
the model, as is the case with the SM itself. Besides, assuming
that the changes required to solve the neutrino problem would be
appropriately included, leaving the charged lepton mass matrix in
a diagonal basis, we have surveyed the contributions to the
$M-\overline{M}$ conversion so as to determine the actual role of
scalars in softening the upper bound on $M_{U^{++}}$, as
suggested in Ref.~\cite{remark}. The unfortunate result was that
scalars are very suppressed compared to the vector bilepton, and
we have to agree with Ref.~\cite{muoexp}, stressing that the
minimal version of $3-3-1$ is inconsistent with $M-\overline{M}$,
unless the self-consistency of the model implies a very different
upper bound on $U^{++}$ mass as obtained in Ref.~\cite{jain}.

It is important to point out that although this model has other
nice features when compared to SM, it loses one of the most
appealing one which is the capability of naturally explaining the
neutrino masses and mixing. This does not make it worse than SM,
but certainly pushes it to a less privileged post, and if the
conflict with $M-\overline{M}$ persists, it can become hard to
sustain its position.

Finally, we would like to call the attention to the fact that the
required minimal $3-3-1$ extension should take into account the
neutrino mass and mixing and also the $M-\overline{M}$ conversion
altogether. Concerning the $M-\overline{M}$ conversion a solution
is already known in which the sextet is substituted by an octet
of scalars~\cite{muoexp}. This change allows the possibility of
keeping the charged lepton sector diagonal, while the leptonic
mixing would manifest itself only among the neutrinos. Here we
would like to point out another interesting way out, where the
change for the octet is not necessary. It requires neutrinos and
charged lepton masses in a non-diagonal basis. In this case the
extended $3-3-1$ might generate the pattern of mixing as presented
in the second case of Sec.~\ref{sec1}, where the charged lepton
mixing takes the form given in Eq.~(\ref{clepton}) and the mixing
among the neutrinos is that of Eq.~(\ref{neutpart}). As we have
shown, this would imply that $V^T_{eR}$ is given by
Eq.~(\ref{VeR2}). It is easy to see that the product
$O^U=V^T_{eR}V_{eL}$ is such that the second element in the
diagonal is null, $O^U_{\mu \mu}=0$, avoiding the $M-\overline{M}$
conversion in the model.


{\it Acknowledgements.}  We would like to thank Vicente Pleitez
for reading the manuscript and for his useful suggestions. This
work was supported by Funda\c c\~ao de Amparo \`a Pesquisa do
Estado de S\~ao Paulo (FAPESP).
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\end{document}

