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\title{Bi-maximal Neutrino Mixing Pattern Reexamined}
\author{Zhi-zhong Xing}
\address{Sektion Physik, Universit$\it\ddot{a}$t M$\it\ddot{u}$nchen,
Theresienstrasse 37A, 80333 M$\it\ddot{u}$nchen, Germany \\
{\it Electronic address: xing@theorie.physik.uni-muenchen.de} }
\maketitle

\begin{abstract}
I propose a simple extension of the bi-maximal neutrino mixing
pattern, allowing slight coupling between solar and
atmospheric neutrino oscillations as well as large $CP$ violation. 
The new scenario is compatible with either the vacuum oscillation
solution or the large-angle MSW solution to the solar neutrino problem.
\end{abstract}

\pacs{PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt}

\newpage

To interpret current experimental data on 
atmospheric and solar neutrino oscillations, a lot of interest 
has been paid to the ``bi-maximal'' neutrino mixing pattern 
(up to a trivial sign rearrangement) \cite{Barger98}:
\begin{equation}
U \; =\; \left ( \matrix{
\frac{1}{\sqrt{2}}	& \frac{1}{\sqrt{2}}	& 0 \cr\cr
-\frac{1}{2}	& \frac{1}{2}	& \frac{1}{\sqrt{2}} \cr\cr
\frac{1}{2}	& -\frac{1}{2}	& \frac{1}{\sqrt{2}} \cr} \right ) \;\; .
%		(1)
\end{equation}
Note that $U$ links the neutrino mass eigenstates
$(\nu_1, \nu_2, \nu_3)$ to the neutrino flavor eigenstates
$(\nu_e, \nu_\mu, \nu_\tau)$. 

Different from the ``tri-maximal'' \cite{Cabibbo} or ``democratic''
(nearly bi-maximal) \cite{FX96} neutrino mixing scenarios,
the vanishing of $U_{e3}$ in the bi-maximal mixing scenario
assures the absence of $CP$ violation and
an exact decoupling between solar ($\nu_e \rightarrow \nu_\mu$)
and atmospheric ($\nu_\mu \rightarrow \nu_\tau$)
neutrino oscillations with identical 
mixing factors 
(i.e., $\sin^2 2\theta_{\rm sun} = \sin^2 2\theta_{\rm atm} =1$).
At present this special ansatz
seems to be more favored by the vacuum oscillation solution
than by the large-angle MSW solution to the solar neutrino problem.

In this short note I propose a simple extension of the naive
bi-maximal neutrino mixing pattern given above, so as to
accommodate possible large $CP$ violation in the lepton sector
and to allow much flexibility in accounting for the solar neutrino
problem. The new lepton flavor mixing matrix takes the following form
%%%%%%%%%%%%
\footnote{Here only the Dirac-type $CP$ phase is taken
into account, as the Majorana-type ones have no effect in 
neutrino oscillations.}:
%%%%%%%%%%%%
\begin{equation}
V \; =\; \left ( \matrix{
\frac{c}{\sqrt{2}}	& \frac{c}{\sqrt{2}}	& -is \cr\cr
-\frac{A}{2}	& \frac{A^*}{2}	& \frac{c}{\sqrt{2}} \cr\cr
\frac{A^*}{2}	& -\frac{A}{2}	& \frac{c}{\sqrt{2}} \cr} \right ) \;\; ,
%		(2)
\end{equation}
where $s \equiv \sin \theta$, $c \equiv \cos \theta$, and 
$A = 1 + is$. The mixing angle $\theta$ measures a slight coupling
between solar and atmospheric neutrino oscillations, whose
mass-squared differences are 
\begin{eqnarray}
\Delta m^2_{\rm sun} & = & \left |m^2_2 - m^2_1 \right | \;\; ,
\nonumber \\
\Delta m^2_{\rm atm} & = & \left |m^2_3 - m^2_2 \right | \;\; ,
%		(3)
\end{eqnarray}
respectively. The ``observed''
hierarchy $\Delta m^2_{\rm sun} \ll \Delta m^2_{\rm atm}$
represents our today's understanding of the neutrino mass spectrum
%%%%%%%%%%%%%
\footnote{Throughout this work we do not take the LSND evidence
for neutrino oscillations \cite{LSND}, which was not confirmed by
the KARMEN experiment \cite{KARMEN}, into account.}.
%%%%%%%%%%%%

Clearly
$U$ can be reproduced from $V$ with $\theta =0$. 
The rephasing-invariant strength of $CP$ violation turns out to be
\begin{eqnarray}
{\cal J} & = & {\rm Im} \left (V_{i\alpha} V_{j\beta}
V^*_{i\beta} V^*_{j\alpha} \right ) \nonumber \\
& = & \frac{s ~ c^2}{4} \;\; ,
%		(4)
\end{eqnarray}
in which $(i, j)$ run over $(e, \mu)$, $(\mu, \tau)$ or $(\tau, e)$ and
$(\alpha, \beta)$ over $(1, 2)$, $(2, 3)$ or $(3, 1)$.
The mixing factors of solar ($\nu_e \rightarrow \nu_e$ disappearance)
and atmospheric ($\nu_\mu \rightarrow \nu_\mu$ disappearance)
neutrino oscillations read
\begin{eqnarray}
\sin^2 2\theta_{\rm sun} & = & c^4 \;\; , \nonumber \\
\sin^2 2\theta_{\rm atm} & = & 1- s^4 \;\; .
%		(5)
\end{eqnarray}
The correlation between ${\cal J}$ and $|V_{e3}|^2$ is illustrated
in Fig. 1(a) for $|V_{e3}|^2 \leq 0.1$, and that between 
$\sin^2 2\theta_{\rm sun}$ (or $\sin^2 2\theta_{\rm atm}$) and
$|V_{e3}|^2$ is shown in Fig. 2(b). Some comments are in order.
\begin{itemize}

\item	The strength of 
$CP$ violation in this new neutrino mixing scenario can
be as large as few percent. A signal of $CP$ (or $T$) violation
could be measured from the probability asymmetry between
$\nu_\mu \rightarrow \nu_e$ and $\bar{\nu}_\mu \rightarrow
\bar{\nu}_e$ (or $\nu_e \rightarrow \nu_\mu$) in the future
long-baseline neutrino experiments with $L \sim E/\Delta m^2_{\rm sun}$.
Such a measurment is feasible if and only if the large-angle
MSW oscillation is the true solution to the solar neutrino problem.

\item	The mixing factor of solar neutrino oscillations is 
sufficiently large, compatible with either the large-angle MSW solution
or the vacuum oscillation solution to the solar neutrino 
problem \cite{Petcov}.

\item	The mixing factor of atmospheric neutrino oscillations
is nearly maximal for reasonable values of $|V_{e3}|^2$. This 
result is particularly favored by the Super-Kamiokande \cite{SK} and CHOOZ
\cite{CHOOZ} experiments.
\end{itemize}

It is also worth pointing out that the lepton flavor mixing matrix $V$
is symmetric about its axis $V_{e3}$-$V_{\mu 2}$-$V_{\tau 1}$.
In comparison, the quark flavor mixing matrix (i.e., the CKM matrix)
is approximately
symmetric about its axis $V_{ud}$-$V_{cs}$-$V_{tb}$ \cite{Xing95}. 
This qualitative difference in the textures of quark and lepton
mixing matrices could have a yet unknown dynamical reason. 

There are of course some other possibilities, depending on the number
of free parameters to be introduced into $U$, towards a slight
modification of the exactly bi-maximal neutrino mixing pattern.
In Ref. \cite{Sumino}, for instance, $U$ has been modified to
$U' = R^{\rm T}_{12} (\theta_{12}) ~ U$, where $R_{12}(\theta_{12})$
describes a small rotation in the real (1,2) plane to diagonalize
the charged lepton mass matrix 
($\theta_{12} \approx \arcsin \sqrt{m_e/m_\mu} \approx 4^{\circ}$).
This scenario, in which $CP$ symmetry remains conserved,
predicts $|U'_{e3}|^2 \approx 0.002$,
$\sin^2 2\theta_{\rm sun} \approx 0.985$, and $\sin^2 2\theta_{\rm atm}
\approx 1.000$. It is therefore distinguishable from the new mixing
pattern $V$ proposed in Eq. (2), through the delicate detection of
leptonic $CP$ violation in the long-baseline neutrino experiments
or through the accurate determination of mixing factors in the atmospheric
and solar neutrino experiments. In Ref. \cite{Stech} the author
presented a different nearly bi-maximal neutrino mixing
ansatz, based on the arguments of lepton-quark similarities and the
assumed textures of charged lepton and neutrino mass matrices. 
Such an ansatz involves several free parameters, whose values
are completely unknown. Hence its
interpretation of current neutrino oscillation data relies
somehow on the fine-tuning of those parameters, and its consequences
on the long-baseline neutrino oscillations are not as unique
as those of the simpler and more instructive scenarios discussed above.

Finally I should emphasize that the bi-maximal mixing pattern $U$ 
in Eq. (1) and its revisited version $V$ in Eq. (2) 
require some special flavor symmetries
to be imposed on the charged lepton and neutrino mass matrices \cite{FX96}.
It is more likely that $V$ (or $U$) serves as the leading-order approximation
of a more complicated flavor mixing matrix. At present, however, such
a simplified scenario is very instructive and useful to explore the
main features of lepton flavor mixing and $CP$ violation through
neutrino oscillations.


%%%%%%%%%%%%% FIG. 1 %%%%%%%%%%%%
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\end{picture}
\vspace{0.5cm}
\caption{Illustrative plots for the correlation between
${\cal J}$, $\sin^2 2\theta_{\rm sun}$ or $\sin^2 2 \theta_{\rm atm}$ 
and $|V_{e3}|^2$ in the new neutrino mixing scenario.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
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\end{document}



