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\begin{document}
\title{Flavor Mixing, Neutrino Masses and Neutrino Oscillations}

\classification{11.30.Hv; 14.60.Pq} \keywords
{flavor mixing, neutrino oscillations}

\author{H. Fritzsch}{address={University of Munich, Physics Department, 
Munich, Germany}}

\begin{abstract}
We study a model for the mass matrices of the leptons. We are able to
relate the mass eigenvalues of the charged leptons and of the neutrinos
to the mixing angles and can predict the masses of the neutrinos. We
find a normal hierarchy {}- the masses are 0.004 eV, 0.01 eV and 0.05
eV. The atmospheric mixing angle is given by the mass ratios of the
charged leptons and the neutrinos. We find 38 degrees, consistent with
the experiments. The mixing element, connecting the first neutrino with
the electron, is found to be 0.05. 
\end{abstract}
\maketitle

We observe three lepton-quark families in nature. The first family
consists of the electron, its neutrino and the u- and d-quarks. The
members of the second family are the muon, its neutrino and the c-
and s-quarks. The third family consists of the tauon, its neutrino
and the t- and b-quarks. It is unknown, whether there is a
connection between the number of families and the number of colors in
quantum chromodynamics.

In the Standard Model 28 fundamental constants have to be introduced.
Their values cannot be calculated - they have to be measured in the
experiments. Theses constants are:

the constant of gravity G,


the fine structure constant,


the coupling constant of the weak interactions,

the coupling constant of the strong interactions or the scale of QCD,


the mass of the W-boson,


the mass of the ``Higgs''-boson,


the masses of the three charged leptons,

the masses of the three neutrinos,


the masses of the six quarks,

the four parameters, describing the flavour mixing of the quarks and 

the six parameters, describing the flavour mixing  of the leptons.



For the masses of the quarks we assume the following values:



u: 5.3 MeV, d: 7.8 MeV, s: 146 MeV, c: 1050 MeV, b: 4 600 MeV, t: 174
000 MeV.



The quark masses are scale dependent. The values given above, except the
one for the t-mass, are normalized at an energy scale of 1 GeV. The
experimental errors are not given, but are at least 10 , except for
the t-mass, which is known to about 1 percent. 



The quarks of the same charge do mix. If the u-quark interacts with a
W-boson, a mixture of d, s and b appears. These mixtures are
described by the CKM matrix [1]. I prefer a description, which we
introduced some time ago [2]:




\begin{equation}
V=\left[\begin{matrix}c_{{u}}&s_{{u}}&0\\-s_{{u}}&c_{{u}}&0\\0&0&1\end{matrix}\right]\left[\begin{matrix}
e^{{-\text{i${\varphi}$}}}&0&0\\0&c&s\\0&-s&c\end{matrix}\right]\left[\begin{matrix}c_{{d}}&-s_{{d}}&0\\s_{{d}}&c_{{d}}&0\\0&0&1\end{matrix}\right]
\end{equation}



In the case of three families there are three mixing angles and one
phase parameter. The latter describes the CP-violation. The angles
with the index u or d describe the mixing in the u-c sector or the
d-s sector, the angle with no index describes the mixing between the
(t,c)-system and the (b,s)-system. 



I proposed years ago a simple texture 0 mass matrix for the quarks,
given by the following matrix, valid both for the (u,c,t)-system and
the  (d,s,b)-system [3]:


\begin{equation}
M=\left[\begin{matrix}0&A&0\\A&C&B\\0&B&D\end{matrix}\right]
\end{equation}




Such mass matrices are obtained, if in the electroweak theory special
symmetries are present, either discrete reflection symmetries or
continuous symmetries [3]. After diagonalization one finds the
following relations for the mixing angles [4]:


\begin{equation}
\theta_{d}={\rm arctan}\sqrt{\frac{m_{d}}{m_{s}}},\ 
\theta_{u}={\rm arctan}\sqrt{\frac{m_{u}}{m_{c}}}.
\end{equation}





Taking as input the quark masses given above, we obtain for the mixing
angle in the d-s- sector 13 degrees, and in the u-c-sector 4
degrees. The first angle is identical to the measured Cabibbo angle.
These angles agree very well with the experimental data on the flavor
mixing, if the phase angle is assumed to be close to 90 degrees. 



Analogously we can describe the flavour mixing in the lepton sector,
which can be studied in the neutrino oscillations. The experiments,
carried out in Japan (Kamiokande detector, [5]) and in Canada (SNO
detector, [6]), give the following results for the mass-squared
differences of the three neutrino mass eigenstates:



\begin{equation}
\begin{matrix}\text{${\Delta}$m}_{{\text{21}^{2}}}\approx
8\cdot \text{10}^{{-5}}\text{eV}^{{2}}\hfill\null
\\\text{${\Delta}$m}_{{\text{32}^{2}}}\approx
2\text{.}5\cdot
\text{10}^{{-3}}\text{eV}^{{2}}\hfill\null
\end{matrix}.
\end{equation}



In neutrino oscillations only mass squared differences can be measured.
No information can be obtained for the absolute magnitude of the
neutrino masses. There is the possibility that the neutrino masses are
nearly degenerate, e.g. masses like m(1) = 0.94 eV, m(2) = 0.95 eV,
m(3) = 1 eV are possible. If one neutrino remains massless, one would
have the masses m(1) = 0 eV, m(2) = 0.009 eV, m(3) = 0.05 eV. In the
latter case a hierarchy of the masses is present, but this hierarchy is
much weaker than the mass hierarchy for the charged leptons. Below we
shall calculate the neutrino masses. 



Neutrino oscillations arise, since the neutrinos, produced by the weak
interactions, are not mass eigenstates, but mixtures of mass
eigenstates. Like for the quarks on has a 3x3 mixing matrix, which can
be written as a product of three simple matrices, and a phase matrix,
which is present only, if neutrinos are Majorana particles:

\vspace{1cm}

$V=U\cdot P$ 


\begin{equation}
\text{U=}\left[\begin{matrix}c_{{l}}&s_{{l}}&0\\-s_{{l}}&c_{{l}}&0\\0&0&1
\end{matrix}\right]\left[\begin{matrix}e^{{-\text{i${\varphi}$}}}&0&0\\0&c&s\\0&-s&c\end{matrix}\right]
\left[\begin{matrix}c_{{\mathit{{\nu}}}}&-s_{{\mathit{{\nu}}}}&0\\s_{{\mathit{{\nu}}}}&c_{{\mathit{{\nu}}}}&0\\0&0&1\end{matrix}\right]
\end{equation}

$\text{P=}\left[\begin{matrix}e^{\text{i${\rho}$}}
&0&0\\0&e^{\text{i${\sigma}$}}&0\\0&0&1\end{matrix}\right]$




Here $s_{\nu}$ stands for $\sin \theta_{\nu}$ ($\theta_{\nu}$: solar mixing angle), $s$ stands for $\sin \theta$ 
($\theta$ : atmospheric mixing angle), and $s_l$ stands for $\sin \theta_{l}$ ($\theta_l$ : reactor mixing angle). 
The latter has not been measured. The experimental
results for the solar and the atmospheric mixing angles are
[5,6]):


\begin{equation}
\begin{matrix}\text{30}^{{o}}\le
\mathit{{\theta}}_{{\mathit{{\nu}}}}\le \text{39}^{{o}}\hfill\null
\\\hfill\null \\\text{37}^{{o}}\le \mathit{{\theta}}\le
\text{53}^{{o}}\hfill\null \end{matrix}.
\end{equation} 


We assume for the lepton mass matrices
the same texture 0 pattern as for the quarks [7]. Thus we obtain
the same relations between the mixing angles and the mass
eigenvalues:


\begin{equation}
{\text{tan}\mathit{{\theta}}_{{l}}=\sqrt{\frac{m_{{e}}}{m_{{\mathit{{\mu}}}}}}\approx
0\text{.}\text{07}}
\end{equation} 
${\text{tan}\mathit{{\theta}}_{{\mathit{{\nu}}}}=\sqrt{\frac{m_{{1}}}{m_{{2}}}}}$




Since the solar angle has been measured (we take 33 degrees), we find
for the mass ratio of the first two neutrinos:


\begin{equation}
m_{{1}}/m_{{2}}\approx 0\text{.}\text{42}.
\end{equation}




The oscillation experiments provide us with the mass squared
differences. The new relation allows us to determine the neutrino
masses. We find (in eV):



${\begin{matrix}m_{{1}}\approx 0\text{.}\text{004}\hfill\null
\\m_{{2}}\approx 0\text{.}\text{01}\hfill\null \\m_{{3}}\approx
0\text{.}\text{05}\hfill\null \end{matrix}}$ 



These neutrino masses are very small. We observe that the masses show a
rather weak hierarchy, but the mass spectrum is not inverted. The first
neutrino has the smallest mass.



If the neutrino masses are Majorana masses, one expects a neutrinoless
double beta decay. The present limit on the Majorana mass is about 0.23
eV [8]. This limit needs to be improved by about a factor 5. At
this level the decay should be seen. 



The atmospheric mixing angle is consistent with 45 degrees. The
parameter C in eq. (2) might be zero, as originally assumed (see ref.
[3]). But the high mass of the t-quark does not allow this possibility
for the quarks. It might work for the leptons. In this case the
atmospheric mixing angle is related to the two angles, which are given
by the corresponding mass ratios:


\begin{eqnarray}
&&\begin{matrix}\text{tan}\mathit{{\theta}}_{{1}}=\sqrt{\frac{m_{{\mathit{{\mu}}}}}{m_{{\mathit{{\tau}}}}}}\hfill\null
\\\text{tan}\mathit{{\theta}}_{{2}}=\sqrt{\frac{m_{{2}}}{m_{{3}}}}\hfill\null
\end{matrix}\nonumber \\ 
&&\begin{matrix}\mathit{{\theta}}_{{1}}\approx
\text{14}^{{o}}\hfill\null \\\hfill\null
\\\mathit{{\theta}}_{{2}}\approx \text{24}^{{o}}\hfill\null
\end{matrix}
\end{eqnarray} 



The atmospheric mixing angle is given by the absolute value of the sum
of the two angles, including a relative phase between the two terms. In
order to get the direct sum, this phase must be 180 degrees. In this
case we have for the atmospheric mixing angle:


\begin{equation}
\text{${\theta}$=${\theta}$}_{1}\text{+${\theta}$}_{2}\approx
\text{38}^{{o}}
\end{equation} 



We cannot obtain a maximal mixing (45 degrees), but our result is
consistent with the experiment. 

We can predict the matrix element 
$V_{3e}$ of the mixing matrix $V$:


\begin{equation}
V_{{3e}}=\text{sin}\mathit{{\theta}}\text{sin}\mathit{{\theta}}_{{l}}\approx
0\text{.}\text{707}\cdot \sqrt{m_{{e}}m_{{\mathit{{\mu}}}}}\approx
0\text{.}\text{05}.
\end{equation} 



A matrix element of this magnitude could be observed in the upcoming
reactor neutrino experiments.


\begin{thebibliography}{99}
\bibitem{1}
M. Kobayashi and T. Maskawa, \emph{Prog. Theor. Phys.} \textbf{49}, 652 (1973).
\bibitem{2}
H. Fritzsch and Z. Xing, \emph{Phys. Letters B} \textbf{413}, 396 (1997);
H. Fritzsch and Z. Xing, \emph{Phys. Rev. D} \textbf{57}, 594 (1998).
\bibitem{3}
H. Fritzsch, \emph{Nucl. Phys. B} \textbf{155}, 189 (1979).
\bibitem{4}
H. Fritzsch and Z. Xing, \emph{Nucl. Phys. B} \textbf{556}, 49 (1999).
\bibitem{5}
J. P. Cravas et al., arXiv: 0803.4312
\bibitem{6}
B. Aharmim et al., \emph{Phys. Rev. C} \textbf{72}, 055502 (2005).
\bibitem{7}
H. Fritzsch and Z. Xing, \emph{Phys. Lett. B} \textbf{634}, 514 (2006).
\bibitem{8}
C. Arnaboldi et al. arXiv: 0802.3439
\end{thebibliography}
\end{document}