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\begin{document}

\title{Theoretical implications of neutrino oscillations.}
\author{G. G. Ross \\
%EndAName
Department of Physics, Theoretical Physics, University of Oxford,\\
1 Keble Road, Oxford OX1 3NP, U.K.\\
and \\
Theory Division, CERN, CH-1211, Geneva 23,\\
Switzerland}
\date{}
\maketitle

\begin{abstract}
We review the possible origins of neutrino mass and the progress that has
been made towards understanding the pattern of neutrino masses and mixings.
The relation of the masses and mixings of the leptons to the quarks is
explored.
\end{abstract}

\section{Introduction}

The observation of neutrino oscillation has raised several crucial questions
concerning the origins of the theory of the fundamental interactions and the
origin of mass \cite{ross}. The first question is how many light neutrinos
and is this number related to the numer of quark and charged leptons? The
second immediate question is why are they so light in comparison to the
other states of the Standard Model and is the mass a Dirac mass, conserving
lepton number or a Majorana mass, violating lepton \ number? Thirdly what is
the spectral pattern; is there near degeneracy, an hierarchy or an inverted
hierarchy? Finally what is the underlying origin of fermion mass generating
the hierarchical pattern of quark, charged lepton and neutrino masses and
giving rise to small mixing angles in the quark sector and large mixing
angles in the lepton sector.

\section{How many light neutrinos?}

Given the fact that in field theory radiative corrections communicate
between different sectors of the theory there are only two ways of
protecting a light state from large radiative corrections to its mass. The
first is if the state is composite in which case the virtual momentum of the
intermediate states in radiative corrections are cut-off at the scale that
the strong binding force becomes weak. However neutrinos are so much lighter
than the energy scales we have already probed without finding composite
structure that it looks extremely improbable that they be composite.
Therefore their mass should be protected by a symmetry from getting large
radiative corrections.

In the Standard Model the neutrinos, $\nu _{Li},$ are protected by Lepton
number conservation from getting Majorana mass (there are no right-handed
neutrino states so there are no Dirac masses). There are three conserved
lepton numbers corresponding to the three generations. What about the
possibility of additional light fermion states? There cannot be a normal
fouth family neutrino because it would contribute to the Z width at an
unacceptable level. However there could be additional light
\textquotedblleft sterile\textquotedblright\ states not transforming under
the Standard Model, provided they too are protected by a symmetry. A simple
example is if the sterile state transforms under an additional Abelian
symmetry for the Majorana mass term for this neutrino breaks this symmetry.
Supersymmetry also provides candidates for additional light fermion states.
The vector supermultiplet has gaugino fermion partners of the gauge bosons
which cannot get a mass until supersymmetry or the gauge symmetry is broken.
The Higgs bosons have Higgsino fermion superpartners which cannot be
different in mass from the Higgs until supersymmetry is broken. Any of the
neutral members of these new fermion states, \textquotedblleft
neutralinos\textquotedblright , are candidates to mix with the Standard
Model neutrino states. The masses of these new states is not very small (of
the order of the supersymmetry breaking scale) but they can generate small
neutrino masses via a see-saw mechanism in which they couple to $\nu _{Li}$
via small R-parity violating couplings. Another possible source of new light
states motivated by string theories are modulini - the superpartners of the
numerous moduli fields. The vacuum expectation values of the moduli fix the
couplings and the compactification characteristics of the theory. They and
their partner modulini are massless until supersymmetry is broken. In
summary, the message is that there is no theoretical obstruction to having
additional sterile neutrino states which may play a role in generating
neutrino masses.

\section{Why so light?}

Given that the neutrinos are protected from getting mass the question why
they are so light is equivalent to the question why the symmetry is broken
so weakly. There are several possible explanations for this. One possibility
is the see-saw mechanism communicating the breaking from a very high scale
such as the scale of Grand Unification via large couplings or from a much
lower scale, such as the supersymmetry breaking scale, via smaller R-parity
violating couplings. A second possibility occurs in theories with additional
space dimensions in which the coupling communicating the breaking of the
symmetry can be extremely small. A third possibility is that the breaking of
the symmetry occurs in radiative order only and thus is suppressed by the
smallnes of such corrections.

\subsection{Small neutrino masses : the see-saw mechanism}

The \textquotedblleft see-saw\textquotedblright\ mechanism \cite{seesaw}
follows naturally in the case that right-handed neutrinos exist. Suppose
that there is no weak isospin 1 Higgs field and hence there are no mass
terms of the $\nu _{L}\nu _{L}$ type. In this case there are just two
possible neutrino masses

\begin{equation}
m_{Dirac}\overline{\nu _{L}}\nu _{R}+M_{Majorana}\nu _{R}\nu _{R}\equiv
\left( 
\begin{array}{cc}
\overline{\nu _{L}} & \nu _{R}%
\end{array}
\right) \left( 
\begin{array}{cc}
0 & m_{D} \\ 
m_{D} & M_{M}%
\end{array}
\right) \left( 
\begin{array}{c}
\overline{\nu _{L}} \\ 
\nu _{R}%
\end{array}
\right)  \label{seesaw}
\end{equation}
The Dirac mass is similar to an up quark mass and one's naive expectation is
that they should have similar magnitude. On the other hand, the Majorana
mass term is invariant under the Standard Model gauge group and does not
require a stage of electroweak breaking to generate it. For this reason, one
expects the Majorana mass to be much larger than the electroweak breaking
scale, perhaps as large as the scale of the new physics beyond the Standard
Model; for example, the Grand Unified scale or even the Planck scale.
Diagonalising the mass matrix gives the eigenvalues 
\begin{eqnarray}
m_{Heavy} &\simeq &M_{M}  \label{eigenvalues} \\
m_{Light} &\simeq &\frac{m_{D}^{2}}{M_{M}}  \nonumber
\end{eqnarray}
The see-saw mechanism generates an effective Majorana mass for the light
neutrino (predominantly $\nu _{L}$) by mixing with the heavy state
(predominantly $\nu _{R})$ of mass $M_{M}.$ It is driven by an effective
Higgs $\Phi ^{I_{W}=1}$ made up of $H^{I_{W}=1/2}$ $H^{I_{W}=1/2}$ $/$ $%
M_{M} $ (hence the two factors of $m_{D}$ in eq.(\ref{eigenvalues})$~).$ Eq.(%
\ref{eigenvalues}) shows that a large scale for the Majorana mass gives a
very light neutrino. For example with $M_{M}=10^{16}~{\rm GeV}$ and $m_{D}$
taken to be the top quark mass gives 
\[
m_{Light}~\simeq ~3.10^{-3}~{\rm eV} 
\]

This estimate shows that it is quite natural to have neutrinos in a mass
range appropriate to give, for example, solar neutrino oscillations.
However, in many cases, larger masses capable of explaining the other
oscillation phenomena are possible because the Majorana mass for the right
handed neutrinos is often smaller than the Grand Unified mass. A Majorana
mass for the right-handed neutrino requires a Higgs carrying right-handed
isospin 1 (in analogy with the left-handed case when it needed left-handed
isospin 1). If this field is not present (for example in level one string
theory this is always the case) one may get a double see-saw because the
Majorana mass for the right-handed neutrino is also generated by an
effective Higgs, made up of $H^{I_{W,R}=1/2}H^{I_{W,R}=1/2}/M^{{\prime }}$,
where $M^{\prime }$ denotes a scale of physics beyond the Grand Unification
scale. Taking this to be the Planck scale (probably the largest reasonable
possibility) and $<H^{I_{W,R}=1/2}>$ to be the Grand Unified scale (it
breaks any Grand Unified group) one finds 
\[
M_{M}~\simeq ~\frac{(10^{16})^{2}}{10^{19}}~{\rm GeV} 
\]
giving 
\[
m_{Light}~\simeq ~1~{\rm eV} 
\]
Thus, one sees that the see saw mechanism naturally gives neutrino masses in
the range relevant to neutrino oscillation measurements. Moreover, as the
neutrino mass is proportional to the Dirac mass squared, taking the Dirac
mass of each family of neutrinos to be of the order of the equivalent up
quark mass, one obtains a large hierarchy between different families of
neutrino. This is what is required if one is to explain several oscillation
phenomena, for it allows the existence of several mass differences.

\subsection{Small neutrino masses - large new dimensions}

It is interesting to see how a very small coupling responsible for neutrino
mass arises in a theory with additional space dimensions. In such theories
the solution to the hierarchy problem is supposed to be due to the absence
of any fundamental mass scale much larger than the electroweak breaking
scale $\cite{dim}$ and so there is no large mass scale to suppress the
communication of the symmetry breaking as happens in the see-saw mechanism.
However, in such theories, the weakness of the gravitational interactions
does not require a fundamental Planck scale of $O(10^{18}GeV).$ The reason
is readily seen if we compute the Newtonian potential between two point
masses in the underlying (4+$\delta )$ dimensional theory. It is given by 
\[
V_{gravity}(r)=\frac{m_{1}m_{2}}{M_{Planck,4+\delta }^{\delta +2}}.\frac{1}{%
r^{\delta +1}} 
\]
where $M_{Planck,4+\delta }$ is the fundamental mass scale in the (4+$\delta
)$ dimensional theory (the string tension in a string theory). This form
applies at distances much smaller than the size, $R,$ of the $\delta $
additional compact dimensions. At distances larger than their size we have 
\[
V_{gravity}(r)|_{r>>R}=\frac{m_{1}m_{2}}{M_{Planck,4+\delta }^{\delta +2}}.%
\frac{1}{R^{\delta }}.\frac{1}{r} 
\]
This shows that the Planck scale in our 4 dimensional theory is made larger
by the spreading of flux in the new dimensions. It is given by 
\begin{equation}
M_{Planck,4}^{2}=M_{Planck,4+\delta }^{\delta +2}\;R^{\delta }
\end{equation}
For a large new dimension (measured in the fundamental unit $%
M_{Planck,4+\delta })$ we have $M_{Planck,4+\delta }\;R\gg 1$ and so we can
see that the large size of the Planck mass (and the associated feebleness of
the gravitational force) may be due to the existence of a new large
dimension.

In these theories the smallness of Newton's coupling is due to flux
spreading of the graviton in the additional dimension. If the right-handed
neutrino is also able to propagates in the new large dimension its flux too
will spread in the extra dimensions and this will lead to a suppression of
the Yukawa coupling of the neutrino to the Higgs. As a result the neutrino
mass will be naturally small, having a similar see-saw form to that in eq(%
\ref{eigenvalues}) suppressed by an inverse power of the four dimensional
Planck mass\cite{neut}. Indeed it is known that candidate sterile neutrino
must exist in supersymmetric versions of such theories, namely the modulini
associated with the moduli setting the size of the new dimension. To make
this idea more explicit consider a five dimensional (5D) space with
coordinates $(x_{i},y).$ The interaction of the right-handed bulk neutrino, $%
\nu _{R}(x.y),$ with the Standard Model lepton doublet and Higgs states, $%
l(x),$ $h(x),$ confined to a 4-D brane is given by 
\[
S=\lambda \int d^{4}xl(x)h(x)\nu _{R}(x,y=0) 
\]
If the fifth dimension is compact (here, compactified on a circle) we may
expand the neutrino wave function in terms of its Kaluza Klein tower of
states 
\[
\nu _{R}(x,y)=\frac{1}{\sqrt{2\pi RM_{\ast }}}\sum_{n=-\infty }^{\infty }\nu
_{R}^{n}(x)e^{iny/R} 
\]
where the normalisation factor has been chosen so that the states, $\nu
_{R}^{n}(x),$ have canonical normalisation for their kinetic terms - this
factor accounts for the flux spreading into the fifth dimension. Here $R$ is
the radius of compactification and $M_{\ast }$ is the fundamental mass
scale. The neutrino mass is given by 
\begin{eqnarray*}
m_{\nu } &=&\frac{\lambda <h>}{\sqrt{2\pi RM_{\ast }}} \\
&=&\frac{\lambda <h>M_{\ast }}{M_{Planck,4}} \\
&\approx &10^{-4}\lambda eV.(\frac{M_{\ast }}{1TeV})
\end{eqnarray*}
where we have used the fact that the 5D mass scale is related to the 4D
Planck scale by $2\pi RM_{\ast }=$ $M_{Planck,4}/M_{\ast }.$ Thus we see the
effect of flux spreading is to suppress the neutrino coupling in the same
way as the gravitational coupling is suppressed since both propagate in the
extra dimension(s). As a result one naturally obtains a see-saw mechanism
suppressed by the large Planck mass even though there are no fundamental
mass scales of this magnitude in the theory. For $M_{\ast }$ of order $1TeV,$
a scale that avoids the hierarchy problem, one readily obtains neutrino
masses of the correct order.

\subsection{Small neutrino masses - radiative generation}

As discussed above, yet another possibility is that neutrino masses are zero
at tree level and are generated at some order in perturbation theory. A
simple example is given in (non-MSSM) supersymmetric extensions of the
Standard Model in which there are $R-$parity violating couplings generate
the lepton-number violating couplings needed to give a Majorana mass. For
example the coupling $h\nu d\widetilde{d}^{c}$ generates a neutrino mass at
one loop order via a $d$ quark $\widetilde{d}^{c}$ squark intermediate state
given by \cite{kingdavidson} 
\[
m_{\nu }=\frac{h^{2}}{16\pi ^{2}}\frac{m_{d}^{2}A_{SUSY}}{m_{\widetilde{d}%
}^{2}} 
\]
where $A_{SUSY}$ is a soft trilinear supersymmetry breaking parameter and $%
m_{\widetilde{d}}$.is the down squark mass. For a reasonable choice of the
supersymmetry breaking parameter this gives 
\[
\frac{m_{\nu }}{m_{d}}\approx 10^{-5}h^{2} 
\]
small without the need for a GUT mass.

\section{What is the spectral pattern?}

\FRAME{ftbpFU}{521.8125pt}{490.6875pt}{0pt}{\Qcb{Four possible spectral
patterns capable of explaining the experimental indications of neutrino
mass. The composition of the neutrino mass eigenstates in terms of the
current neutrino eigenstates is shown,}}{\Qlb{spectral}}{Figure }{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "T";width 521.8125pt;height
490.6875pt;depth 0pt;original-width 517.0625pt;original-height
486.0625pt;cropleft "0";croptop "1";cropright "1";cropbottom
"0";tempfilename 'H68AOG00.wmf';tempfile-properties "XNPR";}}In Figure \ref%
{spectral} four viable mass patterns are shown together with the composition
of the states. The implications of each of these possibilities is different
so we will discuss tham in turn.

\subsection{Near degeneracy}

In this case the neutrinos are very nearly degenerate. Since the oscullation
experiments only measure mass (squared) differences the average mass can be
much larger than is the case if the masses are hierarchical. To generate the
degeneracy needs a family symmetry to ensures it is stable against radiative
corrections \cite{brs}. An example of such a symmetry \cite{bhrk} is
provided by $SO(3)$ in which the three left-handed lepton doublets of the
Standard Model make up a triplet of $SO(3).$ In the case that the symmetry
is unbroken the neutrinos must be degererate and the mass splitting is
generated by small symmetry breaking effects. Given the near degeneracy the
large mixing angles in the neutrino sector is quite natural. Making the mass
as large as possible cosmologically interesting abundances are possible. 
\[
\Omega _{\nu }\leq (0.05-0.5)\left( \frac{0.5}{h}\right) ^{2} 
\]
It is easy to avoid conflict with double beta decay bounds by arranging that
the degenerate mass corresponds to a Pseudo Dirac mass in the $e\mu $ sector
ensuring that $e-\mu $ number is conserved and eliminating double beta decay
processes.

\subsection{Normal hierarchy\label{nh}}

The second spectral pattern of Figure \ref{spectral} has a \textquotedblleft
normal\textquotedblright\ distribution in the sense that the heaviest
neutrino has a mass $m_{@}\simeq \sqrt{\Delta m_{@}^{2}\text{ }}$ and the
next has mass $m_{\odot }\simeq \sqrt{\Delta m_{\odot }^{2}}$ where $\Delta
m^{2}$ denotes the mass squared difference relevant to the oscillation
measurement. For the case of three generations the see-saw mechanism gives
the effective neutrino masses of the form 
\begin{equation}
M_{\nu }=M_{D}^{\nu }.M_{M}^{-1}.M_{\nu }^{D}  \label{matseesaw}
\end{equation}
where the Ms are $3\times 3$ matrices in family space. We will work in a
basis where the charged leptons and Majorana mass are diagonal. Then we have 
\[
M_{D}^{\nu }=\left( {%
\begin{array}{*{20}c}
   {x_1 } & {x_2 } & {x_3 }  \\
   {y_1 } & {y_2 } & {y_3 }  \\
   {z_1 } & {z_2 } & {z_3 }  \\
\end{array}}\right) \quad M_{M}=\left( {%
\begin{array}{*{20}c}
   {M_1 } & {} & {}  \\
   {} & {M_2 } & {}  \\
   {} & {} & {M_3 }  \\
\end{array}}\right) 
\]
An interesting structure emerges if the Majorana masses are hierarchical
such that the first mass is so light it dominates eq.(\ref{matseesaw}) \cite%
{king}. In this case (Single Right Handed Dominance SRHD) we have 
\begin{equation}
m_{@}\simeq {\textstyle{\frac{1}{{M_{1}}}}}  \label{srhd}
\end{equation}
The condition that this should give near maximal neutrino mixing corresponds
to $x_{2}\sim x_{3}>>x_{1}.$ The interest in this is that, as we shall see,
such a pattern occurs in the quark mass matrix. Thus, due to the see-saw
mechanism, the smallness of the quark mixing angles does not imply small
neutrino angles even though the neutrino Dirac mass matrix has the same form
as the quark mass matrix.

What about the solar mixing angle? Again a particularly interesting
structure emerges if the hierarchical form of the Majorana masses continues
to dominate in the sense that the second Majorana state dominates the
contribution to the second heaviest neutrino state \cite{king}. In this case
(Double Right Handed Dominance DRHD) 
\[
m_{\odot }\simeq {\textstyle{\frac{1}{{M_{2}}}}}\sum\limits_{i}{%
y_{i}^{2},\quad y_{2}\sim y_{3}\sim y_{1}} 
\]
and the near maximal measurement of neutrino masses implies ${y_{2}\sim
y_{3}\sim y_{1}.}$ We shall discuss shortly whether this is consistent with
the Dirac mass matrix of the same form as the quark mass matrix. In this
scheme we have a bound on the remaining mixing angle \cite{savoy,kingp}

\[
|U_{e3}|\geq {\textstyle{\frac{1}{2}}}\sin 2\theta _{12}{\textstyle{\frac{{%
m_{\odot }}}{{m_{@}}}}} 
\]
This implies $|U_{e3}|$ is close to the present bound. Such a value is
encouraging for the prospect of measuring CP\ violation using very long
baseline experiments.

\subsection{Inverted hierarchy}

In the case of the inverted hierarchy the two heaviest neutrinos are nearly
degenerate with a Pseudo Dirac mass equal to $m_{@}$ while the third
eigenstate has mass $m_{\odot }$ corresponding to the form 
\[
M_{\nu }\simeq m_{@}\left( {%
\begin{array}{*{20}c}
   0 & 1 & 1  \\
   1 & 0 & {}  \\
   1 & {} & 0  \\
\end{array}}\right) +m_{\odot }\left( {%
\begin{array}{*{20}c}
   0 & {} & {}  \\
   {} & 0 & {}  \\
   {} & {} & 1  \\
\end{array}}\right) 
\]
This structure gives a small contribution to $|U_{e3}|\cite{savoy}$

\[
|U_{e3}|\leq {\textstyle{\frac{{m_{\odot }^{2}}}{{2m_{@}^{2}}}}} 
\]
but there is a contribution from the diagonalisation of the charged lepton
masses which can increase it near to the experimental bound again \cite%
{kingp}.

\section{What is the origin of fermion mass structure?}

Of course the fundamental question raised by the pattern of gauge couplings,
fermion masses and mixing in the Standard Model is what is the underlying
origin of the structure? There are several promising indications that it may
be due to an extension of the symmetries of the Standard Model. The most
significant is the unification of gauge couplings which suggests an
underlying Grand Unified structure such as $SU(5)$ or $SO(10).$ In them both
the quarks and leptons of a single family are related. If this is the case
it seems difficult to understand why quark mixing angles are small while
neutrino mixing angles are large \cite{lola}. In SU(5) the left handed
lepton doublet together with the (charge conjugate ) right handed quarks
make up a $5$ dimensional representation. Thus the mixing in the neutrino
sector is related to the mixing of the right handed down quarks which is not
measured by charged weak processes and so might be large. In $SO(10)$ this
seems unlikely because all the members of a single family, plus the right
handed neutrino belong to a single $16$ dimensional representation and so,
if $SO(10)$ breaking effects are small, the mixing in the neutrino sector
are related to the mixing in both the left- and right- handed sectors. \ 

However, as we have seen, due to the see-saw mechanism, our naive
expectations about the relation between mixing angles may not be correct and
equality of mixing in the Dirac mass matrices of quarks and leptons may be
quite consistent with neutrino oscillations! This is easy to see in the
limit of a strong hierarchy of neutrino Majorana masses, $M_{2}<<M_{1,3}$
which gives (c.f eq.(\ref{srhd}) - the difference is because we have chosen $%
M_{2}$ rather than $M_{1}$ to be the lightest state) 
\[
m_{@}\simeq {{\frac{1}{{M_{2}}}}}\sum\limits_{i}y{_{i}^{2},\quad } 
\]
In the linit ${y_{2}\sim y_{3}>>y_{1}}$ the mixing angle $\left(
U_{MNS}\right) _{23}$ relevant for atmospheric neutrino mixing is large.
This pattern for the elements of the neutrino Dirac mass matrix can be
compared with the measured values of the down quark mass matrix (I\ use the
down quark matrix since the up quark is poorly determined)) \cite{liliana} 
\begin{equation}
{\textstyle{\frac{{M^{d}}}{{m_{b}}}}}=\left( {\begin{array}{*{20}c} 0 &
{1.5\varepsilon ^3 } & {0.4e^{i20} \varepsilon ^3 } \\ {1.5\varepsilon ^3 }
& {\varepsilon ^2 } & {1.3\,\varepsilon ^3 } \\ {0.4e^{i20} \varepsilon ^3 }
& {1.3\,\varepsilon ^3 } & 1 \\ \end{array}}\right) \quad \quad \varepsilon
=0.15  \label{md}
\end{equation}
One sees that the $(2,2)$ element is quite close to the $(2,3)$ element,
just as is wanted in the neutrino sector!

This illustrates the point that there may be a close connection between the
\ mass patterns of the quark and lepton mass matrices consistent with large
neutrino mixing angles. To illustrate how far this connection can be taken
and the implications for an underlying Grand Unified and family symmetry I\
will summarise the form that results from a $SU(3)$ family symmetry \cite%
{kingross,liliana1} in which the three generations of left-handed quarks and
leptons are in the triplet representation as are the charge conjugate of
their right handed partners. In addition there is a Grand Unified symmetry ($%
SO(10))$ which relates quarks and leptons. The symmetry constrains the Dirac
mass matices of down quarks and charged leptons to have the form 
\begin{equation}
\frac{{M^{d}}}{{m_{b}}}=\left( {\begin{array}{*{20}c} {\varepsilon ^8 } &
{\varepsilon ^3 (z + (x + y)\varepsilon )} & {\varepsilon ^3 (z + (x -
y)\varepsilon } \\ { - \varepsilon ^3 (z + y\varepsilon )} & {\varepsilon ^2
(w + u\varepsilon )} & {\varepsilon ^2 (w - u\varepsilon )} \\ { -
\varepsilon ^3 (z + y\varepsilon )} & {\varepsilon ^2 (w - u\varepsilon )} &
1 \\ \end{array}}\right)  \label{mdd}
\end{equation}
\begin{equation}
\frac{{M^{l}}}{{m_{\tau }}}=\left( {\begin{array}{*{20}c} {\varepsilon ^8 }
& {\varepsilon ^3 (z + (x + y)\varepsilon )} & {\varepsilon ^3 (z + (x -
y)\varepsilon } \\ { - \varepsilon ^3 (z + y\varepsilon )} & {\varepsilon ^2
(a_lw + u\varepsilon )} & {\varepsilon ^2 (a_lw - u\varepsilon )} \\ { -
\varepsilon ^3 (z + y\varepsilon )} & {\varepsilon ^2 (a_lw - u\varepsilon
)} & 1 \\ \end{array}}\right)  \label{ml}
\end{equation}
Here $z$, $w$ and $u$ are real coefficients and $x$ and $y$ complex
coefficients of order 1. The parameter $\varepsilon \simeq 0,15$ is the $%
SU(3)$ symmetry breaking parameter and an additional Abelian family symmetry
orders the breaking in the different matrix elements in a way that
reproduces the form of eq.(\ref{md}). We have also included subdominant
contributions developing the first terms in a power series in $\varepsilon .$
The important thing to note is that the underlying $SU(3)$ symmetry requires
the equality of the leading terms of order $\varepsilon ^{3}$ in the $(1,2)$
and $(1,3)$ elements and the leading terms of order $\varepsilon ^{2}$ in
the $(2,2)$ and $(2,3)$ elements. The same applies to the nonleading terms.
This will be important in generating near bi-maximal neutrino mixing. The
form of the lepton mass matrix, eq(\ref{ml}), is the same as that of the
quark mass matrix, eq.(\ref{mdd}) and the coefficients are related due to
the $SO(10)$ Grand Unified symmetry. In particular the $(1,2)$ and the $%
(1,3) $ elements are equal leading to the relations $Det(M_{d})=Det(M_{l})$
and $m_{b}=m_{\tau }$ at the Grand Unified scale. After including radiative
corrections to continue to the laboratory scale, these relations are in
excellent agreement with experiment. The coefficient $a_{l}$ also comes from 
$SO(10)$ but in this case the fields generating the masses have a Clebsch
Gordon coefficient giving $\left| a_{l}\right| =3.$ This gives an excellent
value for $m_{\mu }$, $m_{\mu }=3m_{s}$ at the GUT scale leading, after
radiative corrections, to $m_{\mu }\simeq m_{s}$ at low energies \cite%
{georgi} which, together with the two previous relations, generates the
charged lepton masses from the down quark masses.

The up quark and netrino mass matrices are also related by the GUT giving 
\[
\quad M^{d}\leftrightarrow M^{u}\quad M^{l}\leftrightarrow M^{\nu }\quad
\varepsilon \leftrightarrow \varepsilon ^{\prime }\quad a_{l}\leftrightarrow
a_{\nu } 
\]
The appearance of the different expansion parameter $\varepsilon =0.05$ is
due to the $SO(10)$ breaking effects which are necessarily present. Finally
the Majorana mass matrix is very nearly diagonal and strongly hierarchical.
This leads to the DRHD discussed in Section \ref{nh} with the two heaviest
neutrinos having the form 
\[
\nu _{a}\propto \quad z(\nu _{\mu }+\nu _{\tau })+y\varepsilon ^{\prime
}(\nu _{\mu }-\nu _{\tau }) 
\]
\[
\nu _{b}\propto \quad z\nu _{e}+r(\nu _{\mu }-\nu _{\tau }-y\varepsilon
^{\prime }(\nu _{\mu }+\nu _{\tau })/z) 
\]
where 
\[
r=\sqrt{2}(u-a_{\nu }wy/z)\quad =O(1) 
\]
This naturally gives near bi-maximal mixing in the neutrino sector. There
are two choices for $a_{\nu }$ consistent with $\left| a_{l}\right| =3,$
namely $\left| a_{\nu }\right| =3$ and $a_{\nu }=0.$ The former gives the $%
LOW$ solar solution while the latter gives the $LMA$ solution \cite{liliana1}%
. This example shows that, contrary to our first impressions, near bimaximal
mixing is consistent with a GUT relation between neutrino masses and quark
masses.

\section{Summary and Conclusions.}

The new data on quark and lepton masses and mixing is testing our ideas for
the origin of fermion masses and mixings and raises interesting questions.
The multiplicity of light neutrino-like states sheds light on the symmetries
of the low energy theory and additional sterile neutrinos would indicate new
symmetries. There are several ideas for the origin of the hierarchy of
masses of the neutrinos. These include approximate symmetries such as lepton
numbers, new family symmetries supersymmetry, or new space dimensions giving
rise to anomalously small Yukawa couplings responsible for fermion mass or
the see-saw mechanism in which the large mass of the messenger state makes
the neutrinos anomalously light. The pattern of quark and lepton masses and
mixings suggests there may be a relation between quarks and leptons. This
may be due to an underlying Grand unified symmetry such as $SU(5)$ or $%
SO(10) $ or directly due to the underlying string structure which may not
have a GUT symmetry below the compactification scale. The connection between
the mixing angles in the quark and lepton sectors and the underlying mass
matrix structure is masked by the see-saw mechanism, In one particular limit
with a very strong hierarchy between the Majorana masses the connection
becomes simple and the large mixing angles measured in the lepton sector
correspond to small mixing angles. In this scheme the quark and lepton Dirac
matrices may have quite similar forms, with differences due to simple
Clebsch Gordon coefficients coming from an underlying GUT. An encouraging
feature of the models of neutrino mass for all the spectral patterns is that
the remaining mixing angle $U_{e3}$ is likely to be large, close to the
present bounds and so CP violation in the lepton sector may be measurable in
very long baseline neutrino experiments.

We acknowledge support from the RTN European project HPRN-CT-2000-0148.

\begin{thebibliography}{99}
\bibitem{ross} For reviews of neutrino mass models and extensive references
see

G.G.Ross, ``Models of fermion mass''\ , Published in *Boulder 2000, Flavor
physics for the millennium 775-824;

H.~Fritzsch and Z.~z.~Xing, 
%``Mass and flavor mixing schemes of quarks and leptons,''
Prog.\ Part.\ Nucl.\ Phys.\ {\bf 45}, 1 (2000) [arXiv:hep-ph/9912358];

G.~Altarelli and F.~Feruglio, 
%``Theoretical models of neutrino masses and mixings,''
arXiv:hep-ph/0206077.

\bibitem{seesaw} T.Yanagida in Proc. of the workshop in Unified Theory and
baryon number in the Universe, KEK, 1979; M. Gell-Mann, P. Ramond and R.
Slansky, proceedings of the Supergravity Stony Brook Workshop, New York,
1979, ed. by P. Van Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam).

\bibitem{FN} C.~D.~Froggatt and H.~B.~Nielsen, 
%``Hierarchy Of Quark Masses, Cabibbo Angles And CP Violation,''
Nucl.\ Phys.\ {\bf B147}, 277 (1979).

\bibitem{dim} N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett.
B429(1998) 263;Phys. Rev.D59(1999) 086004.;I.Antoniadis, N. Arkani-Hamed, S.
Dimopoulos and G. Dvali, Phys.Lett B426(1998)257.

\bibitem{neut} K.R.Dienes, E.Dudas and T.Gherghetta, Nucl. Phys. B557 (1999)
25 [hep-ph/9811428]; N.Arkani-Hamed, S.Dimopoulos, G.Dvali and
J.March-Russell, hep-ph/9811448.

\bibitem{kingdavidson} S.~Davidson and S.~F.~King, 
%``Bi-maximal neutrino mixing in the MSSM with a single right-handed  neutrino,''
Phys.\ Lett.\ B {\bf 445} (1998) 191 [arXiv:hep-ph/9808296].

\bibitem{brs} R.~Barbieri, G.~G.~Ross and A.~Strumia, 
%``Vacuum oscillations of quasi degenerate solar neutrinos,''
JHEP {\bf 9910} (1999) 020 [arXiv:hep-ph/9906470].

Phys.\ Lett.\ {\bf B445} (1998) 191 [hep-ph/9808296].

\bibitem{bhrk} R. Barbieri, L.J. Hall, G.L. Kane and G.G. Ross,
hep-ph/9901228. %\cite{King:1999cm}
A. Yu Smirnov, Phys. Rev. D48 (1993) 3264

\bibitem{king} A. Yu Smirnov, Phys. Rev. D48 (1993) 3264; S.~F.~King, 
%``Atmospheric and solar neutrinos from single right-handed neutrino  dominance and U(1) family symmetry,''
Nucl.\ Phys.\ B {\bf 562}, 57 (1999) [arXiv:hep-ph/9904210].;%
%%CITATION = HEP-PH 9904210;%%
%\cite{Davidson:1998bi}

G.~Altarelli, F.~Feruglio and I.~Masina, 
%``Large neutrino mixing from small quark and lepton mixings,''
Phys.\ Lett.\ B {\bf 472}, 382 (2000) [arXiv:hep-ph/9907532]; 
%%CITATION = HEP-PH 9907532;%%

Q.~Shafi and Z.~Tavartkiladze, 
%``Neutrino mixings and fermion masses in supersymmetric SU(5),''
Phys.\ Lett.\ B {\bf 451}, 129 (1999) [arXiv:hep-ph/9901243]; 
%%CITATION = HEP-PH 9901243;%%
%\cite{King:1999mb}

S.~F.~King, Nucl.\ Phys.\ B {\bf 576}, 85 (2000) [arXiv:hep-ph/9912492]. 
%%CITATION = HEP-PH 9912492;%%

\bibitem{savoy} S.~Lavignac, I.~Masina and C.~A.~Savoy, 
%``Large solar angle and seesaw mechanism: A bottom-up perspective,''
Nucl.\ Phys.\ B {\bf 633} (2002) 139 [arXiv:hep-ph/0202086].

\bibitem{kingp} S.~F.~King, 
%``Constructing the large mixing angle MNS matrix in see-saw models with  right-handed neutrino dominance,''
JHEP {\bf 0209} (2002) 11 [arXiv:hep-ph/0204360].

\bibitem{lola} S.~Lola and G.~G.~Ross, 
%``Neutrino masses from U(1) symmetries and the Super-Kamiokande data,''
Nucl.\ Phys.\ B {\bf 553} (1999) 81 [arXiv:hep-ph/9902283].

\bibitem{liliana} R.~G.~Roberts, A.~Romanino, G.~G.~Ross and
L.~Velasco-Sevilla, %``Precision test of a fermion mass texture,''
Nucl.\ Phys.\ B {\bf 615}, 358 (2001). %[arXiv:hep-ph/0104088].
%%CITATION = HEP-PH 0104088;%%

\bibitem{kingross} S.~F.~King and G.~G.~Ross, 
%``Fermion masses and mixing angles from SU(3) family symmetry,''
Phys.\ Lett.\ B {\bf 520}, 243 (2001) [arXiv:hep-ph/0108112].

\bibitem{liliana1} G.~G.~Ross and L.~Velasco-Sevilla, 
%``Symmetries and fermion masses,''
arXiv:hep-ph/0208218.

\bibitem{georgi} H.~Georgi and C.~Jarlskog, 
%``A New Lepton - Quark Mass Relation In A Unified Theory,''
Phys.\ Lett.\ B {\bf 86}, 297 (1979).
\end{thebibliography}

\end{document}
