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\begin{document}
\pagestyle{empty}
\begin{flushright}
{CERN-TH/2000-004}
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\vspace*{5mm}
\begin{center}
{\bf NEUTRINO MASSES AND GRAND UNIFICATION} \\
\vspace*{1cm} 
{\bf G. Altarelli} \\
\vspace{0.3cm}
Theoretical Physics Division, CERN \\
1211 Geneva 23, Switzerland \\
and\\
Universit\`a di Roma Tre, Rome, Italy
\vspace*{2cm}  
 \end{center}
\vspace*{5mm}
\noindent
\begin{center}
Abstract
\end{center}
We discuss some models of neutrino masses and mixings in the context
 of fermion masses in Grand Unified Theories.
 
\vspace*{3cm} 
\noindent 
%
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{\it Talk given at the 6th International Workshop on\\
Topics in Astroparticle and Underground Physics (TAUP 99)\\
6--10 September 1999, Paris, France}
\end{center}

\vspace*{2.0cm}
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 CERN-TH/2000-004\\
January 2000
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\title{Neutrino Masses and Grand Unification}

\author{G. Altarelli \address{Theoretical Physics Division, CERN, \\ 
 1211 Geneva 23, Switzerland \\ 
 and\\
Universit\`a di Roma Tre, Rome, Italy \\ }%
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        %X.-Y. Wang\address{Economics Department, University of Winchester, \\
        %2 Finch Road, Winchester, Hampshire P3L T19, United Kingdom}
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\begin{document}

\begin{abstract}
We discuss some models of neutrino masses and mixings in the context
 of fermion masses in Grand Unified Theories.
\end{abstract}

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%\begin{document} 
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%\begin{center}
 %{\bf Neutrino Masses and Grand Unification} \\ 
 %\vspace*{1cm}  
 %{\bf G.Altarelli} \\ 
 %\vspace{0.3cm} 
 %Theoretical Physics Division, CERN \\ 
 %CH - 1211 Geneva 23 \\ 
 %and\\
%Universit\`a di Roma Tre, Rome, Italy \\ 
%\vspace*{2cm}   
%{\bf Abstract} \\
%\end{center} 
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%We discuss some models of neutrino masses and mixings in the context
 %of fermion masses in Grand Unified Theories.\\


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%CERN-TH/2000-004 \\ 
%January 2000 
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%28/12

Recent data from Superkamiokande \cite{SK} have provided a more solid experimental basis for neutrino
oscillations as an explanation of the atmospheric neutrino anomaly \cite{Ronga}. In addition the solar neutrino deficit
\cite{bel}, observed by several experiments, is also probably an indication of a different sort of neutrino oscillations.
Results from the laboratory experiment by the LSND collaboration \cite{LSND} can be considered as a possible indication of
yet another type of neutrino oscillation.  Neutrino oscillations imply neutrino masses. The extreme smallness of neutrino
masses in comparison with quark and charged lepton masses indicate a different nature of neutrino masses, linked to
lepton number non conservation and the Majorana nature of neutrinos. Thus neutrino masses provide a window on the very large
energy scale where lepton number conservation is violated and on GUTs. The new experimental evidence on
neutrino masses could also give an important feedback on the problem of quark and charged lepton masses, as all these
masses are possibly related in GUTs. In particular the observation of a nearly maximal mixing angle for
atmospheric neutrinos is particularly significant. Perhaps also solar neutrinos may occur with
large mixing angle. At present solar neutrino mixings can be either large or very small, depending on which particular
solution will eventually be established by the data. Large mixings are very interesting because a first
guess was in favour of small mixings in the neutrino sector in analogy to what is observed for quarks. If confirmed, single
or double maximal mixings can provide an important hint on the mechanisms that generate neutrino masses  \cite{ram}.

The experimental status of neutrino oscillations is still very preliminary. While the evidence for the
existence of neutrino oscillations from solar and atmospheric neutrino data is rather convincing by now, the values of
the mass squared differences $\Delta m^2$ and mixing angles are not firmly established. For solar neutrinos, for example,
three or even four possible solutions are still possible \cite{fogli}\cite{hall}.  Another issue which is still open is the
claim by the LSND collaboration of an additional  signal of
neutrino oscillations in an accelerator experiment \cite{LSND}. This claim was not so-far supported by a second recent
experiment, Karmen \cite{Karmen}, but the issue is far from being closed. Given the present experimental uncertainties the
theorist has to make some assumptions on how the data will finally look like in the end. Here we tentatively assume
that the LSND evidence will disappear (for the alternative option, see, for example, refs.\cite{cald}). If so then we only 
have two oscillations frequencies, which can be given in terms of the three known species of light neutrinos without
additional sterile kinds (i.e. without weak interactions, so that they are not excluded by LEP). We then take for granted
that the frequency of atmospheric neutrino oscillations will remain well separated from the solar neutrino frequency, even
for the MSW large angle solution. We also assume that the electron neutrino does not participate in the atmospheric
oscillations, which (in absence of sterile neutrinos) are interpreted as nearly maximal
$\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations as indicated by the Superkamiokande \cite{SK} and Chooz
\cite{Chooz} data. However the data do not exclude a non-vanishing $U_{e3}$ element. In the Superkamiokande allowed
region the bound by Chooz
\cite{Chooz} amounts to  $|U_{e3}|\lappeq 0.2$ \cite{fogli,hall}.

In summary, by now we have a substantial evidence that neutrinos are massive. From a strict minimal standard model point
of view neutrino masses could vanish if no right handed neutrinos existed (no Dirac mass) and lepton number was
conserved (no Majorana mass). In GUTs both these assumptions are violated. The right handed neutrino is required in all
unifying groups larger than SU(5). In SO(10) the 16 fermion fields in each family, including the right handed neutrino,
exactly fit into the 16 dimensional representation of this group. This is really telling us that there is something in
SO(10)! Thus SO(10) must at least appear as a classification group at $M_{Planck}$, if not as a symmetry group at $M_{GUT}$.
The breaking of
$|B-L|$, B and L conservation is also a generic feature of GUTs. In fact, the see-saw mechanism \cite{ssm} explains
the smallness of neutrino masses in terms of the large mass scale where $|B-L|$ and L conservation laws are violated. Thus,
neutrino masses are important as a probe into the physics at the GUT scale, as would be proton decay, although in a less
direct way. For example, heavy Majorana neutrinos could be part of the explanation of baryogenesis. If baryogenesis at the
weak scale is excluded by the data it can occur at or just below the GUT scale, after inflation. But only that part with
$|B-L|>0$ would survive and not be erased at the weak scale by instanton effects. Thus baryogenesis at $kT\sim
10^{12}-10^{15}~GeV$ needs B-L non conservation at some stage like for $m_\nu$ with Majorana neutrinos. The two
effects could be related if baryogenesis arises from leptogenesis via $\nu$ decay \cite{lg} then converted into baryogenesis
by instantons. Present results on neutrino masses are compatible with this picture \cite{buch}. Thus the possibility of
baryogenesis at a large energy scale has been boosted by the recent results on neutrinos. 

Oscillations only determine squared mass differences and not masses. The case of three nearly degenerate neutrinos
is the only one that could in principle accomodate neutrinos as hot dark matter together with solar and atmospheric
neutrino oscillations. For a cosmologically significant fraction of hot dark matter, the common mass should be around 1-3 eV.
The solar frequency could be given by a small 1-2 splitting, while the atmospheric frequency could be given by a still small
but much larger 1,2-3 splitting.  Note that we
are assuming only two frequencies, given by $\Delta_{sun}\propto m^2_2-m^2_1$ and
$\Delta_{atm}\propto m^2_3-m^2_{1,2}$. A strong constraint arises in the degenerate case from neutrinoless double beta
decay which requires that the ee entry of
$m_{\nu}$ must obey
$|(m_{\nu})_{11}|\leq 0.2-0.5~{\rm eV}$ \cite{dbeta}. As observed in ref. \cite{GG}, this bound can only be 
satisfied if
double maximal mixing is realized, i.e. if also solar neutrino oscillations occur with nearly maximal mixing.
Note that for degenerate masses with $m\sim 1-3~{\rm eV}$ we need a relative splitting $\Delta m/m\sim
\Delta m^2_{atm}/2m^2\sim 10^{-3}-10^{-4}$ and an even smaller one for solar neutrinos. It is not simple
to imagine a natural mechanism compatible with unification and the see-saw mechanism to arrange such a
precise near symmetry, stable under running down from the GUT to the weak scale \cite{ello},\cite{BHKR}.

If neutrino masses are smaller than for cosmological relevance, we can have the hierarchies $|m_3| >> |m_{2,1}|$
or $|m_1|\sim |m_2| >> |m_3|$. We prefer the first case, because for quarks and charged leptons one
mass eigenvalue, the third generation one, is largely dominant. Thus the dominance of $m_3$ for neutrinos
corresponds to what we observe for the other fermions.  In this case, $m_3$ is determined by the atmospheric
neutrino oscillation frequency to be around $m_3\sim0.05~eV$. By the see-saw mechanism $m_3$ is related to some
large mass M, by $m_3\sim m^2/M$. If we identify m with the Higgs vacuum expectation value or the top mass
then M turns out to be around $M\sim 10^{15}~GeV$, which is indeed consistent with the
connection with GUTs.

Here we concentrate on models
with three light neutrinos, large light neutrino mass splittings and large mixings \cite{us}. In general large
splittings correspond to small mixings because normally only close-by states are strongly mixed. The requirement of large
splitting and large mixings imposes a condition of a vanishing determinant. For example, in a 2 by 2 context, the matrix
\beq 
m\propto 
\left[\matrix{ x^2&x\cr x&1    } 
\right]~~~~~. 
\label{md0}
\eeq has eigenvalues 0 and $1+x^2$ and for $x$ of 0(1) the mixing is large. Thus in the limit of neglecting small mass
terms of order $m_{1,2}$ the demands of large atmospheric neutrino mixing and dominance of $m_3$ translate into the
condition that the 2 by 2 subdeterminant 23 of the 3 by 3 mixing matrix vanishes in some approximate limit. The problem is to
show that this vanishing can be arranged in a natural way without fine tuning \cite{us12}, \cite{all}.

One possible mechanism is based on asymmetric neutrino Dirac matrices, with  a large
left-handed mixing already present in the Dirac matrix. In ref.\cite{us3}, we argued that in a SU(5)
GUT left-handed mixings for leptons tend to correspond to right-handed mixings for d quarks (in a basis where u quarks are
diagonal). Since large right-handed mixings for quarks are not in contrast with experiment, viable GUT models can be
constructed following this mechanism that correctly reproduce the data on fermion masses and mixings \cite{achi}. 

If, for some reason, one prefers symmetric matrices (for example, one could want to
preserve left-right symmetry at the GUT scale) then assuming that $m_D$ is nearly diagonal in the basis where
charged leptons are diagonal, large mixings could arise from the Majorana sector (for example, by dominance of a light right
handed neutrino, with large couplings to $\nu_{mu}$ and $\nu_{tau}$\cite{bar},\cite{king}) . In a recent paper \cite{us4}, we
have presented examples where a nearly maximal mixing is created from almost nothing: all relevant matrices entering in the
see-saw mechanism are diagonal, yet the resulting mixing is large. Or large neutrino mixings could be generated by an
enhancement of formally small terms \cite{Lola}.  This is because a typical small term in quark or charged lepton mass
matrices is of the order of the Cabibbo angle
$\lambda\sim 0.22$ which is not that small.  

In conclusion the fact that some neutrino mixing angles are large, while surprising at the start, was eventually found to
be well be compatible, without any major change, with our picture of quark and lepton masses within GUTs. Rather it
provides us with new important clues that can become sharper when the experimental picture will be further clarified.

\begin{thebibliography}{199}


\bibitem{SK}
Y. Fukuda et al.,   and  M Nakahata, these Proceedings.
\bibitem{Ronga} F. Ronga,these Proceedings.
\bibitem{bel} T. Kirsten, these Proceedings, S. Turck-Chieze, these Proceedings.
\bibitem{LSND}  C. Athanassopoulos et al., Phys. Rev. Lett. 77 (1996) 3082;  .
\bibitem{ram} P. Ramond, these Proceedings; P. Fisher, B. Kaiser and K. Mc Farland, .
\bibitem{Karmen}
B. Armbruster et al., Phys. Rev. C57 (1998) 3414 and G. Drexlin, talk at
Wein'98.
\bibitem{fogli}  G.L.Fogli, these Proceedings, G.L.Fogli et al, .
\bibitem{hall} R. Barbieri, L. J. Hall, D. Smith, A. Strumia and N. Weiner, hep/G; L. Fogli, E. Lisi and D.
Montanino, ; J.N. Bahcall, P.I. Krastev and A. Yu Smirnov, .
\bibitem{Chooz} M. Apollonio et al., Phys. Lett. B 420 (1998) 397.
\bibitem{cald} D.O. Caldwell, these Proceedings; C. Giunti,these Proceedings.
\bibitem{ssm} M.Gell-Mann, P.Ramond and R.Slansky, in Supergravity, ed. by D.Freedman et al, North Holland, 1979;
T.Yanagida, Prog. Theo. Phys. B135(1978)66. See also R. Mohapatra
and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
\bibitem{lg} See, for example, M. Fukugita and T.Yanagida, Phys. Lett. B174(1986) 45; G.Lazarides and Q.Shafi,
Phys. Lett. B258(1991)305.
\bibitem{buch} See, for example, W. Buchmuller and T. Yanagida,  W. Buchmuller,  and
references therein, R. Barbieri et al, . 
\bibitem{dbeta} L. Baudis et al., Phys. Lett. B407 (1997) 219.
\bibitem{GG} F. Vissani, ; H. Georgi and S.L. Glashow, .
\bibitem{ello} J. Ellis and S. Lola, ; J.A. Casas et al,   ; R.
Barbieri, G.G. Ross and A. Strumia, ; E. Ma, .
\bibitem{BHKR} R. Barbieri, L. J. Hall, G. L. Kane and G. G. Ross, .
\bibitem{us} For a review, see, for example, G. Altarelli and F. Feruglio, Phys. Rep. 320(1999)295  .
\bibitem{us12} G. Altarelli and F. Feruglio, Phys. Lett. B439(1998)112;
JHEP 11(1998)21.
\bibitem{all} E. Akhmedov,  . 
\bibitem{us3} G. Altarelli and F. Feruglio,  Phys.Lett. B451 (1999) 388.
\bibitem{achi} Y. Achiman, these Proceedings; Z. Berezhiani and Z. Tavartkiladze, Phys. Lett. B409 (1997) 220;
C. H. Albright and S. M. Barr, Phys. Rev. D58 (1998) 013002, ; C. H. Albright, K. S. Babu and S.
M. Barr, Phys. Rev. Lett. 81 (1998) 1167, ; C. H. Albright and S. M. Barr, ; Z. Berezhiani and A.
Rossi, ; R. Barbieri, L. Giusti, L. Hall and A. Romanino, ;  G. C. Branco and J. I. Silva-Marcos,
Phys. Lett. B331 (1994) 390, see also G. C. Branco, L. Lavoura and F. Mota, Phys. Rev. D39(1989)3443.
\bibitem{bar} R. Barbieri et al, .
\bibitem{king} S. F. King, Phys. Lett. B439 (1998) 350, ; S. F. King, ; S. Davidson
and S. F. King, Phys. Lett. B445 (1998) 191,  Q.Shafi and Z. Tavartkiladze, .
\bibitem{us4}  G. Altarelli, F. Feruglio and I. Masina, .
\bibitem{Lola} S. Lola and G. G. Ross, ; K. Babu, J. Pati and F. Wilczek, .

\end{thebibliography}

\end{document}

	

		

			 



	

	


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\caption{Biologically treated effluents (mg/l)}
\label{tab:effluents}
\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}rrrr}
\hline
                 & \multicolumn{2}{l}{Pilot plant} 
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\cline{2-3} \cline{4-5}
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\hline
Total cyanide    & $ 6.5$ & $0.35$ & $  2.0$ & $  0.30$ \\
Method-C cyanide & $ 4.1$ & $0.05$ &         & $  0.02$ \\
Thiocyanide      & $60.0$ & $1.0?$ & $ 50.0$ & $ <0.10$ \\
Ammonia          & $ 6.0$ & $0.50$ &         & $  0.10$ \\
Copper           & $ 1.0$ & $0.04$ & $  1.0$ & $  0.05$ \\
Suspended solids &        &        &         & $<10.0?$ \\
\hline
\multicolumn{5}{@{}p{120mm}}{Reprinted from: G.M. Ritcey,
                             Tailings Management,
                             Elsevier, Amsterdam, 1989, p. 635.}
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throughout.

Horizontal lines should be placed above and below table headings, above
the subheadings and at the end of the table above any notes. Vertical
lines should be avoided.

If a table is too long to fit onto one page, the table number and
headings should be repeated above the continuation of the table. For
this you have to reset the table counter with
\verb|\addtocounter{table}{-1}|. Alternatively, the table can be turned
by $90^\circ$ (`landscape mode') and spread over two consecutive pages
(first an even-numbered, then an odd-numbered one) created by means of
\verb|\begin{table}[h]| without a caption. To do this, you prepare the
table as a separate \LaTeX{} document and attach the tables to the
empty pages with a few spots of suitable glue.

\subsection{Line drawings}

Line drawings should be drawn in India ink on tracing paper with the
aid of a stencil or should be glossy prints of the same; computer
prepared drawings are also acceptable. They should be attached to your
manuscript page, correctly aligned, using suitable glue and {\em not
transparent tape}. When placing a figure at the top of a page, the top
of the figure should be at the same level as the bottom of the first
text line.

All notations and lettering should be no less than 2\,mm high. The use
of heavy black, bold lettering should be avoided as this will look
unpleasantly dark when printed.

\subsection{Black and white photographs}

Photographs must always be sharp originals ({\em not screened
versions\/}) and rich in contrast. They will undergo the same reduction
as the text and should be pasted on your page in the same way as line
drawings.

\subsection{Colour photographs}

Sharp originals ({\em not transparencies or slides\/}) should be
submitted close to the size expected in publication. Charges for the
processing and printing of colour will be passed on to the author(s) of
the paper. As costs involved are per page, care should be taken in the
selection of size and shape so that two or more illustrations may be
fitted together on one page. Please contact the Technical Editor in the
Camera-Ready Publications Department at Elsevier for a price quotation
and layout instructions before producing your paper in its final form.

\begin{figure}[htb]
\vspace{9pt}
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Good sharp prints should be used and not (distorted) photocopies.}
\label{fig:largenenough}
\end{figure}
%
\begin{figure}[htb]
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Remember to keep details clear and large enough.}
\label{fig:toosmall}
\end{figure}

\section{EQUATIONS}

Equations should be flush-left with the text margin; \LaTeX{} ensures
that the equation is preceded and followed by one line of white space. 
\LaTeX{} provides the document-style option {\tt fleqn} to get the
flush-left effect.

\begin{equation}
H_{\alpha\beta}(\omega) = E_\alpha^{(0)}(\omega) \delta_{\alpha\beta} +
                          \langle \alpha | W_\pi | \beta \rangle 
\end{equation}

You need not put in equation numbers, since this is taken care of
automatically. The equation numbers are always consecutive and are
printed in parentheses flush with the right-hand margin of the text and
level with the last line of the equation. For multi-line equations, use
the {\tt eqnarray} environment. For complex mathematics, use the
\AmS-\LaTeX{} package.

\begin{thebibliography}{9}
\bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
\bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
                 Phase Separation in Glass, North-Holland, Amsterdam, 1984.
\bibitem{Dimi75} Y. Dimitriev and E. Kashchieva, 
                 J. Mater. Sci. 10 (1975) 1419.
\bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
                 US Patent No. 3 904 422 (1975).
\end{thebibliography}

References should be collected at the end of your paper. Do not begin
them on a new page unless this is absolutely necessary. They should be
prepared according to the sequential numeric system making sure that
all material mentioned is generally available to the reader. Use
\verb+\cite+ to refer to the entries in the bibliography so that your
accumulated list corresponds to the citations made in the text body. 

Above we have listed some references according to the
sequential numeric system \cite{Scho70,Mazu84,Dimi75,Eato75}.
\end{document}

