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\begin{document}

\title{GUT Models}

\author{Tom\'{a}\v{s} Bla\v{z}ek
 \footnote[1]{On leave of absence from the Comenius University,
              Bratislava, Slovakia
             }
       }

\address{Department of Physics and Astronomy,\\
         University of Southampton,
         Southampton SO17 1BJ, U.K.}

%\ead{blazek@soton.ac.uk}


\begin{abstract}
I emphasize that a GUT based on the minimal supersymmetric SO(10)
remains a possibility in nature. The yukawa couplings of the 
third generation fermions can all unify to a point provided 
strong constraints are placed on the SUSY parameters entering
the analysis via the low-energy SUSY threshold corrections.
This makes the minimal SO(10) a very predictive and testable GUT.

In the second part of the talk the focus is on the relation
%  between the see-saw mechanism for the generation of light 
%  neutrino masses and lepton flavour violating processes. 
between the neutrino masses, SUSY, and lepton flavour violation.
In a class of GUT models, the large atmospheric mixing angle
induces significant $\tau\to\mu\gamma$ branching ratio.
This can be tested in present experiments.
Chances are that the lepton flavour violation in $\tau$ decay will
become the first signal of new physics observed at B factories. 
\end{abstract}

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%%%%%\section{Introduction}

Although the minimal 
supersymmetric (SUSY) grand unified theory (GUT) involving 
gauge coupling unification appears to be based on SU(5), once the remaining 
dimensionless couplings of the theory, the yukawa couplings,
are included into the analysis the minimal choice is a theory
based on the SO(10) gauge group. 
A 16-dimensional spinorial representation of SO(10) readily 
describes the fifteen Weyl fermions of a single generation
plus a singlet right-handed neutrino state. It can then be argued that the 
SO(10) unification predicts neutrino masses  (in fact, in the right ballpark)
as the right-handed neutrino state is no longer protected by the
symmetry after the breaking of the 
rank five SO(10) unification gauge group down to the rank four Standard Model 
gauge group. Thus it is important for neutrino physics whether
minimal SO(10) models are phenomenologically viable or not.

Recent studies \cite{so10.2001,so10.2002} have addressed this 
question in a detailed top-down analysis. As a starting point it
was assumed that the sign of the $\mu$ parameter is positive
as suggested by the BNL experiments on muon $g\!-\!2$ \cite{BNL},
the third generation yukawa couplings exactly unify and GUT threshold
corrections lead to a discrepancy between $\alpha_s(M_{GUT})$ and
$\alpha_1(M_{GUT})\equiv\alpha_2(M_{GUT})$ at the level of $\le 6$\% .
This regime was not studied in the previous work \cite{deboer}.
%
In our global analysis the $\chi^2$ was computed based on nine 
low-energy observables related, at tree level, to the gauge couplings 
($\alpha_{EM}$, $G_\mu$, and $\alpha_s(M_Z)$), 
third family yukawa couplings ($M_{top}$, $m_b(m_b)$ and $M_\tau$)
and electroweak symmetry breaking ($M_Z$ and $M_W$), respectively.
The measure of the SU(2)$_V$ breaking, the $\rho$ parameter, was chosen
as the ninth observable. Each quantity was fully computed at one loop
taking into account 2-loop (1-loop) RGE running of the dimensionless 
(dimensionful) MSSM parameters \cite{bcrw}.



\begin{figure}
\begin{center}
\begin{tabular}{cc}
\input{m_2000_chi2.pstex_t}
\label{figure:chi2}   &
\hspace{.5cm}  \input{m_M_chi2.pstex_t}
\end{tabular}
\end{center}
\caption{$\chi^2$ contours for $m_{16} = 2000$ GeV (Left) and $\mu = 150$ GeV (Right).}
\end{figure}


The results in figure 1 suggest that 
the minimal SO(10) is consistent
with the low-energy data if the dimensionful SUSY parameters satisfy 
phenomenologically interesting constraints. 
Most notably, a non-universal just-so Higgs splitting is required
\be
    m_{(H_u, \; H_d)}^2 = m_{10}^2 \;( 1 \mp \Delta m_H^2) 
\label{eq:JustSo}
\ee
at the GUT scale with $m_{10}/m_{16} = 1.3  - 1.5$ and 
                      $\Delta m_H^2  = 0.10 - 0.14$ 
together with low positive value
of the $\mu$ parameter at the weak scale and large values for
the scalar mass $m_{16} > 1200\,$GeV and trilinear coupling 
$A_0 \approx -1.9\:m_{16}$ at the GUT scale. 
The fits were  obtained without taking the right-handed 
neutrinos into account.
Several sources for the splitting in (\ref{eq:JustSo}) are
suggested in \cite{splittings} and \cite{so10.2002}. It was encouraging
to find in the discussion in  \cite{so10.2002} 
that the decoupling of the right-handed $\ol{\nu}_\tau$ at scale 
$10^{13}\,$GeV typically induces $\Delta m_H^2\approx 0.10$.
%
The primary constraint comes from the fit to the bottom quark mass.
For the positive sign of $\mu$ there is a positive non-logarithmic 
gluino correction to $m_b$ enhanced by large $\tan\!\beta$. 
However, for large and negative $A_0(M_{GUT})$
this can be balanced out by a negative non-logarithmic chargino  
loop correction. As a result a large left-right stop mixing is present.
In the preferred SUSY region neutralino is the LSP. In the best fits 
$\mu(M_Z)\approx 150$GeV and thus the LSP, as well as the chargino, are
higgsino-like. The mass of the latter is close to the 
current experimental limit. In the Higgs sector, the Standard-Model-like
Higgs has naturally mass $114$--$120\,$GeV, while for the rest of the
Higgs spectrum there is no clear prediction.

In the second part of the talk I would like to emphasize the
relation between the neutrino and SUSY sectors of GUT models. 
Assuming the see-saw mechanism with masses of the heavy Majorana 
$\ol{\nu}$ states below the GUT scale there emerges a
possibility of observing lepton flavour violating (LFV) decays
$\ell_i^\pm \to \ell_j^\pm + \gamma$ \cite{BorzMas}.
The decay rates are enhanced by $(\tan\!\beta)^2$ and 
are only suppressed by the sparticle masses because
the LFV is induced in the slepton mass matrix from
the loop processes at scales between $M_{GUT}$ and $M_{\ol{\nu}}$
(even if there is no LFV at the unification scale 
$M_{GUT}$).
%
In a recent work \cite{BlaSFK} it was pointed out that the large
atmospheric mixing angle may lead to a potentially large $\tau\to\mu\gamma$
decay rate. This prediction is rather insensitive to the details 
of a particular GUT model. In fact, to avoid fine tuning many GUT 
models with hierarchical neutrino masses lead to the Yukawa matrix 
\be
      Y_{\nu} \sim 
                   \left( \begin{array}{ccc}
0 & 0 & 0    \\
0 & 0 & 1    \\
0 & 0 & 1   
                          \end{array}
                   \right)
\label{Yuk}
\ee
in the basis where charged lepton Yukawa matrix and heavy Majorana 
matrix are both diagonal. 
The prediction for BR($\tau\to\mu\gamma$) then does not depend
on the subleading entries denoted in (\ref{Yuk}) as zeros.
The results 
in figure 2, taken from \cite{BlaSFK}, clearly show that if $\tan\!\beta$
is large the decay could be observed in the current experiments at B factories.
They were obtained for the best fits of a particular string motivated
model \cite{OliSFK, BlaPerSFK}.
%  These results were obtained for the best fits 
%  of a particular string motivated
%  model \cite{OliSFK, BlaPerSFK} but are, as explained, quite general 
%  \cite{BlaSFK2}. 
Note that the results for $\mu\to e\gamma$ are necessarily dependent
on the subleading model dependent entries in (\ref{Yuk}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\begin{center}
\epsfxsize=10cm
\epsfysize=5cm
\epsfbox{2cd.eps}
\end{center}
\caption{\label{fig2} 
 Contour lines of the branching ratio 
 BR$(\tau\to\mu\gamma)\times 10^6$ 
 for two different values of the $\mu$ parameter fixed at scale
 $M_Z$. The data is computed for the best fits of the model described 
 in \cite{OliSFK}. In these fits $\tan\!\beta=50$ and $A_0=0$.
 $m_F$ is the universal sfermion SUSY breaking mass at the GUT scale.
 The current experimental bound is 
 BR$(\tau\to\mu\gamma)<2\times 10^{-6}$ at 95\% C.L..  }
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In summary, GUT models can be very predictive and can combine 
constraints from the neutrino and SUSY sectors. 
%  Thus the neutrino data can constrain SUSY searches and {\em 
%  vice versa}. 
A highlighted example is the prediction for
the observation of the LFV decay $\tau\to\mu\gamma$ at the
current B factories. 






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{References}
\begin{thebibliography}{99}

\bibitem{so10.2001}
  T.~Bla\v{z}ek, R.~Derm\' \i \v sek  and S.~Raby,
                  Phys.\ Rev.\ Lett. {\bf 88} (2002) 111804 
                                     [arXiv:hep-ph/0107097].

\bibitem{so10.2002}
  T.~Bla\v{z}ek, R.~Derm\' \i \v sek  and S.~Raby,
                  Phys.\ Rev.\ D {\bf 65} (2002) 115004
                                     [arXiv:hep-ph/0201081].


\bibitem{BNL}
  H.~N.~Brown,   et al., Muon g-2 Collaboration,
                  Phys.\ Rev.\ Lett. {\bf 86} (2001) 2227
                                     [arXiv:hep-ex/0102017]
  and the update
  G.~W.~Bennett, et al., Muon g-2 Collaboration,
                  Phys.\ Rev.\ Lett. {\bf 89} (2002) 101804;
                       Erratum-ibid. {\bf 89} (2002) 129903;
                                     [arXiv:hep-ex/0208001].

\bibitem{deboer}
  W.~de Boer, M.~Huber, A.V.Gladyshev and D.I.~Kazakov,
                  Eur.\ Phys.\ J.\ C {\bf 20}, 689 (2001)
                                    [arXiv:hep-ph/0102163]
  and the update at the SUSY02 conference at DESY, Hamburg, June 2002,
  unpublished. 
% Low Energies, Flavours, and CP parallel session


\bibitem{bcrw}
  T.~Bla\v{z}ek, M.~Carena, S.~Raby and C.~Wagner,
                  Phys.\ Rev.\ D {\bf 56} (1997) 6919.
%                                    [arXiv:hep-ph/9611217]



\bibitem{splittings}
  N.~Polonsky and A.~Pomarol,
                  Phys.\ Rev.\ D {\bf 51} (1995) 6532,
%                                     [arXiv:hep-ph/9410231],
  H.~Murayama, M.~Olechowski and S.~Pokorski,
                  Phys.\ Lett.\ B {\bf 371}, 57 (1996)
                                     [arXiv:hep-ph/9510327].
           


\bibitem{BorzMas}
  F.~Borzumati and A.~Masiero, 
                  Phys.\ Rev.\ Lett. {\bf 57} (1986) 961;
  J.~Hisano, et al.,
                  Phys.\ Lett.\ B {\bf 357}, 579 (1995)
                                     [arXiv:hep-ph/9501407].




\bibitem{BlaSFK}
  T.~Bla\v{z}ek and S.~F.~King,
                  Phys.\ Lett.\ B {\bf 518}, 109 (2001)
                                     [arXiv:hep-ph/0105005].


\bibitem{OliSFK}
  S.~F.~King and M.~Oliveira,
                  Phys.\ Rev.\ D {\bf 63} (2001) 095004
                                     [arXiv:hep-ph/0009287].



\bibitem{BlaPerSFK}
  T.~Bla\v{z}ek, J.~Perry and S.~F.~King, in preparation.


%  \bibitem{BlaSFK2}
%    T.~Bla\v{z}ek and S.~F.~King, in preparation.



%  
%  
%  
%  \bibitem
%  
%  \bibitem
%  
%   zzz


%  \bibitem{Apollonio:1999ae}
%  M.~Apollonio {\it et al.}  [CHOOZ Collaboration],
%  %``Limits on neutrino oscillations from the CHOOZ experiment,''
%  Phys.\ Lett.\ B {\bf 466}, 415 (1999)
%  [arXiv:hep-ex/9907037].


%  \bibitem{Loredo:2001rx}
%  T.~J.~Loredo and D.~Q.~Lamb,
%  %``Bayesian analysis of neutrinos observed from supernova SN 1987A,''
%  Phys.\ Rev.\ D {\bf 65} (2002) 063002
%  [arXiv:astro-ph/0107260].

\end{thebibliography}
\end{document}



