\subsection{Proton decay}

Proton decay is one of the few predictions of Grand Unified Theories
that can be tested in low-energy experiments. Its discovery would
definitely testify for a more fundamental structure beyond the Standard
Model.

In the past twenty years, the first generation (IMB, Fr{\'e}jus, Kamiokande)
and second generation (Super-Kamiokande) proton decay experiments have
already put stringent lower limits on the partial proton lifetimes,
qualitatively ruling out non-supersymmetric $SU(5)$ theories (first generation)
and the minimal supersymmetric $SU(5)$ theory (second generation). 
A megaton-scale
water \v{C}erenkov detector would improve further the experimental sensitivity
to proton decay by more than one order of magnitude and allow to probe
non-minimal $SU(5)$ models as well as other types of GUTs, such as $SO(10)$,
flipped $SU(5)$ and higher-dimensional GUTs.
Indeed, recent experimental and theoretical progresses point
towards smaller values of the partial lifetime of the proton into
$\pi^0 e^+$, implying that this decay mode -- the most
model-independent one -- is not out of reach, contrary to previous
expectations. Using the new, more accurate lattice calculation of
the nucleon decay matrix element one can estimate 
$\tau (p \rightarrow \pi^0 e^+) 
\approx 10^{35}\, \mbox{yrs}\, (M_X / 10^{16}\, 
\mbox{GeV})^4\,
((1/25) / \alpha_{GUT})^2$, 
where $M_X$ is the mass of the
superheavy gauge bosons mediating proton decay, 
$\alpha_{GUT}
\equiv g^2_{GUT} / 4 \pi$ and $g_{GUT}$ is the value of the GUT
gauge coupling at the unification scale. This is to be compared
with the present Super-Kamiokande lower limit ($5 \times 10^{33}\,
\mbox{yrs}$), and with the expected sensivity of a megaton water
\v{C}erenkov detector ($10^{35}\, \mbox{yrs}$ after 10 years 
of data taking for MEMPHYS). 

The dominant decay channel in supersymmetric GUTs,
$p \rightarrow K^+ \bar \nu$, is much more model-dependent. The corresponding
decay rate indeed depends on the couplings and masses of the supersymmetric
partners of the heavy colour-triplet Higgs bosons, and on the details of the
sparticle spectrum. The effective triplet mass, in particular, is extremely
dependent on the GUT model.
%In the
%simplest models, $\tau (p \rightarrow K^+ \bar \nu)$ strongly depends on
%$\tan \beta$, the ratio of the vevs of the two Higgs bosons of the MSSM;
%larger values of $\tan \beta$ yield shorter values of
%$\tau (p \rightarrow K^+ \bar \nu)$.
In many models, one finds an upper limit $\tau (p \rightarrow K^+ \bar \nu)
\leq \mbox{few}\, 10^{34}\, \mbox{yrs}$ 
\cite{Dermisek:2000hr}\cite{Babu:1998wi}\cite{Altarelli:2000fu},
to be compared with the present Super-Kamioka nde lower limit
($1.6 \times 10^{33}\, \mbox{yrs}$), and with the expected sensivity of
a megaton water \v{C}erenkov detector ($2 \times 10^{34}\, \mbox{yrs}$ after
10 years for MEMPHYS). 

There are many more decay channels that could be accessible to a megaton
water \v{C}erenkov detector. The measurement of several partial lifetimes
would allow to discriminate between different Grand Unified models, at a time
when, after several years of LHC running, 
the supersymmetry landscape will be drastically clarified, 
through discovery or severe exclusion limits. Therefore the predictions 
of proton lifetime, in constrained or more general supersymmetric models, 
will be sharpened even further. 


