\section{Proton decay sensitivity}
%\REDBLA{version 0 by JEC 6/3/06}
%\REDBLA{update Introduction by Pavel F. Perez 8/3/06}
%\REDBLA{update by A. Bueno 23/3/06}
%\REDBLA{update by T. Marrodan Undagoitia 10/4/06}
%\REDBLA{update by JEC 20/4/06}
%\REDBLA{update by JEC 21/6/06}
%\REDBLA{update by JEC 10/10/06: this is a section now}
%\REDBLA{update by A. Bueno 20/1/07}
For all relevant aspects of the proton stability in grand unified theories, 
in strings and in branes see reference~\cite{Nath:2006ut}.   

Since proton decay is the most dramatic prediction coming from theories 
where the matter is unified, we hope to test those scenarios at future experiments.
For this reason, a theoretical upper bound on the lifetime of the proton is very 
important to know about the possibilities of future experiments.   

Recently a model-independent upper bound on the total proton decay lifetime has 
been pointed out~\cite{Dorsner:2004xa}:
\begin{widetext}
\begin{equation}
	\tau_p^{upper} = 	
		\left\{\begin{array}{lr}
	6.0 \times 10^{39} & (\text{Majorana case}) 
         \\ 
         2.8 \times 10^{37}  & (\text{Dirac case})
	\end{array}\right\}
                 \times 
	 \frac{\left(M_X/10^{16}GeV\right)^4}{\alpha_{GUT}^2} \times \left( \frac{0.003GeV^3}{\alpha} \right)^2 \ \text{yrs}         
\end{equation}
\end{widetext}
where $M_X$ is the mass of the superheavy gauge bosons. The parameter $\alpha_{GUT}= g_{GUT}^2 / 4 \pi$, 
where $g_{GUT}$ is the gauge coupling at the grand unified scale. $\alpha$ is the matrix element. 
See \refFig{fig:Phys-PDK-Majorana} and \refFig{fig:Phys-PDK-Dirac} for the present parameter space allowed by the experiments. 

Most of the models (Supersymmetric or non-Supersymmetric) predict a lifetime $\tau_p$ below 
those upper bounds $10^{33-37}$~years, which are very interesting since it is the possible 
range of the proposed detectors. 

In order to have an idea of the proton decay predictions, let us list in \refTab{tab:Phys-PDK-Models} 
the results in different models.
%
\begin{table*}
		\caption{\label{tab:Phys-PDK-Models}Summary of some recent predictions on proton partial lifetimes. 
%JEC 31/01/07 START put here the references		
		Some references for the different models are: (1) \cite{Georgi:1974sy}, (2) \cite{Dorsner:2005fq,Dorsner:2005ii}, (3) \cite{Lee:1994vp},  (4)  \cite{Murayama:2001ur,Bajc:2002bv,Bajc:2002pg,Emmanuel-Costa:2003pu}, (5) \cite{Babu:1992ia,Aulakh:2003kg,Fukuyama:2004pb,Goh:2003nv}, (6) \cite{Friedmann:2002ty}
		}
		\begin{tabular}{cccc} \hline\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Model       &   Decay modes     &  Prediction   &  References \\ \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Georgi-Glashow model & - &  ruled out      &        (1)        \\ 
%\hline 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{4cm}{\center{Minimal realistic\\ non-SUSY $SU(5)$}} & all channels & $\tau_p^{upper} = 1.4 \times 10^{36}$ & (2)
%\REDBLA{JEC:BibTex pb}
\\[8mm]
% \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Two Step Non-SUSY SO(10) &  $p \to e^+ \pi^0$ &  $\approx 10^{33-38}$ & (3)  \\[5mm] 
%\hline 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Minimal SUSY $SU(5)$   &   $p \to  \bar{\nu}K^+$  &  $\approx 10^{32-34}$  & (4)
\\ 
%\hline 
\\[-5mm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{4cm}{\center{SUSY $SO(10)$ \\ with $10_H$, and $126_H$}} & $p \to \bar{\nu} K^+$ & $\approx 10^{33-36}$ &  (5)  
\\[8mm]
% \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
M-Theory($G_2$)   & $p \to e^+\pi^0$    &  $\approx 10^{33-37}$     & (6)  \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline\hline
		\end{tabular}
\end{table*}

\begin{figure}
\includegraphics[width=\columnwidth]{./figures/PavelMajoranaNew.eps}
\caption{\label{fig:Phys-PDK-Majorana} Isoplot for the upper bounds on the total
proton lifetime in years in the Majorana neutrino case in the
$M_X$--$\alpha_{GUT}$ plane. The value of the unifying coupling
constant is varied from $1/60$ to $1/10$. The conventional values
for $M_X$ and $\alpha_{GUT}$ in SUSY GUTs are marked in thick
lines. Experimentally excluded region is given in black~\cite{Dorsner:2004xa}.}
\end{figure}


\begin{figure}
\includegraphics[width=\columnwidth]{./figures/PavelDiracNew.eps}
\caption{\label{fig:Phys-PDK-Dirac} Isoplot for the upper bounds on the total
proton lifetime in years in the Dirac neutrino case in the
$M_X$--$\alpha_{GUT}$ plane. The value of the unifying coupling
constant is varied from $1/60$ to $1/10$. The conventional values
for $M_X$ and $\alpha_{GUT}$ in SUSY GUTs are marked in thick
lines. Experimentally excluded region is given in black~\cite{Dorsner:2004xa}.}
\end{figure}

%
No specific simulation for MEMPHYS has been carried out yet. We therefore rely on the study done by UNO, adapting the results to MEMPHYS (which has an overall better coverage) when possible. 

% Antonio Bueno 20/01/07 START 
To study the physics potentialities
of very large underground Liquid Argon Time Projection Chambers (LAr TPC), 
a detailed simulation of signal efficiency and
background sources, including atmospheric neutrinos and cosmogenic
backgrounds was carried out \cite{Bueno:2007um}. Liquid Argon TPCs,
offering good granularity and energy resolution, low particle detection threshold,
and excellent background discrimination, should  
yield  very good signal over background ratios in many possible
decay modes, allowing to reach partial lifetime sensitivities in
the range of $10^{34}-10^{35}$~years with exposures up to 1000~kton$\times$year, 
often in quasi-background-free conditions optimal for discoveries 
at the few events level, corresponding
to atmospheric neutrino background rejections of the order of $10^5$.
Multi-prong decay modes like e.g. $p\rightarrow \mu^- \pi^+ K^+$
or $p\rightarrow e^+\pi^+\pi^-$ and channels involving kaons like
e.g. $p\rightarrow K^+\bar\nu$, $p\rightarrow e^+K^0$ and $p\rightarrow \mu^+K^0$
are particularly suitable, since liquid
Argon imaging
provides typically an order of magnitude improvement in efficiencies for similar
or better background conditions compared to Water Cerenkov detectors.
Up to a factor 2 improvement in efficiency is expected for modes like $p\rightarrow e^+\gamma$
and $p\rightarrow \mu^+\gamma$ thanks to the clean photon identification
and separation from $\pi^0$. Channels like $p\rightarrow e^+\pi^0$ or $p\rightarrow \mu^+\pi^0$,
dominated by intrinsic nuclear effects,
yield similar efficiencies and backgrounds as in Water Cerenkov detectors. 
An extremely important feature of GLACIER is that thanks to the self-shielding 
and 3D-imaging properties of the liquid Argon TPC,
this result remains valid even at shallow depths where
cosmogenic background sources are important.
The possibility of a very large area annular active muon veto shield in order to
further suppress cosmogenic backgrounds at shallow depths is also a very promising 
option to complement the GLACIER detector.
% Antonio Bueno 20/01/07 END

%T. Marrodan Undagoitia  10/4/06 START
In order to quantitatively estimate the potential of the LENA detector
for measuring the proton lifetime, a Monte Carlo simulation for the
decay channel $p\to K^{+}\overline{{\nu}}$ has been performed. For
this purpose, the Geant4 simulation toolkit has been
used \cite{Agostinelli:2002hh}. Not only all default Geant4 physics lists were
included but  also optical processes as scintillation, Cherenkov light
production, Rayleigh scattering and light absorption. From these
simulations a light yield  of $\sim 110$~pe/MeV for an event in the
center of the detector results. In  addition, to take into account the so
called quenching effects, the  semi-empirical Birk's formula \cite{Birk}
has been introduced into the code.
%T. Marrodan Undagoitia 10/4/06 END
%
\subsection{$p \rightarrow e^+\pi^0$}
%
%JEC MEMPHYS version 0
Following UNO study, the detection efficiency of $p \rightarrow e^+\pi^0$
(3 showering rings event) is $\epsilon=43\%$ 
for a 20 inch-PMT coverage of 40\% or its equivalent, as envisioned for
MEMPHYS. The corresponding estimated
atmospheric neutrino induced background is at the level of $2.25$~events/Mt.yr. 
From these efficiencies and background levels,
proton decay sensitivity as a function of detector exposure can be
estimated. A $10^{35}$ years partial
lifetime ($\tau_p/B$) could be reached at the 90\% C.L. for a 5~Mt.yr exposure (10~yrs) with MEMPHYS
(similar to case A in \refFig{fig:pdk1}). Beyond that exposure, tighter cuts may be envisaged to further reduce the atmospheric neutrino background to $0.15$~events/Mt.yr, by selecting quasi exclusively the free proton decays.
%
\begin{figure}
\includegraphics[width=\columnwidth]{./figures/epi0-WC-Shiozawa.eps}
\caption{\label{fig:pdk1} Sensitivity for $e^+\pi^0$ proton decay
lifetime, as determined by UNO \cite{Jung:1999jq}. MEMPHYS corresponds to case (A).}
\end{figure}

\begin{figure}
\includegraphics[width=\columnwidth]{./figures/Knu-WC-Shiozawa.eps}
\caption{\label{fig:pdk9_jbz}
Expected sensitivity on $\nu K^+$ proton decay as a function of MEMPHYS
exposure \cite{Jung:1999jq} (see text for details).}
\end{figure}
%

%Antonio Bueno 20/01/07 START
The positron and the two photons issued from the $\pi^0$ gives clear events 
in the GLACIER detector. We find that the $\pi^0$ is absorbed by the nucleus
$\sim$45\% of the times. 
Assuming a perfect particle and track identification, 
one may expect 
a $45\%$ efficiency and a background level of $1$~event/Mt.y. 
So, for a 1~Mt.yr (10~yrs) exposure with GLACIER one 
reaches $\tau_p/B > 0.4~10^{35}$~yrs at 90$\%$ C.L. (see Fig.~\ref{fig:GLACIERpdk}). 
%
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{./figures/limit_pdk_expo.eps}
\end{center}
\caption{\label{fig:GLACIERpdk} Expected proton decay lifetime limits ($\tau / B$ at 90\% C.L.) as a function of exposure for GLACIER. In this plot, only atmospheric neutrino background 
has been taken into account.}
\end{figure}
%Antonio Bueno 20/01/07 END

%LENA
%T. Marrodan Undagoitia  10/4/06 START
 In a liquid scintillator detector the decay $p\to e^{+}\pi^{0}$ will
produce a $\sim 938$~MeV signal coming from $e^{+}$ and $\pi^{0}$
showers. Only atmospheric neutrinos are expected to cause background
events in this energy range. Using the fact that showers from both
$e^{+}$ and $\pi^{0}$ propagate $\sim$4~m in opposite directions
before being stopped, atmospheric neutrino background can be
reduced. Applying this method, the current limit for this channel
($\tau_p/B=5.4~10^{33}$~y \cite{Nakaya:2005nk}) could be improved.
%T. Marrodan Undagoitia 10/4/06 END
%
\subsection{$p \rightarrow \overline{\nu}K^+$}
%
%Update by T.M.U see below
%LENA use the pulse shape analysis (rise time) to discriminate the kaon production then decay 18~ns later, from the atmospheric neutrino kaon production and the atmospheric neutrino charged current production of muon and charged pion. The signal efficiency is expected to be $65\%$ keeping the background below $1$~event/Mt.y. Then, one can reach $\tau_p/B > 4~10^{34}$~yrs (90\% CL) in 0.5~Mt.yr exposure (10~yrs).

%T. Marrodan Undagoitia  10/4/06 START
In LENA, proton decay events via the mode $p\to K^{+}\overline{{\nu}}$
have a very clear signature. The kaon causes a prompt monoenergetic
signal (T=105~MeV) and from the kaon decay there is a short-delayed
second monoenergetic signal, bigger than the first one. The kaon has a
lifetime of $\tau(K^{+})=12.8$~ns and two main decay channels: with a
probability of 63.43~$\%$ it decays via $K^{+}\to\mu^{+}{\nu_{\mu}}$
and with 21.13\%, via \mbox{$K^{+}\to\pi^{+} \pi^{0}$}.

Simulations of proton decay events and atmospheric neutrino background
has  been performed and a pulse shape analysis has been applied.
From the analysis an efficiency of 65\% for
the detection of a possible proton decay has been determined and a
background  suppression of $\sim2~10^{4}$ has  been
achieved \cite{Undagoitia:1-2uu}. A detail study of background implying pion and
kaon production in atmospheric  neutrino reactions has been performed
leading to a background rate of $0.064~\mathrm{y}^{-1}$ due to the reaction
${\nu}_{\mu}+p\to \mu^{-}+K^{+}+p$.

For the current proton lifetime limit for the channel considered
($\tau_p/B=2.3~10^{33}$~y) \cite{Kobayashi:2005pe}, about 40.7 proton decay
events would be observed in LENA after a measuring time of ten years
with less than 1 background event. If no signal is seen in the detector
within this ten years, the lower limit for the lifetime of the proton
will be placed at $\tau_p/B>4~10^{34}$~y at $90\%$~C.L. 
%%JEC 21/6/06 START: From T. Marrodán Undagoitia this part is to be removed 
%\REDBLA{Although interesting, it may be too detailed here (comment by JEC): If one candidate is observed, the lower limit will be reduced to
%$\tau>3~10^{34}$~y at $90\%$~C.L. and the probability of
%this event being background would be 32\%.}
%%JEC 21/6/06 END
%T. Marrodan Undagoitia 10/4/06 END

%Antonio Bueno 20/01/07 START
For GLACIER, this is a quite clean
channel due to the presence of a strange meson and no other particle in
the final state. Using $dE/dx$ versus range as discriminating variable 
in a Neural Net, we can determine the particle identity. We expect 
less than $1\%$ of kaons mis-identified as protons. 
In this channel, the selection efficiency is high ($97\%$) 
for a low atmospheric neutrino background $< 1$~event/Mt.y. 
In case of absence of signal and for a detector location at a depth of 
1 km w.e., we expect for 1~Mt.y (10~years) exposure one event background 
due to cosmogenic sources. This translates into a limit 
$\tau_p/B > 0.6~10^{35}$~yrs at 90\% C.L. This result remains 
valid even at shallow depths where
cosmogenic background sources are a very important limiting factor for proton 
decay searches. 
A very large area annular active muon veto shield could be used in order to
further suppress cosmogenic backgrounds at shallow depths.
For example, the study done by \cite{Bueno:2007um} shows that 
a three plane active veto at a shallow
depth of about 200~m rock overburden in the {\it under a hill configuration} yields
similar sensitivity for $p\rightarrow K^+\bar\nu$ as a 3~km~w.e. deep detector.
%Antonio Bueno 20/01/07 END

For the MEMPHYS detector, one should rely on the detection of the decay products of the $K^+$
since its momentum ($360$~MeV) is below the water \v{C}erenkov threshold (ie. $570$~MeV): a 256~MeV/c muon and its decay electron (type I) or a 205~MeV/c $\pi^+$ and $\pi^0$
(type II), with the possibility of a delayed (12~ns) coincidence
with the 6~MeV ${}^{15}\mathrm{N}$ de-excitation prompt $\gamma$ (Type III).
Using the imaging and timing capability of Super-Kamiokande, the efficiency for the reconstruction of
$p \rightarrow \overline{\nu}K^+$ is $\epsilon=$ 33\% (I), 6.8\% (II)
and 8.8\% (III), and the background is at 2100, 22 and 6 events/Mt.yr level. For the
prompt $\gamma$ method, the background is dominated by
mis-reconstruction. As stated by UNO, there are good 
reasons to believe that this background can be lowered by at least a factor 2 corresponding
to the atmospheric neutrino interaction $\nu p \rightarrow \nu
\Lambda K^+$. In these conditions, and using Super-Kamiokande performances,
a 5~Mt.yr MEMPHYS exposure would allow to reach $\tau_p/B > 2~10^{34}$~yrs (see \refFig{fig:pdk9_jbz}).
%
%Antonio Bueno 20/01/07 START
\subsection{Comparison between the detectors}
%
Preliminary comparisons have been done between the detectors 
(Tab.~\ref{tab:Phys-PDK-Summary}). 
For the $e^+ \pi^0$ channel, the \v{C}erenkov detector gets a better limit due to their
higher mass. However it should be noted that GLACIER, although five times smaller 
in mass than MEMPHYS,  
gets an expected limit that is only a factor two smaller. 
Liquid argon TPCs and liquid scintillator detectors get better results for the
$\bar{\nu} K^+$ channel, due to their higher detection efficiency.
The two techniques look therefore quite complementary. 
We have also seen that GLAICER does not necessarily requires very deep underground 
laboratories, like those currently existing or future planned sites, to perform very 
sensitive nucleon decay searches.
% and it would be worth to 
%investigate deeper the pro and cons of each techniques with other 
%channels not yet addressed by 
%the present study as $e^+ (\mu^+) +\gamma$ and neutron decays.
\begin{table}
\caption{\label{tab:Phys-PDK-Summary}Summary of the $e^+\pi^0$ and $\bar{\nu}K^+$ discovery potential by the three detectors. The $e^+\pi^0$ channel is not yet simulated in LENA.}
\begin{tabular}{lccc}\hline\hline
						& GLACIER             &      LENA              &  MEMPHYS \\ \hline
$e^+\pi^0$	&                     &                        &          \\
$\epsilon (\%)
/ \mathrm{Bkgd (Mt.y)}$ & $45/1$  &         -               &   $43/2.25$ \\
$\tau_p/B$ (90\% C.L., 10~yrs) & 	$0.4\times 10^{35}$ & -           &  $1.0\times 10^{35}$ \\ \hline

$\bar{\nu}K^+$                    &                         &              \\
$\epsilon (\%)
/ \mathrm{Bkgd (Mt.y)}$ & $97/1$  &         $65/1$               &   $8.8/3$ \\
$\tau_p/B$ (90\% C.L., 10~yrs) & 	$0.6\times 10^{35}$ & $0.4\times 10^{35}$            &  $0.2\times 10^{35}$ \\
 \hline\hline
\end{tabular}
\end{table}
%Antonio Bueno 20/01/07 END
