\section{Introduction}


Underground water Cherenkov detectors have 
found unambiguous evidence for
neutino oscillations and therefore beyond-the Standard Model
physics.
%  focused much attention on  neutrino physics.
The atmospheric neutrino results of Super- Kamiokande(SK),
followed by the solar observations of SK, SNO and KamLAND,
have confirmed that neutrinos have mass and
two large mixing angles.
However, there remain many  questions
about the parameters and properties of leptons,
some of which could be addressed by a larger (megatonne)
underground neutrino detector.
%nonetheless there are questions
%remaining. More statistics are required to increase
%the sensitivity to unknown neutino parameters, 
If the  location of such a detector was
judiciously selected,  it could be 
a suitable distance along  the  path of 
a new  high intensity
$\nu_\mu$ beam (superbeam), and/or or $\nu_e$ beam ($\beta$ beam).
%source = beam, not astro
%{\it build beam and detector so can do an accelerator expt}.

The observation of neutrinos from SN1987A forshadowed
the linked results on astrophysics and neutrino physics
that can be obtained from a supernova.   Such an exploding
star is an extraordinary source, for which it
would be reasonable to have a detector.
A megatonne detector could perhaps even 
 see relic neutrinos 
accumulated from past supernovae.

Originally,  large underground detectors were built
to look for proton decay, a prediction of
Grand Unified Theories.   Nucleon decay is
 a ``smoking gun'' for quark lepton
unification,  observation of which would 
confirm many years of theoretical speculation.
The current lower bound on the proton lifetime from SK  has
ruled out the simplest non-supersymmetric GUT, 
 a megatonne detector would
cover a substantial area of interesting parameter
space.



\section{Bread and Butter: $\nu$ Physics}

A megatonne detector  would have improved sensitivity to 
currently unknown parameters of neutrino mixing.
The neutrinos could be of astrophysical origin---
solar, atmospheric or from supernovae--- or $\nu$
beams of specific flavour and energy could be directed
at the detector.   
%The solar and atmospheric  neutrino fluxes would
%arrive for free. 
A  high intensity $\nu_\mu$ ``superbeam'', 
could be produced by increasing the intensity of the
proton driver at the source, 
or a very pure $\nu_e$ beam could be produced
in the $\beta$ decay of an ion beam. 



\subsection{status}

A review of our current knowledge of neutrino parameters
was presented by G. Fogli. 

Information
\footnote{The numerical values are from the global fit
presented by Fogli} on $\sin ^2 \theta_{23} = 0.45 
\pm \stackrel{0.18}{_{0.11}}$,
$\Delta m_{23}^2 = 2.4 \pm \stackrel{0.5}{_{0.6}} \times 10^{-3}$ eV$^{2}$ 
and  $\sin ^2 \theta_{13} \leq 0.035$ 
is obtained from SuperKamiokande, K2K and CHOOZ.
  The evidence for atmospheric neutrino
oscillations  with large, or maximal mixing
is robust, and confirmed with neutrinos
from the K2K beam.  
SK has found  evidence  for  a decrease
in $\nu_\mu$ flux at the location
expected for the first dip in the oscillation 
probability---this despite the smearing in
energy and path length.
As discussed by Fogli, the data sets can be
combined in various ways to determine the parameters.
The results quoted were obtained 
from the combined data of all three experiments, by 
using a three-dimensional simulation for
the atmospheric neutrino fluxes,   by including
subleading effects due to 
$\Delta m_{12}^2$ and  $\sin ^2 \theta_{12}$,
and leaving $\sin ^2 \theta_{13}$ free.
Letting 
$\sin ^2 \theta_{13}$ float has little effect
because the data prefers it small.


SNO, SK and KamLAND are  sensitive to  the solar mass
difference $\Delta m_{12}^2 = 8.0 
\pm \stackrel{0.8}{_{0.7}} \times
10^{-5} $ eV$^2$ and 
a large but not maximal mixing
angle  $\sin ^2 \theta_{23} = 0.31 \pm \stackrel{0.05}{_{ 0.04}} $. 
These data also prefer 
$\sin ^2 \theta_{13} \sim 0$  (a non-trivial
consistency check with atmospheric and CHOOZ),
so the allowed ranges for 
 $\Delta m_{12}^2 $ and 
$\sin ^2 \theta_{23} $  are not significantly
affected when $\theta_{13}$ is allowed to float.



\subsection{ agenda for future experiments}

The current bounds on the unknown  neutrino  parameters,
and future  prospects for  measuring them were discussed by
J. Ellis and G. Fogli, and T Schwetz. Some of these unknowns
(items 4-7 of the following list)
could be determined from more precise oscillation
experiments.
%---in particular from neutrino beams
%directed at a megatonne detector. 
\begin{enumerate}
\item the number of light neutrinos participating
in oscillations is usually taken to be the three 
active neutrinos expected in the Standard Model. 
However, the LSND experiment found evidence
for $\Delta m^2 \sim$ eV$^2$, which  would require
one (or more)
additional light sterile neutrinos.
MiniBoone  is searching for oscillations
in the LSND window; their results,
expected in 2005,  will confirm or
rule out the LSND claim. 
\item The absolute neutrino  mass scale
is probed in three ways.
Firstly, the endpoint spectrum of electrons in  nucleon
($^3H$)  $\beta$ decay is sensitive to the ``effective
electron neutrino mass''
$$ m_e^2 = [c^2_{13} c_{12}^2 m_1^2 +
c^2_{13} s_{12}^2 m_2^2 +
s^2_{13} m_3^2 ]^2  \leq 1.8 ~{\rm eV}~~.$$ 

Cosmological Large Scale  Structure  is affected
by neutrino masses, because neutrino free-streaming
in the early Universe would suppress density fluctuations
on small scales. Current cosmological data sets the constraint:
$$ m_1 + m_2 + m_3 \leq 0.47 - 1.4 {\rm eV}$$
The range of the bound is representative of different
results in the literature, which are based on
inequivalent data sets. The strong bound uses
Ly$\alpha$ data to probe small scale structure;
this data is sometimes left out because of
uncertain systematic errors. 

The final observable to which neutrino masses
 could contribute---if they are majorana---
is  lepton number violating neutrino-less
double $\beta$ decay ($0 \nu 2 \beta$). 
The amplitude can be written as  a nuclear
matrix element,
$\times$ the coefficient of a $\Delta L = 2$
non-renormalisable operator. This coefficient
can be calculated perturbatively from the
new physics that permits the decay.
When this new physics is
majorana neutrino masses, the coefficient 
is proportional to
$ m_{ee}$, where
$$
m_{ee}  = [c_{13}^2c_{12}^2m_1 + c_{13}^2s_{12}^2m_2e^{i \phi_2}
  + s_{13}^2m_3e^{i \phi_3}  ]
$$
The PMNS matrix has be taken
$U  = V  P$, with $V$ CKM-like with one phase $\delta$ 
($V_{13} = \sin \theta_{13}e ^{-i \delta}$), and $P = diag
\{ 1, e^{ \phi_2/2}, e^{i (\phi_3/2 + \delta)} \}
$ (See talk by G. Fogli.)

There is a controversial claim that $0 \nu 2 \beta$
has been detected in $^{76}Ge$, with a rate corresponding
to $|m_{ee}| \simeq  0.23 \pm 0.18 $ eV. A
disagreement with the cosmological bound
 can be avoided by not using  Ly$\alpha$ data.
\item  Are neutrinos Majorana or Dirac?
 Oscillation experiments are sensitive to
mass$^2$ differences, so do not distinguish whether
neutrinos are majorana 
or dirac.
The majorana nature of neutrinos, which is 
``natural'' in the popular seesaw mechanism, 
can be tested in processes that violate lepton number,
such as $0 \nu 2 \beta$.
\item Is the  mass pattern hierarchical
($\Delta m_{13}^2 >0)$ or inverted ($\Delta m_{13}^2<0$)?
Oscillation probabililities in matter,
for neutrinos and antineutrinos,  depend on this sign,
because the matter contribution to the  mass matrix 
changes  sign between  neutrinos and anti-neutrinos.
Long baseline neutrino beams and the flux of
neutrinos from supernovae are sensitive to this sign. 
\item  What is $\theta_{13}$? There are
only upper bounds on this remaining angle of the PMNS matrix,
It can be probed by looking for a $\nu_e$ 
contribution to  $\Delta m_{13}^2$ oscillations. 
This angle controls ``three flavour'' effects, like
CP violation. 
\item What is $\delta$, the  ``Dirac phase'' of the PMNS
matrix, which  contributes to CP violation in neutrino
oscillations (multiplied by $\sin \theta_{13}$)?
\item is $\theta_{23}$ maximal? 
\end{enumerate}
%The sensitivity of various beam and
%detector combinations is illustrated in figure 
%\ref{Ellis}.
 
% \begin{figure}[ht]
%\vspace{4cm}
%\epsfxsize=7cm\epsfbox{Fig2.ps}
%\hspace{1cm}
%\epsfxsize=7cm\epsfbox{fig3a.ps}
%\caption{ plots shown in the presentation of J Ellis,
%showing the sensitivity to   $\theta_{13}$,  $\Delta m_{12}^2$,
%and $\delta$ of various beams.  }
%%\vspace{4cm}
%\protect\label{Ellis}
%\end{figure}




\subsection{$\theta_{13}$, $\delta$ and and the sign of $\Delta m_{13}^2$ }
\label{TS}


%Summary of discussions by Kajita, Nakahata, elsewhere?

Determining items 4-6 (of the above list)
at  a future megatonne detector  was
discussed by T. Schwetz, and 
J Ellis  presented  prospects for beams from  CERN.

  It is known that 
the 3-flavour oscillation probability has degeneracies,
as can be
seen from
\beq
P_{\mu e} \simeq   \sin^2 2\theta_{13} \sin^2 \theta_{23} \sin^2 \Delta_{ 31} 
 +  \alpha^2  \sin^2 \theta_{12} \cos^2 \theta_{23} 
 \Delta^2_{31}       +     
\alpha \sin 2\theta_{12} 
\sin 2\theta_{13} \sin2\theta_{23}   \Delta_{ 31} \sin  \Delta_{ 31} \cos(
 \Delta_{ 31} \pm \delta).
\eeq
where $\alpha = \Delta_{21}/ \Delta_{31}$, and
$ \Delta_{31} = (m_3^2 - m_1^2)L/4 E_\nu$.
For instance,  a measured $P_{\mu e}$ could corresponds
 to several solutions in the  ($\delta, \theta_{13}$) plane.
This is refered to as the ``intrinsic'' degeneracy.
There are additional degeneracies associated with
the sign of $\Delta m_{13}^2$ (``hierarchy'' degeneracy), and  with 
the sign of $\pi/4 - \theta_{23}$ (``quadrant'' degeneracy), if 
$\theta_{23}$ is not maximal. 

The degeneracies can be resolved  with
spectral information, and by looking at
different channels.  Having a $\beta$-beam and
superbeam is helpful in this second respect.
Spectral information is available with 
an off-axis beam, so the   ($\delta, \theta_{13}$) 
degeneracy wouuld be absent  at T2K-II
(T2K to HyperK).

T Schwetz discussed   using atmospheric neutrino data to  
address the degeneracies, by measuring sub-dominant
effects due to three-flavour mixing. He showed that
there is an enhancement in the  $\nu_e$ (or $\bar{\nu}_e$)
flux, for multi-GeV events, due to $\theta_{13}$. 
The enhancement is for neutrinos in the
normal hierarchy, and anti-neutrinos in the
inverted case. Since the $\nu_e$ and $\bar{\nu}_e$
detection cross-sections are different, 
mesuring this enhancement would give information
on $\theta_{13}$ and the sign of $\Delta m_{13}^2$. 
Sub-GeV events could be  sensitive to
the octant of $\theta_{23}$ via 
contributions arising due to $\Delta m_{12}^2$.


The   hierarchy and octant  degeneracies could be reduced at T2K-II
by using the the atmospheric neutrino data of HyperK.
This was shown by combining
a numerical 3-flavour atmospheric analysis,
with  long baseline simulation of
the beam and detector using  with 
the GloBES software
( http://www.ph.tum.de/globes/ ).
An example figure is shown on the right below
(figure \ref{TSfig}). 
Preliminary results, assuming a superbeam and
$\beta$-beam from CERN, and  including atmospheric data
at  a 450 kt Cherenkov detector at Frejus, were also
 shown. 

In  summary, the combined analysis of atmospheric and
long baseline neutrino data at a megatonne detector
could resolve parameter degeneracies---with the advantage
that atmospheric neutrinos arrive ``for free''.



 \begin{figure}[ht]
%\vspace{4cm}
 \epsfxsize=17cm
\epsfbox{TS.eps}
%\epsfxsize=7cm\epsfbox{delta.eps}
\caption{ Resolving hierarchy(H) and octant (O) 
degeneracies using atmospheric neutrinos. The
figures compare $\beta$-beam  and SPL from CERN to Fr\'ejus,
(details of the experiments can be found
in the NuFact05  talks of Mezzetto and  Campagne), and  T2K 
to HK
The detector in all cases is 450 kt water Cherenkov. }
%\vspace{4cm}
\protect\label{TSfig}
\end{figure}





\subsection{ Theoretical interest} 

One of the outstanding puzzles for particle theorists
is the origin of Yukawa couplings. There are many models,
 which fit the masses
and mixing angles observed in the quark and lepton sector,
%with a variety of free parameters,
%However, 
but none are particularily compelling. Additional hints from
the data ---  symmetries respected by the masses,
constraints on the Yukawa parameters--- would be particularily
welcome. Measuring the third  leptonic mixing angle $\theta_{13}$,
and determining whether $\theta_{23}$ is maximal,
are  both important in this respect.


A popular mechanism to explain the smallness of
neutrino masses is the seesaw, which has 18 parameters 
in its simplest  form (type I) with three $\nu_R$.  
Twelve of these parameters appear among
the light leptons (although not all are realistically
measurable), and some of the remaining unknowns
affect $\mu$ and $\tau$ decays in SUSY. So
measuring  many neutrino parameters with 
good accuracy  would reduce the parameter space of seesaw models.


If $\theta_{13}$ is found to be large ($\gappeq .01$),
%, seefigure \ref{Ellis},
 the phase $\delta$ of the PMNS matrix
could be experimentally accessible. Observing CP violation
in the leptons, for the first time,  would be an exciting 
phenomenological novelty.
%\footnote{
%The PMNS matrix  contains one
%unremoveable phase, so CP violation in oscillations
%is phenomenologically ``expected''. But it is
%important to verify expectations---we also ``expected''
%mixing angles in the lepton sector to be small.} 
It is also tempting to relate $\delta$ to
the CP violation required in the generation of
the matter excess of the Universe (baryo/lepto-genesis).
Various leptogenesis mechanisms 
can be implemented in  the seesaw model,
and  depend on some combination
of the seesaw's complex couplings. Observing
$\delta \neq 0$ would  demonstrate that at least one
combination of couplings is complex, thereby
suggesting that the phases relevant for leptogenesis
might also be present. 





\section{Theory Dreams: Nucleon Decay}

Nucleon decay  was 
 the original motivations for large underground detectors,
ancestors of the megatonne, and
attracted attention from many speakers during the
workshop.
The theoretical expectations for
the proton's lifetime were  discussed in some
detail in the talks of  of J. Ellis and L. Covi.

Our concept of theoretical progress is
that we advance by unifying apparently diverse
concepts.  An example of 
successful unification is the Standard Model, which
united electromagnetism with the weak interactions. 
Some hints that quarks and leptons might be united
in a larger theory are the curious anomaly cancellation
among known fermions---where 
the quarks and leptons cancel each others contributions
to dangerous operators which would destroy
the consistency (and experimental accuracy)
of the SM. Another tantalising hint is
that the strong, and electroweak gauge couplings  become equal
at $\Lambda \sim 10^{16}$ GeV, suggesting a
unique gauge interaction at this scale. 

Unifying  the quarks and leptons into
a multiplet  means that  there are particles
in the theory that turn quarks  into leptons,
so  baryons can decay.  Observing proton decay would
be a smoking gun for such theories, 
 confirming that  our theoretical preference
for unified theories is reflected in nature---and
it could probe higher energy scales,
or shorter distances, than any previous observation.
It also could give some information  on mixing
angles in the right-handed quark sector, about which
the Standard Model says nothing.


\subsection{SU(5)}


The  simplest GUT is SU(5), the 
lowest rank (``smallest'') group  capable
of accomodating all the SM particles. % is SU(5),
%of rank 4, which was much studied at the birth of GUTS.
SO(10) is the  one possibility at rank 5, and it
has the advantage over SU(5) of accomodating 
 the  right-handed neutrino (SM gauge singlet)
in its 16-dimensional multiplets. At rank six
is $E_6$, which appears in some string models.
 
In the minimal SU(5) GUT,
the colour-triplet $d^c = \overline{d_R}$ are combined with
the lepton SU(2) doublet $\ell_L$ into a 
$\bar{5}$, and the $e^c$ shares a  10
with the $q_L$ and $u^c$.
 The X and Y  gauge bosons,
which acquire masses $\sim M_{GUT}$ when
SU(5) is broken,  have Baryon + Lepton
number violating gauge interactions because
they mix different multiplet members.  
They mediate proton decay
via dimension six operators such as
\beq
\frac{ g_5^2}{M_X^2} \epsilon_{\alpha \beta \gamma}
(\overline{d^c}_{\alpha,k} 
\overline{u^c}_{\beta,j} q_{\gamma , j} \ell _k - 
\overline{e^c}_{k} 
\overline{u^c}_{\alpha,j} q_{\beta , j} q_{\gamma ,k}
)
\eeq
There are also operators induced by GUT Higgses,
with baryon number violating Yukawa-strength couplings.

Proton decay is expected 
at rates
\beq
\Gamma_{p} = C \frac{\alpha_{5}^2 m_p^5}{M_X^4}
\eeq
where $C$ is a constant englobing  mixing angles,
renormalisation group running, and  strong
interaction effects.  The dominant decay channel in
non-supersymmetric SU(5) is $ p \rightarrow \pi^0 e^+$.
The experimental limit 
$\tau_{p \rightarrow \pi e} > 6.9 \times 10^{33}$ years,
imposes $M_X \geq 7.3 \times 10^{15}$ GeV,
 so non-SUSY SU(5) is
ruled out  because this is above
the mass scale where the gauge couplings approximately
unify.




Proton decay in  supersymmetric SU(5) 
is different in many respects. The GUT scale 
(determined from gauge coupling unification) is
higher,  so decays mediated by
$X$ and $Y$ are slower. However, there are new
{\it dimension 5}  operators, induced by
 the coloured triplet Higgsino 
that shares a 5 with SM-type doublet Higgsinos, and which
 has Yukawa couplings to SM fields. Schematically
these operators can be written
$$
\frac{Y^{ij}_{qq} Y^{km}_{ql}}{2 M_c }
  Q_iQ_jQ_kL_m + 
\frac{ Y^{ij}_{ue} Y^{km}_{ud} }{ M_c } U^c_i E^c_j U^c_k D^c_m
$$
where $M_c$ is the triplet
Higgsino mass $\leq M_X$, 
the capitals are superfields, two of which
are scalars and two fermions. 
Dressing this operator with the exchange
of a ``-ino'' gives a 4-fermion operator
$\propto 1/(m_{SUSY} M_{c})$. This is
enhanced with respect to the $X$-boson
exchange, but suppressed by small Yukawa couplings.
In addition, the SM SU(2) and SU(3)
contractions are antisymmetric,  so 
the operator is flavour non-diagonal, giving
a dominant decay $p \rightarrow K^+ \bar{\nu}$.

There are relations among the quark and lepton
Yukawa couplings,
which depend  on the GUT Higgs content of
the model.
The simplest would  be for all the Yukawa matrices
to be equal at the GUT scale, but  some
differences must be included to
fit the  observed fermion masses.
The proton lifetime in SUSY SU(5) depends 
which Yukawa matrices are equal at the GUT scale: 
setting $Y_{ql} = Y_{ud}$ equal to the down
Yukawa matrix $Y_d$ predicts a a proton lifetime shorter
than the current SK limit of $
\tau_{p \rightarrow K \bar{\nu}} >1.9 \times 10^{33}$ years. 
However, setting  $Y_{ql} = Y_{ud}$ equal to the 
charged lepton Yukawa $Y_e$ changes the
dependence of $\tau_p$ on the fermion mixing
angles, so lifetimes  
 in excess 
of the bound
can be found.
 The proton lifetime in SUSY SU(5)
is uncertain due to the non-unification of
Yukawa couplings.



A possible string-motivated GUT model, discussed
by J Ellis,  is
flipped SU(5)$\times U(1)$, where
the SU(2) doublets of the SM  are inverted 
($\nu \leftrightarrow e, u \leftrightarrow d$)
in the GUT multiplets. This extends
the $p \rightarrow K^+ \bar{\nu}$  lifetime
to $\tau \gsim 10^{35} - 10^{36}$ years, 
%CITE ? %\cite{Ellis:2002vk}
%\bibitem{Ellis:2002vk}
%J.~R.~Ellis, D.~V.~Nanopoulos and J.~Walker,
%%``Flipping SU(5) out of trouble,''
%Phys.\ Lett.\ B {\bf 550} (2002) 99
%[arXiv:hep-ph/0205336].
%%%CITATION = HEP-PH 0205336;%%,
potentially testable at a megatonne detector. 

\subsection{ SO(10) in six space dimensions}


In recent years, theorists have 
constructed models in $d>4$ dimensional
space, with the additional dimensions
compactified at some  scale $\ll m_{pl}$.
These models offer a framework to
study new physics possibilities not
included in  the MSSM.  L Covi discussed proton
decay in a 6-dimensional SUSY SO(10) model, where
the extra 2 dimensions are compactified
on a torus (that has additional discrete symmetries).
The four fixed points of this torus correspond
to 4-dimensional branes, where SM
particles can reside. Each
SM generation lives at a different fixed point,
with a different breaking of  SO(10), so the Yukawas
in this model are different  from 4-dimensional
SO(10). The higgsino mixing 
which allowed the dimension 5 proton decay
operators is  suppressed, so
the dimension 6 $X$-mediated diagrams
dominate in this supersymmetric extra-dimensional
model. The proton decay  rates 
are slightly larger than 4-dimensional SU(5) due to
the  sum over the tower of Kaluza-Klein $X$ modes,
but they differ in the flavour
structure. This has characteristic
signatures, such as suppressing
$p \rightarrow K^0 \mu^+$. The 
current bound $\tau_{p \rightarrow \pi^0 e^+} 
\geq 6.9 \times 10^{33}$ years  implies in
this model 
$M_X > 9.6 \times 10^{15}$ GeV $ \sim M_{GUT}$,
suggesting that the proton could 
be discovered to have a lifetime $\sim 10^{34}$ years.



In summary, proton decay is an unmistakable
footprint of Unification, and is just around
the corner in many models.  Looking to the
future, once proton decay is observed, 
the branching ratios  will  open a new
perspective on the structure and origin
of the Yukawa matrices, giving  new
information on the Yukawa puzzle.



\section{From the Sky: Supernova Neutrinos}

Supernova neutrinos were discussed by A Dighe
(galactic supernovae) and S Ando(relic neutrinos),
and also by G Fogli. Astrophysical 
observation of nearby galaxies suggests
that 1-4 supernovae should take place in our galaxy
per century. Neutrinos carry  $ 99 \%$ of the
star's binding energy,
so  these  infrequent events  could
be a fund of information about 
neutrino parameters and supernova astrophysics.


A real-time  SN within 10 kpc  may determine whether the
hierarchy is normal or inverted, and be sensitive to
very small values of $\sin \theta_{13}$.
A megatonne detector is probably required to see
these effects.  
The neutrino signal   could  also  trace
the outward propagation of the shock which powers the optical
explosion. 


%determine the location
%of the SN in the sky to $\sim 10 ^o$ ( this could
%be improved by  a factor of 2 to 3 with  Gadolinium).



While waiting for the next galactic supernova,
detectors could look for ``supernovae relic
neutrinos'' (SRN), the diffuse background of neutrinos
emitted by past supernovae. SK's present limit on
this flux is background-limited, and
just above predictions.  Detecting these neutrinos
could give useful information on neutrinos and the
history of star formation.

\subsection{soon in our galaxy?}


A  star of mass $\gsim 8 {\cal M}_{\odot}$ becomes
unstable at the end of its life.  It  resembles
an onion, with the different layers burning lighter
elements into heavier, the end-products of one
layer serving as fuel for the one underneath.
At the centre develops an iron core, which eventually
cannot support the outer layers, and collapses.
Most of the binding energy is released as
neutrinos.

The SN  neutrino flux has various components.
The neutronisation burst takes place 
in the first 10 ms, as the
heavy nuclei break up.  It consists of $\nu_e$
from $p + e \rightarrow n + \nu_e$, and is
emitted from the ``neutrinosphere'', that is,
the radius from which neutrinos can free-stream
outwards.  The core  density is  near nuclear, above
the $\sim 10^{10}$ g/cm$^3$ required 
  to trap a 10 MeV neutrino.

For the following 10 seconds, the core  cools
by emitting $\nu$ and $\bar{\nu}$ of all flavours.
99 $\%$ of the SN energy is emitted in
these fluxes, refered to as ``initial''
fluxes $F^0$, whose 
characteristics are predicted to be flavour dependent. 
In particular, the average energies
of $\nu_e$, $\bar{\nu}_e$
and $\nu_x$    are predicted to differ:
%with the average energies
$E_0(\nu_e) \sim   10-12$ MeV, 
$E_0(\bar{\nu}_e) \sim  13-16$  MeV,
and $E_0({\nu}_x) \sim  15-25$  MeV.
The more weakly interacting neutrinos are
more energetic because  they escape
from closer to the hot centre of the star. 

As the neutrinos travel outwards, they pass
through ever-decreasing density, so 
 matter effects on the mixing are
crucial.  Level-crossing occurs when
$\Delta m^2 \cos 2 \theta = \pm 2 \sqrt{2}  E_\nu G_F n_e$,
where the $+$ ($-$) refers to (anti) neutrinos. 
Flavour conversion is
possible at two level crossings,
corresponding to the solar and atmospheric
mass differences, and  can
appear in the $\nu$ or the $\bar{\nu}$ 
 depending on the mass hierarchy. This will mix the 
initial neutrino fluxes, which were labelled by flavour.

 Towards the centre of the star, $\nu_e$ is the heaviest neutrino.
In the normal mass hierarchy, $\nu_e$ 
has a level crossing  at the 
H resonance, which  arises at a matter density
$\sim 10^3$ g/cm$^3$,
where  $\nu_3$ can
transform to $\nu_2$  via the atmospheric
mass difference and $\theta_{13}$. % at this
%resonance.
The H resonance takes place in the $\bar{\nu}_e$
channel, for the inverted mass hierachy.
The L resonance  arises at a matter density
$\sim 10$ g/cm$^3$. It is in the $\nu$ channel for
both hierarchies, and crosses  $\nu_2$ with
$\nu_1$ via the solar mass difference and angle.
The level crossing probability  is adiabatic
for the L resonance, and for the H resonance
when $\sin^2 \theta_{13}  \gappeq 10^{-3}$.
%(refered to as ``large'' for the remainder
%of this section.)
It is non-adiabatic 
at the H resonance if
$\sin^2 \theta_{13}  \lappeq 10^{-3}$.
%(``small, for the remainder of this section.)
The fluxes arriving at the earth ($F$) depend on
the initial fluxes ($F^0$) and the oscillation probabilities
($p$ and $\bar{p}$):
$$
F_{\nu_e}     =    pF^0_{\nu_e} + (1 - p)F^0_{\nu_x}
~~~
F_{\bar{\nu}_e}     =    \bar{p} F^0_{\bar{\nu}_e} + (1 - \bar{p})F^0_{\nu_x}  
$$
(There is a related formula for  $F_{{\nu}_x}$.) 
There are three interesting  cases:
\begin{itemize}
\item Case A: normal hierarchy, $\sin^2 \theta_{13}  \gappeq 10^{-3}$,
($p = 0$, $\bar{p} = \cos^2 \theta_{\odot}$)
\item Case B: inverted hierarchy, $\sin^2 \theta_{13}  \gappeq 10^{-3}$
(($p = \sin^2 \theta_{\odot}$,  $\bar{p} = 0$)
\item Case C: any hierarchy, $\sin^2 \theta_{13}  \lappeq 10^{-3}$
($p = \sin^2 \theta_{\odot}$, $\bar{p} = \cos^2 \theta_{\odot}$)
\end{itemize}


A Dighe discussed whether these cases could be distinguished
in the observable signal, given that the initial
spectra are poorly known, and only the final spectra for
$\bar{\nu}_e$ are cleanly available.  It is
difficult to find observables that do not
depend on assumptions about the initial spectra.
a possibility, if the SN neutrino flux crosses
the earth, is to look for  oscillations in the
signal due to matter effects in the earth.
This would contribute high frequency
wiggles to the spectrum, which could be
extracted form the data at a megatonne
detector.
For  the normal  hierarchy or small
$\theta_{13}$,  these earth effects would
appear in the $\bar{\nu}_e$ channel, so
observing such wiggles would eliminate case B.

It could also be possible to  identify
earth effects if the SN is observed with two detectors,
where one is in  the earth's shadow and
the other not. As A. Dighe discussed, IceCube could
be the second detector, which would be complementary
to Hyper-K.


Neutrinos have a crucial role in the explosion of supernovae,
for instance the energy they deposit in the shock may
be the critical contribution that allows 
the star to explode.  The interactions between
the shock and the outgoing neutrinos may also
provide information on the neutrino parameters. As the shock passes
through the $H$ resonance region, it can
make  adiabatic transitions non-adiabatic,
thereby temporarily turning scenarios A and B,
into scenario C. One can therefore hope  to
to track the shock fronts  through the
star in the time-dependent neutrino signal.


A nearby supernova would illuminate 
the earth with  neutrinos.  This flux  can be
used to simultaneously obtain information about
the source, and about neutrino properties.
At a megatonne detector,  
``earth effects'' in the
neutrino spectra could be observed,
which would give  SN-model
independent information on the hierarchy
(inverted vs normal) and  whether   $\theta_{13}$
is large or small. Alternatively, if
the SN neutrinos do not cross the earth,
information about  neutrino parameters
could be extracted from shock wave
propagation effects in the neutrino
spectra.


\subsection{relics}


Most of the energy of a supernova is released
as neutrinos. The diffuse background of
these neutrinos, today, depends on the
neutrino spectrum emitted from each explosion,
 on the oscillation of those neutrinos in
the SN and in the earth, and  on the
supernova rate over the past history of 
the Universe.

As discussed in the previous section, the neutrino
fluxes emitted from the SN core are expected to
be flavour dependent, and to oscillate
due to matter effects as they leave the star. For
instance, in the normal hierarchy, a $\bar{\nu}_e$
emitted from the core is the lightest $\bar{\nu}$,
due to matter effects, so it will exit
the star as $\bar{\nu}_1$. The observed $\bar{\nu}_e$
flux will therefore be
$$ F_{\bar{\nu}_e} = | U_{ei}|^2  F_{\bar{\nu}_i} 
=  | U_{e1}|^2  F^0_{\bar{\nu}_e }
+ (1 - | U_{e1}|^2)  F^0_{\bar{\nu}_x}
$$
so $ (1 - | U_{e1}|^2)  \sim 30 \% $   comes from the
harder $\nu_x$ spectrum.  The oscillations
enhance the high-energy tail, but not dramatically
in the detectable energy range ($< 30$ MeV).


The SN rate is infered from  the star formation rate, 
which can be  extracted from other cosmological observables.
Using the recent Galactic Evolution Explorer data,
the event rate at SK  can be calculated, and is
found to be mostly due to SN at $z < 1$. 
A few $\bar{\nu}_e p \rightarrow n e^+$  events
per year are predicted in the $E > 18$ MeV window
where the flux exceeds the solar and armospheric
neutrinos. Unfortunately,  in this range there
is a background from the decays of slowly moving muons, 
which are produced
by atmospheric $\nu_\mu$   and are invisible at SK.
So SK can set an upper limit on the SRN flux,
which can then be inverted into a constraint
on the supernova rate. The bound is just above
theoretical predictions, so  SRN might  be seen
using 5-10 years of data.

The background could be reduced by 
adding Gadolinium to a water Cherenkov
detector. This  would  tag the neutrons produced
in $\bar{\nu}_e p \rightarrow n e^+$,
and therefore    distinguish the  $\bar{\nu}_e$
from other neutrinos. Liquid Argon detectors
are sensitive to $\nu_e$, so would be complementary
to a water detector.

S. Ando also discussed the possibility of observing,
at a megatonne detector,  a few neutrinos from  SN
in nearby galaxies ($\sim$  Mpc away). This would give
the time of the collapse, helpful for gravitational
wave searches.

In summary,  the SK limit on   supernovae relic
neutrinos is just above the theoretical prediction;
 a future  megatonne  detector should therefore
have a good chance to see them.
 At a megatonne Cerenkov detector, a 5 $\sigma$ detection could
be possible with pure water after a few years,
($\sim$ 300 events/yr would be expected with Gd).
A 100 kt liquid Argon detector would expect
$\sim 57 \pm 12 $ events after 5 years.




%\section*{Acknowledgements}

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