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%   	 	Super-Kamiokande atmospheric neutrinos:			%
%		Status of subdominant oscillations			%
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\documentstyle[aps,prd,preprint,epsfig,tighten]{revtex}
\begin{document}
\draft
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\title{		Super-Kamiokande atmospheric neutrinos:\\ 
		Status of subdominant oscillations}
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\author{ 	G.L.\ Fogli, E.\ Lisi, and A.\ Marrone}
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\address{	Dipartimento di Fisica and Sezione INFN di Bari\\
             	Via Amendola 173, I-70126 Bari, Italy \\ }
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\maketitle
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\begin{abstract}%............................................................
In the context of the recent (79.5 kTy) Super-Kamiokande atmospheric neutrino
data, we concisely review the status of  muonic-tauonic flavor oscillations and
of the subdominant electron or sterile neutrino mixing, in schemes with three
or four families and one dominant mass scale. In the three-family case, where
we include the full CHOOZ spectral data, we also show, through a specific
example, that ``maximal'' violations of the one-dominant mass scale
approximation are not ruled out yet.
\end{abstract}%.............................................................
\medskip
\pacs{PACS: 14.60.Pq, 13.15.+g, 95.85.Ry}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The Super-Kamiokande (SK) Collaboration has recently presented an updated set
of atmospheric neutrino data for a detector exposure  of 79.5 kTy \cite{re01}.
The corresponding statistics is about twice as large as compared to the one
considered in our earlier published analysis of three-flavor mixing \cite{re02}
and of two-flavor mixing with nonstandard dynamics \cite{re03}, and is about
12\% larger than in our previous analysis of four-family schemes in
\cite{re04}. In addition, the CHOOZ collaboration presented in \cite{re05}
final  {\em spectral} data, which we now include in $3\nu$ analyses
\cite{re06}, improving the accuracy of our previous results \cite{re02} based
on the CHOOZ {\em total rate\/} \cite{re07}.


Therefore, we think it useful to present a concise update of our  $2\nu$,
$3\nu$, and $4\nu$ oscillation studies, performed under the hypothesis of one
mass-scale dominance, so as to elucidate the current status and implications of
subdominant (electron or sterile) neutrino mixing in $SK$. Finally, we also
discuss a specific $3\nu$ example which maximally violates the assumption of
one-dominant mass scale.

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two neutrinos}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The standard $2\nu$ case of $\nu_\mu\leftrightarrow\nu_\tau$ oscillations can
be parametrized by one squared mass difference between two states
$(\nu_1,\nu_2)$,
%.......................................................................
\begin{equation}
m^2=|m^2_2-m^2_1|\ ,
\end{equation}
%.........................................................}
and by one mixing angle $\psi$, describing the flavor content of $\nu_2$,
%...............
\begin{eqnarray}
|\langle \nu_2|\nu_\mu\rangle|&=&s_\psi\ ,\\
|\langle \nu_2|\nu_\tau\rangle|&=&c_\psi\ ,
\end{eqnarray}
%...............
where $s=\sin$ and $c=\cos$.


Figure~1 shows our $2\nu$ best fit to the latest SK data, reached at
$(m^2,\sin^22\psi)=(3\times 10^{-3}{\rm\ eV}^2,0.97)$, and corresponding to
$\chi^2_{\min}=38.5$ for $55-2$ degrees of freedom---a very good fit.%
%----------
\footnote{The overall reduction of $\chi^2_{\min}$ with respect to the $2\nu$
(subcase) analysis presented in \protect\cite{re04} is mainly due to a better
agreement of the latest SK electron distributions with their no-oscillation
expectations (especially in the multi-GeV sample).}
%----------
The SK collaboration finds the $\chi^2$ minimum at a slightly different  point,
$(m^2,\sin^22\psi)_{\rm SK}=(2.5\times 10^{-3}{\rm\ eV}^2,1.0)$ \cite{re01}.
However, the  difference is not statistically significant, since the $\chi^2$
function turns out to be rather flat around the minimum, and the ``distance''
between our best-fit point and the SK one is only about one unit in $\Delta
\chi^2$. Concerning the $2\nu$ bounds on $(m^2,\psi)$ from parameter
estimation, they will be discussed later as limits of  $3\nu$ and $4\nu$ cases.


The striking evidence in favor of standard $\nu_\mu\leftrightarrow\nu_\tau$
oscillations (Fig.~1) strongly constrains nonstandard explanations. By using
the approach described in \cite{re03}, we parametrize a wide class of scenarios
involving nonstandard dynamics through three (free) parameters: an oscillation
amplitude $\alpha$, an overall phase factor $\beta$, and an energy exponent
$n$, the standard mass-mixing dynamics being recovered for $n=-1$. 


Figure~2 shows the results of such a three-parameter fit in terms of the
projection $\chi^2(n)$. The corresponding bounds on $n$ give $n=-1.03\pm 0.31$
at 90\% C.L.\ ($\Delta \chi^2=6.25$ for three free parameters, $N_{\rm DF}=3$),
in perfect agreement with the standard case. Such results strengthen our
previous bounds obtained with smaller (45 kTy) SK statistics  ($n=0.9\pm 0.4$
\cite{re03}), and definitely exclude nonstandard dynamics with integer $n\neq
-1$ in the $\nu_\mu\leftrightarrow\nu_\tau$ channel.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Three neutrinos}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Oscillations of three neutrinos $(\nu_1,\nu_2,\nu_3)$, under the hypothesis of
one mass scale dominance for atmospheric $\nu$'s (equivalent to set
$m^2_1\simeq m^2_2$), are characterized by one squared mass difference,
%.......................................................................
\begin{equation}
m^2=m^2_3-m^2_{1,2}\ ,
\end{equation}
%.........................................................
(the cases $m^2>0$ and $m^2<0$ being physically different \cite{re08}), and by
two mixing angles ($\psi=\theta_{23}\in [0,\pi/2]$ and $\phi=\theta_{13}\in
[0,\pi/2]$), describing the flavor content of the state $\nu_3$,
%...............
\begin{eqnarray}
|\langle \nu_3|\nu_e\rangle|&=&s_\phi\ ,\\
|\langle \nu_3|\nu_\mu\rangle|&=&c_\phi s_\psi\ ,\\
|\langle \nu_3|\nu_\tau\rangle|&=&c_\phi c_\psi\ ,
\end{eqnarray}
%...............
the pure $\nu_\mu\leftrightarrow\nu_\tau$ case being recovered for  $\phi=0$
(see \cite{re02} and references therein).


In order to show the bounds in the mixing parameter space, we used in
\cite{re02} a triangular representation \cite{re09,re10}, which is basically a
linear mapping in the variables ($\sin^2\psi,\sin^2\phi$). Such a
representation has the advantage of embedding unitarity by construction, but
has the disadvantage of not showing in detail the phenomenologically
interesting case of small $\phi$. We use here the alternative representation in
terms of $(\tan^2\psi,\tan^2\phi)$ in logarithmic scale (also introduced in
\cite{re09,re10}), which expands the small $\phi$ region while preserving
octant symmetry (when applicable).


Using SK data only, and assuming $m^2>0$, we find the best fit
$(\chi^2_{\min}=38.1)$ at $(m^2,\tan^2\psi,\tan^2\phi)= (3\times 10^{-3}{\rm\
eV}^2, 0.9, 0.01)$. The slight deviation of the best fit mixing  from the pure
$\nu_\mu\leftrightarrow\nu_\tau$ maximal mixing 
[$(\tan^2\psi,\tan^2\phi)=(1,0)$], although intriguing%
%--------
\footnote{There is no reason to have {\em exactly\/}
$(\tan^2\psi,\tan^2\phi)=(1,0)$.},
%-------
is---unfortunately---not statistically significant  $(\Delta \chi^2\lesssim
1)$. This also implies that there is no significant indication for possible
matter effects related to $\nu_e$ mixing $(\tan^2\phi>0)$ in the SK data.


Figure~3 shows the $3\nu$ volume allowed at 90\% and 99\% C.L.\ ($\Delta
\chi^2=6.25$ and 11.36 for $N_{\rm DF}=3$, respectively) in the
$(m^2,\tan^2\psi,\tan^2\phi)$ parameter space, through its projections onto the
coordinate planes. The upper limit on $\tan^2\phi$ ($\lesssim 0.35$ at 90\%
C.L.) improves the one found in \cite{re02} ($\tan^2\phi\lesssim 1$ at 90\%
C.L.\ from 33 kTy SK data), showing the steady progress of SK in confirming
dominant $\nu_\mu\leftrightarrow\nu_\tau$ mixing and in constraining additional
$\nu_e$ mixing. The 90\% C.L.\ range for $m^2$ is  $(1.6$--$7.2)\times 10^{-3}$
eV$^2$. The bounds on $\tan^2\psi$ in Fig.~3 are octant-symmetric only in the
$2\nu$ limit $\tan^2\phi\to0$ (as they should), and show a slight preference
for $\psi$ in the second octant when $\tan^2\phi>0$. Correspondingly, slightly
higher values of $m^2$ are preferred. The (weak) positive correlation between
$\tan^2\phi$ and $\tan^2\psi$ or $m^2$, however, is largely suppressed by the
inclusion of CHOOZ data, as we now discuss.


As described in \cite{re06}, we can now include the CHOOZ  reactor spectral
data (14 bins minus one adjustable normalization parameter) through a $\chi^2$
statistics reproducing the bounds of the so-called ``CHOOZ analysis A''
\cite{re05}. This improvement provides more accurate bounds in the $m^2$ region
of interest for atmospheric neutrinos. Our best fit to SK+CHOOZ data
($\chi^2_{\min}=45.7$) is reached at $(m^2,\tan^2\psi,\tan^2\phi)= (3\times
10^{-3}{\rm\ eV}^2, 0.75, 0.003)$. Once again,  the small deviation of the best
fit mixing  from  $(\tan^2\psi,\tan^2\phi)=(1,0)$ is not statistically
significant ($\Delta \chi^2\lesssim 1$). 


Figure~4 shows the projections of the  $(m^2,\tan^2\psi,\tan^2\phi)$ volume
allowed by SK+CHOOZ. By comparing Fig.~3 with Fig.~4, the tremendous impact of
CHOOZ on $\nu_e$ mixing bounds becomes evident  (one order of magnitude
difference in the upper bound on $\tan^2\phi$). As expected, at the small
values of $\tan^2\phi$ allowed by the fit in Fig.~4, both the octant-asymmetry
in $\psi$ and the upper limit on $m^2$ are reduced, and the 90\% C.L.\ range
for $m^2$ becomes $(1.6$--$5.3)\times 10^{-3}$ eV$^2$. 


We have also repeated the fit for the case $m^2<0$ (not shown), corresponding
to a state $\nu_3$ lighter than $\nu_{1,2}$. For negative $m^2$, we get
somewhat weaker bounds on $\tan^2\phi$ ($\lesssim 0.5$ at 90\% C.L.) for the
fit to SK data only, while the fit to SK+CHOOZ data gives results almost
identical to those in Fig.~4. This fact shows that, unfortunately, current
atmospheric+reactor data are basically unable to discriminate the sign of $m^2$
in $3\nu$ scenarios, as it was the case for pre-SK and pre-CHOOZ data
\cite{re08}.


Finally, Fig.~5 shows the SK zenith distributions computed for three
representative cases at $\tan^2\phi=0.025$ (allowed at 90\% C.L.\ by SK+CHOOZ)
and for $\psi$ both maximal ($\tan^2\psi=1$) and nonmaximal ($\tan^2\psi=1/2$
and 2). Within statistical errors, the $3\nu$ zenith distributions in Fig.~5
are hardly distinguishable from the $2\nu$ one in Fig.~1 (even more so for
$m^2<0$, not shown), the differences being at most $\sim 1.5\sigma$ in a few
bins. Therefore, there is little hope to unambiguously discover $\phi\neq 0$
(i.e., $\nu_e$ mixing) from SK atmospheric data in the near future. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Four neutrinos}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We consider the $4\nu$ (3 active + 1 sterile) scenario described in 
\cite{re04}, characterized by a 2+2 mass spectrum with well-separated
atmospheric and solar doublets. We also make the simplifying assumption
\cite{re04} that the atmospheric doublet $(\nu_3,\nu_4)$ is almost decoupled
from $\nu_e$, and that the solar doublet $(\nu_1,\nu_2)$ is almost decoupled
from $\nu_\mu$. The dominant mass scale for atmospheric neutrinos is then
%.......................................................................
\begin{equation}
m^2=m^2_4-m^2_3\ ,
\end{equation}
%.........................................................
and two mixing angles ($\psi=\theta_{23}\in [0,\pi/2]$ and $\xi=\theta_{13}\in
[0,\pi/2]$) are sufficient to describe the flavor contents of the state
$\nu_4$,
%...............
\begin{eqnarray}
|\langle \nu_4|\nu_e\rangle|&\simeq & 0\ ,\\
|\langle \nu_4|\nu_\mu\rangle|&=&s_\psi\ ,\\
|\langle \nu_4|\nu_\tau\rangle|&=&c_\psi c_\xi\ ,\\
|\langle \nu_4|\nu_s\rangle|&=&c_\psi s_\xi\ ,
\end{eqnarray}
%...............
the pure $\nu_\mu\leftrightarrow\nu_\tau$  and $\nu_\mu\leftrightarrow\nu_s$ 
cases being recovered for  $s_\xi=0$ and $s_\xi=1$, respectively. In such
scenario, the cases $m^2>0$ and $m^2<0$ are not physically different (being
equivalent under octant inversion, $\psi\to\pi/2-\psi$ \cite{re04}), and we
take $m^2>0$.


Figure~6 shows the bounds obtained by a fit to SK data at 90\% and 99\% C.L.\
($N_{\rm DF}=3$) in the parameter space  $(m^2,\tan^2\psi,\tan^2\xi)$. The best
fit point $(\chi^2_{\min}=38.1)$ is reached at  $(m^2,\tan^2\psi,\tan^2\xi)=
(3\times 10^{-3}{\rm\ eV}^2, 0.76, 0.1)$ but,  once again,  the preferred
mixing differs from  $(\tan^2\psi,\tan^2\phi)=(1,0)$ by less than one unit in
$\Delta \chi^2$. Notice that in the $(m^2,\tan^2\psi)$ plane, the $4\nu$
projected bounds of Fig.~6 are very similar to the $3\nu$ ones in Fig.~4,
implying that the current limits on the ``$2\nu$'' subset of parameters
$(m^2,\tan^2\psi)$ are rather stable even by making allowance for additional
$\nu_s$ or $\nu_e$ mixing. 


Concerning the $(\tan^2\psi,\tan^2\xi)$ plane in Fig.~6, the upper bounds on
$\tan^2\xi$ indicate that pure  $\nu_\mu\leftrightarrow\nu_s$ oscillations
$(\tan^2\xi\to\infty)$ are disfavored as compared with pure 
$\nu_\mu\leftrightarrow\nu_\tau$ oscillations $(\tan^2\xi\to 0)$, in agreement
with \cite{re11,re12}, although large $\nu_s$ mixing is not excluded yet. In
particular, the current upper limit from Fig.~6 ($\tan^2\xi\lesssim 4$ at 90\%
C.L.) is even slightly {\em weaker\/} than the one we found with smaller
statistics in \cite{re04} ($\tan^2\xi \lesssim 2$). The reason can be traced to
a  peculiar feature of the latest UP$\mu$ data, namely, the flatness of the
muon suppression pattern in the four UP$\mu$ bins at
$\cos\theta\in[-0.7,-0.4]$, as described in Fig.~7.


Figure~7 shows the SK zenith distributions for three representative $4\nu$
cases with sizable $\nu_s$ mixing ($\tan^2\xi=1$). As it is well known (and
evident from a comparison of Fig.~7 with Fig.~1), additional $\nu_s$ mixing for
atmospheric neutrinos tends to reduce the muon suppression and, in particular,
tends to flatten the normalized UP$\mu$ distribution. Although the SK UP$\mu$
data do prefer a mean positive slope rather than a flat suppression, the four
bins in the zenith range  $\cos\theta\in[-0.7,-0.4]$ happen to favor a locally
flat distribution. This current feature might be just a statistical fluctuation
but, at present, it plays some role in global fits, where it tends to weaken
the rejection of ``flat'' distributions (i.e., of sizable $\nu_s$ mixing), as
compared with previous UP$\mu$ data \cite{re04}.


The SK Collaboration has also presented additional (preliminary)  indication in
favor of $\nu_\mu\leftrightarrow\nu_\tau$ mixing coming from statistical
$\nu_\tau$ appearance in selected event samples \cite{re01}. It seems possible
to isolate an excess of about $100\pm 50$ $\tau$-like events, to be compared
with standard $\nu_\mu\leftrightarrow\nu_\tau$ expectations of $\sim 100$
\cite{re01}. Taken at face value, such numbers imply an additional $\sim
2\sigma$ evidence ($\Delta \chi^2 \simeq 4$) in favor of pure
$\nu_\mu\leftrightarrow\nu_\tau$  ($s^2_\xi=0$) as compared with pure
$\nu_\mu\leftrightarrow\nu_s$  ($s^2_\xi=1$).  We have then roughly
parametrized the SK ``tau appearance''  signal by adding a penalty function
$\Delta \chi^2=4 s^2_\xi$ in the $4\nu$ fit. We get an ``improved'' upper bound
$\tan^2\xi\lesssim 1.5$ at 90\% C.L.\ ($N_{\rm DF}=3$), to be compared with
$\tan^2\xi\lesssim 4$ in Fig.~6 (without penalty function). On the one hand,
this seems to indicate that there is certainly room to refine current bounds on
additional $\nu_s$ mixing in SK; on the other hand, our analysis shows that
large $\nu_s$ mixing  (e.g., a fifty-fifty admixture of $\nu_\tau$ and $\nu_s$
at $\tan^2\xi\simeq 1$) is not yet excluded at present. Therefore,
compatibility with complementary solar neutrino bounds on $\tan^2\xi$
\cite{re13} is still possible, as it was the case for previous SK data
\cite{re04}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two, three, and four neutrino summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The bounds on dominant and subdominant mixing found in the previous sections
can be conveniently summarized in one single plot, as shown in Fig.~8. The left
panel shows the bounds on the $(m^2,\tan^2\psi)$ parameters for pure
$\nu_\mu\leftrightarrow\nu_\tau$ mixing (equivalent to our $3\nu$ scheme for
$\tan^2\phi=0$ or to our $4\nu$ scheme for  $\tan^2\xi=0$). As discussed
before, such bounds are not significantly altered by additional $\nu_e$ mixing
($\tan^2\phi>0$) or by additional $\nu_s$ mixing ($\tan^2\xi>0$), and thus they
hold also in the global $3\nu$ and $4\nu$ fits with good accuracy. For such
reason, the $2\nu$ bounds in the left panel are formally obtained for $N_{\rm
DF}=3$, so as to match those in the middle and right panels. The middle panel
shows the  $3\nu$ bounds on additional $\nu_e$ mixing ($\tan^2\phi>0$), with
and without CHOOZ.  Finally, the left panel shows the $4\nu$ bounds on
additional $\nu_s$ mixing ($\tan^2\xi>0$). Such synthetic figure represents the
main result of our analysis.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two mass scales}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The analyses in the previous sections  are based on the assumption that
atmospheric neutrino oscillations are driven by only one mass scale $(m^2)$.
This is not necessarily the case, especially if one takes the ``solar'' squared
mass difference in the upper range allowed by the data (see, e.g.,
\cite{re14}), provided that one accepts an averaged or quasiaveraged solar 
neutrino survival probability \cite{re06}.  Concerning atmospheric neutrinos,
two mass scales in the range $\sim 10^{-3}$ eV$^2$ were shown to provide
acceptable fits to previous SK+CHOOZ data  \cite{re15}. Here we show, through a
specific example, that such possibility is not yet excluded by the latest data.


Let us consider the specific $3\nu$ case shown in Fig.~9, characterized by two
equal squared mass differences ($\Delta m^2_{32}=\Delta m^2_{21}=0.7\times
10^{-3}$) eV$^2$, and by mixing angles%
%-----------------
\footnote{In standard notation, $\phi=\theta_{13}$, $\psi=\theta_{23}$,
and $\omega=\theta_{12}$.}
%-------------------
 $(\tan^2\phi,\tan^2\psi,\tan^2\omega)=(0,1,2)$, giving the flavor composition
of mass eigenstates shown in the same figure. The spectrum in Fig.~9 might be
called ``democratic'', since it maximally violates the usual ``hierarchical''
approximation ($\Delta m^2_{21}\ll \Delta m^2_{32}$). Notice that $\nu_e$
oscillations are driven only by $\Delta m^2_{21}$, which is purposely chosen
just below the current CHOOZ bounds \cite{re05}. The solar neutrino survival
probability is then $P(\nu_e\to\nu_e)\simeq 1-\frac{1}{2}\sin^2 2\omega \simeq
5/9\sim 1/2$, up to small (quasiaveraged \cite{re06}) corrections. 


Figure~10 shows the SK zenith distributions computed for the democratic
scenario in Fig.~9, and corresponding to $\chi^2=50.9$ for the fit to SK data
only ($\chi^2=61.2$ if CHOOZ data are also included). Although such value is
significantly higher than in the best-fit $2\nu$ case of Fig.~1, it is still
acceptable from the point of view of goodness of fit. All in all, the curves in
Fig.~10 provide a globally acceptable ``fit-by-eye'',   with moderate
departures from the data along the horizontal direction ($\cos\theta\sim 0$)
for the MG$\mu$ and US$\mu$ samples. Therefore, if one accepts an almost
constant $(\sim 1/2)$ suppression as explanation of the solar neutrino deficit,
$3\nu$ scenarios with two comparable mass scales appear to represent a viable
possibility in the current atmospheric neutrino phenomenology. Needless to say,
the most general analysis of such cases (depending on all the $3\nu$
mass-mixing parameters) would be rather intricated, and is postponed to a
future work.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In the context of the latest (79.5 kTy) SK atmospheric $\nu$ data, we have
concisely reviewed the status of dominant  $\nu_\mu\leftrightarrow\nu_\tau$
oscillations (including the case of nonstandard dynamics, Fig.~2) and of
subdominant $\nu_e$ and $\nu_s$ mixing (in $3\nu$ and $4\nu$ schemes,
respectively). In the $3\nu$ case we have applied an improved CHOOZ analysis.
The main $2\nu$, $3\nu$, and $4\nu$ results are discussed separately and then
summarized in Fig.~8. Finally, we have shown (through a specific example)  that
current atmospheric neutrino data are also compatible with  oscillations driven
by two comparable mass scales.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments

This work was supported by INFN and by  the Italian MURST within the
``Astroparticle Physics'' project.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 			R E F E R E N C E S 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{references}

\bibitem{re01}	Super-Kamiokande presentations in winter conferences:
		C.\ McGrew in {\em Neutrino Telescopes 2001},
		9th International Workshop on Neutrino Telescopes
		(Venice, Italy, March 2001), to appear;
		T.\ Toshito in {\em Moriond 2001}, XXXVI Rencontres de
		Moriond on Electroweak Interactions and Unified Theories
		(Les Arcs, France, March 2001), to appear.

\bibitem{re02}	G.L.\ Fogli, E.\ Lisi, A.\ Marrone, and G.\ Scioscia,
		Phys.\ Rev.\ D {\bf 59}, 033001 (1999).

\bibitem{re03}	G.L.\ Fogli, E.\ Lisi, A.\ Marrone, and G.\ Scioscia, 
		Phys.\ Rev.\ D {\bf 60}, 053006 (1999).

\bibitem{re04}	G.L.\ Fogli, E.\ Lisi, and A.\ Marrone,
		Phys.\ Rev.\ D {\bf 63}, 053008 (2001).

\bibitem{re05}	CHOOZ Collaboration, M.\ Apollonio {\em et al.},
		Phys.\ Lett.\ B {\bf 466}, 415 (1999).

\bibitem{re06}	G.L.\ Fogli, E.\ Lisi, and A.\ Palazzo,
		.

\bibitem{re07}	CHOOZ Collaboration, M.\ Apollonio {\em et al.},
		Phys.\ Lett.\ B {\bf 420}, 397 (1998).

\bibitem{re08}	G.L.\ Fogli, E.\ Lisi, D.\ Montanino, and G.\ Scioscia, 
		Phys.\ Rev.\ D {\bf 55}, 4385 (1997).

\bibitem{re09}	G.L.\ Fogli, E.\ Lisi, and G.\ Scioscia, 
		Phys.\ Rev.\ D {\bf 52}, 5334 (1995).

\bibitem{re10}	G.L.\ Fogli, E.\ Lisi, and D.\ Montanino,  
		Phys.\ Rev.\ D {\bf 54}, 2048 (1996).

\bibitem{re11}	Super-Kamiokande Collaboration, S.\ Fukuda {\em et al.},
		Phys.\ Rev.\ Lett.\ {\bf 85}, 3999 (2000).	

\bibitem{re12}	MACRO Collaboration, talk by B.C.\ Barish in
		{\em Neutrino 2000}, Proceedings of the
		19th International Conference 
		on Neutrino Physics and Astrophysics, edited by 
		J.\ Law, R.W.\ Ollerhead, and J.J.\ Simpson,
		Nucl.\ Phys.\ B (Proc. Suppl.) {\bf 91}, 141 (2001).

\bibitem{re13}	C.\ Giunti,
		M.C.\ Gonzalez-Garcia, and C.\ Pe{\~n}a-Garay,
		Phys.\ Rev.\ D {\bf 62}, 013005 (2000);
		M.C.\ Gonzalez-Garcia and C.\ Pe{\~n}a-Garay,
		Phys.\ Rev.\ D {\bf 63}, 073013 (2001).

\bibitem{re14}	S.\ Choubey, S.\ Goswami, N.\ Gupta, and D.P.\ Roy,
		.

\bibitem{re15}	A.\ Strumia,
		J.\ of High Energy Physics, {\bf 04} 26 (1999).

\end{references}


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%..............................................................................
\InsertFigure{f01.ps}%
{Fig.~1. Super-Kamiokande zenith distributions (79.5 kTy \protect\cite{re01})
used in the analysis, normalized to no-oscillation expectations. The data set
includes sub-GeV electrons (SG$e$, 10 bins), multi-GeV electrons (MG$e$, 10
bins), sub-GeV muons (SG$\mu$, 10 bins), multi-GeV muons  (MG$\mu$, 10 bins),
upward stopping muons (US$\mu$, 5 bins), and upward through-going muons
(UT$\mu$, 10 bins), for a total of 55 data points. The error bars are
statistical only ($\pm 1\sigma$); systematic (correlated) uncertainties are
treated as in \protect\cite{re02}. The solid line is our best fit for $2\nu$
oscillations ($\chi^2=38.5$).}
%..............................................................................
\InsertFigure{f02.ps}%
{Fig.~2. Dependence of the $\chi^2$ function on the neutrino energy exponent
$n$, assuming an oscillation phase proportional to $E^n$, with unconstrained
factors for the overall phase and amplitude. The only integer $n$ compatible
with the SK data is $n=-1$, corresponding to standard mass-mixing dynamics.}
%..............................................................................
\InsertFigure{f03.ps}%
{Fig.~3. Projections of the regions allowed  in the $3\nu$ parameter space
$(m^2,\tan^2\psi,\tan^2\phi)$  at 90\% and 99\% C.L.\  ($\Delta \chi^2=6.25$
and 11.36 for $N_{\rm DF}=3$) onto the coordinate planes. The fit includes SK
data only (79.5 kTy). The pure $2\nu$ case of $\nu_\mu\leftrightarrow\nu_\tau$
oscillations is recovered for  $\tan^2\phi\to 0$. Nonzero values of $\phi$
parametrize $\nu_e$ mixing. }
%..............................................................................
\InsertFigure{f04.ps}%
{Fig.~4. As in Fig.~3, but including final CHOOZ positron spectra
\protect\cite{re05} (14 data points minus one adjustable normalization
factor).}
%..............................................................................
\InsertFigure{f05.ps}%
{Fig.~5. Zenith distributions for three representative $3\nu$ cases with 
$\tan^2\phi=2.5\times 10^{-2}$, allowed at 90\% C.L.\ by SK+CHOOZ. Notice the
distortion of the MG$e$ distribution. Such distortion would be somewhat smaller
for negative $m^2=-3\times 10^{-3}$ eV$^2$ (not shown).}
%..............................................................................
\InsertFigure{f06.ps}%
{Fig.~6. Projections of the regions allowed  in the $4\nu$ parameter space
$(m^2,\tan^2\psi,\tan^2\xi)$ at 90\% and 99\% C.L.\  ($\Delta \chi^2=6.25$ and
11.36 for $N_{\rm DF}=3$)  onto the coordinate planes. The fit includes SK data
only (79.5 kTy). The pure  $\nu_\mu\leftrightarrow\nu_\tau$ case 
($\tan^2\xi\to 0$) is clearly preferred over  the pure
$\nu_\mu\leftrightarrow\nu_s$ case  ($\tan^2\xi\to \infty$). However, sizable
$\nu_s$ mixing  [$\tan^2\sim O(1)$] is allowed in addition to
$\nu_\mu\leftrightarrow\nu_\tau$ oscillations.}
%..............................................................................
\InsertFigure{f07.ps}%
{Fig.~7. Zenith distributions for three representative $4\nu$ cases with 
sizable $\nu_s$ mixing ($\tan^2\xi=1$), allowed at 90\% C.L.\ by SK data.
Notice the reduced suppression in the muon samples, as compared with  pure
$\nu_\mu\leftrightarrow\nu_\tau$ mixing in Fig.~1.}
%..............................................................................
\InsertFigure{f08.ps}%
{Fig.~8. Summary of $2\nu$, $3\nu$, and $4\nu$ bounds at 90\% and 99\% C.L.\ on
the mass-mixing parameters from SK data. Left panel: Bounds on
$(m^2,\tan^2\psi)$ for pure $\nu_\mu\leftrightarrow\nu_\tau$ mixing (i.e., for
$\tan^2\phi=0=\tan^2\xi$). Middle panel: Bounds on additional $\nu_e$ mixing
(parametrized by $\tan^2\phi>0$) in $3\nu$ scenarios, both without CHOOZ (see
also Fig.~3) and with CHOOZ (see also Fig.~4). Right panel: Bounds on
additional $\nu_s$ mixing  (parametrized by $\tan^2\xi>0$) in $4\nu$ scenarios
(see also Fig.~6). All the bounds are derived for $N_{\rm DF}=3$, including
those in the left panel.}
%..............................................................................
\InsertFigure{f09.ps}%
{Fig.~9. A ``democratic'' (i.e., nonhierarchical) $3\nu$ scenario with two
equal squared mass differences ($\Delta m^2_{32}=\Delta m^2_{21}= 0.7\times
10^{-3}$ eV$^2$) and with $(\nu_e,\nu_\mu,\nu_\tau)$ flavor content as follows:
$(1/3,1/3,1/3)$ for $\nu_1$, $(1/6,1/6,1/6)$ for $\nu_2$, and $(0,1/2,1/2)$ for
$\nu_3$. Notice that $\nu_e$ disappearance is driven only by $\Delta m^2_{21}$
(just below the CHOOZ sensitivity \protect\cite{re05}), and gives
$P(\nu_e\to\nu_e)\simeq 5/9$ for solar neutrinos (up to small quasiaveraged
oscillation corrections).}
%..............................................................................
\InsertFigure{f10.ps}%
{Fig.~10. Zenith distributions for the democratic scenario in Fig.~9, giving
$\chi^2=50.9$ for the fit to SK data ($\chi^2=61.2$ for the fit to SK+CHOOZ).}
%..............................................................................

\eject
\end{document}
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