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\title{
\vspace*{-2.0cm}
\begin{flushright}
\normalsize{
FERMILAB-Pub-02/329-T
}
\end{flushright}
\vspace*{1.0cm}
{$CPT$ violating neutrinos
in the light of KamLAND}
\vspace*{0.8cm}
\author{\large\textbf
{G.~Barenboim$^a$, L.~Borissov$^b$, J.~Lykken$^{a,c}$}\\ 
\\
$^a$\normalsize\emph{Fermi National Accelerator Laboratory,
P.O. Box 500, Batavia, IL 60510, USA }\\
$^b$\normalsize\emph{Columbia University, New York, NY, 10027, USA}\\
$^c$\normalsize\emph{Enrico Fermi Institute, Univ. of Chicago, 5640
S. Ellis Ave., Chicago, IL 60637, USA}\\ }
}


\begin{document}
\maketitle

\vspace*{2cm}

\begin{abstract}
The KamLAND collaboration has observed a medium baseline
oscillation signal for reactor antineutrinos.
We show that a hierarchical $CPT$ violating
neutrino spectrum can simultaneously accommodate the
oscillation data from LSND, atmospheric, solar and KamLAND,
as well as the nonobservation of antineutrino
disappearance in short baseline reactor experiments. In our scenario
the KamLAND experiment is not
observing an LMA solar oscillation signal.
Instead the KamLAND oscillation signal is due to an
independent mass splitting in the antineutrino spectrum.
A larger antineutrino mass splitting accounts for the LSND
signal and also contributes to atmospheric oscillations.
\end{abstract}


\thispagestyle{empty}
\newpage

\section{Introduction}

$CPT$ violating neutrino masses allow the possibility 
\cite{Murayama} - \cite{Nos3}
of reconciling the LSND \cite{lsnd}, atmospheric \cite{atm_sk}, 
and solar oscillation \cite{solar, SNO_solar}
data without resorting to sterile neutrinos. As argued in
\cite{Nos1}, 
there are good reasons to imagine that $CPT$ violating
dynamics couples directly to the neutrino sector, but not
to other Standard Model degrees of freedom. 
An explicit $CPT$ violating model of this type was presented
in \cite{Nos3}.

KamLAND \cite{kamland}, a medium baseline reactor antineutrino disappearance
experiment, is sensitive to antineutrino mass-squared splittings 
in the $10^{-4}$ eV$^2$ range characteristic of the large mixing
angle (LMA) solar neutrino scenario. The KamLAND collaboration has
recently reported \cite{kamland2}
an electron antineutrino survival probability
which is significantly less than one:
\be
P(\overline{\nu}_e \rightarrow \overline{\nu}_e) = 
0.611 \pm 0.085 \pm 0.041 \; .
\label{kamdata}
\ee

If the neutrino mass spectrum conserves $CPT$, then this result
is consistent with the LMA interpretation of solar neutrino
oscillations. If the neutrino mass spectrum violates $CPT$,
however, the KamLAND result provides no information about
solar oscillations, but rather constrains the splittings
in the antineutrino spectrum.

In this paper we show that a hierarchical $CPT$ violating
neutrino spectrum can simultaneously accommodate the
oscillation data from LSND, atmospheric, solar and KamLAND,
as well as the nonobservation of antineutrino
disappearance in short baseline reactor experiments. In our scenario
the KamLAND experiment is not observing an LMA solar oscillation signal.
Instead the KamLAND oscillation signal is due to an
independent mass splitting in the antineutrino spectrum.
A larger antineutrino mass splitting accounts for the LSND
signal and also contributes to atmospheric oscillations.


\section{The spectrum}

To analyze all the possible $CPT$ violating spectra is not an easy job.
With four mass differences and six mixing angles (not taking into
account the two $CP$ violating phases which participate in oscillations)
a complete scan of the whole parameter space is impractical. 
However, thanks to
the available experimental data, it is possible to reduce the
allowed regions to two sets of well-differentiated spectra with
(quasi) orthogonal experimental signatures.

The easiest way to make contact with the experimental results 
is in terms of the neutrino survival and transition probabilities,
which are given by
\bea
P(\nu_\alpha \rightarrow \nu_\beta) = \delta_{\alpha \beta} -
 4 \sum_{i>j=1 }^3 U_{\alpha i} U_{\beta i} U_{\alpha j} 
U_{\beta j}
\;\sin^2 \left[ \frac{\Delta m_{ij}^2 L}{4 E} \right]
\label{pro}
\eea
for neutrinos and
\bea
P(\overline{\nu}_\alpha \rightarrow \overline{\nu}_\beta) = 
\delta_{\alpha \beta} -
 4 \sum_{i>j=1 }^3 \overline{U}_{\alpha i} \overline{U}_{\beta i} 
\overline{U}_{\alpha j} \overline{U}_{\beta j}
\;\sin^2 \left[ \frac{\Delta \overline{m}_{ij}^2 L}{4 E} \right]
\label{apro}
\eea
for antineutrinos.
Here the matrix $U=\left\{ U_{\alpha i}\right\}$  
($\overline{U}=\left\{ \overline{U}_{\alpha i}\right\}$)
describes the weak
interaction neutrino (antineutrino) states, $\nu_\alpha$, in terms of the 
neutrino (antineutrino) mass eigenstates,
$\nu_i$. That is,
\bea
\nu_\alpha = \sum_i U_{\alpha i} \nu_i \;\;\;\;\;\; \mbox{and} \;\;\;\;\;\;
\overline{\nu}_\alpha = \sum_i \overline{U}_{\alpha i} \overline{\nu}_i
\eea
where we have ignored the possible $CP$ violation phases in both matrices and 
took them to be real. The matrices can be
parametrized as follows:
\bea 
U=\pmatrix{c_{12} c_{13} & s_{12}c_{13} & s_{13} \cr
       -s_{12}c_{23} - c_{12}s_{23}s_{13} & c_{12}c_{23}- s_{12}s_{23}s_{13} & s_{23}c_{13} 
\cr
        s_{12}s_{23} - c_{12}c_{23}s_{13} & - c_{12}s_{23} - s_{12}c_{23}s_{13} 
&c_{23}c_{13}}
\eea
and similarly for $\overline{U}$.
In Eq.~(\ref{pro}) $L$ denotes the neutrino flight path, \ie\ 
the distance 
between the neutrino source and the detector, and
$E$ is the energy of the neutrino in the laboratory 
system. 

Regarding the mass spectrum of the three neutrinos we 
assume that it is hierarchical and thus
characterized by two different  squared masses
\[ \Delta m_{12}^2 = m^2_2 
-m^2_1\quad\mbox{and } \Delta m_{13}^2 = m^2_3 -m^2_1 \]
whose numerical values are rather different, \ie\ $
\Delta m_{13}^2 \gg  \Delta m_{12}^2$  and similarly for the 
antineutrinos. Having said that, it becomes apparent that
the larger mass-squared difference in the neutrino sector
will be related to the atmospheric neutrino signal observed
by SuperKamiokande, while the smaller one will drive the
solar neutrino oscillations. In the antineutrino sector,
the largest mass difference will provide an explanation
to the signal observed in LSND, while the smaller one is
the one which might have been (mis)identified by KamLAND as a confirmation
of LMA.


\begin{figure}[ht]
\vspace{1.0cm}
\centering
\epsfig{file=uno.eps,width=10cm}
\caption{\it Possible neutrino mass spectrum with almost all the 
electron content in the heavy state.  Although the figure shows an 
example of large mixing, our approach is agnostic about the mixing matrix.
The flavor content is distributed as follows: electron flavor (red),
muon flavor (brown) and tau flavor (yellow)}
\label{neutrino-spectrum}
\end{figure}

The key ingredient to sort out the antineutrino spectra are reactor 
experiments. Their results indicate \cite{chooz, bugey}
that electron antineutrinos produced
in reactors remain electron antineutrinos on short baselines. As the 
distance traveled by our antineutrinos is small we can forget about
the smallest mass difference and average the other two, thus
the survival
probability can be expressed as
\bea
P(\overline{\nu}_e \rightarrow \overline{\nu}_e) = 
1 - 2 \overline{U}_{e3}^2 (1- \overline{U}_{e3}^2) \; .
\eea
It is clear that there are two possible ways to achieve a
survival probability close to one, \ie\ $\overline{U}_{e3}$ can
be almost one or almost zero. Physically this means that we
can choose between having almost all the antielectron flavor
in the heavy state (or in the furthest away state) or just
leave in this state almost no antielectron flavor. 
The first possibility (which is depicted in Fig. 1) 
is the one we explored in our previous works. This spectrum predicts 
for KamLAND a survival probability consistent with one.
Since this is strongly disfavored by the KamLAND result
(\ref{kamdata}), we instead pursue the second possibility,
which is represented by the spectrum shown in Figure 2.

\begin{figure}[htb]
\vspace{1.0cm}
\centering
\epsfig{file=dos.eps,width=10cm}
\caption{\it Possible neutrino mass spectrum with almost no  
electron content in the heavy state.  Although the figure shows an 
explict mixing pattern, there is a whole family of mixing
matrices that can do an equally good job.
The flavor content is distributed as follows: electron flavor (red),
muon flavor (brown) and tau flavor (yellow)}
\label{neutrino2-spectrum}
\end{figure}




This second family of spectra is characterized by a
strong violation of $CPT$ in the mass differences
but a much slighter effect in the mixing matrix. This is
seen in Fig.~2 where  the flavor distribution in the neutrino and
antineutrino spectra is rather similar. The most
distinctive feature of this family of solutions is its  $\theta_{23}$,
which lives far away from maximal mixing, 
or in other words which has  a large component
of antitau neutrino in the heavy state.
The small antimuon neutrino component in the heavy state
is not bounded by the non observation of
muon neutrino disappearance over short
baselines in the CDHS experiment\cite{dydak}, as the antineutrino
component in this experiment was minimal.

KamLAND could have observed an oscillation signal driven
by the smaller antineutrino mass splitting
and misinterpreted it as LMA oscillations. To explicitly see how this
might have
happened, we will choose two sample points in our parameter space and
calculate the transition probabilities for it. Let us emphasize 
that we have not performed a chi-squared fit and therefore the points
we are selecting (by eye and not by chi) are not optimized to give
the best fit to the existing data. Instead, they must be regarded 
as two among the many equally good sons in this family of solutions.  

The point we have chosen has $\overline{\theta}_{13}=.08\;$,
$\overline{\theta}_{23}=.5\;$, $\overline{\theta}_{12}=.6\;$, 
$\Delta \overline{m}_{12}^2 = 5 \cdot 10^{-4}$ eV$^2$ and 
$\Delta \overline{m}_{13}^2 = \cal{O}$(1) eV$^2$.
Since we are dealing with an antineutrino signal, we
do not need to identify either the flavor distribution
or the mass eigenstates of the neutrino sector. We will do
it later, when showing the zenith angle dependence this model
predicts for SuperKamiokande atmospheric neutrinos.

The survival probability measured by KamLAND is given by
\bea
P_{\mbox{\tiny{KamLAND}}} &= &1 -4 \overline{U}_{e3}^2  (1-
\overline{U}_{e3}^2) \,\sin^2 \left[ \frac{\Delta \overline{m}_{13}^2  L}{4 E} 
\right]
 -4  \overline{U}_{e1}^2  \overline{U}_{e2}^2
\,\sin^2 \left[ \frac{\Delta \overline{m}_{12}^2  L}{4 E} \right] \; ,
\eea
where the second term (proportional to $\overline{U}_{e3}^2$) is negligible.
Plugging our numbers in, it is straightforward to see that
$P_{\mbox{\tiny{KamLAND}}} \approx .6 $ regardless of whether
the mass difference that drives the solar neutrino oscillations
belongs to the LMA region.

By the same token, we can calculate the probability associated with
the LSND signal. It is given by
\bea
P_{\mbox{\tiny{LSND}}}= 4 \overline{U}_{\mu3}^2 \overline{U}_{e3}^2  
\,\sin^2 \left[ \frac{\Delta \overline{m}_{13}^2 L}{4 E} \right] \; ,
\eea
where we have neglected terms proportional to $\Delta \overline{m}_{12}^2 $
which are irrelevant for such small distances. As the reader can easily
verify, we predict a $P_{\mbox{\tiny{LSND}}} \simeq .0022$ in excellent
agreement with the LSND final analysis:
\bea
P_{\mbox{\tiny{LSND-final}}}= 0.00264 \pm .00081 \;.
\eea

The only piece of experimental evidence involving antineutrinos
which remains to be checked is the signal found for SuperKamiokande
atmospheric neutrinos. As we are introducing an antineutrino mass
difference roughly two orders of magnitude larger than the SuperK best
fit point (for an analysis with two generations and conserving
$CPT$), there is cause for concern.
In fact we pass this test as successfully as we did the others.
To see this, we have first to state the parameters in
the neutrino sector. Once more they have been chosen almost randomly
from the different analyses available in the literature and are
given by $ \theta_{13}=.08\;$,
$ \theta_{23}=.78\;$, $ \theta_{12}=.52\;$, 
$\Delta  m_{12}^2 = 1 \cdot 10^{-4}$ eV$^2$ and 
$\Delta  m_{13}^2 = 2.8 \cdot 10^{-3}$ eV$^2$. 
We stress that although we have chosen a
point in the LMA region, the particular election of both
$\Delta  m_{12}^2 $ and  $ \theta_{12}$ does not affect the 
quality of the agreement with the data.

With these parameters we have calculated the zenith angle dependence of the 
ratio (observed/expected in the no oscillation case) for muon and
electron neutrinos for the sub-GeV and multi-GeV energy ranges 
(remember that since SuperK is a water Cherenkov
detector it does not distinguish neutrinos from antineutrinos and
washes out any possible difference between the conjugated 
channels). The results are shown in Fig. 3 where we have also included
the experimental data for the sake of comparison.
As we have closely followed the spirit of the 
calculation in  \cite{amol, Nos2},
we refer the reader to this article for details and skip the technicalities.
We worked in a complete three generation
framework and  included matter effects.


\begin{figure}[htb]
\vspace{1.0cm}
\centering
\epsfig{file=tres.eps,width=12cm}
\caption{\it SK zenith angle distributions normalized to no-oscillations
expectations, for our first $CPT$ violating example.
Circles with error bars correspond to SK data.}
\end{figure}

\begin{figure}[htb]
\vspace{1.0cm}
\centering
\epsfig{file=cuatro.eps,width=12cm}
\caption{\it SK zenith angle distributions normalized to no-oscillations
expectations, for our second $CPT$ violating example.
Circles with error bars correspond to SK data.}
\end{figure}

In Fig.~4 we show the comparison to SuperK for our second
example point. For this point we have chosen
$\overline{\theta}_{13}=.08\;$,
$\overline{\theta}_{23}=.5\;$, $\overline{\theta}_{12}=.785\;$, 
$\Delta \overline{m}_{12}^2 = 7 \cdot 10^{-5}$ eV$^2$ and 
$\Delta \overline{m}_{13}^2 = \cal{O}$(1) eV$^2$.
Note that this point is consistent with the best-fit point
of KamLAND \cite{kamland2}.

In order to understand the results it is important to remember that
due to production and cross section effects SuperK is dominated by neutrinos,
with antineutrinos a minor (but not negligible) contribution. 
One might wonder though why the analysis done by the SuperK collaboration
allowing for $CPT$ violation does not allow (at 99\% C.L.) a 
mass difference in the
antineutrino sector so drastically different from the one in the
neutrino sector. The answer comes from a variety of sources. 
The SuperK
analysis was not only done in a two generation context but also
forcing the two mixing angles to be maximal. This latter fact indeed maximizes 
the antineutrino contribution and compels the antineutrino mass
difference to take the closest possible value to the neutrino
one. Leaving the mixing angles and the mass differences free
indeed complicates the analysis a lot (to the point where we
do not even consider making a complete chi-squared fit to
the whole parameter space) but does not risk losing solutions.
Instead, we have tried to make an educated guess and search for
a point/region that survives all the cuts, so that others (more brave
people) will take the following step.

The two vs three generation analysis has also an impact, as is
seen by inspecting the transition probability for muon antineutrinos
into tau antineutrinos, which is given by,
\bea
P(\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau) =
&= & 4 \overline{U}_{\mu 3}^2 \overline{U}_{\tau 3}^2 
\sin^2 \left[ \frac{\Delta \overline{m}^2_{23} L}{4 E} \right] 
- 4 \overline{U}_{\mu 2} \overline{U}_{\tau 2} 
\overline{U}_{\mu 1} \overline{U}_{\tau 1} 
\sin^2 \left[ \frac{\Delta \overline{m}^2_{12}  L}{4 E} \right]
\eea
From this formula it becomes apparent that for
neutrinos coming from above only the largest mass difference contributes.
However, for those neutrinos which have travelled through sizeable
portions of the Earth and have covered distances of the order of
10$^4$ km, the second mass difference also plays a role. This 
contribution (which does affect the final result, especially
for sub-GeV neutrinos) is neglected if only one mass difference is taken
into account. 

Our analysis agrees with the spirit of the findings in
Ref~\cite{solveig} where a two generation approximation that 
didn't include matter effects was used. 
Also a simplified analysis based only on the up/down asymmetry in the
number of multi-GeV events (in the $CPT$ violating case) is available in the
literature \cite{strumia}, which used an older SuperK data set.
If one uses (as we do) the result
from the full 1490 day of SK-I data, \ie \
$ A_\mu= -.288 \pm .030 $ \cite{Todd} the $CPT$ violating case (which gives
for the sample points we have being using  
$ A_\mu = -.27$) is clearly favored
over the $CPT$ conserving one ($ A_\mu= -.32 $).  Indeed with the new 
experimental numbers this is clear also 
from the discussion in Ref~\cite{strumia}. 
In all the cases
the electron neutrino asymmetry is consistent (within experimental
errors) with zero.

\section{Discussion}

Once we have established that a $CPT$ violating mass spectrum
as the one shown in Fig. 2 can account for all the available
experimental evidence (including the KamLAND
result), it is time to ask how
we might confirm $CPT$ violation in future data.

The most straightforward answer is through experiments able to
run in both modes (neutrino and antineutrino), by simple 
comparison of the conjugated channels. 
The first of them is MiniBooNE, which is meant to
close the discussion about LSND one way or the other.
MiniBooNE started taking data last summer and is expected to give
a definite answer to the $CPT$ question after some years of running
in each mode. Needless to say we expect MiniBooNE
to confirm LSND only when running in the antineutrino mode.

For our type of spectrum, the observation of atmospheric
neutrinos using the MINOS detector \cite{minoscosmic} is
also ideal. Because the MINOS detector discriminates
positive and negative charge, this experiment can
disentangle the neutrino and antineutrino components
of atmospheric oscillations in a straightforward way.
As the mass differences in the atmospheric sectors
differ by orders of magnitude in our scenario,
MINOS will be able to tell them apart easily.

A positive oscillation signal at
KamLAND (here assumed to be 
a misidentification of a $CPT$ violating spectrum
as LMA) and Borexino \cite{borexino} finding a day/night asymmetry (evidence
of a LOW solution \cite{pdg}) or a seasonal variation
(an indication of VAC \cite{pdg})
will point towards $CPT$ violation.
Indeed a conflict between KamLAND
and Borexino results would constitute strong evidence for
$CPT$ violation even if LSND is disconfirmed by MiniBooNE.
Note that the best-fit point reported by KamLAND has
maximal mixing, which is clearly disfavored by SNO data; more
data will be required to determine if this is a real inconsistency.

All in all, $CPT$ violation has the potential to explain all the existing
evidence about neutrinos with oscillations to active flavors. 
Such a scenario makes distinctive predictions that 
will be tested in the present round of neutrino experiments.
One should always bear in mind that so far we have no evidence 
of $CPT$ conservation in the neutrino
sector. Indeed as we have shown, all the existing data,
including the zenith angle dependence of the atmospheric muon neutrinos 
(and antineutrinos) seen by SuperKamiokande, are equivalently explained
if $CPT$ is broken in a rather drastic way. 
The true status of $CPT$ in the neutrino
sector might be established by the
combined results of KamLAND, Borexino and SNO, and certainly by MiniBooNE.
In the atmospheric sector MINOS is the ideal experiment for 
such a test. 

\subsection*{Acknowledgments}
\noindent
We are grateful to Andr\'e de Gouv\^ea, Bill Louis and Steve Mrenna
for comments and assistance.
This research was supported by the U.S.~Department of Energy
Grant DE-AC02-76CHO3000.


\begin{thebibliography}{99}

\bibitem{Murayama}
H.~Murayama and T.~Yanagida,
Phys.\ Lett.\ B {\bf 520}, 263 (2001).

\bibitem{Nos1}
G.~Barenboim, L.~Borissov, J.~Lykken and A.~Y.~Smirnov,
%``Neutrinos as the messengers of $CPT$ violation,''
JHEP {\bf 0210}, 001 (2002)
.
%%CITATION = ;%%

\bibitem{Nos2}
G.~Barenboim, L.~Borissov and J.~Lykken,
%``Neutrinos that violate $CPT$, and the experiments that love them,''
Phys.\ Lett.\ B {\bf 534}, 106 (2002)
.
%%CITATION = ;%%

\bibitem{Nos3}
G.~Barenboim and J.~Lykken,
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%\cite{Strumia:2002fw}
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%%CITATION = ;%%

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\end{thebibliography}

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