arXiv:hep-ph/0609286v1 27 Sep 2006
Resolving Eight-Fold Neutrino Parameter Degeneracy by
Two Identical Detectors with Different Baselines
Takaaki Kajita1,∗Hisakazu Minakata2,†Shoei Nakayama1,‡and Hiroshi Nunokawa3§
1Research Center for Cosmic Neutrinos,
Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa, Chiba 277-8582, Japan
2Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
3Departamento de F´ ısica, Pontif´ ıcia Universidade Cat´ olica do Rio de Janeiro,
C. P. 38071, 22452-970, Rio de Janeiro, Brazil
(Dated: September 27, 2006)
We have shown in a previous paper that two identical detectors with each fiducial mass of 0.27
megaton water, one in Kamioka and the other in Korea, which receive the (anti-) muon neutrino
beam of 4 MW power from J-PARC facility have potential of determining the neutrino mass
hierarchy and discovering CP violation by resolving the degeneracies associated with them. In this
paper, we point out that the same setting has capability of resolving the θ23octant degeneracy in
region where sin22θ23<∼0.97 at 2 standard deviation confidence level even for very small values
of θ13. Altogether, it is demonstrated that one can solve all the eight-fold neutrino parameter
degeneracies in situ by using the Tokai-to-Kamioka-Korea setting if θ13is within reach by the next
generation superbeam experiments. We also prove the property called “decoupling between the
degeneracies”, which is valid to first order in perturbation theory of the earth matter effect, that
guarantees approximate independence between analyses to solve any one of the three different type
PACS numbers: 14.60.Pq,14.60.Lm,13.15.+g
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
‡Electronic address: email@example.com
§Electronic address: firstname.lastname@example.org
Typeset by REVTEX1
Physics of neutrinos has entered into a new stage after establishment of the mass-induced
neutrino oscillation due to the atmospheric , the accelerator
neutrino  experiments, confirming the earlier discovery [5, 6, 7] and identifying the nature
of the phenomenon. In the new era, the experimental endeavors will be focused on search
for the unknowns in neutrino masses and the lepton flavor mixing, θ13, the neutrino mass
hierarchy, and CP violation. On the theory side, various approaches toward understanding
physics of lepton mixing and the quark-lepton relations are extensively pursuit , which
then further motivate precision measurement of the lepton mixing parameters. We will use
the standard notation  of the lepton mixing matrix, the Maki-Nakagawa-Sakata (MNS)
matrix , throughout this paper.
It was recognized sometime ago that there exists problem of parameter degeneracy which
would act as an obstacle against precision measurement of the lepton mixing parameters.
The nature of the degeneracy can be understood as the intrinsic degeneracy , which is
duplicated by the unknown sign of atmospheric ∆m2 (hereafter, “sign-∆m2degeneracy”
for simplicity) and by the possible octant ambiguity of θ23  that exists if θ23 is not
maximal. For an overview of the resultant eight-fold degeneracy, see e.g., [14, 15].
In a previous paper , we have shown that the identical two detector setting in Kamioka
and in Korea with each fiducial mass of 0.27 Mton water, which receives the identical
neutrino beam from the J-PARC facility can be sensitive to the neutrino mass hierarchy
and CP violation in a wide range of the lepton mixing parameters, θ13and the CP phase δ.
It is the purpose of this paper to point out that the same setting has capability of resolving
the θ23octant degeneracy to a value of θ23which is rather close to the maximal, sin22θ23<
0.97(0.94) at 2 (3) standard deviation confidence level (CL). It is achieved by detecting
solar-∆m2scale oscillation effect in the Korean detector. Together with the sensitivities to
resolution of the degeneracy related to the mass hierarchy and the CP phase discussed in the
previous paper, we demonstrate that the Kamioka-Korea two detector setting is capable of
solving the total eight-fold parameter degeneracy. We stress that resolving the degeneracy
is crucial to precision measurement of the lepton mixing parameters on which we make
further comments at appropriate points in the subsequent discussions. We also emphasize
that it is highly nontrivial that one can formulate such a global strategy for resolving all
the known degeneracies (though in a limited range of the mixing parameters) only with the
experimental apparatus using conventional muon neutrino superbeam.1
In some of the previous analyses including ours [16, 18], people often tried to resolve the
degeneracy of a particular type without knowing (or addressing) the solutions of the other
types of degeneracies. But, then, the question of consistency of the procedure immediately
arises; Can one solve the degeneracy of type A without knowing the solutions of the other
degeneracies B and C? Does the obtained solution remain unchanged when the assumed
solutions for the other type of degeneracies are changed to the alternative ones?, etc. We
answer to these questions in the positive in experimental settings where the earth matter
effect can be treated as perturbation. We do so by showing that the resolution of the
degeneracy of a particular type decouples from the remaining degeneracies, the property
[2, 3], and the reactor
1It may be contrasted to the method for resolving the degeneracy based on neutrino factory examined
in ; It uses a 40 kton magnetized iron calorimeter and a 4 kton emulsion chamber, and conventional
νµbeam watched by a 400 kton water Cherenkov detector.
called the “decoupling between the degeneracies” in this paper.
In Sec. II, we present a pedagogical discussion of how the eight-fold degeneracy can be
lifted by measurement with the Kamioka-Korea two detector setting. In Sec. III, we prove
the “decoupling” and make a brief comment on its significance. In Sec. IV, we discuss some
characteristic features of the νeand ¯ νe appearance probabilities that allow the Kamioka-
Korea identical two detector setting to resolve the θ23 octant degeneracy. In Sec. V, the
actual analysis procedure and the obtained sensitivities for solving the θ23degeneracy are
described in detail. In Sec. VI, we reexamine the sensitivities to the mass hierarchy and
CP violating phase by using our new code with disappearance channels and additional
systematic errors. In Sec. VII, we give a summary and discussions.
HOW THE IDENTICAL TWO DETECTOR SYSTEM SOLVES THE EIGHT-
We describe in this section how the eight-fold parameter degeneracy can be resolved by
using two identical detectors, one placed at a medium baseline distance of a few times 100
km, and the other at ∼1000 km or so. We denote them as the intermediate and the far
detectors, respectively, in this paper. Whenever necessary we refer the particular setting
of Kamioka-Korea two detector system, but most of the discussions in this and the next
sections are valid without the specific setting.
To give the readers a level-one understanding we quote here, ignoring complications,
which effect is most important for solving which degeneracy:
• The intrinsic degeneracy; Spectrum information solves the intrinsic degeneracy.
• The sign-∆m2degeneracy; Difference in the earth matter effect between the interme-
diate and the far detectors solves the sign-∆m2degeneracy.
• The θ23 octant degeneracy; Difference in solar ∆m2oscillation effect (which is pro-
portional to c2
23) between the intermediate and the far detectors solves the θ23octant
To show how the eight-fold parameter degeneracy can be resolved, we present in Fig 1 a
comparison between the sensitivities achieved by the Kamioka only setting and the Kamioka-
Korea setting by taking a particular set of true values of the mixing parameters which are
quoted in caption of Fig 1. The left four panels of Fig 1 show the expected allowed regions of
oscillation parameters in the Tokai-to-Kamioka phase-II (T2K II) setting, while the right four
panels show the allowed regions by the Tokai-to-Kamioka-Korea setting. For both settings
we assume 4 years of neutrino plus 4 years of anti-neutrino running2and the total fiducial
volume is kept to be the same, 0.54 Mton. Some more information of the experimental
setting and the details of the analysis procedure are described in the caption of Fig 1 and
in Sec. V.
2It was shown in the previous study  that the sensitivity obtained with 2 years of neutrino and 6 years
of anti-neutrino running in the T2K II setting  is very similar to that of 4 years of neutrino and 4
years of anti-neutrino running.
Kamioka 0.54Mton detector, ν 4yr + ν
– 4yr 4MW beams
Kamioka 0.27Mton + Korea 0.27Mton detectors, ν 4yr + ν
– 4yr 4MW beams
FIG. 1: The region allowed in δ − sin22θ13 and sin2θ23− sin22θ13 spaces by T2K II (left four
panels) and by the Kamioka-Korea two detector setting (right four panels) in both of which 4
years of neutrino plus 4 years of anti-neutrino running are assumed. The upper (lower) four panels
show the allowed region for the positive (negative) sign of ∆m2
T2K II and Kamioka-Korea settings are assumed to be 0.54 Mton and each 0.27 Mton, respectively,
and the beam power of J-PARC is assumed to be 4 MW. The baseline to the Kamioka and Korea
detectors are, 295 km and 1050 km, respectively. The true solution is assumed to be located at
sin22θ13=0.01, sin2θ23=0.60 and δ=π/4 with positive sign of ∆m2
indicated by the green star. The solar mixing parameters are fixed as ∆m2
sin2θ12=0.31. Three contours in each figure correspond to the 68% (blue line), 90% (black line)
and 99% (red line) C.L. sensitivities, which are defined as the difference of the χ2being 2.30, 4.61
and 9.21, respectively.
31. The detector fiducial volumes of
31(= +2.5 × 10−3eV2), which is
21= 8 × 10−5eV2and
Let us first focus on the left four panels of Fig 1. In the left-most two panels labeled as
(aN) and (aI), one observes some left-over degeneracies of the total eight-fold degeneracy; If
we plot the result of a rate only analysis without spectrum information we would have seen
8 separate (or overlapped) allowed parameter regions. The θ23octant degeneracy remains
unresolved as seen in panels (bN) and (bI). Note that the overlapping two regions in (aN) and
(aI) are nothing but the consequence of unresolved θ23degeneracy. The intrinsic degeneracy,
horizontal pair seen in (aN), is almost resolved apart from 99% CL region at the particular
set of values of the mixing parameters indicated above. The corresponding pair in (aI) is
missing because the intrinsic degeneracy is completely lifted. Since the matter effect plays
minor role in the T2K II setting it is likely that the spectral information is mainly responsible
for lifting the intrinsic degeneracy. See Sec. IIIC for more about it.
Here is a brief comment on the property of the intrinsic and the sign-∆m2degeneracies.
Because the degenerate solutions of CP phase δ satisfy approximately the same relationship
δ2= π − δ1in both the intrinsic and the sign-∆m2degeneracies [11, 12] (see Eqs. (5) and
(6) in Sec. III), the would-be four (one missing) regions in the panels (aN) and (aI) in Fig. 1
forms a cross (or X) shape, with crossing connection between a pair of solutions of the
In the right four panels of Fig. 1 it is exhibited that the intrinsic degeneracy as well as θ23
octant degeneracy are completely resolved by the Kamioka-Korea two-detector setting at
the same values of the mixing parameters, indicating power of the two detector method .
Namely, the comparison between the spectral shapes in Kamioka and in Korea located at
the first and nearly the second oscillation maxima, respectively, supersedes a single detector
measurement in Kamioka with the same total volume despite much less statistics in the
Korean detector. We will give a detailed discussion on how θ23octant degeneracy can be
resolved by the Kamioka-Korea setting in Sec. IV, and present the details of the analysis in
It should be noted that the sign-∆m2degeneracy is also lifted though incompletely at
the particular set of values of the mixing parameters as indicated in the panels (cI) of Fig. 1
where only the 99% CL regions remain. In fact, we have shown in our previous paper
that the Kamioka-Korea identical two detector setting is powerful in resolving the sign-∆m2
degeneracy in a wide range of the mixing parameters . We note that resolution of the
degeneracy in turn leads to an enhanced sensitivity to CP violation than that of T2K II
setting in a region of relatively large θ13. See  for comparison with T2K II sensitivity.
Altogether, we verify that the identical two detector setting in Kamioka-Korea with neutrino
beam from J-PARC solves all the eight-fold parameter degeneracy in situ if θ13 is within
reach by the next generation superbeam experiments such as T2K  and NOνA .
III.DECOUPLING BETWEEN DEGENERACIES
In this section we discuss the property called the “decoupling between degeneracies”
which arises due to the special setting of baselines shorter than ∼1000 km. The content of
this section is somewhat independent of the main line of the discussion in this paper, and
the readers can skip it to go directly to the analysis of θ23octant degeneracy in Secs. IV
and V. Nonetheless, the property makes the structure of analysis for resolving the eight-fold
degeneracy transparent, and therefore it may worth to report.
The problem of decoupling came to our attention via the following path. In most part of
the previous paper , we have discussed how to solve the sign-∆m2degeneracy without
worrying about the θ23octant degeneracy. Conversely, the authors of  analyzed the latter
degeneracy without resolving the former one. Are these correct procedure? The answer is
yes if the analysis procedure and the results for the θ23degeneracy is independent of which
solutions we take for the sign-∆m2degeneracy, and vice versa. We call this property the
“decoupling between the degeneracies”.3Though discussion on this point was partially given
in , we present here a complete discussion of the decoupling.
Under the approximation of lowest nontrivial order in matter effect, we prove that the
decoupling holds between the above two degeneracies, and furthermore that it can be gen-
eralized, though approximately, to the relation between any two pair of degeneracies among
the three types of degeneracies. To our knowledge, leading order in matter perturbation
theory appears to be the only known circumstance that the argument goes through. For-
tunately, the approximation is valid for the setting used in this paper with baseline up to
∼1000 km, in particular in the Kamioka-Korea setting. In the following treatment we make
a further approximation that the degenerate solutions are determined primarily by the mea-
3Here is a concrete example for which the decoupling does not work; In the method of comparison between
νµ and ¯ νµ disappearance measurement for lifting θ23 octant degeneracy  one in fact determines the
combined sign of cos2θ23× ∆m2
31as noticed in , and hence no decoupling.
surement at the intermediate detector. It is a sensible approximation because the statistics
is about 10 times higher at the intermediate detector, and its validity is explicitly verified
in the analysis performed in .
A.Approximate analytic treatment of the parameter degeneracy
To make the discussion self-contained, we start from the derivation of the degenerate
solutions by using the matter perturbation theory , in which the matter effect is kept
to its lowest nontrivial order. Namely, the matter effect can be ignored in leading order
in the disappearance channel whose oscillation probability is order of unity. Then, the
disappearance probability P(νµ→ νµ) can be given by the vacuum oscillation approximation
with leading order in s2
13and the solar ∆m2
1 − P(νµ→ νµ) =
The probability for anti-neutrino channel is the same as that for neutrino one in this approx-
imation. One can show  that the other solar ∆m2
by either a small sin22θ13<∼0.1, or the Jarlskog factor J ≡ c12s12c2
In fact, the validity of the approximation is explicitly verified in  where the matter ef-
fect terms are shown to be of the order of 10−3even at L = 1000 km. A disappearance
measurement, therefore, determines s2
21corrections are suppressed further
23to first order in s2
23)(0)(1 + s2
the two solutions of (s2
form of the θ23 octant degeneracy. For example, (s2
For the appearance channel, we use the νµ(¯ νµ) → νe(¯ νe) oscillation probability with
first-order matter effect 
23)(0)is the solution obtained by ignoring s2
23)(0)is determined as (s2
13. From the first term in Eq. (1),
23)(0)= 0.4 or 0.6 (0.45 or 0.55) for
1 − sin22θ23
, the simplest
P[νµ(¯ νµ) → νe(¯ νe)] = c2
where the terms of order s13
√2GFNe where GF is the Fermi constant, Nedenotes the averaged electron number
are neglected. In Eq. (3), a ≡
density along the neutrino trajectory in the earth, Jr (≡ c12s12c2
reduced Jarlskog factor, and the upper and the lower sign ± refer to the neutrino and
anti-neutrino channels, respectively. We take constant matter density approximation in this
paper. The first term of Eq. (3) is due to the oscillation driven by the solar ∆m2
essentially negligible in the intermediate detector but not at the far detector and is of key
importance to resolve the θ23octant degeneracy.
We make an approximation of ignoring terms of order (∆m2
Note that keeping only the leading order in this quantity is reasonable because Jr<∼0.04,
solutions obey an approximate relationship
13s13c23s23) denotes the
21, which is
31)Jrcos2θ23in Eq. (3).
31| ≃ 1/30, and cos2θ23= ±0.2 for sin22θ23= 0.96. Then, the two degenerate
correction in s2
Analytic treatment of the intrinsic and the sign-∆m2degeneracies is given in . In an
environment where the vacuum oscillation approximation applies the solutions corresponding
to the intrinsic degeneracy are given by 
23 ignoring higher order terms in s13. We can neglect the leading order
23in these relations because it gives O(s4
13,δ(2)= π − δ(1),(5)
where the superscripts (1) and (2) label the solutions due to the intrinsic degeneracy. Under
the same approximation the solutions corresponding to the sign-∆m2degeneracy are given
13,δnorm= π − δinv,(∆m2
where the superscripts “norm” and “inv” label the solutions with the positive and the
negative sign of ∆m2
exchange of these two solutions through which the degeneracy is uncovered . The va-
lidity of these approximate relationships in the actual experimental setup in the T2K II
measurement is explicitly verified in . It should be noticed that even if sizable matter
effect is present the relation (5) holds at the energy corresponding to the vacuum oscillation
maximum, or more precisely, the shrunk ellipse limit .
31. The degeneracy stems from the approximate symmetry under the
B.Decoupling between degeneracies
Resolution of the degeneracy can be done when a measurement distinguishes between
the values of the oscillation probabilities with the two different solutions corresponding to a
degeneracy. Therefore, we define the probability difference
where the superscripts a and b label the degenerate solutions. Suppose that we are discussing
the degeneracy A. The decoupling between the degeneracies A and B holds if ∆Pabdefined
in (7) for the degeneracy A is invariant under the replacement of the mixing parameters
corresponding to the degeneracy B, and vice versa.
The best example of the decoupling is given by the one between the θ23 octant and
the sign-∆m2degeneracies. By noting that J1st
cos2θ23, the difference between probabilities with the first and the second octant solutions
can be given by
in leading order in
∆P1st 2nd(νµ→ νe) = cos2θ1st
The remarkable feature of (8) is that the leading-order matter effect terms drops out com-
pletely. Therefore, our approximated treatment remains valid until the second order matter
effect starts to become sizable in the appearance oscillation probability. More importantly,
∆P1st 2nd, being composed only of the vacuum oscillation terms, is obviously invariant under
the replacement normal ↔ inverted solutions with different signs of ∆m2
Therefore, the resolution of the θ23octant degeneracy can be carried out without worrying
about the presence of the sign-∆m2
Next, we examine the inverse problem; Does the determination of mass hierarchy decouple
with the resolution of the θ23degeneracy? One can show, by using Eq. (3), that the similar
probability difference between the solutions in Eq. (6) with the normal and the inverted
hierarchies is given by
31given in Eq. (6).
∆Pnorm inv(νµ→ νe)
where the superscripts “norm” and “inv” can be exchanged if one want to start from the
inverted hierarchy. We notice that most of the vacuum oscillation terms, including the solar
term, drop out because of the invariance under δ → π − δ and ∆m2
observe that ∆Pnorm invis invariant under the transformation θ1st
because ∆Pnorm invdepends upon θ13 and θ23 only through the combination sin22θ13s2
within our approximation. Therefore, the sign-∆m2
In our previous paper, we have shown that the sign-∆m2
Kamioka-Korea two detector setting. The above argument for decoupling guarantees that
our treatment is valid irrespective of the solutions assumed for θ23degeneracy.4
31. Now, we
31and the θ23degeneracies decouple with
31degeneracy can be lifted by the
4We remark that in most part of , we have assumed that θ23= π/4 so that this problem itself does
not exist. The above discussion implies that even in the case of non-maximal value of θ23 the similar
analysis as in  must go through without knowing in which octant θ23lives. The resultant sensitivity
for resolving the mass hierarchy will change as θ23goes away from π/4 but only slightly as will be shown
in Sec. V.
C.Including the intrinsic degeneracy
Now we turn to the intrinsic degeneracy for which the situation is somewhat different.
First of all, in our setting, the intrinsic degeneracy is already resolved by spectrum infor-
mations at the intermediate detector if θ13is relatively large, sin22θ13>∼0.02 as illustrated
in Fig. 1 before the information from the far detector is utilized. It means that there is no
intrinsic degeneracy from the beginning in the analysis with spectrum informations. Because
of this feature, the intrinsic degeneracy decouple from the beginning from the task of resolv-
ing the eight-fold degeneracy in our setting for the relatively large values of θ13. Because of
a further enhanced sensitivity, the intrinsic degeneracy may be resolved to a smaller values
of θ13in the Kamioka-Korea setting.
In fact, powerfulness of the spectral information for resolving the intrinsic degeneracy
can be understood easily by noting that ∆Pabdefined in (7) is given by using the intrinsic
degeneracy solution (5) as
∆P12(νµ→ νe) = 4Jr
with notable feature that the matter effect cancels out. Notice that even if the matter effect
cannot be negligible the solution (5) holds for measurement at energies around the vacuum
oscillation maximum. The right-hand side of Eq. (10) is proportional to E−2at energies
near the vacuum oscillation maximum,
used to lift the degeneracy. Hence, the spectrum analysis is a powerful tool for resolving the
= π, and the steep energy dependence can be
D. The case that the intrinsic degeneracy is not solved
Even if θ13is too small, or if the energy resolution is too poor for the spectrum analysis
to resolve the intrinsic degeneracy, we can show that the intrinsic degeneracy approximately
decouples from the other degeneracies. ∆P12in Eq. (10) is not exactly but approximately
invariant under the transformation first-octant ↔ second-octant solutions. The difference
between ∆P12(1st) and ∆P12(2nd) is of order cos2θ23Jr∆m2
0.4 and sin22θ13 = 0.1 apart from the further suppression by sin
oscillation maximum. Being the vacuum oscillation term ∆P12is obviously invariant under
the replacement normal ↔ inverted solutions with different signs of ∆m2
resolution of the intrinsic degeneracy can be done, to a good approximation, independent of
the presence of the sign-∆m2
The remaining problem we need to address is the inverse problem, whether the resolution
of the sign-∆m2
lutions of the intrinsic degeneracy. The sign-∆m2
one because ∆Pnorm invin (9) is invariant under the exchange of two intrinsic degeneracy
solutions. The θ23octant degeneracy also approximately decouples from the intrinsic one.
∆P1st 2ndin (8) changes under the interchange of two intrinsic degeneracy solutions only by
the same amount as the difference between ∆P12of the first and the octant θ23solutions,
Here is a clarifying comment on what the decoupling really means; Because of the cross-
shaped structure of the degenerate solutions of the intrinsic and the sign-∆m2
31≃ 3 × 10−4for s2
at around the
31and the θ23octant degeneracies.
31and the θ23octant degeneracies can be carried out without knowing so-
31degeneracy decouples from the intrinsic
31≃ 3 × 10−4(for s2
23= 0.4 and sin22θ13= 0.1).
(as was shown in Sec. II) the decoupling of the former from the latter does not imply that the
correct value of δ can be extracted from the measurement without knowing the correct sign
related by δ ↔ π − δ for a given sign of ∆m2
hierarchy, the true sign of ∆m2
always resolved by the spectrum analysis in region of not too small θ13, as is the case in our
setting, is particularly transparent one from this viewpoint.
To sum up, we have shown that to leading order in the matter effect the intrinsic, the
except for between the intrinsic and the θ23octant degeneracies for which the decoupling
is approximate but sufficiently good to allow one-by-one resolution of all the three types of
degeneracies. The decoupling implies that in analysis for lifting the eight-fold degeneracy
the structure of the χ2minimum is very simple in multi-dimensional parameter space, and
it may be of use in discussions of how to solve the degeneracy in much wider context than
that discussed in this paper.
31.5It means that the elimination of one of the “intrinsic” degenerate pair solutions
31can be done without knowing the mass
31. Therefore, the situation that the intrinsic degeneracy is
31, and the θ23octant degeneracies decouples with each other. They do so exactly
HOW IDENTICAL TWO DETECTOR SETTING SOLVES θ23OCTANT DE-
Now, we turn to the problem of how the identical two detector setting can resolve the θ23
degeneracy, the unique missing link in a program of resolving eight-fold parameter degen-
eracy in the Kamioka-Korea two detector setting. The solar ∆m2oscillation term, the first
term in Eq. (3) with the coefficient of c2
was argued on very general ground  that the θ23degeneracy is hard to resolve only by ac-
celerator experiments with baseline of <∼1000 km or so, the argument can be circumvented
if the solar term can be isolated. We emphasize that the accuracy of the determination of
θ23is severely limited by the octant degeneracy, as discussed in detail in .
Therefore, the question we must address first is the relative importance of the solar
term to the remaining terms in ∆P1st 2ndin Eq. (8). We note that the ratio of the so-
lar term to the δ-dependent solar-atmospheric interference term in ∆P1st 2ndis given by
tio is roughly given by ≃ 3(1/30)(π/2)0.86(1/4Jr) ≃ 0.9 (0.16/s13) with beam energy having
the first oscillation maximum in Kamioka. Therefore, the solar term is indeed comparable
or lager for smaller θ13in size with the interference terms in ∆P1st 2ndat the far detector.
Obviously, the solar term is independent of θ13, which suggests that the sensitivity to resolve
the θ23degeneracy is almost independent of θ13, as will be demonstrated in Sec. V. We note
that while the solar term is the key to resolve the θ23degeneracy, the interference terms also
contributes to lift the degeneracy. In particular, as shown in (8), the sinδ term has opposite
sign when the polarity of the beam is switched from the neutrino to the anti-neutrino runs.
The next question we must address is how the solar term can be separated from the
other terms to have enhanced sensitivity to the θ23degeneracy. To understand the behavior
of the solar term and its difference from that of the atmospheric terms in the oscillation
probability, we plot in Fig. 2 a comparison between them in Kamioka (left panels) and in
23, may be of key importance to do the job. While it
21L/4E)/4Jr, assuming the square parenthesis in (8) is of order unity. The ra-
5Notice, however, that it does not obscure the CP violation, because the ambiguity is only two-fold;
δ ↔ π − δ.
Patm or Psolar [%]
δ = 0
δ = π/2
Neutrino Energy (GeV)
Patm or Psolar [%]
Neutrino Energy (GeV)
Normal Hierarchy, sin22θ13 = 0.05, sin2θ23 = 0.5
of atmospheric plus interference terms in the νeappearance oscillation probabilities with various
values of CP phase δ; δ = 0 (dotted line), δ = π/2 (dashed line), δ = π (dash-dotted line), and
δ = 3π/2 (double-dash-dotted line). For this plot, we used analytic expression in Eq. (3); Psolaris
defined to be the first term in Eq. (3) whereas Patmis defined to be the rest in Eq. (3).¯Psolarand
¯Patmrefer to the corresponding terms for anti-neutrinos.
The energy dependence of the solar term (red solid line) is contrasted with the ones
Korea (right panels) for various values of δ. As one can observe in the right panels, the
energy dependence of the solar oscillation term, a monotonically decreasing (approximately
1/E2) behavior with increasing energy, is quite different from the oscillating behavior of the
atmospheric ones. It is also notable that the ratio of the solar term to the atmospheric-
solar interference term is quite different between the intermediate and the far detectors.
Due to the differing relative importance of the solar term in the two detectors and the
clear difference in the energy dependences between the solar and the atmospheric terms,
the spectrum analysis, the powerful method for resolving the intrinsic degeneracy, must be
able to isolate the solar term from the remaining ones. This will be demonstrated in the
quantitative analysis in the next section.
We note that several alternative methods are proposed to resolve the θ23 degeneracy.
They include: the atmospheric neutrino method [27, 28, 29], and the reactor accelerator
combined method [18, 30], the atmospheric accelerator combined method . The atmo-
spheric neutrino method discussed in [27, 28] is closest to ours in physics principle of utilizing
the solar mass scale oscillation effect. Possible advantage of the present method may be in
a clean detection of the solar term by the intermediate-versus-far two detector comparison.
V.SENSITIVITY FOR RESOLVING θ23OCTANT DEGENERACY
In this section, we describe details of our analysis for resolving the θ23octant degeneracy.
They include treatment of experimental errors, treatment of background, and the statistical
procedure which is used to investigate the sensitivity of the experiment. Then, the results
of our analysis are presented.
A. Assumptions and the definition of χ2
In order to understand the sensitivity of the experiment with the two detector system at
295 km (Kamioka) and 1050 km (Korea), we carry out a detailed χ2analysis. To address
the θ23 octant degeneracy, it is of course necessary to include νµ and ¯ νµ disappearance
channels in addition to the appearance ones in our treatment. In short, the definition of
the statistical procedure is similar to the one used in Ref.  with necessary extension for
including muon events. The assumption on the experimental setting is also identical to that
of the best performance setting identified in Ref. . Namely, 0.27 Mton fiducial masses for
the intermediate site (Kamioka, 295 km) and the far site (Korea, 1050 km). The neutrino
beam is assumed to be 2.5 degree off-axis one produced by the upgraded J-PARC 4 MW
proton beam. It is assumed that the experiment will continue for 8 years with 4 years of
neutrino and 4 years of anti-neutrino runs.
We use various numbers and distributions available from references related to T2K , in
which many of the numbers are updated after the original proposal . Here, we summarize
the main assumptions and the methods used in the χ2analysis. We use the reconstructed
neutrino energy for single-Cherenkov-ring electron and muon events. The resolution in the
reconstructed neutrino energy is 80 MeV for quasi-elastic events. We assume that |∆m2
should be known precisely by the time when the experiment we consider in this report will be
carried out. We take ∆m2
of the beam is the one expected by the 2.5 degree off-axis-beam in T2K. The shape of the
energy spectrum for the anti-neutrino beam is assumed to be identical to that of the neutrino
beam. The event rate for the anti-neutrino beam in the absence of neutrino oscillations is
smaller by a factor of 3.4 due mostly to the lower neutrino interaction cross sections and
partly to the slightly lower flux. The signal to noise ratio is worse for the anti-neutrino beam
than that for the neutrino beam by a factor of about 2.
28 background electron events are expected for the reconstructed neutrino energies be-
tween 350 and 850 MeV for (0.75×0.0225×5)MW·Mton·yr measurement with the neutrino
beam. The energy dependence of the background rate and the rate itself are taken from .
The background rate is expected to be higher in the lower neutrino energies. The expected
number of electron events is assumed to be 122 for sin22θ13=0.1 with the same detector
exposure and beam, assuming the normal mass hierarchy and δ = 0.
We assume that the experiment is equipped with a near detector which measures the
31= ±2.5×10−3eV2. Hence, we assume that the energy spectrum
rate and the energy dependence of the background for electron events, un-oscillated muon
spectrum, and the signal detection efficiency. These measurements are assumed to be carried
out within the uncertainty of 5%. We already demonstrated that the dependence on the
assumed value of the experimental systematic errors is rather weak . We stress that in
the present setting the detectors located in Kamioka and in Korea are not only identical but
also receive neutrino beams with essentially the same energy distribution (due to the same
off-axis angle of 2.5 degree) in the absence of oscillations. However, it was realized recently
that, due to a non-circular shape of the decay pipe of the J-PARC neutrino beam line, the
flux energy spectra viewed at detectors in Kamioka and in Korea are expected to be slightly
different even at the same off-axis angle, especially in the high-energy tail of the spectrum
. The possible difference between fluxes in the intermediate and the far detectors is
newly taken into account as a systematic error in the present analysis.
We compute neutrino oscillation probabilities by numerically integrating neutrino evolu-
tion equation under the constant density approximation. The average density is assumed
to be 2.3 and 2.8 g/cm3for the matter along the beam line between the production target
and Kamioka and between the target and Korea, respectively . We assume that the
number of electron with respect to that of nucleons to be 0.5 to convert the matter density
to the electron number density. In our χ2analysis, we fix the absolute value of |∆m2
be 2.5 × 10−3eV2, and fix solar parameters as ∆m2
Fig. 3 shows an example of the energy spectrum of electron and muon events to be
observed in Kamioka and Korea for 4 years of neutrino beam plus 4 years of anti-neutrino
beam. The two sets of parameters give very similar spectrum for both the electron and muon
events at the Kamioka detector and the muon events at the Korean detector. However, due
to the long baseline distance, the solar term plays some role in the(ν)
probability at the Korean detector. Therefore, the two sets of parameters give slightly
different oscillation probabilities in Korea. Since the solar term is proportional to c2
this feature to obtain information on sin2θ23.
The statistical significance of the measurement considered in this paper was estimated
by using the following definition of χ2:
21= 8 × 10−5eV2and sin2θ12=0.31.
· (1 +
j· ǫj) + Nsignal
· (1 +
j· ǫj) , (12)
· (1 +
j· ǫj) + NQE
· (1 +
j· ǫj) .(13)
The first and second terms in Eq. (11) are for the number of observed single-ring electron
and muon events, respectively. N(e or µ)obs
is the number of events to be observed for the
given oscillation parameter set, and N(e or µ)exp
assumed oscillation parameters in the χ2analysis. k = 1,2,3 and 4 correspond to the four
combinations of the detectors in Kamioka and in Korea with the neutrino and anti-neutrino
beams, respectively. The index i represents the reconstructed neutrino energy bin for both
electrons and muons. For electron events, both N(e)obs
is the expected number of events for the
Number of electron events/bin
Reconstructed Eν (GeV)
Number of muon events/bin
FIG. 3: Examples of electron and muon events to be observed in Kamioka and Korea for 4 years of
neutrino plus 4 years of anti-neutrino running are presented as a function of reconstructed neutrino
energy. The fiducial masses are taken to be 0.27 Mton for both the detectors in Kamioka and Korea.
The dashed histograms for electron events show the background events. The open circles show the
expected energy spectrum of signal events with sin2θ23=0.40 and sin22θ13=0.01. The solid circles
show the expected energy spectrum of signal events with sin2θ23=0.60 and sin22θ13=0.0067. In
both cases, δ = 3π/4 and normal mass hierarchy are assumed in simulating the events.
events. The energy ranges of the five energy bins for electron events are respectively 400-
500 MeV, 500-600 MeV, 600-700 MeV, 700-800 MeV and 800-1200 MeV. The energy range
for the muon events covers from 200 to 1200 MeV. Each energy bin has 50 MeV width. σi
denotes the statistical uncertainties in the expected data. The third term in the χ2definition
collects the contributions from variables which parameterize the systematic uncertainties in
the expected number of signal and background events.
is the number of background events for the ithbin for electrons. Nsignal
number of electron appearance events that are observed, and depends on neutrino oscillation
parameters. The uncertainties in NBG
to 3 and 7).Similarly, Nnon−QE
are the number of non-quasi-elastic events for the ith
are represented by 4 parameters ǫj(j = 1
bin for muons.
quasi-elastic and quasi-elastic muon events separately, since the neutrino energy cannot be
properly reconstructed for non-quasi-elastic events. Both Nnon−QE
neutrino oscillation parameters. The uncertainties in Nnon−QE
4 parameters ǫj(j = 4 to 7).
During the fit, the values of N(e or µ)exp
are recalculated for each choice of the oscillation
parameters which are varied freely to minimize χ2, and so are the systematic error parame-
ters ǫj. The parameter f(e or µ)i
in the ithbin due to a variation of the parameter ǫj. The overall background normalization
for electron events is assumed to be uncertain by ±5% (˜ σ1=0.05). It is also assumed that the
background events for electron events have an energy dependent uncertainty with the func-
tional form of f(e)i
in ǫ2(˜ σ2=0.05). The functional form of f(µ)i
define the uncertainty in the spectrum shape for muon events (˜ σ4=0.05). The uncertainties
in the signal detection efficiency are assumed to be 5% for both electron and muon events
(˜ σ3 = ˜ σ5=0.05). The uncertainty in the separation of quasi-elastic and non-quasi-elastic
interactions in the muon events is assumed to be 20% (˜ σ6=0.20). These systematic errors
are assumed to be not correlated between the electron and muon events. In addition, for
the number of events in Korea, the possible flux difference between Kamioka and Korea is
taken into account in f(e or µ)i
the 1 σ uncertainty in the flux difference (˜ σ7).
are the number of quasi-elastic muon events.We treat the non-
are represented by
jrepresents the fractional change in the predicted event rate
2= ((Eν(rec)−800 MeV)/400 MeV). 5% is assumed to be the uncertainty
4= (Eν(rec) − 800 MeV)/800 MeV is used to
7. The predicted flux difference  is simply assumed to be
B.Sensitivity with two-detector complex
Now we present the results of the sensitivity analysis for the θ23octant degeneracy. The
results for the mass hierarchy as well as CP violation sensitivities will be discussed in the next
section. Fig. 4 shows the sensitivity to the θ23octant determination as a function of sin22θ13
and sin2θ23. The areas shaded with light (dark) gray of this figure indicate the regions of
parameters where the octant of θ23can be determined at 2 (3) standard deviation confidence
level, which is determined by the condition χ2
(9). The upper (lower) panels correspond to the case where the true hierarchy is normal
(inverted). Note, however, that the fit was performed without assuming the mass hierarchy.
Since the sensitivity mildly depends on the CP phase δ, we define the sensitivity to resolving
the octant degeneracy in two ways: the left (right) panels correspond to the case where the
sensitivity is defined such that the octant is determined for any value of delta (half of the δ
space). From this figure, we conclude that the experiment we consider here is able to solve
the octant ambiguity, if sin2θ23< 0.38(0.42) or > 0.62(0.58) at 3 (2) standard deviation
confidence level. This conclusion depends weakly on the value of sin22θ13, as well as the
value of the CP phase δ and the mass hierarchy.
The sensitivity of lifting the octant degeneracy by this setting is quite high even for rather
small values of θ13to sin22θ13∼ 10−3where the mass hierarchy is not determined, a possible
consequence of the decoupling. See Figs. 5 in the next section. The sensitivity depends very
weakly on θ13in relatively small values of sin22θ13where the dominant atmospheric terms
are small. The feature of almost independence of the sensitivity to θ13should be contrasted
with that of the accelerator-reactor combined method in which a strong dependence on θ13
is expected . Very roughly speaking the sensitivity by the present method is better than
min(wrong octant) − χ2
min(true octant) > 4
FIG. 4: 2 (light gray area) and 3 (dark gray area) standard deviation sensitivities to the θ23octant
degeneracy for 0.27 Mton detectors both in Kamioka and Korea. 4 years running with neutrino
beam and another 4 years with anti-neutrino beam are assumed. In (a), the sensitivity is defined
so that the experiment is able to identify the octant of θ23for any values of the CP phase δ. In
(b), it is defined so that the experiment is able to identify the octant of θ23for half of the CP δ
the latter method in a region sin22θ13<∼0.05 − 0.06 according to the result given in Fig. 8
of . The sensitivity of our method is also at least comparable to that could be achieved
by the high statistics observation of atmospheric neutrinos [27, 28, 29].
CHY AND CP VIOLATION
REEXAMINATION OF SENSITIVITIES TO NEUTRINO MASS HIERAR-
In this section we reexamine the problem of sensitivities to the neutrino mass hierarchy
and CP violation achievable by the Kamioka-Korea identical two detector complex. We
want to verify that the sensitivities do not depend on which octant θ23lives, as indicated
by our discussion of the decoupling given in Sec. III. It is also interesting to examine how
the sensitivities depend upon sin22θ23. Furthermore, the inclusion of the new systematic
error which accounts for difference in the spectral shapes of the neutrino beam between the
intermediate and the far detectors makes the reexamination worth to do.
2(thin lines) and 3(thick lines) standard deviation sensitivities to the mass hierarchy
determination for several values of sin22θ23 (red, yellow, black, green and blue lines show the
results for sin2θ23= 0.40, 0.45, 0.50, 0.55 and 0.60, respectively). The sensitivity is defined in the
plane of sin22θ13versus CP phase δ. The top and bottom panels show the cases for positive and
negative mass hierarchies, respectively. The experimental setting is identical to that in Fig.4.
sin2θ23 = 0.40
sin2θ23 = 0.45
sin2θ23 = 0.50
sin2θ23 = 0.55
sin2θ23 = 0.60
In Figs. 5 and 6 the regions sensitive to the mass hierarchy and CP violation, respectively,
are presented. In both figures, the thin-lines and the thick-lines indicate the sensitivity
region at 2 and 3 standard deviations, respectively. As in the previous work , 2 (3)
standard deviation sensitivity regions are defined by the conditions, χ2
mass hierarchy and CP violation, respectively.
The sensitivities to the mass hierarchy and CP violation at sin2θ23 = 0.5 are almost
identical to those obtained in . It is evident that the sensitivities do not depend strongly
on sin2θ23 as far as the value is between 0.40 and 0.60. In fact, the mass hierarchy can
be determined even if the θ23 octant degeneracy is not resolved. But, the sensitivity to
mass hierarchy resolution gradually improves as sin2θ23becomes larger, as seen in Fig. 5.
It is natural because ∆Pnorm invin (9), or the appearance probability itself is proportional
min(true hierarchy) > 4 (9) and χ2
min(δ = 0 or π) − χ2
min(true value ofδ) > 4 (9) for the
Sensitivities to the CP violation, sinδ ?= 0. The meaning of the lines and colors are
identical to that in Fig. 5.
to sin2θ23. An alternative way of presenting the same result is to use s2
ordinate. An approximate scaling behavior is observed as expected by ∆Pnorm invin (9).
VII.SUMMARY AND DISCUSSION
In this paper, we have shown that a setting with two identical water Cherenkov detectors
of 0.27 Mton fiducial mass, one in Kamioka and the other in Korea, which receive almost
the same neutrino beam from J-PARC has capability of resolving the θ23octant degeneracy
in situ by observing difference of the solar oscillation term between both detectors. The
feature of the sensitivity region indicates that the present method is quite complementary
to the reactor-accelerator combined method explored in . Together with the potential for
resolution of the intrinsic and the sign-∆m2
confirmation in Sec. VI by an improved treatment), we have demonstrated that the Kamioka-
Korea two detector complex can resolve all the eight-fold neutrino parameter degeneracy
31degeneracies previously reported in  (with
under the assumption that θ13is within reach by the next generation accelerator experiments
and θ23is not too close to π/4.
As an outcome of these studies, the strategy toward determination of the remaining
unknowns in the lepton flavor mixing can be discussed. It is nice to see that such program can
be defined only with the single experiment based on the conventional superbeam technology
which does not require long-term R&D efforts, and the well established detector technology.
It opens the possibility of accurate determination of the neutrino mixing parameters, θ23,
θ13, δ, as well as the neutrino mass hierarchy, by lifting all the eight-fold degeneracy which
should merit our understanding of physics of lepton sector.
Our treatment in this paper includes a new systematic error which accounts for possible
difference in spectral shape of the neutrino beam received by the two detectors in Kamioka
and in Korea. We have shown that, despite the existence of such new uncertainty which
might hurt the principle of near-far cancellation of the systematic errors, the capability of
determining neutrino mass hierarchy and sensitivity to CP violation are kept intact.
We have also reported a progress in understanding the theoretical aspect of the problem
of how to solve the parameter degeneracy. Because of the property phrased as “decoupling
between degeneracies” which is shown to hold in a setting that allows perturbative treat-
ment of matter effect, one can try to solve a particular degeneracy without worrying about
the presence of other degeneracies. This feature may be contrasted to those of the very
long baseline approaches, such as the neutrino factory, in which one would not expect the
discussion in this paper to hold.
An alternative but closely related approach toward determination of the global structure
of lepton flavor mixing in a single experiment is to utilize an on-axis wide band neutrino
beam to explore the multiple oscillation maxima, which may be called the “BNL strategy”
[34, 35]. This strategy can be applied to the far detector in Korea, as examined by several
authors [36, 37, 38].6In this case, however, one needs to understand the energy dependence
of the background and the signal efficiency as well as the neutrino interaction cross section
precisely for both the intermediate and the far detectors. In particular, since the low energy
bins are enriched with neutral current background contamination that comes from events
with higher neutrino energies  the cancellation of the systematic errors between the
two detectors, which is the key ingredient in our analysis, does not hold. Nonetheless, we
emphasize that the potentially powerful method is worth to examine further with realistic
estimate of the detector performance.
Finally, we remark that the J-PARC 2.5 degree off-axis beam with the baseline length of
1,000 to 1,250 km should be available in the Korean Peninsula. Therefore, it may be possible
to further enhance the sensitivity to the θ23octant by taking a longer baseline length for
the Korean detector. The best baseline length and the detector location should be decided
so that the experiment has the best sensitivities to the oscillation parameters, especially to
the CP phase δ, mass hierarchy and the octant of θ23.
6Very roughly speaking ignoring the issue of backgrounds and assuming the same baseline length, one
would expect that wide band beam option is better in sensitivity to the neutrino mass hierarchy, while
the same off-axis angle option studied in this paper is advantageous to resolve the θ23octant degeneracy
for which low energy bins are essential.
We would like to thank M. Ishitsuka and K. Okumura for the assistance in the analysis
code. H.N. thanks Stephen Parke and Olga Mena for useful discussion. This work was
supported in part by the Grant-in-Aid for Scientific Research, Nos. 15204016 and 16340078,
Japan Society for the Promotion of Science, Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de
Rio de Janeiro (FAPERJ) and by Conselho Nacional de Ciˆ encia e Tecnologia (CNPq).
 Y. Ashie et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 93, 101801 (2004)
[arXiv:hep-ex/0404034]. Y. Ashie et al. [Super-Kamiokande Collaboration], Phys. Rev. D 71,
112005 (2005) [arXiv:hep-ex/0501064].
 E. Aliu et al. [K2K Collaboration], Phys. Rev. Lett. 94, 081802 (2005) [arXiv:hep-ex/0411038].
 D. G. Michael et al. [MINOS Collaboration], arXiv:hep-ex/0607088.
 T. Araki et al. [KamLAND Collaboration],
 Y. Fukuda et al. [Kamiokande Collaboration], Phys. Lett. B 335, 237 (1994); Y. Fukuda et al.
[Super-Kamiokande Collaboration], Phys. Rev. Lett. 81, 1562 (1998) [arXiv:hep-ex/9807003].
 B. T. Cleveland et al., Astrophys. J. 496, 505 (1998); J. N. Abdurashitov et al. [SAGE Collabo-
ration], Phys. Rev. C 60, 055801 (1999) [arXiv:astro-ph/9907113]; W. Hampel et al. [GALLEX
Collaboration], Phys. Lett. B 447, 127 (1999); M. Altmann et al. [GNO Collaboration], Phys.
Lett. B 616, 174 (2005) [arXiv:hep-ex/0504037]; J. Hosaka et al. [Super-Kamiokande Collab-
oration], Phys. Rev. D 73, 112001 (2006) [arXiv:hep-ex/0508053]; Q. R. Ahmad et al. [SNO
Collaboration], Phys. Rev. Lett. 87, 071301 (2001) [arXiv:nucl-ex/0106015]; ibid. 89, 011301
(2002) [arXiv:nucl-ex/0204008]; B. Aharmim et al. [SNO Collaboration], Phys. Rev. C 72,
055502 (2005) [arXiv:nucl-ex/0502021].
 K. Eguchi et al. [KamLAND Collaboration],
 For a review, see e.g., R. N. Mohapatra and A. Y. Smirnov, arXiv:hep-ph/0603118.
 W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
 Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
 J. Burguet-Castell, M. B. Gavela, J. J. Gomez-Cadenas, P. Hernandez and O. Mena, Nucl.
Phys. B 608, 301 (2001) [arXiv:hep-ph/0103258].
 H. Minakata and H. Nunokawa, JHEP 0110, 001 (2001) [arXiv:hep-ph/0108085]; Nucl. Phys.
Proc. Suppl. 110, 404 (2002) [arXiv:hep-ph/0111131].
 G. Fogli and E. Lisi, Phys. Rev. D54, 3667 (1996); [arXiv:hep-ph/9604415].
 H.Minakata,H. NunokawaandS.
 M. Ishitsuka, T. Kajita, H. Minakata and H. Nunokawa, Phys. Rev. D 72, 033003 (2005)
 A. Donini, AIP Conf. Proc. 721, 219 (2004) [arXiv:hep-ph/0310014].
 K. Hiraide, H. Minakata, T. Nakaya, H. Nunokawa, H. Sugiyama, W. J. C. Teves and
R. Z. Funchal, Phys. Rev. D 73, 093008 (2006) [arXiv:hep-ph/0601258].
Phys. Rev. Lett. 94,081801 (2005)
Phys. Rev. Lett. 90,021802 (2003)
 Y. Itow et al., arXiv:hep-ex/0106019.
For an updated version, see: http://neutrino.kek.jp/jhfnu/loi/loi.v2.030528.pdf
 H. Minakata and H. Nunokawa, Phys. Lett. B 413, 369 (1997) [arXiv:hep-ph/9706281].
 D. Ayres et al. [Nova Collaboration], arXiv:hep-ex/0503053.
 S. Choubey and P. Roy, Phys. Rev. Lett. 93, 021803 (2004) [arXiv:hep-ph/0310316].
 J. Arafune, M. Koike and J. Sato, Phys. Rev. D 56 (1997) 3093 [Erratum-ibid. D 60 (1997)
 L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). S. P. Mikheyev and A. Yu. Smirnov, Yad.
Fiz. 42, 1441 (1985) [ Sov. J. Nucl. Phys. 42, 913 (1985)]; Nuovo Cim. C 9, 17 (1986).
 H. Minakata,M. Sonoyama and H. Sugiyama,
 O. L. G. Peres and A. Y. Smirnov, Phys. Lett. B 456, 204 (1999) [arXiv:hep-ph/9902312];
Nucl. Phys. B 680, 479 (2004) [arXiv:hep-ph/0309312]; M. C. Gonzalez-Garcia, M. Maltoni
and A. Y. Smirnov, Phys. Rev. D 70, 093005 (2004) [arXiv:hep-ph/0408170].
 M. Shiozawa, T. Kajita, S. Nakayama, Y. Obayashi, and K. Okumura, in Proceedings of the
RCCN International Workshop on Sub-dominant Oscillation Effects in Atmospheric Neutrino
Experiments, Kashiwa, Japan, Dec. 2004, p.57; T. Kajita, Nucl. Phys. Proc. Suppl. 155, 87
 S. Choubey and P. Roy, Phys. Rev. D 73, 013006 (2006) [arXiv:hep-ph/0509197].
 H. Minakata, H. Sugiyama, O. Yasuda, K. Inoue and F. Suekane, Phys. Rev. D 68, 033017
(2003) [Erratum-ibid. D 70, 059901 (2004)] [arXiv:hep-ph/0211111].
 P.Huber, M.Maltoni andT.Schwetz,
 T. Kobayashi, J. Phys. G29, 1493 (2003); S. Mine, Talk presented at the Neutrino Session of
NP04 workshop, Aug. 2004, KEK, Tsukuba, Japan (http://jnusrv01.kek.jp/jhfnu/NP04nu/).
 A. Rubbia and A. Meregaglia, Talk at the 2nd International Workshop on a Far Detector in
Korea for the J-PARC Neutrino Beam, Seoul National University, Seoul, July 13-14, 2006.
Web page: http://t2kk.snu.ac.kr/
 D. Beavis et al., arXiv:hep-ex/0205040; M. V. Diwan et al., Phys. Rev. D 68, 012002 (2003)
 V. Barger, M. Dierckxsens, M. Diwan, P. Huber, C. Lewis, D. Marfatia and B. Viren,
 K.Hagiwara, N.OkamuraandK.i.
 F. Dufour, Talk at the 2nd International Workshop on a Far Detector in Korea for the J-PARC
Neutrino Beam, Seoul National University, Seoul, July 13-14, 2006.
 A. Rubbia, Talk at the 2nd International Workshop on a Far Detector in Korea for the J-PARC
Neutrino Beam, Seoul National University, Seoul, July 13-14, 2006.
Phys. Rev. D 70,113012 (2004)