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
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
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supported in part by the Grant-in-Aid for Scientific Research, Nos. 15204016 and 16340078,
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