Erratum: Reactor measurement of θ13 and its complementarity to long-baseline experiments [ Phys. Rev. D 68, 033017 (2003)]
ABSTRACT A possibility to measure $\sin^22\theta_{13}$ using reactor neutrinos is examined in detail. It is shown that the sensitivity $\sin^22\theta_{13}>0.02$ can be reached with 20 ton-year data by placing identical CHOOZ-like detectors at near and far distances from a giant nuclear power plant whose total thermal energy is 24.3 ${\text{GW}_{\text{th}}}$. It is emphasized that this measurement is free from the parameter degeneracies which occur in accelerator appearance experiments, and therefore the reactor measurement plays a role complementary to accelerator experiments. It is also shown that the reactor measurement may be able to resolve the degeneracy in $\theta_{23}$ if $\sin^22\theta_{13}$ and $\cos^22\theta_{23}$ are relatively large. Comment: 25 pages, 8 figures, uses revtex4 and graphicx. Several modifications added to make the text easier to understand. Two more figures added. To be published in Phys. Rev. D
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ABSTRACT: Preliminary results of MINOS experiment indicate that theta23 is not maximal. Global fits to world neutrino data suggest two nearly degenerate solutions for theta23: one in the lower octant (LO: theta23 < 45 degree) and the other in the higher octant (HO: theta23 > 45 degree). numu to nue oscillations in superbeam experiments are sensitive to the octant and are capable of resolving this degeneracy. We study the prospects of this resolution by the current T2K and upcoming NOvA experiments. Because of the hierarchy-deltacp degeneracy and the octant-deltacp degeneracy, the impact of hierarchy on octant resolution has to be taken into account. As in the case of hierarchy determination, there exist favorable (unfavorable) values of deltacp for which octant resolution is easy (challenging). However, for octant resolution the unfavorable deltacp values of the neutrino data are favorable for the anti-neutrino data and vice-verse. This is in contrast to the case of hierarchy determination. In this paper, we compute the combined sensitivity of T2K and NOvA to resolve the octant ambiguity. If sin^2\theta23 =0.41, then NOvA can rule out all the values of theta23 in HO at 2 sigma C.L., irrespective of the hierarchy and deltacp. Addition of T2K data improves the octant sensitivity. If T2K were to have equal neutrino and anti-neutrino runs of 2.5 years each, a 2 sigma resolution of the octant becomes possible provided sin^2\theta23 \leq 0.43 or \geq 0.58 for any value of deltacp.01/2013; - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: Precision measurement of the leptonic CP violating phase δ will suffer from the, then surviving, large uncertainty of sin2θ23 of 10–20% in the experimentally interesting region near maximal mixing of θ23. We advocate a new method for determination of both θ23 and δ at the same time using only the νe and ν̅ e appearance channels and show that sin2θ23 can be determined automatically with much higher accuracy, approximately a factor of six, than sinδ. In this method, we identify a new degeneracy for the simultaneous determination of θ23 and δ, the θ23 intrinsic degeneracy, which must be resolved in order to achieve precision measurement of these two parameters. Spectral information around the vacuum oscillation maxima is shown to be the best way to resolve this degeneracy.Physical review D: Particles and fields 06/2013; 87(11). - SourceAvailable from: Udit Raha[Show abstract] [Hide abstract]
ABSTRACT: For the low-energy anti-neutrino reaction, $\bar{\nu}_e + p \to e^+ + n$, which is of great current interest in connection with on-going high-precision neutrino-oscillation experiments, we calculate the differential cross section in a model-independent effective field theory (EFT), taking into account radiative corrections of order $\alpha$. In EFT, the short-distance radiative corrections are subsumed into well-defined low-energy constants the values of which can in principle be determined from the available neutron beta-decay data. In our low-energy EFT, the order-$\alpha$ radiative corrections are considered to be of the same order as the nucleon recoil corrections, which include the "weak magnetism" contribution. These recoil corrections have been evaluated as well. We emphasize that EFT allows for a systematic evaluation of higher order corrections, providing estimates of theoretical uncertainties in our results.Physical Review C 12/2011; 85(4). · 3.72 Impact Factor
Page 1
Reactor measurement of ?13and its complementarity to long-baseline experiments
H. Minakata,* H. Sugiyama,†and O. Yasuda‡
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
K. Inoue§and F. Suekane?
Research Center for Neutrino Science, Tohoku University, Sendai, Miyagi, 980-8578, Japan
?Received 8 November 2002; revised manuscript received 18 April 2003; published 28 August 2003?
The possibility of measuring sin22?13using reactor neutrinos is examined in detail. It is shown that the
sensitivity sin22?13?0.02 can be reached with 40 tonyr data by placing identical CHOOZ-like detectors at
near and far distances from a giant nuclear power plant whose total thermal energy is 24.3 GWth. It is
emphasized that this measurement is free from the parameter degeneracies that occur in accelerator appearance
experiments, and therefore the reactor measurement is complementary to accelerator experiments. It is also
shown that the reactor measurement may be able to resolve the degeneracy in ?23if sin22?13and cos22?23are
relatively large.
DOI: 10.1103/PhysRevD.68.033017PACS number?s?: 14.60.Pq, 25.30.Pt, 28.41.?i
I. INTRODUCTION
Despite the accumulating knowledge of neutrino masses
and lepton flavor mixing from atmospheric ?1?, solar ?2,3?,
and accelerator ?4? neutrino experiments, the ?1-3? sector of
the Maki-Nakagawa-Sakata ?MNS? matrix ?5? is still unclear.
At the moment, we know only that ?Ue3??sin?13?s13is
small, s13
reactor experiment ?6?. In this paper we assume that the light
neutrino sector consists of three active neutrinos only. One of
the challenging goals in the attempt to explore the full struc-
ture of lepton flavor mixing would be measuring the leptonic
CP or T violating phase ? in the MNS matrix. If the Kam-
LAND experiment ?7? confirms the large-mixing-angle
?LMA? Mikheyev-Smirnov-Wolfenstein ?MSW? ?8,9? solu-
tion of the solar neutrino problem, the one most favored by
recent analyses of solar neutrino data ?3,10?, we will have an
open route toward this goal. Yet there might still exist a last
impasse, namely, the possibility of a too small value of ?13.
Thus, it has recently been emphasized more and more
strongly that the crucial next step toward the goal is the
determination of ?13.
In this paper, we raise the possibility that a ?¯edisappear-
ance experiment using reactor neutrinos could be potentially
the fastest ?and the cheapest? way to detect the effects of a
nonzero ?13. In fact, such an experiment using the Krasno-
yarsk reactor complex was described earlier ?11?, in which
the sensitivity to sin22?13can be as low as ?0.01, an order
of magnitude lower than in the CHOOZ experiment. We also
briefly outline basic features of our proposal and reexamine
the sensitivity to sin22?13in this paper.
It appears that the most popular way of measuring ?13is
the next generation long-baseline ?LBL? neutrino oscillation
2?0.03, from the bound imposed by the CHOOZ
experiments MINOS ?12?, OPERA ?13?, and JHF phase I
?14?. It may be followed either by conventional superbeam
?15? experiments ?the JHF phase II ?14? and possibly others
?16,17?? or by experiments at neutrino factories ?18,19?. It is
pointed out, however, that the measurement of ?13in LBL
experiments with only a neutrino channel ?as planned in JHF
phase I? would suffer from large intrinsic uncertainties, on
top of the experimental errors, due to the dependence on an
unknown CP phase and the sign of ?m31
it is noticed that an ambiguity remains in the determination
of ?13and other parameters even if precise measurements of
the appearance probabilities in neutrino as well as an-
tineutrino channels are carried out, that is, the problem of the
parameter degeneracy ?20–26?. ?For a global overview of
parameter degeneracy, see ?26?.? While some ideas toward a
solution have been proposed, the problem is hard to solve
experimentally, and it is not likely to be resolved in the near
future.
We emphasize in this paper that reactor ?¯edisappearance
experiments provide a particularly clean environment for the
measurement of ?13; namely, it can be regarded as a dedi-
cated experiment for determination of ?13; it is insensitive to
the ambiguity due to all the remaining oscillation parameters
as well as to the matter effect. This is in sharp contrast with
the features of LBL experiments described above. Thus, the
reactor measurement of ?13will provide us with valuable
information complementary to that from LBL experiments
and will play an important role in resolving the problem of
parameter degeneracy. We show here that reducing the sys-
tematic errors is crucial for the reactor measurement of ?13to
be competitive in accuracy with LBL experiments. We
present a preliminary analysis of its possible role in this con-
text. It is then natural to think about the possibility that one
has better control by combining the two complementary way
of measuring ?13, the reactor and the accelerator methods. In
fact, we show in this paper that nontrivial relations exist
between the ?13measurements by the two methods thanks to
their complementary nature, so that in the luckiest case one
may be able to derive constraints on the value of the CP
violating phase ? or determine the neutrino mass hierarchy.
2?20?. Furthermore,
*Electronic address: minakata@phys.metro-u.ac.jp
†Electronic address: hiroaki@phys.metro-u.ac.jp
‡Electronic address: yasuda@phys.metro-u.ac.jp
§Electronic address: inoue@awa.tohoku.ac.jp
?Electronic address: suekane@awa.tohoku.ac.jp
PHYSICAL REVIEW D 68, 033017 ?2003?
0556-2821/2003/68?3?/033017?12?/$20.00©2003 The American Physical Society
68 033017-1
Page 2
II. REACTOR EXPERIMENT AS A CLEAN LABORATORY FOR ?13MEASUREMENT
Let us examine in this section how clean the measurement of ?13by a reactor experiments is. To define our notation, we
note that the standard notation ?27?
U??
c12c13
s12c13
s13e?i?
s23c13
?s12c23?c12s23s13ei?
s12s23?c12c23s13ei?
c12c23?s12s23s13ei?
?c12s23?s12c23s13ei?
c23c13?
?1?
is used for the MNS matrix throughout this paper, where cij
and sij(i,j?1–3) imply cos?ijand sin?ij, respectively.
The mass squared difference of the neutrinos is defined as
?mij
We examine possible ‘‘contamination’’ by ?, the matter
effect, the sign of ?m31
one. We first note that, due to the low neutrino energy of a
few MeV, reactor experiments are inherently disappearance
experiments, which can measure only the survival probabil-
ity P(?¯e→?¯e). It is well known that the survival probability
does not depend on the CP phase ? in arbitrary matter den-
sities ?28?.
In any reactor experiment on the Earth, short or long
baseline, the matter effect is very small because the energy is
quite low and can be ignored to a good approximation. This
can be seen by comparing the matter and the vacuum effects
?as the matter correction comes in only through this combi-
nation in the approximate formula in ?18??:
??31??2.8?10?4?
??
2?mi
2?mj
2, where miis the mass of the ith eigenstate.
2, and the solar parameters one by
aL
??m31
2?
2.5?10?3eV2?
2.3 gcm?3??
?1?
E
4 MeV?
?
Ye
0.5?,
?2?
where
?ij?
?mij
2E
2L
?3?
with E being the neutrino energy and L the baseline length.
The best fit value of ??m31
?10?3eV2from the Super-Kamiokande atmospheric neu-
trino data ?29?, and we use this as the reference value for
??m31
of refraction in matter with GFbeing the Fermi constant and
Nethe electron number density on Earth, which is related to
the Earth matter density ? as Ne?Ye?/mpwhere Yeis the
proton fraction. Once we know that the matter effect is neg-
ligible we immediately recognize that the survival probabil-
ity is independent of the sign of ?m31
Therefore, the vacuum probability formula applies. The
general probability formula in vacuum is analytically written
as ?27?
2? is given by ??m31
2??2.5
2? throughout this paper. a??2GFNedenotes the index
2.
P(??→??)
P(?¯?→?¯?)??????4?
j?kRe?U?jU?j* U?k* U?k?
4E??2?
2E?,
?sin2?
?sin?
?mjk
2L
j?kIm?U?jU?j* U?k* U?k?
?mjk
2L
?4?
where ?,??e,?,?, and the minus and plus signs in front of
the Im(U?jU?j* U?k* U?k) term correspond to neutrino and an-
tineutrino channels, respectively. From Eq. ?4? the exact ex-
pression for P(?¯e→?¯e) is given by
j?k?Uej?2?Uek?2sin2?
1?P??¯e→?¯e??4?
?mjk
2L
4E?
?sin22?13sin2?31
2
?1
2c12
2sin22?13sin?31sin?21
??c13
4sin22?12?c12
2sin22?13cos?31?
?sin2?21
2
,
?5?
where the parametrization ?1? has been used in the second
equality. The last three terms in the second equality of Eq.
?5? are suppressed relative to the main depletion term, the
first term of the right-hand side of Eq. ?5?, by ?, ?2/sin22?13,
and ?2, respectively, where ???m21
??m31
LMA MSW solar neutrino solution ?3,10?. Then, the second
and fourth terms in the second equality can be ignored, al-
though the third term can be of order unity compared with
the main depletion term provided that ??0.1. ?Notice that
we are considering the measurement of sin22?13in the range
of 0.1–0.01.? Therefore, assuming that ??m31
by LBL experiments with good accuracy, the reactor ?¯edis-
appearance experiment gives us a clean measurement of ?13
which is independent of any solar parameters except for the
case of high ?m21
If the high ?m21
LMA solution with ?m21
turns out to be the right one, we need to take special care of
2/??m31
2?. Assuming that
2??(1.6–3.9)?10?3eV2?29?, ??0.1–0.01 for the
2? is determined
2LMA solutions.
22?10?4eV2
MINAKATA et al.
PHYSICAL REVIEW D 68, 033017 ?2003?
033017-2
Page 3
the second term of the second equality of Eq. ?5?. In this
case, the determination of ?13and the solar angle ?12are
inherently coupled,1and we would need a joint analysis from
a near-far detector complex ?see the next section? and Kam-
LAND.
III. NEAR-FAR DETECTOR COMPLEX: BASIC
CONCEPTS AND ESTIMATION OF SENSITIVITY
In order to obtain good sensitivity to sin22?13, the selec-
tion of an optimized baseline and having small statistical and
systematic errors are crucial. For instance, the baseline
length that gives the oscillation maximum for reactor ?¯e’s
which have typical energy 4 MeV is 1.7 km for ?m2?2.5
?10?3eV2. Along with this baseline selection, if systematic
and statistical errors can be reduced to the 1% level, which is
2.8 times better than the CHOOZ experiment ?6?, an order of
magnitude improvement for the sin22?13sensitivity is pos-
sible at ?m2?2.5?10?3eV2. In this section we demon-
strate that this kind of experiment is potentially possible if
we place a CHOOZ-like detector with a baseline of 1.7 km
200 m underground near a reactor of 24.3 GWththermal
power. The reactor can be regarded as a simplified version of
the Kashiwazaki-Kariwa nuclear power plant, which consists
of seven reactors whose maximum energy generation is
24.3 GWth.
The major part of the systematic error is caused by uncer-
tainties in the neutrino flux calculation, the number of pro-
tons, and the detection efficiency. For instance, in the
CHOOZ experiment, the uncertainty of the neutrino flux is
2.1%, that of the number of protons is 0.8%, and that of the
detection efficiency 1.5%, as is shown in Table I. The uncer-
tainty of the neutrino flux includes ambiguities of the reactor
thermal power generation, the reactor fuel component, the
neutrino spectra from fission, and so on. The uncertainty of
the detection efficiency includes a systematic shift in defin-
ing the fiducial volume. These systematic uncertainties, how-
ever, cancel out if identical detectors are placed near and far
from the reactors and data taken at the different detectors are
compared.2
To estimate how good the cancellation will be, we study
the case of the Bugey experiment, which uses three identical
detectors to detect reactor neutrinos at 14, 40, and 90 m. For
the Bugey case, the uncertainty of the neutrino flux improved
from 3.5% to 1.7% and the error on the solid angle remained
the same (0.5%→0.5%). If each ratio of the improvement
for the Bugey case is directly applicable to our case, the
systematic uncertainty will improve from 2.7% to 0.8% as
shown in Table I. The ambiguity in the solid angle will be
negligibly small because the absolute baseline is much
longer than in the Bugey case. We are thinking of the case
where the front detector is located 300 m away from the
1The effect of nonzero ?13for measurement of ?12at KamLAND
is discussed in ?30?.
2This is more or less the strategy taken in the Bugey experiment
?31?. The Krasnoyarsk group also plans in their Kr2Det proposal
?11? to construct two identical 50 ton liquid scintillators at 1100 m
and 150 m from the Krasnoyarsk reactor. They indicate that the
systematic error can be reduced to 0.5% by comparing the near and
far detectors.
TABLE I. Systematic errors in the Bugey and CHOOZ-like ex-
periments. Relative errors in the CHOOZ-like experiment are those
expected with the same reduction rates of errors as those of Bugey.
Bugey
Absolute
normalization
Relative
normalization
Relative/
absolute
Flux
Number of protons
Solid angle
Detection efficiency
2.8%
1.9%
0.5%
3.5%
0.0%
0.6%
0.5%
1.7%
0
0.32
1
0.49
Total4.9%2.0%
CHOOZ-like
Absolute
normalization
Relative
normalization
?expected?
Relative/
absolute
Flux
Number of protons
Detection efficiency
2.1%
0.8%
1.5%
0.0%
0.3%
0.7%
0
0.38
0.47
Total
For bins
2.7%
8.1%
0.8%
2.4%
0.0001
0.001
0.01
0.1
0.01 0.11
|∆m13
2 /eV2|
sin2 2θ13
90%CL
CHOOZ
2d.o.f.
σsys = 2%
10t•yr, 2d.o.f.
σsys = 2%
10t•yr, 1d.o.f.
σsys = 0.8%, 40t•yr, 2d.o.f.
σsys = 0.8%, 40t•yr, 1d.o.f.
FIG. 1. Shown are the 90% C.L. exclusion limits on sin22?13
that can be placed by the reactor measurement as described in Sec.
III. From left to right, the dash-dotted and dotted ?the long-dashed
and short-dashed? lines are based on analyses with one and two
degrees of freedom ?see the text?, respectively, for ?sys?0.8%, 40
tonyr (?sys?2%, 10 tonyr?. The solid line is the CHOOZ result,
and the 90% C.L. interval 1.6?10?3eV2??m31
of the Super-Kamiokande atmospheric neutrino data is shown as a
shaded strip.
2?3.9?10?3eV2
REACTOR MEASUREMENT OF ?13AND ITS . . .PHYSICAL REVIEW D 68, 033017 ?2003?
033017-3
Page 4
reactor we consider. In the actual setting with the
Kashiwazaki-Kariwa power plant, two near detectors may be
necessary due to the extended array of seven reactors. Here-
after, we take 2% and 0.8% as the reference values for the
relative systematic error ?sysfor the total number of ?¯e
events in our analysis. Let us examine the physics potential
of such a reactor experiment, assuming these reference val-
ues for the systematic error. We take, for concreteness, the
Kashiwazaki-Kariwa reactor of 24.3 GWththermal power
and assume its operation with 80% efficiency. Two identical
liquid scintillation detectors are located at 300 m and 1.7 km
away from the reactor and assumed to detect ?¯eby delayed
coincidence with 70% detection efficiency. ?¯e’s of 1–8 MeV
visible energy, Evisi?E?¯e?0.8 MeV, are used and the num-
ber of events is counted in 14 bins of 0.5 MeV. Without
oscillation, a 10 ?40? tonyr measurement at the far detector
yields 20000 ?80000? ?¯eevents, which is naively compa-
rable to a 0.7% ?0.35%? statistical error.
First, let us calculate how much we could constrain
sin22?13. Unlike the analysis in ?31?, which uses the ratio of
the numbers of events at the near and far detectors, we use
the differenceof thenumbers
?(L1/L2)2Ni(L1), because statistical analysis with ratios is
complicated ?see, e.g., ?32??. The definition of ??2, which
stands for the deviation from the best fit point ?nonoscillation
point?, is given by
of events
Ni(L2)
??2?sin22?13,??m31
2????
i?1
14
??Ni(0)?L2???L1/L2?2Ni(0)?L1????Ni?L2???L1/L2?2Ni?L1???2
Ni(0)?L2???L1/L2?4Ni(0)?L1????sys
bin?2Ni(0)
2?L2?
,
?6?
Ni?Lj??Ni?sin22?13,??m31
2?;Lj?,
Ni(0)?Lj??Ni?0,0;Lj?,
where ?sys
denotes the theoretical number of ?¯eevents within the ith energy bin. In principle both the systematic errors ?abs sys
normalization? and ?sys
have (1??abs sys
)?(1??sys
L2)2Ni(L1)?, which indicates that the systematic error is dominated by the relative error ?sys
?(L1/L2)2Ni(L1)? is supposed to be small. In fact we have explicitly verified numerically that the presence of
(?abs sys
)2?Ni(L2)?(L1/L2)2Ni(L1)?2in the denominator of Eq. ?6? does not affect any of our results. From the assumption
that the relative systematic error for each bin is distributed equally into the bins, ?sys
error ?sysfor the total number of events by
binis the relative systematic error for each bin, which is assumed to be the same for all bins, and Ni(sin22?13,??m31
2?)
bin
?absolute
bin?relative normalization? appear in the denominator of Eq. ?6?, but by taking the difference, we
bin)Ni(L2)?(L1/L2)2Ni(L1)???Ni(L2)?(L1/L2)2Ni(L1)???sys
bin binNi(L2)??abs sys
bin, as the second term ?Ni(L2)
bin
?Ni(L2)?(L1/
bin
binis estimated from the relative systematic
??sys
bin?2??sys
2
?N(0)
?
i
tot?L2??2
Ni(0)
2?L2?
,
N(0)
tot?L2???
i
Ni(0)?L2?,
?7?
since the uncertainty squared of the total number of events is obtained by adding up the bin-by-bin systematic errors
(?sys
??2?2.7 for one degree of freedom ?DOF?, are presented for two cases: a 10 tonyr measurement with 2% systematic error
of the total number of events and a 40 tonyr measurement with 0.8% error. The figure shows that it is possible to measure
sin22?13down to 0.02 at the maximum sensitivity with respect to ??m31
measurement, provided the quoted values of the systematic errors are realized. The CHOOZ result ?6? is also depicted in Fig.
1. For a fair comparison with the CHOOZ contour, we also present in Fig. 1 the results of an analysis with two degrees of
freedom, which correspond to ??2?4.6 for 90% C.L., without assuming any precise knowledge of ??m31
Next, let us examine how precisely we could measure sin22?13. The definition of ??2is
bin)2Ni(0)
2(L2); the ratio ?sys
bin/?sysis about 3 in our analysis. In Fig. 1, the 90% C.L. exclusion limits, which correspond to
2?, and to 0.04 for larger ??m31
2?, by a 40 tonyr
2?.
??2?sin22?13,??m31
2????
i?1
14
??Ni(best)?L2???L1/L2?2Ni(best)?L1????Ni?L2???L1/L2?2Ni?L1???2
Ni(best)?L2???L1/L2?4Ni(best)?L1????sys
bin?2Ni(best)
2
?L2?
,
?8?
where Ni(best)denotes Nifor the set of best fit parameters
(sin22?13
Eq. ?7? by replacing Ni(0)with Ni(best)and the ratio ?sys
is about 3 again. We assume that the value of ??m31
(best), ??m31
2(best)?) given artificially. ?sys
binis obtained in
bin/?sys
2? is
known to a precision of 10?4eV2from JHF phase I by the
time the reactor measurement is actually utilized to solve the
degeneracy. Then we rely on the analysis with one degree of
freedom, fixing ??m31
2? as ??m31
2(best)??2.5?10?3eV2. The
MINAKATA et al.
PHYSICAL REVIEW D 68, 033017 ?2003?
033017-4
Page 5
90% C.L. allowed regions for one degree of freedom, whose
bounds correspond to ??2?2.7, are presented in Fig. 2 for
the values of sin22?13
units of 0.01 in the case of a 10 tonyr ?40 tonyr? measure-
ment with systematic error ?sys?2.0% ?0.8%?. We can read
off the error at 90% C.L. in sin22?13and it is almost inde-
pendent of the central value sin22?13
(best)from 0.05 to 0.08 ?0.02 to 0.08? in
(best). Thus, we have
sin22?13?sin22?13
(best)?0.043
?at 90% C.L., DOF?1? for sin22?13
(best)?0.05
in the case of ?sys?2% with a 10 tonyr measurement, and
sin22?13?sin22?13
(best)?0.018
?at 90% C.L., DOF?1? for sin22?13
(best)?0.02
in the case of ?sys?0.8% with a 40 tonyr measurement.
IV. THE PROBLEM OF THE „?13,?,?23,?m31
PARAMETER DEGENERACY
2…
We explore in this and the following sections the possible
significance of reactor measurements of ?13in the context of
the problem of parameter degeneracy. We show that a reactor
measurement of ?13can resolve the degeneracy at least
partly if the measurement is sufficiently accurate. Toward
this goal, we first explain the problem of parameter degen-
eracy in long-baseline neutrino oscillation experiments. It is
a notorious problem; a set of measurements of the ??disap-
pearance probability and the appearance oscillation prob-
abilities of ??→?eand ?¯?→?¯e, no matter how accurate
they may be, does not allow unique determination of ?13, ?,
and ?23. The problem was first recognized in the form of
intrinsic degeneracy between the two sets of solutions of
(?23,?13) for a given set of measurements in two different
channels ??→?eand ??→???21?. It was then observed
independently that a similar degeneracy of solutions of
(?13,?) exists in measurements of ?eappearance in the neu-
trino and antineutrino channels ?22?. The authors of ?22?
made the first systematic analysis of the degeneracy problem.
It was noticed that the degeneracy is further duplicated pro-
vided that two neutrino mass patterns, the normal (?m31
?0) and the inverted (?m31
?23?. Finally, it was pointed out that the degeneracy can be
maximally eightfold ?24?. The analytical structure of the de-
generate solutions was worked out in a general setting in
?26?.
To illuminate the point, let us first restrict our treatment to
a relatively short-baseline experiment such as the CERN
Frejus project ?16?. In this case, one can use the vacuum
oscillation approximation for the disappearance and appear-
ance probabilities. From the general formula ?4? we have
2
2?0) hierarchies, are allowed
1?P???→????4?
j?k?U?j?2?U?k?2sin2?
?mjk
2L
4E?
?sin22?23sin2?31
2??
1
2c12
2sin22?23?s13s23
2sin2?23sin2?12cos??sin?21sin?31?O??2??O?s13
2?, ?9?
P???→?e?
P??¯?→?¯e????4?
j?kRe?U?jUej*U?k* Uek?sin2?
?mjk
2L
4E??2?
j?kIm?U?jUej*U?k* Uek?sin?
?mjk
2L
2E?,
?s23
2sin22?13sin2?31
2?1
2Jrsin?21sin?31cos??Jrsin?21sin2?31
2sin??O??s13
2?,
?10?
where ???m21
parametrization ?1? has been used in the second equality in
each formula. The minus and plus signs in front of the sin?
term in Eq. ?10? correspond to the neutrino and antineutrino
channels, respectively. The explicit perturbative computation
2/??m31
2?, Jr?sin2?23sin2?12c13
2s13, and the
in ?33? indicates that the matter effect enters into the expres-
sion in a particular combination with other quantities ?in the
form of s13
appearance measurement at JHF, for example, sin22?23and
??m31
2aL/?31), so that the effect is small. By the dis-
2? will be determined with accuracies of 1% for 0.92
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?sin22?23?1.0 ?Fig. 11 in ?14?? and 4%, respectively ?14?.3
If ?23is not maximal, then we have two solutions for ?23
(?23and ?/2??23), even if we ignore the uncertainty in the
determination of sin22?23. For example, if sin22?23?0.95,
which is perfectly allowed by the most recent atmospheric
neutrino data ?29?, then s23
the dominant term in the appearance probability depends
upon s23
ence in the number of appearance events in this case. On the
other hand, in the case of maximal mixing, it still leaves a
2can be either 0.39 or 0.61. Since
2instead of sin22?23, this leads to a ?20% differ-
rather wide range of ?23, despite the fantastic accuracy of
the measurement. 1% accuracy in sin22?23implies about
10% uncertainty in s23
sin22?13from the appearance measurement, we have to face
the ambiguity due to the twofold nature of the solution
for s23
Let us discuss the simplest possible case, the low ?m2or
the vacuum oscillation solution of the solar neutrino prob-
lem. ?See, e.g., ?34? for a recent discussion.? In this case, one
can safely ignore terms of order ? in Eqs. ?9? and ?10?. Then
we are left with only the first terms in the second equality of
these equations, the one-mass-scale dominant vacuum oscil-
lation probabilities. Now let us define the symbols x
?sin22?13and y?s23
forms y?y1or y2?corresponding to two solutions of s23
and xy?const, respectively, for given values of the prob-
abilities. It is then obvious that there are two crossing points
of these curves. This is the simplest version of the (?13,?23)
degeneracy problem. We next discuss what happens if ? is
not negligible although small: the case of the LMA solar
neutrino solution. In this case, the appearance curve xy
?const is split into two curves ?although they are in fact
connected at their maximum value of s23
degenerate solutions of the set (?,?13) that are allowed for a
given set of values of s23
Then, we have, in general, four crossing points on the x-y
plane for a given value of sin22?23, the fourfold degeneracy.
Simultaneously, the two y?const lines are slightly tilted and
the split between the two curves becomes larger at larger
sin22?13, although the effect is too tiny to be clearly seen. If
the baseline distance is longer, the Earth matter effect comes
in and further splits each appearance contour into two, de-
pending upon the sign of ?m31
?or two continuous contours, each of which intersects twice
with the y?const line? and hence there are eight solutions as
displayed in Fig. 3.4This is a simple pictorial representation
of the maximal eightfold parameter degeneracy ?24?. To
draw Fig. 3, we have calculated disappearance and appear-
ance contours by using the approximate formula derived by
Cervera et al. ?18?. We take the baseline distance and neu-
trino energy as L?295 km and E?400 MeV with possible
relevance to the JHF project ?14?. The Earth matter density is
taken to be ??2.3 gcm?3based on the estimate given in
?35?. The electron fraction Yeis taken to be 0.5. We assume,
for definiteness, that a long-baseline disappearance measure-
menthas resultedin sin22?23?0.92
?10?3eV2. For the LMA solar neutrino parameters we take
tan2?12?0.38 and ?m21
values of these parameters and the matter density throughout
this paper unless otherwise stated. The qualitative features of
the figure remain unchanged even if we employ values of the
parameters obtained by other analyses.
2. Thus, whenever we try to determine
2.
2. Then, Eqs. ?9? and ?10? take the
2)
2) because of the two
2, P(??→?e), and P(?¯?→?¯e).
2. Then we have four curves
and
?m31
2?2.5
2?6.9?10?5eV2?36?. We take the
3Usually one thinks of determining not ??m31
disappearance measurement. But it does not appear possible to re-
solve the difference between these two quantities because one has
to achieve a resolution of order ? for the reconstructed neutrino
energy.
2? but ??m32
2? by a
4The readers might be curious about the feature that the two con-
tours are connected with each other at a large s23
is a phase variable, the contours must be closed as ? varies.
2point. Because ?
FIG. 2. Shown is the accuracy of determination of sin22?13at
90% C.L. for the case of positive evidence based on analysis with
one degree of freedom, ??2?2.7. Figures ?a? and ?b? are for ?sys
?2%, 10 tonyr, and ?sys?0.8%, 40 tonyr, respectively. The lines
correspond to the best fit values of sin22?13, from left to right, 0.05
to 0.08 in units of 0.01 in ?a?, and 0.02 to 0.08 in units of 0.01 in
?b?. The reference value of ??m31
?10?3eV2, which is indicated by a gray line.
2(best)? is taken to be 2.5
MINAKATA et al.
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V. RESOLVING THE PARAMETER DEGENERACY
BY REACTOR MEASUREMENT OF ?13
Now we discuss how reactor experiments can contribute
to resolving the parameter degeneracy. To make our discus-
sion as concrete as possible we use a particular long-baseline
experiment, the JHF experiment ?14?, to illuminate the
complementary role played by reactor and long-baseline ex-
periments. It is likely that the experiment will be carried out
at around the first oscillation maximum (??31???) for a
number of reasons: the dip in energy spectrum in the disap-
pearance channel is the deepest, the number of appearance
events is nearly maximal ?14?, and the twofold degeneracy in
? becomes simple (?↔???) for each mass hierarchy
?20,24?.5With the distance L?295 km, the oscillation maxi-
mum is at around E?600 MeV. We take the same mixing
parameters as those used in Fig. 3.
A. Illustration of how reactor measurement helps resolve
the „?13,?23… degeneracy
Let us first give an illustrative example showing how re-
actor experiments could help resolve the (?13,?23) degen-
eracy. To present a clear step-by-step explanation of the re-
lationship between LBL and reactor experiments, we first
plot in Fig. 4 the allowed regions in the sin22?13-s23
separate measurements of P(??→?e) alone and P(? ¯?
→? ¯e) alone. The former are indicated by the regions
bounded by black lines and the latter by gray lines. The solid
and dashed lines are used for cases with positive and nega-
tive ?m31
probabilities are chosen arbitrarily for illustrative purposes
and are given in the caption of Fig. 4. Notice that the nega-
tive ?m31
curve in the neutrino ?antineutrino? channel. The plot with
measurements in only the neutrino mode has more than aca-
demic interest because the JHF experiment is expected to run
only with the neutrino mode in its first phase. We observe
that there is large intrinsic uncertainty in the ?13determina-
tion due to the unknown ?, the problem addressed in ?20?.
The two regions corresponding to positive and negative
?m31
two measurements of the ? and ? ¯ channels are combined, the
allowed solution becomes a line which lies inside the overlap
of the ? and ? ¯ regions for each sign of ?m31
In Fig. 5 we have plotted such solutions as two lines, one for
2plane by
2. The values of disappearance and appearance
2curve is located right ?left? of the positive ?m31
2
2heavily overlap due to the small matter effect. When
2
in Fig. 4.6
5In order to have this reduction, one has to actually tune the en-
ergy spectrum so that the cos? term in Eq. ?10? averaged over the
energy with the neutrino flux times the cross section vanishes,
which is shown to be possible in ?20?.
6In the absence of the matter effect, the reason why the closed
curve shrinks to a line at the oscillation maximum can be seen as
follows. By eliminating ? in Eq. ?10?, it is easy to show that there
are two solutions of sin2?13?0 for given values of P, P¯, and ?23
off the oscillation maximum (?31??), whereas there is only one
solution of sin2?13?0 at the oscillation maximum (?31??). Even
if we switch on the matter effect, one can easily show by using the
approximate formula in ?18? that the same argument holds.
FIG. 3. Depicted in the sin22?13-s23
termined by arbitrarily given values of the appearance probabilities
P?P(??→?e)?0.01 and P¯?P(?¯?→?¯e)?0.015 with E/L off the
oscillation maximum (??31???) at the JHF experiment. Here, s23
?sin2?23. The solid and the dashed lines correspond to positive and
negative ?m31
boundary of the region 0.36?s23
by the atmospheric neutrino data, 0.92?sin22?23?1. As indicated
in the figure, there are four solutions for each s23
there are eight solutions as denoted by blobs for any values of ?23
??/4. The oscillation parameters are taken as follows: ?m31
?2.5?10?3eV2,
?m21
Earth density is taken to be ??2.3 g/cm3.
2plane are the contours de-
2
2, respectively. The dash-dotted lines represent the
2?0.64 which is presently allowed
2, and altogether
2
2?6.9?10?5eV2, tan2?12?0.38. The
FIG. 4. The allowed regions are shown in the sin22?13-s23
determined with a given value of P?P(??→?e) alone ?in this case
P?0.025), or P¯?P(?¯?→?¯e) alone ?in this case P¯?0.035) at the
oscillation maximum ??31??? of the JHF experiment. Each al-
lowed region is the area bounded by the black solid ?for ?m31
with P only?, the black dashed ?for ?m31
solid ?for ?m31
?0 with P¯only? line, respectively, where the line with a definite
value of the CP phase ? sweeps out each region as ? varies from 0
to 2?. The oscillation parameters and the Earth density are the
same as those in Fig. 3.
2plane
2?0
2?0 with P only?, the gray
2?0 with P¯only?, and the gray dashed ?for ?m31
2
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Page 8
positive ?m31
?m31
?the dashed curve? at the first oscillation maximum
??31???. It may appear curious that the two curves with
positive and negative ?m31
in Fig. 5. In fact, the slight split between the solid (?m31
?0) and dashed (?m31
? and the matter effect in the case of the JHF experiment are
small. Thus, the degeneracy in the set (?13,?23) is effectively
twofold in this case.
To get a feeling as to whether the reactor experiment de-
scribed in Sec. III will be able to resolve the degeneracy, we
plot in Fig. 5 two sets of degenerate solutions by taking a
particular value of ?23, sin22?23?0.92, the lower end of the
region allowed by Super-Kamiokande. We denote the true
and fake solutions as (sin22?13,s23
spectively, assuming that the true ?23satisfies ?23??/4. We
overlay in Fig. 5 a shadowed region to indicate the accuracy
to be achieved by the reactor measurement of ?13. If the
experimental error ?re(sin22?13) in the reactor measurement
of sin22?13is smaller than the difference
2?the solid curve? and the other for negative
2
2almost overlap with each other
2
2?0) lines is due to the fact that both
2) and (sin22?13? ,s23
2?), re-
?de?sin22?13???sin22?13? ?sin22?13?
?11?
due to the (?13,?23) degeneracy, then the reactor experiment
may resolve the degeneracy. Notice that once the ?23degen-
eracy is lifted one can easily obtain four allowed sets of
(?,?m31
same point on the sin22?13-s23
ship between them has been given analytically in a com-
pletely general setting ?26?.
2) ?although they are still degenerate at almost the
2plane? because the relation-
B. Resolving power of the „?13,?23… degeneracy
by a reactor measurement
Let us make a semiquantitative estimate of how powerful
the reactormethod isfor
degeneracy.7For this purpose, we compare in this section the
difference of the two ?13solutions due to the degeneracy
with the resolving power of the reactor experiment. We con-
sider, for simplicity, the special case ??31???, i.e., energy
tuned at the first oscillation maximum. The simplest case
seems to be indicative of features of more generic cases.
As we saw in the previous section, there are two solutions
of ?13due to the doubling of ?23for a given sin22?23for in
each sign of ?m31
due to the degeneracy
resolving the(?13,?23)
2. Then we define the fractional difference
?de?sin22?13?
sin22?13
.
?12?
It is to be compared with ?re(sin22?13)/sin22?13of the reactor
experiment, where ?re(sin22?13) denotes the experimental
uncertainty estimated in Sec. III, i.e., 0.043 or 0.018. In Fig.
6?a? we plot the normalized error ?re(sin22?13)/sin22?13
which is expected to be achieved in the reactor experiment
described in Sec. III. We restrict ourselves to an analysis
with one degree of freedom, because we expect that the JHF
phase I experiment will provide us with accurate information
on ?m31
generacy in JHF phase II. The fractional difference ?12? can
be computed from the relation ?24?
sin22?13? ?sin22?13tan2?23??
2by the time the issue is really focused on the de-
?m21
?m31
2
2?
2tan2?aL/2?
?aL/??2
??1??aL/??2?sin22?12?1?tan2?23?, ?13?
and the result for ?de(sin22?13)/sin22?13is plotted in Fig.
6?b? as a function of sin2?23for two typical values of ?. We
notice that the fractional differences differ by up to a factor
of ?2 in the small sin22?23region between the first (?23
??/4) and the second octants (?23??/4). For the best fit
value of the two mass squared differences ?m21
?10?5eV2) and ??m31
??m21
case with sin22?13?0.03 and the one with sin22?13?0.09. In
2(6.9
2? (2.5?10?3eV2), for which ?
2??0.028, there is little difference between the
2/??m31
7The possibility of resolving the (?13,?23) by a reactor experi-
ment was qualitatively mentioned in ?21,34?. An alternative way to
resolve the ambiguity is to look at the ?e→??channel because the
main oscillation term in the probability P(?e→??) depends upon
c13
in detail, although it is briefly mentioned in ?24,25?.
2. Unfortunately, this idea does not appear to have been explored
FIG. 5. The allowed region in the sin22?13-s23
line when both P(??→?e) and P(?¯?→?¯e) are given ?in this case
P(??→?e)?0.025, P(?¯?→?¯e)?0.035] at the oscillation maxi-
mum (??31???, E?0.6 GeV for the JHF experiment?, as indicated
in the figure. The solid and the dashed lines are for the ?m31
and ?m31
tions of (sin22?13,s23
0.92. It is assumed arbitrarily that the solution of ?23in the first
octant (?23??/4) is the genuine one, while the one in the second
octant (?23??/4) with primes is the fake one. Superimposed in the
figure as a shaded region is the anticipated error in the reactor
measurement of ?13estimated in Sec. III. If the error ?re(sin22?13)
is smaller than the difference ?de(sin22?13)??sin22?13? ?sin22?13?
due to the degeneracy, then the reactor experiment may be able to
resolve it.
2plane becomes a
2?0
2?0 cases, respectively. Assuming ?23??/4, two solu-
2) are plotted; in this figure sin22?23is taken as
MINAKATA et al.
PHYSICAL REVIEW D 68, 033017 ?2003?
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Page 9
this case they are all approximated by the first term in Eq.
?13?, and ?de(sin22?13)/sin22?13depends approximately only
on ?23, making the analysis easier. On the other hand, if the
ratio ???m21
point, then the second term in Eq. ?13? is not negligible. In
Fig. 6?b?, ?de(sin22?13)/sin22?13is plotted in the extreme
case of ??1.9?10?4eV2/1.6?10?3eV2?0.12, which is
allowed at the 90% C.L. ?atmospheric? or the 95% C.L. ?so-
lar?, with sin22?13?0.03, 0.06, 0.09. From this, we observe
that the suppression in the first term in Eq. ?13? is compen-
sated by the second term for sin22?13?0.03, i.e., the degen-
eracy is small and therefore resolving the degeneracy is dif-
ficult in this case. To clearly illustrate the resolving power of
the degeneracy in the reactor measurement, assuming the
best fit value ??0.028, we plot in Fig. 7 the region where
the degeneracy can be lifted in the sin22?13-sin22?23plane. It
is evident that the reactor measurement will be able to re-
solve the (?13,?23) degeneracy in a wide range inside its
sensitivity region, in particular for ?23in the second octant.
A quantitative estimation of the significance of the fake
solution requires a detailed analysis of accelerator experi-
ments which includes the statistical and systematic errors as
well as the correlations of errors and the parameter degen-
eracies, and it will be worked out in future research.
2/??m31
2? is much larger than that at the best fit
VI. MORE ABOUT REACTOR VS
LONG-BASELINE EXPERIMENTS
The discussions in the previous section implicitly assume
that the sensitivities of the reactor and LBL experiments with
both ? and ?¯channels are good enough to detect the effects
of nonzero ?13. However, this need not be true, in particular,
in the coming decade. To further illuminate the complemen-
tary roles played by reactor and LBL experiments, we exam-
ine their possible mutual relationship, including the cases
where there is a signal in the former but none in the latter
FIG. 6. ?a? The normalized error at 90% C.L. in the reactor
measurement of ?13is given for ?sys?2%, 10 tonyr ?DOF?1,
?re(sin22?13)?0.043] in gray and for ?sys?0.8%, 40 tonyr ?DOF
?1, ?re(sin22?13)?0.018] in black, respectively. Notice that the
number of degrees of freedom becomes 1 once the value of ??m31
isknown from JHF.
?b?
?de(sin22?13)/sin22?13due to the degeneracy is plotted as a function
of sin22?23. Here, ?de(sin22?13)??sin22?13? ?sin22?13? stands for the
difference between the true solution sin22?13and the fake one
sin22?13? ,and
???m21
?10?3eV2?0.028 is for the best fit, and an extreme case with ?
?1.9?10?4eV2/1.6?10?3eV2?0.12, which is allowed at 90%
C.L. ?atmospheric? or 95% C.L. ?solar?, is also shown for illustra-
tion. The horizontal axis is suitably defined so that it is linear in
sin22?23, where the left half is for ?23??/4 whereas the right half
is for ?23??/4. The solar mixing angle is taken as tan2?12?0.38.
sin22?23?0.92 has to be satisfied due to the constraint from the
super-Kamiokande atmospheric neutrino data. If the value of
cos22?23is large enough, the value of ?de(sin22?13)/sin22?13in-
creases and lies outside the normalized error of the reactor experi-
ment; then the reactor result may resolve the ?23ambiguity.
2?
Thefractionaldifference
2/??m31
2?;
??6.9?10?5eV2/2.5
FIG. 7. The shadowed area stands for the region in which
?re(sin22?13)??de(sin22?13) is satisfied for ?sys?0.8%, 40 tonyr,
DOF?1, and for the best fit values of the solar and atmospheric
oscillation parameters. In this shadowed region, the (?13,?23) de-
generacy may be solved. The vertical axis is the same as the hori-
zontal axis of Fig. 6?b?.
REACTOR MEASUREMENT OF ?13AND ITS . . .PHYSICAL REVIEW D 68, 033017 ?2003?
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experiments, or vice versa. For ease of understanding by the
readers, we restrict our presentation in this section to a very
intuitive level by using a figure. It is, of course, possible to
make it more precise by deriving inequalities based on the
analytical approximate formulas ?18?. Throughout this sec-
tion LBL experiments at the oscillation maximum and ?23
??/4 are assumed.
If a reactor experiment sees affirmative evidence for the
disappearance in ?¯e→?¯e?the case of reactor affirmative?, it
will be possible to determine ?13up to certain experimental
errors. In this case, the appearance probability in LBL ex-
perimentmust fall into
?P(?)?
? (?) sign refers to ?m31
refers to the maximum ?minimum? value of the allowed re-
gion for P?P(??→?e), respectively. ?See Fig. 8.? Without
knowledge of the mass hierarchy, the probability is within
the region P(?)?
are present also for the antineutrino appearance channel. In
Fig. 8 we present the allowed regions in the cases of ?m31
?0 and ?m31
P(?¯?→?¯e) by taking the two best fit values sin22?13?0.08
and 0.04 ?labeled as a and b) as reactor affirmative cases.
They are inside the sensitivity region of the reactor experi-
ment discussed in Sec. III. We have used a one-dimensional
?2analysis ?i.e., the only parameter is sin22?13) to obtain the
allowed regions in Fig. 8. In doing this we have used the
theregion
P(?)?
min?P(?)
maxif the mass hierarchy is known, where the
2?0 (?m31
2?0) and max ?min?
min?P(?)?P(?)?
max. Similar inequalities
2
2?0 on a plane spanned by P(??→?e) and
same systematic error of 0.8% and the statistical errors cor-
responding to 40 tonyr measurement by the detector consid-
ered in Sec. III. For sin22?13?0.02, this particular reactor
experiment would fail ?the case of reactor negative? but the
allowed region can be obtained by the same procedure, and
is presented in Fig. 8, the region labeled as c. We use the
same LMA parameters as used earlier for Fig. 3 and Fig. 4.
We discuss four cases depending upon the two possibili-
ties of affirmative and negative evidence in each disappear-
ance and appearance search in the reactor and long-baseline
accelerator experiments. However, it is convenient to orga-
nize our discussion by classifying the possibilities into two
categories, reactor affirmative and reactor negative.
A. Reactor affirmative
We have two alternative cases, the LBL appearance search
affirmative or negative.
1. LBL affirmative
The implications of affirmative evidence in the appear-
ance search in LBL experiments differ depending upon the
region in which the observed appearance probability P(?)
falls.
?1? P?
?2? P?
two intervals that are given by the projection on the P axis of
the whole shadowed region (a or b) minus the projection on
the P axis of the darker shadowed region (a or b) in Fig. 8.
It is remarkable that in these cases not only is the sign of
?m31
nonvanishing. If P(?) is in the former region then ?m31
negative and sin? is positive, whereas if P(?) is in the latter
then ?m31
?3? P?
terval that is given by the projection on the P axis of the
darker shadowed region (a or b) in Fig. 8. In this case,
neither the sign of ?m31
mined.
It may be worth noting that if the reactor determination of
?13is accurate enough, it could be advantageous for LBL
appearance experiments to run only in the neutrino mode
?where the cross section is larger than that for antineutrinos
by a factor of 2–3? to possibly determine the sign of ?m31
depending upon the region in which P(?) falls.
min?P(?)?P?
max?P(?)?P?
minor
max. These cases correspond to the
2determined, but also the CP phase ? is known to be
2is
2is positive and sin? is negative.
min?P(?)?P?
max. This case corresponds to the in-
2nor the sign of sin? can be deter-
2
2. LBL negative
In principle, it is possible to have no appearance event
even though the reactor sees evidence for disappearance.
This case corresponds to the left edge of the analogous shad-
owed region in the case of sin22?13?0.02 in Fig. 8, i.e., the
allowed region with sin22?13?0.02 for which P?
axis falls below P?0.005. In order for this case to occur the
sensitivity limits P(?)limitof the LBL experiment must sat-
isfy P?
?m31
of ?m31
minon the P
min?P(?)limit, assuming our ignorance of the sign of
2. If it occurs that P?
2is determined to be minus.
min?P(?)limit?P?
min, then the sign
FIG. 8. Predicted allowed regions are depicted in the P-P¯plane
for the JHF experiment at the oscillation maximum after an affir-
mative ?a negative? result of the reactor experiment is obtained,
where P?P(??→?e) and P¯?P(?¯?→?¯e) are the appearance prob-
abilities, and ?23??/4 is assumed. The cases a,b, and c correspond
to sin22?13?0.08?0.018, sin22?13?0.04?0.018, and sin22?13
?0.019, respectively. The regions bounded by the solid lines and
the dashed lines are for the normal hierarchy (?m31
inverted hierarchy (?m31
the maximum (P?
hierarchy (? for the normal and ? for the inverted hierarchy?,
although P?
2?0) and the
2?0), respectively. Each region predicts
max) and the minimum (P?
min) values of P for each
minof the region c are zero.
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The P(?)limitof the JHF experiment in its phase I is esti-
mated to be 3?10?3?14?.8Therefore, by using the mixing
parameters typical for the LMA solution, the case of LBL
negative cannot occur unless the sensitivity of the reactor
experiment becomes sin22?13?0.01. However, in the inter-
mediate stage of the JHF experiment, where P(?)limitis
larger than 3?10?3, this situation may occur.
B. Reactor negative
If the reactor experiment does not see disappearance of ?¯e
one obtains the bound ?13??13
native cases, the LBL appearance search affirmative or nega-
tive.
RL. We have again two alter-
1. LBL affirmative
If a LBL experiment measures the oscillation probability
P(?), then, for a given value of P(?) the allowed region of
sin2?13is given by sin2??
of ?m31
erwise. We denote below the maximum and the minimum
values of ?13collectively as ?maxand ?min, respectively. In
Fig. 4, theregionbounded
sin2??
bounded by the solid ?dashed? black line for a given value of
s23
Then, there are two possibilities which we discuss one by
one.
?i? ?13
obtained by nonobservation of disappearance of ?¯ein the
reactor experiment.
?ii? ?min??13
constraint ?min??13??13
min?sin2?13?sin2??
min?sin2?13?sin2??
maxif the sign
maxoth-
2is known, and by sin2??
by sin2??
min
and
max(sin2??
minand sin2??
max) is indicated as the region
2.
RL??max: In this case no additional information is
RL??max: In this case we have the nontrivial
RL.
2. LBL negative
In this case, we obtain an upper bound on ?13, which,
however, depends on the assumed values of ? and the sign of
?m31
?min??13
2. A ?-independent bound can also be derived: ?13
RL,?max?.
VII. DISCUSSION AND CONCLUSION
In this paper, we have explored in detail the possibility of
measuring sin22?13using reactor neutrinos. We stressed that
this measurement is free from the problem of parameter de-
generacies from which accelerator appearance experiments
suffer, and that the reactor measurement is complementary to
accelerator experiments. We showed that sensitivity to
sin22?13?0.02 ?0.05? is obtained with a 24.3 GWthreactor
with identical detectors at near and far distances and with a
data size of 40 ?10? tonyr, assuming that the relative system-
atic error is 0.8% ?2%? for the total number of events. In
particular, if the relative systematic error is 0.8%, the error in
sin22?13is 0.018, which is smaller than the uncertainty due to
the combined ?intrinsic and hierarchical? parameter degen-
eracies expected in accelerator experiments. We also showed
that the reactor measurement can resolve the degeneracy in
?23↔?/2??23and determine whether ?23is smaller or
larger than ?/4 if sin22?13and cos22?23are relatively large.
We took 2% and 0.8% as the reference values for the
relative systematic error for the total number of events. 2% is
exactly the same figure as in the Bugey experiment while
0.8% is what we naively expect in the case where we have
two identical detectors, near and far, which are similar to that
of the CHOOZ experiment. It is also technically possible to
dig a 200 m depth shaft hole with diameter wide enough to
place a CHOOZ-like detector in. Therefore, the discussions
in this paper are realistic. We hope the present paper stimu-
lates the interest of the community in reactor measurements
of ?13.
ACKNOWLEDGMENTS
We thank Yoshihisa Obayashi for correspondence and
Michael Shaevitz for useful comments. H.M. thanks Andre
de Gouvea for discussions and the Theoretical Physics De-
partment of Fermilab for hospitality. H.S. acknowledges the
hospitality of Professor Atsuto Suzuki and the members of
the Research Center for Neutrino Science, Tohoku Univer-
sity, where a core part of the sensitivity analysis was carried
out. This work was supported by Grants-in-Aid for Scientific
Research in Priority Areas No. 12047222 and No. 13640295,
Japan Ministry of Education, Culture, Sports, Science, and
Technology.
?1? Kamiokande Collaboration, Y. Fukuda et al., Phys. Lett. B
335, 237 ?1994?; Super-Kamiokande Collaboration, Y. Fukuda
et al., Phys. Rev. Lett. 81, 1562 ?1998?; Super-Kamiokande
Collaboration, S. Fukuda et al., ibid. 85, 3999 ?2000?.
?2? B.T. Cleveland et al., Astrophys. J. 496, 505 ?1998?; SAGE
Collaboration, J.N. Abdurashitov et al., Phys. Rev. C 60,
055801 ?1999?; GALLEX Collaboration, W. Hampel et al.,
Phys. Lett. B 447, 127 ?1999?; Super-Kamiokande Collabora-
tion, S. Fukuda et al., Phys. Rev. Lett. 86, 5651 ?2001?; 86,
5656 ?2001?; SNO Collaboration, Q.R. Ahmad et al., ibid. 87,
071301 ?2001?; 89, 011301 ?2002?.
?3? SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 89,
011302 ?2002?.
?4? K2K Collaboration, S.H. Ahn et al., Phys. Lett. B 511, 178
?2001?; Super-Kamiokande and K2K Collaborations, T. Na-
kaya, hep-ex/0209036; K. Nishikawa, Nucl. Phys. B ?Proc.
Suppl.? 118, 129 ?2003?.
?5? Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28,
870 ?1962?.
?6? CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B 420,
8The sensitivity limit of sin22?13quoted in ?14?, sin22?13?6
?10?3, obtained by using the one-mass-scale approximation (?
?1), may be translated into this limit for P(?).
REACTOR MEASUREMENT OF ?13AND ITS . . .PHYSICAL REVIEW D 68, 033017 ?2003?
033017-11
Page 12
397 ?1998?; 466, 415 ?1999?.
?7? J. Shirai, Nucl. Phys. B ?Proc. Suppl.? 118, 15 ?2003?.
?8? L. Wolfenstein, Phys. Rev. D 17, 2369 ?1978?.
?9? S.P. Mikheyev and A.Yu. Smirnov, Yad. Fiz. 42, 1441 ?1985?
?Sov. J. Nucl. Phys. 42, 913 ?1985??; Nuovo Cimento Soc. Ital.
Fis., C 9, 17 ?1986?.
?10? M.B. Smy, in Neutrino Oscillations and Their Origin,
Kashiwa, 2001, edited by Y. Suzuki et al. ?World Scientific,
Singapore, 2003?, p. 40, hep-ex/0202020; J.N. Bahcall, M.C.
Gonzalez-Garcia, and C. Pena-Garay, J. High Energy Phys. 07,
054 ?2002?; V. Barger, D. Marfatia, K. Whisnant, and B.P.
Wood, Phys. Lett. B 537, 179 ?2002?; A. Bandyopadhyay, S.
Choubey, S. Goswami, and D.P. Roy, ibid. 540, 14 ?2002?; P.C.
de Holanda and A.Y. Smirnov, Phys. Rev. D 66, 113005
?2002?; G.L. Fogli, E. Lisi, A. Marrone, D. Montanino, and A.
Palazzo, ibid. 66, 053010 ?2002?; M. Maltoni, T. Schwetz,
M.A. Tortola, and J.W. Valle, ibid. 67, 013011 ?2003?.
?11? Y. Kozlov, L. Mikaelyan, and V. Sinev, Yad. Fiz. 66, 497
?2003? ?Phys. At. Nucl. 66, 469 ?2003??.
?12? MINOS Collaboration, P. Adamson et al., ‘‘MINOS Detectors
Technical Design Report, Version 1.0,’’ Report No. NuMI-L-
337,1998,http://www.hep.anl.gov/ndk/hypertext/
minos_tdr.html
?13? M. Komatsu, P. Migliozzi, and F. Terranova, J. Phys. G 29, 443
?2003?.
?14? Y. Itow et al., hep-ex/0106019.
?15? H. Minakata and H. Nunokawa, Phys. Lett. B 495, 369 ?2000?;
Nucl. Instrum. Methods Phys. Res. A 472, 421 ?2000?; J. Sato,
ibid. 472, 434 ?2000?; B. Richter, hep-ph/0008222.
?16? CERN Working Group on Super Beams Collaboration, J.J.
Gomez-Cadenas et al., hep-ph/0105297.
?17? D. Ayres et al., hep-ex/0210005.
?18? A. Cervera, A. Donini, M.B. Gavela, J.J. Gomez Cadenas, P.
Hernandez, O. Mena, and S. Rigolin, Nucl. Phys. B579, 17
?2000?; B593, 731?E? ?2001?.
?19? C. Albright et al., hep-ex/0008064; V.D. Barger, S. Geer, R.
Raja, and K. Whisnant, Phys. Rev. D 63, 113011 ?2001?; J.
Pinney and O. Yasuda, ibid. 64, 093008 ?2001?; P. Hernandez
and O. Yasuda, Nucl. Instrum. Methods Phys. Res. A 485, 811
?2002?; O. Yasuda, ibid. 503, 104 ?2003?; in Neutrino Oscilla-
tions and Their Origin ?10?, p. 259, hep-ph/0203273;
hep-ph/0209127, and references therein.
?20? T. Kajita, H. Minakata, and H. Nunokawa, Phys. Lett. B 528,
245 ?2002?.
?21? G. Fogli and E. Lisi, Phys. Rev. D 54, 3667 ?1996?.
?22? J. Burguet-Castell, M.B. Gavela, J.J. Gomez-Cadenas, P. Her-
nandez, and O. Mena, Nucl. Phys. B608, 301 ?2001?.
?23? H. Minakata and H. Nunokawa, J. High Energy Phys. 10, 001
?2001?; Nucl. Phys. B ?Proc. Suppl.? 110, 404 ?2002?.
?24? V. Barger, D. Marfatia, and K. Whisnant, Phys. Rev. D 65,
073023 ?2002?; in Proceedings of the APS/DPF/DPB Summer
Study on the Future of Particle Physics, Snowmass, 2001, ed-
ited by R. Davidson and C. Quigg, hep-ph/0108090.
?25? A. Donini, D. Meloni, and P. Migliozzi, Nucl. Phys. B646, 321
?2002?.
?26? H. Minakata, H. Nunokawa, and S. Parke, Phys. Rev. D 66,
093012 ?2002?.
?27? Particle Data Group, K. Hagiwara et al., Phys. Rev. D 66,
010001 ?2002?.
?28? T.K. Kuo and J. Pantaleone, Phys. Lett. B 198, 406 ?1987?; H.
Minakata and S. Watanabe, ibid. 468, 256 ?1999?. In a recent
paper, a similar treatment is generalized into all other channels;
H. Yokomakura, K. Kimura, and A. Takamura, ibid. 544, 286
?2002?.
?29? M. Shiozawa, talk presented at XXth International Conference
on Neutrino Physics and Astrophysics, Neutrino 2002, Mu-
nich, Germany.
?30? M.C. Gonzalez-Garcia and C. Pena-Garay, Phys. Lett. B 527,
199 ?2002?.
?31? Y. Declais et al., Nucl. Phys. B434, 503 ?1995?.
?32? G.L. Fogli and E. Lisi, Phys. Rev. D 52, 2775 ?1995?.
?33? J. Arafune, M. Koike, and J. Sato, Phys. Rev. D 56, 3093
?1997?; 60, 119905?E? ?1999?.
?34? G. Barenboim and A. de Gouvea, hep-ph/0209117.
?35? M. Koike and J. Sato, Mod. Phys. Lett. A 14, 1297 ?1999?.
?36? Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Lett.
B 539, 179 ?2002?.
MINAKATA et al.
PHYSICAL REVIEW D 68, 033017 ?2003?
033017-12