EPJ manuscript No.
(will be inserted by the editor)
The Reliability on the Direction of the Incident Neutrino for the
Fully Contained Events and Partially Contained Events due to
QEL in the Super-Kamiokande
E. Konishi1, Y. Minorikawa2, V.I. Galkin3, M. Ishiwata4, I. Nakamura4, N. Takahashi1, M. Kato5 and A. Misaki6
1 Graduate School of Science and Technology, Hirosaki University, Hirosaki, 036-8561, Japan
2 Department of Science, School of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502, Japan
3 Department of Physics, Moscow State University, Moscow, 119992, Russia
4 Department of Physics, Saitama University, Saitama, 338-8570, Japan
5 Kyowa Interface Science Co.,Ltd., Saitama, 351-0033, Japan
6 Advanced Research Institute for Science and Engineering, Waseda University, Tokyo, 169-0092, Japan
e-mail: misaki@kurenai.waseda.jp
Received: / Revised version: March 30, 2007
Abstract. In the SK analysis of the neutrino events for Fully Contained Events and Partially Contained
Events on their zenith angle distribution, it is assumed that the zenith angle of the incident neutrino is
the same as that of the detected charged lepton. In the present paper, we examine the validity of the SK
assumption on the direction of the incident neutrinos. Concretely speaking, we analyze muon-like events due
to QEL. For the purpose, we develop Time Sequential Monte Carlo Simulation to extract the conclusion on
the validity of the SK assumption. In our Time Sequential Simulation, we simulate every physical process
concerned as exactly as possible without any approximation. From the comparison between the zenith
angle distributon of the emitted muons under the SK assumption on the direction and the corresponding
one obtained under our Time Sequential Simulation, it is concluded that the measurement of the direction
of the incident neutrino for the neutrino events occurring inside the detector in the SK analysis turns out
to be unreliable, which holds irrespective of the existence and/or non-existence of the neutrino oscillation.
PACS. Superkamiokande, QEL, Fully Contained Event, Time Sequential Monte Carlo Simulation
1 Introduction
Superkamiokande have been analyzing Fully Contained
Events and Partially Contained Events which are gen-
erated inside the detector, and Upward Through Going
Events and Stopping Events which are generated outside
the detector, for the studies on the neutrino oscillation in
atmospheric neutrinos. The report of oscillations between
muon and tau neutrinos for atmospheric neutrinos de-
tected with SuperKamiokande (SK, hereafter) is claimed
to be robustly established for the following reasons:
(1) The discrimination between electrons and muons in the
SK energy range, say, several hundred MeV to several
GeV, has been proved to be almost perfect, as demon-
strated by calibration using accelerator beams [1] 1.
1 The SK discrimination procedure between muon and elec-
tron is constructed on the average value theory. In our opinion,
discrimination procedure should be examined, taking into ac-
count the stochastic characters of the physical processes in the
neutrino events concerned. If we take this effect into account,
then, for example, we give uncertainties of 3◦ to 14◦ in the in-
(2) The analysis for the electron-like events and the muon-
like events which give the single-ring structure in Fully
Contained Events and Partially Contained Events with
their zenith angle distribution, based on the well es-
tablished discrimination procedure mentioned in (1),
reveals a significant deficit of muon-like events but
the expected level of electron-like events. It is, thus,
concluded that muon neutrinos oscillate into tau neu-
trinos which cannot be detected due to the small ge-
ometry of SK. As the most new one, the SK collab-
oration published their comprehensive paper[2]. The
analysis of SK data presently yields sin22θ > 0.92 and
1.5× 10−3eV2 < ∆m2 < 3.4× 10−3eV2 at 90% confi-
dence level.
(3) The analysis of Upward Through Going Events and
Stopping Events, in which the neutrino interactions
occur outside the detector, leads to similar results to
cident direction of the charged lepton and uncertainties of 2m
to 7m in the vertex point of the events. See, our papers [2].
However, SK give 1.8◦ to 3.0◦ and 0.3m for the same physical
quantities. See, accompanied two papers.
http://arxiv.org/abs/0704.0190v1
2 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
(2). The charged leptons which are produced in these
categories are regarded as being exclusively muons, be-
cause electrons have negligible probabilities to produce
such events as they lose energy very rapidly in the sor-
rounding rock. Thus for these events the discrimina-
tion procedure described in (1) is not required, and,
therefore, the analysis here is independent of the anal-
ysis in (2). For these events, however, the SK group
obtains the same parameters for neutrino oscillations
as in (2) 2.
(4) Now, SK assert that they have found the oscillatory
signature in atomospheric neutrinos from L/E anal-
ysis, which should be the ultimate evidence for the
exsistence of the neutrino oscillation [3] Our critical
examination on the L/E analysis by SK will be pub-
lished elsewhere.
As for item(3), we have clarified that SK hardly dis-
criminate electron( neutrino) from muon( neutrino ) in
the SK manner and, instead, propose more rigorous and
suitable procedure with theoretical background for the dis-
crimination between them in the preceeding two papers.
Among the neutrino events both occurring inside and out-
side the detector, the most robust evidence for the neu-
trino oscillation, if exists, should have been obtained from
the analysis of both electron-like events and muon-like
events in Fully Contained Events. Because (i) all necces-
sary informations for the physical interpretation are in-
cluded in Fully Contained Events due to their character
and (ii),
furthermore, both electron-like events and muon-like events
give the single structure image free from arbitrary inter-
pretation with the proper electron/muon discrimination
procedure. SK treat neutrino events whose energies cover
from several hundred MeV to several GeV, if the neutrino
events occur inside the detector. In this energy region,
Quasi Elastic Scattering(QEL) [4] is dominant compared
with other physical processes, such as one-pion produc-
tion [5], coherent pion production [6] and deep inelastic
scattering [7]. Events due to other processes, except QEL,
are not free from ambiguities due to multi-ring structure
of the images.
Therefore, SK should have analyzed the muon-
like events and the electron-like events with the
single ring image in Fully Contained Events exculu-
sively where QEL is dominant, without utilizing
poorer quality events, if SK pursue to obtain the
clear cut conclusion on the neutrino oscillation 3.
2 It seems strange that the experimental data with different
qualities give similarly precise results, because Fully Contained
Events whose information is totally inside the detector are of
higher experimental qualities compared with those of both Up-
ward Through Going Events and Stopping Events.
3 In their analysis, they really add Partially Contained
Events to the experimental data as those with the same quality
under the assumption that they belong to muon-like events in
Fully Contained Events to raise the statistics higher. However,
such the assumption lacks in theoretical background. Further-
more, SK utilize to add multi-ring structure events which are
Therefore, it is essential for us to examine single ring
structure events among Fully Contained Events due to
QEL which have the least ambiguities among the neutrino
events concerned to obtain clear cut conclusion as for the
neutrino oscillation. Here, the main concern of the present
paper is devoted to the detailed analysis of the muon-like
events from QEL, focusing on the direction of the incident
neutrino. Situation around the corresponding electron-like
event is the same as in the muon-like event. The exami-
nation on the separation of Fully Contained Events from
Partially Contained Events will be discussed in subsequent
papers.
Here, it should be emphasized that the direction of
the incident neutrino is assumed to be the same as that
of the emitted charged lepton, i.e., the (anti-)muon or
(anti-)electron, in the SK analysis of both Fully Contained
Events and Partially Contained Events [8,9]. The SK De-
tector Simulation is to be constructed without any con-
tradiction with the SK assumption on the direction.
From the point of orthodoxical Monte Carlo Simula-
tion, it seems to be unnatural for SK to impose such the
assumption that the direction of the incident neutrino is
the same as that of the emitted lepton ( hereafter, we call
this assumption simply ”the SK assumption on the di-
rection”) upon their Detector Simulation. Obviously, one
need not any assumption on the relation between the di-
rection of the incident neutrino and that of the emitted
lepton in any sense, if we develop the Monte Carlo Simu-
lation in a rigorous manner, which will be shown later in
the present paper.
In order to avoid any misunderstanding toward the SK
assumption on the direction we reproduce this assumption
from the original SK paper:
”However, the direction of the neutrino must be estimated
from the reconstructed direction of the products of the neu-
trino interaction. In water Cherenkov detectors, the direc-
tion of an observed lepton is assumed to be the direction of
the neutrino. Fig.11 and Fig.12 show the estimated corre-
lation angle between neutrinos and leptons as a function of
lepton momentum. At energies below 400 MeV/c, the lep-
ton direction has little correlation with the neutrino direc-
tion. The correlation angle becomes smaller with increas-
ing lepton momentum. Therefore, the zenith angle depen-
dence of the flux as a consequence of neutrino oscillation
is largely washed out below 400 MeV/c lepton momentum.
With increasing momentum, the effect can be seen more
clearly. ” [8] 4.
On the other hand, Ishitsuka states in his Ph.D thesis
which is exclusively devoted into the L/E analysis of the
caused by one-pion roduction, coherent pion production and
deep inelastic scattering. However, the discrimination among
the multi-ring structures is not so easy, which may lead the
worse estimation of energies as well as directions of the events
concerned.
4 It could be understood from this statement that SK justify
the validity of this assumption above 400 MeV/c. However, it is
not correct, because SK put ”to be proved ” as the proposition.
See, page 101 in their paper [8].
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 3
atmospheric neutrino from Super Kamiokande as follows:
” 8.4 Reconstruction of Lν
Flight length of neutrino is determined from the neu-
trino incident zenith angle, although the energy and the
flavor are also involved. First, the direction of neutrino is
estimated for each sample by a different way. Then, the
neutrino flight lenght is calclulated from the zenith angle
of the reconstructed direction.
8.4.1 Reconstruction of Neutrino Direction
FC Single-ring Sample
The direction of neutrino for FC single-ring sample is
simply assumed to be the same as the reconstructed direc-
tion of muon. Zenith angle of neutrino is reconstructed as
follows:
cosΘrecν = cosΘµ (8.17)
,where cosΘrecν and cosΘµ are cosine of the reconstructed
zenith angle of muon and neutrino, respectively. ” [9] 5.
In our understanding, SK Monte Carlo Simulation is
named usually as the Detector Simulation. It is, however,
noticed that the effect of the azimuthal angles of the emit-
ted leptons in QEL could not be taken into account in their
Simulation. As will be shown in later (see Section 3), this
effect greatly influences over the final zenith angle distri-
bution of the emitted leptons. Also, the back scattering
due to QEL can not be neglected for the rigorous deter-
mination of the direction of the incident neutrino, but this
effect could not be treated in the SK Detector Simulation,
which is beyond the application limitaition 6.
On the other hand, we could take into account these
effects correctly in our Monte Carlo Simulation which is
named as Time Sequential Simulation.
In the present paper, we carry out the full Monte Carlo
Time Sequential Simulation as exactly as possible, with-
out the SK assumption on the direction to clarify the prob-
lematic issue raised by SK. We carry out simulation which
starts from the opposite side of the Earth to the SK de-
tector. A neutrino sampled from the atmospheric neutrino
energy spectrum at the opposite side of the Earth tra-
verses through the medium with different densities in the
interior of the Earth and penetrates finally into the SK
5 It should be noticed that the SK assumption on the di-
rection may hold on the following possible two cases: [1] The
scattering angle of the emitted lepton is so small that the ef-
fect of the scattering angle could be neglected really. However,
in the present case, it could not be true from Fig. 1 and Fig.
2 and Table 1. [2] One may assert that the assumption could
not hold on individual case, but it could hold statistically af-
ter accumulation of large amount of the data. However, such
assertion should be verified. We verify such assumption could
not hold. See, Fig. 11 and Fig. 12 in the present paper.
6 SK have never clarified not only the details, but also the
principle and its validity on their Monte Carlo Simulation. We
hope disclosure of their Detector Simulation for open and fair
scientific discussion.
detector where the neutrino interactions occur. The emit-
ted energy of the individual lepton thus produced and its
direction are simulated exactly based on the probability
function of the cross sections concerned.
We finally show the zenith angle distribution of the
emitted leptons as well as that of the incident neutrinos
are quite different from corresponding ones of the SK. This
indicates that the SK assumption on the direction coud
not be a reliable estimator as for the determination of the
direction of the incident neutrino (See, section 5).
2 Cross Sections of Quasi Elastic Scattering
in the Neutrino Reaction and the Scattering
Angle of Charged Leptons.
We examine the following reactions due to the charged
current interaction (c.c.) from QEL.
νe + n −→ p+ e−
νµ + n −→ p+ µ−
ν̄e + p −→ n+ e+ (1)
ν̄µ + p −→ n+ µ+
The differential cross section for QEL is given as fol-
lows [6].
dσℓ(ℓ̄)(Eν(ν̄))
G2F cos
ν(ν̄)
A(Q2)±B(Q2)
C(Q2)
where
A(Q2) =
+ f1f2
+ g21
B(Q2) = (f1 + f2)g1Q
C(Q2) =
f21 + f
+ g21
The signs + and − refer to νµ(e) and ν̄µ(e) for charged
current (c.c.) interactions, respectively. The Q2 denotes
the four momentum transfer between the incident neu-
trino and the charged lepton. Details of other symbols are
given in [4].
The relation among Q2, Eν(ν̄), the energy of the in-
cident neutrino, Eℓ, the energy of the emitted charged
lepton (muon or electron or their anti-particles) and θs,
the scattering angle of the emitted lepton, is given as
Q2 = 2Eν(ν̄)Eℓ(1− cosθs). (3)
4 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
Fig. 1. Relation between the energy of the muon and its
scattering angle for different incident muon neutrino energies,
0.5 GeV, 1 GeV, 2 GeV, 5 GeV, 10 GeV and 100 GeV.
Also, the energy of the emitted lepton is given by
Eℓ = Eν(ν̄) −
. (4)
Now, let us examine the magnitude of the scattering
angle of the emitted lepton in a quantitative way, as this
plays a decisive role in determining the accuracy of the
direction of the incident neutrino, which is directly related
to the reliability of the zenith angle distribution of both
Fully Contained Events and Partially Contained Events in
By using Eqs. (2) to (4), we obtain the distribution
function for the scattering angle of the emitted leptons
and the related quantities by a Monte Carlo method. The
procedure for determining the scattering angle for a given
energy of the incident neutrino is described in the Ap-
pendix A. Fig. 1 shows this relation for muon, from which
we can easily understand that the scattering angle θs of
the emitted lepton ( muon here ) cannot be neglected.
For a quantitative examination of the scattering angle,
we construct the distribution function for θs of the emit-
ted lepton from Eqs. (2) to (4) by using a Monte Carlo
method.
Fig. 2 gives the distribution function for θs of the muon
produced in the muon neutrino interaction. It can be seen
that the muons produced from lower energy neutrinos are
scattered over wider angles and that a considerable part
of them are scattered even in backward directions. Simi-
lar results are obtained for anti-muon neutrinos, electron
neutrinos and anti-electron neutrinos.
Also, in a similar manner, we obtain not only the dis-
tribution function for the scattering angle of the charged
leptons, but also their average values < θs > and their
standard deviations σs. Table 1 shows them for muon neu-
trinos, anti-muon neutrinos, electron neutrinos and anti-
electron neutrinos. In the SK analysis, it is assumed that
the scattering angle of the charged particle is zero [8,9].
Fig. 2. Distribution functions for the scattering angle of the
muon for muon-neutrino with incident energies, 0.5 GeV, 1.0
GeV and 2 GeV. Each curve is obtained by the Monte Carlo
method (one million sampling per each curve).
3 Influence of Azimuthal Angle of Quasi
Elastic Scattering over the Zenith Angle of
both the Fully Contained Events and
Partially Contained Events
In the present section, we examine the effect of the az-
imuthal angles of the emitted leptons over their own zenith
angles for given zenith angles of the incident neutrinos 7.
For three typical cases (vertical, horizontal and diag-
onal), Fig. 3 gives a schematic representation of the re-
lationship between, θν(ν̄), the zenith angle of the incident
neutrino, and (θs, φ) a pair of scattering angle of the emit-
ted lepton and its azimutal angle.
From Fig. 3(a), it can been seen that the zenith angle
θµ(µ̄) of the emitted lepton is not influenced by its φ in
the vertical incidence of the neutrinos (θν(ν̄) = 0
o), as it
must be. From Fig. 3(b), however, it is obvious that the
influence of φ of the emitted leptons on their own zenith
angle is the strongest in the case of horizontal incidence
of the neutrino (θν(ν̄) = 90
o). Namely, one half of the
emitted leptons are recognized as upward going, while the
other half is classified as downward going ones. The di-
agonal case ( θν(ν̄) = 43
o) is intermediate between the
vertical and the horizontal. In the following, we examine
the cases for vertical, horizontal and diagonal incidence of
the neutrino with different energies, say, Eν(ν̄) = 0.5 GeV,
Eν(ν̄) = 1 GeV and Eν(ν̄) = 5 GeV.
The detailed procedure for the Monte Carlo simulation
is described in the Appendix A.
7 Throughout this paper, we measure the zenith angles of
the emitted leptons from the upward vertical direction of the
incident neutrino. Consequently, notice that the sign of our
direction is oposite to that of the SK ( our cos θν(ν̄) = - cos θν(ν̄)
in SK)
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 5
Table 1. The average values < θs > for scattering angle of the emitted charged leptons and their standard deviations σs for
various primary neutrino energies Eν(ν̄).
Eν(ν̄) (GeV) angle νµ(µ̄) ν̄µ(µ̄) νe ν̄e
(degree)
0.2 < θs > 89.86 67.29 89.74 67.47
σs 38.63 36.39 38.65 36.45
0.5 < θs > 72.17 50.71 72.12 50.78
σs 37.08 32.79 37.08 32.82
1 < θs > 48.44 36.00 48.42 36.01
σs 32.07 27.05 32.06 27.05
2 < θs > 25.84 20.20 25.84 20.20
σs 21.40 17.04 21.40 17.04
5 < θs > 8.84 7.87 8.84 7.87
σs 8.01 7.33 8.01 7.33
10 < θs > 4.14 3.82 4.14 3.82
σs 3.71 3.22 3.71 3.22
100 < θs > 0.38 0.39 0.38 0.39
σs 0.23 0.24 0.23 0.24
Fig. 3. Schematic view of the zenith angles of the charged
muons for diffrent zenith angles of the incident neutrinos, fo-
cusing on their azimuthal angles.
3.1 Dependence of the spreads of the zenith angle for
the emitted leptons on the energies of the emitted
leptons for different incident directions with different
energies
We give the scatter plots between the fractional energies
of the emitted muons and their zenith angle for a definite
zenith angles of the incident neutrino with different ener-
gies in Figs. 4 to 6. In Fig. 4, we give the scatter plots
for vertically incident neutrino with different energies 0.5
GeV, 1 GeV and 5 GeV . In this case, the relations between
the emitted energies of the muon and and their zenith an-
gles are unique, which comes from the definition of the
zenith angle of the emitted lepton. However, the densities
(frequencies of event number) along each curve is differ-
ent in position to position and depend on the energies of
the incident neutrinos. Generally speaking,densities along
curves become higher toward cos θµ(µ̄) = 1. In this case,
cos θµ(µ̄) is never influenced by the azimuthal angel in the
scattering by the definition 8.
Fig. 5 tells us that the horizontally incident neutrinos
give the most widely spread of the zenith angle distribu-
tion of the emitted lepton influenced by the azimuthal an-
gle. The more lower incident neutrino energies, the more
wider spreads of the emitted leptons. The diagonally in-
cident neutrinos give the intermediate distribution of the
emitted leptons between those of vertically incident neu-
trinos and horizontally incident neutrinos.
8 The zenith angles of the particles concerned are measured
from the vertical direction.
6 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=1(θν=0°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=1(θν=0°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=1(θν=0°)
Fig. 4. The scatter plots between the fractional energies of the produced muons and their zenith angles for vertically incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=0(θν=90°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=0(θν=90°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=0(θν=90°)
Fig. 5. The scatter plots between the fractional energies of the produced muons and their zenith angles for horizontally incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=0.731(θν=43°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=0.731(θν=43°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=0.731(θν=43°)
Fig. 6. The scatter plots between the fractional energies of the produced muons and their zenith angles for diagonally incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 7
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=1(θν=0°)
average=0.262
s.d.=0.547
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=1(θν=0°)
average=0.590
s.d.=0.439
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=1(θν=0°)
average=0.978
s.d.=0.067
Fig. 7. Zenith angle distribution of the muon for the vertically incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=0(θν=90°)
average=0.003
s.d.=0.564
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=0(θν=90°)
average=0.001
s.d.=0.480
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=0(θν=90°)
average=0.006
s.d.=0.141
Fig. 8. Zenith angle distribution of the muon for the horizontally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=0.731(θν=43°)
average=0.189
s.d.=0.556
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=0.731(θν=43°)
average=0.432
s.d.=0.463
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=0.731(θν=43°)
average=0.715
s.d.=0.103
Fig. 9. Zenith angle distribution of the muon for the diagonally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
8 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
3.2 Zenith angle distribution of the emitted lepton for
the different incidence of the neutrinos with different
energies
In Figs. 7 to 9, we give the zenith angle distributions of
the emitted muons for the given direction of the incident
neutrinos with different energies of the neutrino. These
figures are obtained through summation on the energies
of the emitted muons for their definite zenith angles in
Figs. 4 to 6.
In Figs. 7(a) to 7(c), we give the zenith angle distri-
bution of the emitted muon for the case of vertically inci-
dent neutrinos with different energies, say, Eν = 0.5 GeV,
Eν = 1 GeV and Eν = 5 GeV.
Comparing the case for 0.5 GeV with that for 5 GeV,
we understand the big contrast between them as for the
zenith angle distribution. The scattering angle of the emit-
ted muon for 5 GeV neutrino is relatively small (See, Table
1) that the emitted muons keep roughly the same direction
as their original neutrino. In this case, the effect of their
azimuthal angle on the zenith angle is also small. However,
in the case for 0.5 GeV which is the dominant energy for
Fully Contained Events in the Superkamiokande, there is
even a possibility for the emitted muon to be emitted in
the backward direction due to the large angle scattering,
the effect of which is enhanced by their azimuthal angle.
The most frequent occurrence in the backward scatter-
ing of the emitted muon appear in the horizontally inci-
dent neutrino as shown in Figs. 8(a) to 8(c). In this case,
the zenith angle distribution of the emitted muon should
be symmetrical to the horizontal direction. Comparing the
case for 5 GeV with those both for ∼0.5 GeV and for ∼1
GeV, even 1 GeV incident neutrinos lose almost the orig-
inal sense of the incidence if we measure it by the zenith
angle of the emitted muon. Figs. 9(a) to 9(c) for the di-
agonally incident neutrino tell us that the situation for
diagonal cases lies between the case for the vertically in-
cident neutrino and that for horizontally incident one.
4 Zenith Angle Distribution of Fully
Contained Events and Partially Contained
Events for a Given Zenith Angle of the
Incident Neutrino, Taking Their Energy
Spectrum into Account
In the previous sections, we discuss the relation between
the zenith angle distribution of the incident neutrino with
a single energy and that of the emited muons produced by
the neutrino for the different incident direction. In order to
apply our motivation around the uncertainty of the SK as-
sumption on the direction for Fully Contained Events and
Partially Contained Events, we must consider the effect of
the energy spectrum of the incident neutrino. The Monte
Carlo simulation procedure for this purpose are given in
the Appendix B.
In Fig. 10, we give the zenith angle distributions of the
sum of µ+(µ̄) and µ− for a given zenith angle of ν̄µ̄ and
νµ, taking into account primary neutrino energy spectrum
at Kamioka site.
In Table 2, the average values for cosθµ+µ̄ and their
standard deviation for different incidences of the incident
neutrinos with different energies are presented 9.
In the SK case, their average values are given by cosθν(ν̄)
themselves by definition and, consequently, the standard
deviations are zero under the assumption, because the SK
assumption is of the delta function for the incidence direc-
tion. They are shown in the bottom line of Table 2. In the
second line from the bottom in this table, we give the av-
erage values and their standard deviations for cos θν+ν̄ ob-
tained under the inclusion of the energy spectrum for pri-
mary neutrinos. Thus, we found these values correspond to
those for incident neutrino with the effective single energy
between 0.5 GeV and 1 GeV. If we compare the average
energies and the standard deviations for the inclusion of
incident neutrino energy spectrum with those under the
SK assumption, it is easily understood that SK assump-
tion does not represent real zenith angle distribution of
the emitted muon.
5 Relation between the Zenth angle
Distribution of the Incident Neutrinos and
that of the emitted leptons
Now, we extend the results for the definite zenith angle
obtained in the previous section to the case in which we
consider the zenih angle distribution of the incident neu-
trinos totally.
Here, we examine the real correlation between cos θν
and cos θµ, by peforming the exact Monte Carlo simula-
tion.
The detail for the simulation procedure is given the
Appendix C.
In Fig. 11 we classsify the correlation between cos θν
and cos θµ according to the different energy range of the
incident muon neutrinos. It should be noticed that the
SK assumption on cos θν = cos θµ is roughly hold only for
Eν ≥ 5 GeV, but the widths in cos θµ for the definite cos θν
near cos θν =0 (for horizontally incident neutrino ) are
much larger than those near cos θν =1 ( for the vertically
incident neutrino). Of course, this is due to the effect of
the azimutal angle in QEL which could not be derived by
the SK simulation (DETECTOR SIMULATION ). Such
tendencies become more remarkable in Eν ≤ 5 GeV and
in these energies the SK assumption on the direction does
not hold any more.
In Fig. 12, we classify the correlation between cos θν and
cos θµ according to the different energy range of Eµ. The
similar argument on Fig. 11 can be done on the case of
Fig.12
9 Notice that the difference in the corresponding quantites
between the case for single energy and the case for the energy
spectrum. The formers are given in the µ−,while th latter is
given in µ− and µ+. However, such the difference does not
change the essential recognition.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 9
Table 2. Average values and their standerd deviations in cosθµ+µ̄ for the zenith angle distributions of the muons with different
primary energies of the insident neutrinos.
Vertical Diagonal Horizontal
cosθν+ν̄ = 1 (0
◦) cosθν+ν̄ = 0.731 (43
◦) cosθν+ν̄ = 0 (90
Eν+ν̄(GeV) cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄
0.5 0.262 0.547 0.189 0.556 -0.003 0.564
1.0 0.590 0.439 0.432 0.463 0.001 0.480
2.0 0.581 0.250 0.623 0.290 0.001 0.325
5.0 0.978 0.067 0.715 0.103 0.006 0.141
Spectrum∗ 0.468 0.531 0.339 0.519 -0.005 0.500
SK∗∗ 1.00 0.00 0.731 0.000 0.000 0.000
(a) (b) (c)
−1 −0.5 0 0.5 1
cosθµ+µ− 
µ+ and µ−
cosθν+ν − =1(θν+ν −=0°)
Avg.=0.81
S.D.=0.30
−1 −0.5 0 0.5 1
µ −  
cosθµ+µ −
µ+ and µ−
cosθν+ν −=0(θν+ν −=90°)
Avg.=0.00
S.D.=0.34
−1 −0.5 0 0.5 1
µ −  
cosθµ+µ −
µ+ and µ−
cosθν+ν −=0.73(θν+ν −=43°)
Avg.=0.60
S.D.=0.33
Fig. 10. Zenith angle distribution of µ− and µ+ for ν and ν̄ for the incident neutrinos with the vertical, horizontal and diagonal
directions, respectively. The overall neutrino spectrum at Kamioka site is taken into account. The sampling number is 10000
for each case. SK stand for the corresponding ones under the SK assumption.
0 0.2 0.4 0.6 0.8 1
cosθν
Eµ < 0.5 GeV
0.5 < Eµ < 1 GeV
Eµ > 1 GeV
0 0.2 0.4 0.6 0.8 1
cosθν
1 < Eµ < 2 GeV
2 < Eµ < 5 GeV
Eµ > 5 GeV
Fig. 12. Correlation diagrams between cos θν and cos θµ for different muon energy ranges.
10 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
0 0.2 0.4 0.6 0.8 1
cosθν
1 < Eν < 2 GeV
2 < Eν < 5 GeV
Eν > 5 GeV
Fig. 11. Correlation Diagram between cos θν and cos θµ for
different neutrino energy regions.
Thus, it could be surely concluded from Fig. 11 and
Fig. 12 that the SK assumption on the direction never
holds as a good estimator for the determination of the
directions of the incident neutrinos.
In order to obtain the zenith angle distribution of the
emitted leptons for that of the incident neutrinos, we di-
vide the cosine of the zenith angle of the incident neutrino
into twenty regular intervals from cos θν = 0 to cos θν = 1.
For the given interval of cos θν , we carry out the exact
Monte Carlo simulation, the detail of which is give in the
Appendix D and obtain the cosine of the zenith angle of
the emitted leptons, taking account of the geometry for
surronding the SK detector.
Thus, for each interval of cos θν , we obtain the corre-
sponding zenith angle distribution of the emitted leptons.
Then, we sum up these corresponding ones over all zenth
angles of the incident neutrinos and we finally obtain the
relation between the zenith angle distribution for the in-
cident neutrinos and that for the emitted leptons.
In a similar manner, we could obtain between cos θν̄
and cos θµ̄ for anti-neutrinos. The situation for anti-neutrinos
is essentially same as that for neutrinos.
Here, we examine the zenith angle distribution of the
muons from both upward neutrinos and downward ones
in the case that neutrino oscillation does not exist.
By performing the procedures described in Appendix
C, a pair of sampling ( cos θν+ν̄ , Eν+ν̄ ) gives a pair of (
cos θµ+µ̄, Eµ+µ̄ ). In Fig. 13, we give the zenith angle dis-
tribution of the upward neutrinos ( the sum of νµ and ν̄µ )
which is constructed from the energy spectra for different
cos θν+ν̄ . (see, Honda et. al. [10] and Appendix B)
Upward neutrinos may produce even downward lep-
tons due to both the backscattering effect and the effect
of azimuthal angle on larger forward scattering for the in-
teraction concerned (see, Figure 3 and Figures 4 to 6 in
−1 −0.5 0 0.5 1
cosθν+ν −
µ+ ,µ−  (pµ>0.4 GeV/c)
upward neutrinos
no oscillation
from upward neutrinos
cosθµ+µ −, 
Fig. 13. The relation between the zenith angle distribution
of the incident neutrino and corresponding ones of the emitted
lepton
the text). As the result of it, the zenith angle distribution
of the emitted muons for the upward neutrino may leak in
the downward direction. From Figure 13, it is very clear
that the shape of the zenith angle distribution for the in-
cident neutrinos is quite different from that of the emitted
muons produced by these neutrinos. If the SK assumption
on the direction statistically holds, the zenith angle dis-
tribution for the emitted muons should coincide totally
with that of the incident neutrinos. In other words, one
may say that the zenith angle distribution for the emitted
muons should be understood as that of the incident neu-
trino under the SK assumption on the direction. However,
the muon spectrum is distinctively different from the real
(computational) incident neutrino spectrum as shown in
the figure. Thus we conclude that SK assumption on the
direction leads to the wrong conclusion on the neutrino
oscillation. The further examination on the experimental
data obtained by SK will be carried out in the subsequent
papers.
It is, further, noticed that upward neutrino energy
spectrum in the figure biggest near cos θν+ν̄ = 0 and the
smallest near cos θν+ν̄ = 1, which reflects from the en-
hancement of the primary incident neutrino energy spec-
trum from the inclined direction and is independent on the
neutrino oscillation, while in SK opinion, such tendency
may be favor of the existence of neutrino oscillation.
6 Discussions and Conclusion
In order to extract the definite conclusion on the neutrino
oscillation from the experiment by cosmic ray neutrinos
whose intensity as well as interaction with the substance
are both very weak, first of all, one should analyze the
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 11
most clear cut and ambiguity free events. Among neutrino
events analyzed by SK, the most clear cut events are Single
Ring Events, such as electron-like events and muon-like
events in Fully Contained Events which are generated by
QEL. These events are provided with simplicity due to
single ring and all possible measureable physical quantities
are confined in the detector.
Furthermore, QEL is the most dominant source for
neutrino events which are generated in the SK detector.
This is the reason why we examine the QEL events excul-
sively in present and subsequent papers.
If the neutrino oscillation really exists, the most clear
cut evidence surely appears in the analysis of single ring
events due to QEL in Fully Contained Events and one does
not need the analysis of any other type of events, such as
single ring events in Partially Contained Events, multi-
ring events in either Fully Contained Events or Partially
Contained Evetns, all of which include inevitably ambigu-
ities for the interpretation and show merely sub-evidences
compared with that from the single ring events due to
QEL in Fully Contained Events.
SK analyze the zenith angle distribution of the inci-
dent neutrinos under the asssumption that the direction
of the incident neutrino is the same as that of the emitted
lepton. We conclude that this assumption is supplemented
by their Monte Carlo Simulation named as Detector Sim-
ulation 10.
In the present paper, we adopt Time Sequential Simu-
lation which starts from the incident neutrino energy spec-
trum on the opposite side of the Earth to the SK detec-
tor and simulate all posible physical processes which are
connected with the zenith angle distribution of the in-
cident neutrinos according to their probability functions
concerned for the examination on the validity of the SK
assumption on the direction.
Concretely speaking, we take the following treatment,
(i) the stochastic treatments of the scattering angle of the
emitted lepton in QEL, including the scattering on the
backward as well as the azimuthal angle, which could not
be treated in Detector Simulation, (ii) the stochastic treat-
ment on the zenith angle distribution of the emitted lep-
ton, considering the incident neutrino energy spectrum,
(iii) the stochastic treatment on the detection of the QEL
events inside the SK detector.
Furthermore, the discrimination between Fully Con-
tained Events and Partially Contained Events is only pos-
sible in the Time Sequential Simulation, because the events
concerned may be classified into different categories by
chance, Fully Contained Events and Partially Contained
Events due to different occurring points and different di-
rections.
10 The SK Detector Simulation for obtaing the zenith angle
distribtuion of the incident neutrino as for the neutrino oscilla-
tiom has never been disclosed in their papers, even in the Ph.D
thesis. Consequently, this is only our onjecture as for utiliza-
tion of the SK Detector Simulation . A clear thing is only that
SK impose the proposittion that the direction of the incident
neutrino is the same as that of the emitted lepton upon the
neutrino oscillation analysis.
The conclusions thus obtained are as follows:
(1) The zenith angle distributions of the emitted lepton
in QEL for the incident neutrino with both the def-
inite zenith angle and the definite energy are widely
spread, particularly, into even the backward region due
to partly pure backscattering and partly the combina-
tion of the azimuthal angle with the slant direction
of the incident neutrinos. However, for every incident
neutrino with a definte zenith angle, SK give the same
definite zenith angle to the emitted lepton. Already in
this stage, the SK assumption on the direction does
not hold.
(2) Taking account of the incident neutrino energy spec-
trum and simulating all physical processes concerned,
we obtain the zenith angle distribution of the emitted
leptons for the incident neutrino with a definite zenith
angle. It is proved that the SK assumption on the di-
rection does not hold again.
(3) The correlation diagrams between cos θν and cos θµ
show that SK assumption does noty hold well even
for higher energies of the incident neutrinos, and it
is shown that the correlation between them become
weaker in more inclined incident neutrinos due to the
effect of the azimuthal angle in QEL.
(4) Taking into account the detection efficiency for the
events concerned in the simulation for upward neu-
trinos and anti-neutrinos, we obtain the zenith angle
distribution of the leptons ( muons plus anti-muons
). According to the SK Assumption on the direction,
the zenith angle distribution is the same as that of the
incident neutrinos. However, the original zenith angle
distribution of incident neutrino is found to be quite
different from that derived from that of leptons. This is
the final conclusion that SK have not measured the di-
rection of the incident neutrinos reliably, which is quite
independent on either the existence or non-existence of
the neutrino oscillation.
(5) The SK assume that the Partially Contained Events
exclusively belong to the muon-like event. However,
such the assumption lacks in theoretical background.
Electron events can also contribute to the Partially
Contained Events under some geometrical condition,
for example, partly coming from the transformation
by Eq.(A.5). The quantitative examination on the Par-
tially Contained Events among the electron-like event
will be published elsewhere.
In subsequent papers, we will give the relation between
the zenith angle distributions of the incident neutrinos
and the corresponding muons in the cases with and with-
out neutrino oscillation, including downward neutrino and
will examine whether it is possible to or not to detect the
neutrino oscillation by using atmospheric neutrino.
12 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
In the following Appendices we give the concrete Monte
Carlo Simulations, namely, the details of our Time Se-
quential Simulation.
A Appendix: Monte Carlo Procedure for the
Decision of Emitted Energies of the Leptons
and Their Direction Cosines
Here, we give the Monte Carlo Simulation procedure for
obtaining the energy and its direction cosines, (lr,mr, nr),
of the emitted lepton in QEL for a given energy and its
direction cosines, (l,m, n), of the incident neutrino.
The relation among Q2, Eν+ν̄ , the energy of the inci-
dent neutrino, Eℓ, the energy of the emitted lepton (muon
or electron or their anti-particles) and θs, the scattering
angle of the emitted lepton, is given as
Q2 = 2Eν(ν̄)Eℓ(ℓ̄)(1− cosθs). (A·1)
Also, the energy of the emitted lepton is given by
Eℓ(ℓ̄) = Eν(ν̄) −
. (A·2)
Procedure 1
We decide Q2 from the probability function for the differ-
ential cross section with a given Eν(ν̄) (Eq. (2) in the text)
by using the uniform random number, ξ, between (0,1) in
the following
Pℓ(ℓ̄)(Eν(ν̄), Q
2)dQ2, (A·3)
where
Pℓ(ℓ̄)(Eν(ν̄), Q
dσℓ(ℓ̄)(Eν(ν̄), Q
dσℓ(ℓ̄)(Eν(ν̄), Q
(A·4)
From Eq. (A·1), we obtain Q2 in histograms together with
the corresponding theoretical curve in Fig. 14. The agree-
ment between the sampling data and the theoretical curve
is excellent, which shows the validity of the utlized proce-
dure in Eq. (A·3) is right.
Procedure 2
We obtain Eℓ(ℓ̄) from Eq. (A·2) for the given Eν(ν̄) and
Q2 thus decided in the Procedure 1.
Procedure 3
We obtain cos θs, cosine of the the scattering angle of the
emitted lepton, for Eℓ(ℓ̄) thus decided in the Procedure 2
from Eq. (A·1) .
Procedure 4
We decide φ, the azimuthal angle of the scattering lepton,
0 0.2 0.4 0.6 0.8 1
1 GeV
2 GeV
Eν=0.5 GeV
Fig. 14. The reappearance of the probability function for QEL
cross section. Histograms are sampling results, while the curves
concerned are theoretical ones for given incident energies.
which is obtained from
φ = 2πξ. (A·5)
Here, ξ is a uniform random number (0, 1).
As explained schematically in the text(see Fig. 3 in the
text), we must take account of the effect due to the az-
imuthal angle φ in the QEL to obtain the zenith angle
distribution of both Fully Contained Events and Partially
Contained Events correctly.
Procedure 5
The relation between direction cosines of the incident neu-
trinos, (ℓν(ν̄),mν(ν̄), nν(ν̄)), and those of the corresponding
emitted lepton, (ℓr,mr, nr), for a certain θs and φ is given
ℓ2 +m2
ℓ2 +m2
ℓν(ν̄)
ℓ2 +m2
ℓ2 +m2
mν(ν̄)
ℓ2 +m2 0 nν(ν̄)
sinθscosφ
sinθssinφ
cosθs,
(A·6)
where nν(ν̄) = cosθν(ν̄), and nr = cosθℓ. Here, θℓ is the
zenith angle of the emitted lepton.
The Monte Carlo procedure for the determination of θℓ
of the emitted lepton for the parent (anti-)neutrino with
given θν(ν̄) and Eν(ν̄) involves the following steps:
We obtain (ℓr,mr, nr) by using Eq. (A·6). The nr is
the cosine of the zenith angle of the emitted lepton which
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 13
Fig. 15. The relation between the direction cosine of the
incident neutrino and that of the emitted charged lepton.
should be contrasted to nν , that of the incident neutrino.
Repeating the procedures 1 to 5 just mentioned above, we
obtain the zenith angle distribution of the emitted leptons
for a given zenth angle of the incident neutrino with a def-
inite energy.
In the SK analysis, instead of Eq. (A·6), they assume
nr = nν(ν̄) uniquely for Eµ(µ̄) ≥ 400 MeV.
B Appendix: Monte Carlo Procedure to
Obtain the Zenith Angle of the Emitted
Lepton for a Given Zentith Angle of the
Incident Neutrino
The present simulation procedure for a given zenith an-
gle of the incident neutrino starts from the atmospheric
neutrino spectrum at the opposite site of the Earth to
the SK detector. We define, Nint(Eν(ν̄), t, cosθν(ν̄)), the in-
teraction neutrino spectrum at the depth t from the SK
detector in the following way
Nint(Eν(ν̄), t, cosθν(ν̄)) = Nsp(Eν(ν̄), cos θν(ν̄))×
λ1(Eν(ν̄), t1, ρ1)
× · · · ×
λn(Eν(ν̄), tn, ρn)
(B·1)
Here, Nsp(Eν(ν̄), cos θν(ν̄)) is the atmospheric (anti-)
neutrino spectrum for the zenith angle at the opposite
surface of the Earth.
Here λi(Eν(ν̄), ti, ρi) denotes the mean free path due
to the neutrino(anti neutrino) with the energy Eν(ν̄) from
QEL at the distance, ti, from the opposite surface of the
Earth inside whose density is ρi.
The procedures of the Monte Carlo Simulation for the
incident neutrino(anti neutrino) with a given energy,Eν(ν̄),
whose incident direction is expressde by (l,m, n) are as fol-
lows.
Procedure A
For the given zenith angle of the incident neutrino, θν(ν̄),
we formulate, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the produc-
tion function for the neutrino flux to produce leptons at
the Kamioka site in the following
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
= σℓ(ℓ̄)(Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄), (B·2)
where
σℓ(ℓ̄)(Eν(ν̄)) =
dσℓ(ℓ̄)(Eν(ν̄), Q
dQ2. (B·3)
Each differential cross section above is given in Eq. (2) in
the text.
Utilizing, ξ, the uniform random number between (0,1),
we determine Eν(ν̄), the energy of the incident neutrino in
the following sampling procedure
∫ Eν(ν̄)
Eν(ν̄),min
Pd(Eν(ν̄), t, cos θν(ν̄)(ν̄))dEν(ν̄), (B·4)
where
Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
. (B·5)
14 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
In our Monte Carlo procedure,
the reproduction of, Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the nor-
malized differential neutrino interaction probability func-
tion, is confirmed in the same way as in Eq. (A·4).
Procedure B
For the (anti-)neutrino concerned with the energy ofEν(ν̄),
we sample Q2 utlizing ξ3, the uniform random number be-
tween (0,1). The Procedure B is exactly the same as in the
Procedure 1 in the Appendix A.
Procedure C
We decide, θs, the scattering angle of the emitted lepton
for given Eν(ν̄) and Q
2. The procedure C is exactly the
same as in the combination of Procedures 2 and 3 in the
Appendix A.
Procedure D
We randomly sample the azimuthal angle of the charged
lepton concerned. The Procedure D is exactly the same as
in the Procedure 4 in the Appendix A.
Procedure E
We decide the direction cosine of the charged lepton con-
cerned. The Procedure E is exactly the same as in the
Procedure 5 in the Appendix A.
We repeat Procedures A to E until we reach the de-
sired trial number.
C Appendix: Correlation between the Zenith
Angles of the Incident Neutrinos and Those
of the Emitted Leptons
Procedure A
By using, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), which is defined
in Eq. (B·2),
we define the spectrum for cos θν(ν̄) in the following.
I(cos θν(ν̄))d(cos θν(ν̄)) =
d(cos θν(ν̄))
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄).(C·1)
By using Eq. (C·2) and ξ, a sampled uniform random num-
ber between (0,1), then we could determine cos θν(ν̄) from
the following equation
∫ cos θν(ν̄)
Pn(cos θν(ν̄))d(cos θν(ν̄)), (C·2)
where
Pn(cos θν(ν̄)) = I(cos θν(ν̄))
I(cos θν(ν̄))d(cos θν(ν̄)).
(C·3)
Procedure B
For the sampled d(cos θν(ν̄)) in the Procedure A, we sam-
ple Eν(ν̄) from Eq.(C·4) by using ξ, the uniform randum
number between (0,1)
∫ Eν(ν̄)
Eν(ν̄),min
Ppro(Eν(ν̄), cos θν(ν̄))dEν(ν̄), (C·4)
where
Ppro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) =
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
. (C·5)
Procedure C
For the sampled Eν(ν̄) in the Procedure B, we sample
Eµ(µ̄) from Eqs. (A·2) and (A·3). For the sampled Eν(ν̄)
and Eµ(µ̄), we determine cos θs, the scattering angle of the
muon uniquely from Eq. (A·1).
Procedure D
We determine, φ, the azimuthal angle of the scattering lep-
ton from Eq. (A·5) by using ξ, an uniform randum number
between (0,1).
Procedure E
We obtain cos θµ(µ̄) from Eq. (A·6). As the result, we ob-
tain a pair of (cos θν(ν̄), cos θµ(µ̄)) through Procedures A
to E. Repeating the Procedures A to E, we finally the cor-
relation between the zenith angle of the incident neutrino
and that of the emitted muon.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 15
D Appendix: Detection of the Neutrino
Events in the SK Detector and Their
Interaction Points
The plane ABCD is always directed vertically to the di-
rection of the incident neutrino with a given zenith an-
gle, which is shown in Fig. 16. The rectangular ABCDE-
FGH encloses the SK detector whose radius and height
is denoted by R and H, respectively. The width and the
height of the plane ABCD for a given zenith angle, θν(ν̄),
is given as, R and Rcos θν(ν̄) + H sin θν ,respectively, which
are shown in Fig. 16-c.
Now, let us estimate the ratio of the number of the neu-
trino events inside the SK detector to that in the rectan-
gular ABCDEFGH. As the number of the neutrino events
inside some material is proportional to the number of the
nucleons in the material concerned. The number of the
nucleons inside the SK detector (ρ = 1) is given as
Nsk =
NavogaR
2H, (D·1)
whereNavoga denotes the Avogadro number, and the num-
ber of the nucleons in the exterior of the SK detector inside
ABCDEFGH is given as
Fig 16-a Fig 16-b
Injection Point
(X,Y)
Neutrino
Injection Point
Fig 16-c Fig 16-d
Fig. 16. Sampling procedure for neutrino evens injected into
the detector
Nextr(cos θν(ν̄) ) = ρNavoga
R2H +
R(H2+R2) sin θν(ν̄) cos θν(ν̄)
, (D·2)
where ρ is the density of the rock which surrounds the SK
detector.
Then, the total number of the target in the rectangu-
larABCDEFGH is given as
Ntot(cos θν(ν̄)) = Nsk +Nextr(cos θν(ν̄)). (D·3)
Here, we take 2.65, as ρ (standard rock).
Then, Rtheor, the ratio of the number of the neutrino
events in the SK detector to that in the rectangular ABCDE-
FGH is given as
Rtheor(cos θν(ν̄)) = Nsk/Ntot(cos θν(ν̄)). (D·4)
We obtain Rtheor for different values of cos θν(ν̄) given in
the Table 3.
Here, we simulate neutrino events occured in the rectangu-
lar ABCDEFH, by using the atmospheric neutrino beam
which falls down on the plane ABCD. Thus, Nsmaple, the
sampling number of the (anti-)neutrino events inside the
rectangular ABCDEFG for a given cos θν(ν̄) is given as
Nsample(cos θν(ν̄)) = Ntot(cos θν(ν̄))×
∫ Eν(ν̄),max
Eν(ν̄),min
ℓ(ℓ̄)
(Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄)
(D·5)
where σℓ(ℓ̄)(Eν(ν̄)) is the total cross section for (anti-)neutrino
due to QEL, and Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄) is the dif-
ferential nutrino energy spectrum for the definite zenith
angle, θν(ν̄), in the plane ABCD. The injection points of
the neutrinos in the plane ABCD are distributed over the
plane randomly and uniformely and the injection points
are determined from a pair of the uniform random num-
bers between (0,1). They penetrate into the rectangular
ABCDEFGH from the injection point in the plane ABCD
and some of them may penetrate into the SK detector or
may not, which depend on their injection point.
In the neutrino events which penetrate into the SK
detectorr, their geometrical total track length, Ttrack, are
devided into three parts
Ttrack = Tb + Tsk + Ta, (D·6)
where Tb denotes the track length from the plane ABCD
to the entrance point of the SK detector, Tsk denotes the
track length inside the SK detector, and Ta denotes the
track length from the escaping point of the SK detector
to the exit point of the rectangular ABCDEF, and thus
16 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
Table 3. Occurrence probabilities of the neutrino events in-
side the SK detector for different cos θν ’s. Comparison between
Rtheor and Rmonte. The sampling numbers for the Monte Carlo
Simulation are, 1000, 10000, 100000, respectively.
cos θν Rtheor Rmonte
Sampling Number
1000 10000 100000
0.000 0.58002 0.576 0.5750 0.57979
0.100 0.41717 0.425 0.4185 0.41742
0.200 0.32792 0.353 0.3252 0.32657
0.300 0.27324 0.282 0.2731 0.27163
0.400 0.23778 0.223 0.2329 0.23582
0.500 0.21491 0.206 0.2063 0.21203
0.600 0.20117 0.197 0.1946 0.19882
0.700 0.19587 0.193 0.1925 0.19428
0.800 0.20117 0.198 0.2002 0.20001
0.900 0.22843 0.230 0.2248 0.22803
1.00 0.58002 0.557 0.5744 0.57936
Ttrack denotes the geometrical length of the neutrino con-
cerned in the rectangular ABCDEFGH.
By the definition, the neutrinos concerned with Ttrack
interact surely somewhere along the Ttrack. Here, we are
interested only in the interaction point which ocuurs along
Tsk. We could determine the interaction point in the Tsk
in the following.
We define the following quantities for the purpose.
Tweight = Tsk + ρ(Tb + Ta), (D·7)
ρav = Tweight/Ttrack, (D·8)
ξρ = ρav/ρ, (D·9)
ξsk = Tsk/Tweight. (D·10)
The flow chart for the choice of the neutrino events in
the SK detector and the determination of the interaction
points inside the SK detector is given in Fig. 17. Thus, we
obtain neutrino events whose occurrence point is decided
in the SK detector in the following.
xf = x0, (D·11)
yf = y0 + ξTsk sin θν(ν̄) (D·12)
zf = z0 + ξTsk cos θν(ν̄). (D·13)
If we carry out the Monte Carlo Simulation, following
the flow chart in Fig. 17, then, we obtain Nevent, the num-
ber of the neutrino events generasted in the SK detector.
The ratio of the selected events to the total trial is given
Rmonte(cos θν(ν̄)) = Nevent(cos θν(ν̄))/Nsample(cos θν(ν̄)).
(D·14)
Comparison between Rtheor and Rmonte in Table 3 shows
that our Monte Carlo procedure is valid.
References
1. Kasuga, S. et al., Phys. Lett. B374 (1996) 238.
N Nsample
N=N+1
N=N+1
Entry
Determination of
Point in the plane ABCD
by using 1 and 2
Judgement on the
Event’s Entering
Determination of
Ta,Tb and Tsk
Determination of Tweight
av, and sk
Determination of e
interaction point of
e events inside SK
Fig. 17. Flow Chart for the determination of the interaction
points of the neutrino events inside the detector
2. Ashie,Y. et al., Phys. Rev. D 71 (2005) 112005.
3. Ashie,Y. et al., Phys. Rev. Lett.93 (2004) 101801.
4. Renton, P., Electro-weak Interaction, Cambridge University
Press (1990). See p. 405.
5. D.Rein and L.M.Sehgal, Ann. of Phys. 133 (1981) 1780.
6. D.Rein and L.M.Sehgal Nucl. Phys. B84 (1983) 29.
7. R.H.Gandhi et. al. Astropart. Phys. 5 (1996) 81.
8. Kajita, T. and Totsuka, Y. Rev. Mod. Phys., 73 (2001) 85.
See p. 101.
9. Ishitsuka, M., Ph.D thesis, University of Tokyo (2004). See
p. 138.
10. Honda, M., et al., Phys. Rev. D 52 (1996) 4985
	Introduction
	Cross Sections of Quasi Elastic Scattering in the Neutrino Reaction and the Scattering Angle of Charged Leptons.
	Influence of Azimuthal Angle of Quasi Elastic Scattering over the Zenith Angle of both the Fully Contained Events and Partially Contained Events
	Zenith Angle Distribution of Fully Contained Events and Partially Contained Events for a Given Zenith Angle of the Incident Neutrino, Taking Their Energy Spectrum into Account
	Relation between the Zenth angle Distribution of the Incident Neutrinos and that of the emitted leptons
	Discussions and Conclusion
	Appendix: Monte Carlo Procedure for the Decision of Emitted Energies of the Leptons and Their Direction Cosines 
	Appendix: Monte Carlo Procedure to Obtain the Zenith Angle of the Emitted Lepton for a Given Zentith Angle of the Incident Neutrino
	Appendix: Correlation between the Zenith Angles of the Incident Neutrinos and Those of the Emitted Leptons
	Appendix: Detection of the Neutrino Events in the SK Detector and Their Interaction Points

  In the SK analysis of the neutrino events for [Fully Contained Events] and
[Partially Contained Events] on their zenith angle distribution, it is assumed
that the zenith angle of the incident neutrino is the same as that of the
detected charged lepton. In the present paper, we examine the validity of [the
SK assumption on the direction] of the incident neutrinos. Concretely speaking,
we analyze muon-like events due to QEL. For the purpose, we develop [Time
Sequential Monte Carlo Simulation] to extract the conclusion on the validity of
the SK assumption. In our [Time Sequential Simulation], we simulate every
physical process concerned as exactly as possible without any approximation.
  From the comparison between the zenith angle distributon of the emitted muons
under [the SK assumption on the direction] and the corresponding one obtained
under our [Time Sequential Simulation], it is concluded that the measurement of
the direction of the incident neutrino for the neutrino events occurring inside
the detector in the SK analysis turns out to be unreliable, which holds
irrespective of the existence and/or non-existence of the neutrino oscillation.

Introduction
Superkamiokande have been analyzing Fully Contained
Events and Partially Contained Events which are gen-
erated inside the detector, and Upward Through Going
Events and Stopping Events which are generated outside
the detector, for the studies on the neutrino oscillation in
atmospheric neutrinos. The report of oscillations between
muon and tau neutrinos for atmospheric neutrinos de-
tected with SuperKamiokande (SK, hereafter) is claimed
to be robustly established for the following reasons:
(1) The discrimination between electrons and muons in the
SK energy range, say, several hundred MeV to several
GeV, has been proved to be almost perfect, as demon-
strated by calibration using accelerator beams [1] 1.
1 The SK discrimination procedure between muon and elec-
tron is constructed on the average value theory. In our opinion,
discrimination procedure should be examined, taking into ac-
count the stochastic characters of the physical processes in the
neutrino events concerned. If we take this effect into account,
then, for example, we give uncertainties of 3◦ to 14◦ in the in-
(2) The analysis for the electron-like events and the muon-
like events which give the single-ring structure in Fully
Contained Events and Partially Contained Events with
their zenith angle distribution, based on the well es-
tablished discrimination procedure mentioned in (1),
reveals a significant deficit of muon-like events but
the expected level of electron-like events. It is, thus,
concluded that muon neutrinos oscillate into tau neu-
trinos which cannot be detected due to the small ge-
ometry of SK. As the most new one, the SK collab-
oration published their comprehensive paper[2]. The
analysis of SK data presently yields sin22θ > 0.92 and
1.5× 10−3eV2 < ∆m2 < 3.4× 10−3eV2 at 90% confi-
dence level.
(3) The analysis of Upward Through Going Events and
Stopping Events, in which the neutrino interactions
occur outside the detector, leads to similar results to
cident direction of the charged lepton and uncertainties of 2m
to 7m in the vertex point of the events. See, our papers [2].
However, SK give 1.8◦ to 3.0◦ and 0.3m for the same physical
quantities. See, accompanied two papers.
http://arxiv.org/abs/0704.0190v1
2 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
(2). The charged leptons which are produced in these
categories are regarded as being exclusively muons, be-
cause electrons have negligible probabilities to produce
such events as they lose energy very rapidly in the sor-
rounding rock. Thus for these events the discrimina-
tion procedure described in (1) is not required, and,
therefore, the analysis here is independent of the anal-
ysis in (2). For these events, however, the SK group
obtains the same parameters for neutrino oscillations
as in (2) 2.
(4) Now, SK assert that they have found the oscillatory
signature in atomospheric neutrinos from L/E anal-
ysis, which should be the ultimate evidence for the
exsistence of the neutrino oscillation [3] Our critical
examination on the L/E analysis by SK will be pub-
lished elsewhere.
As for item(3), we have clarified that SK hardly dis-
criminate electron( neutrino) from muon( neutrino ) in
the SK manner and, instead, propose more rigorous and
suitable procedure with theoretical background for the dis-
crimination between them in the preceeding two papers.
Among the neutrino events both occurring inside and out-
side the detector, the most robust evidence for the neu-
trino oscillation, if exists, should have been obtained from
the analysis of both electron-like events and muon-like
events in Fully Contained Events. Because (i) all necces-
sary informations for the physical interpretation are in-
cluded in Fully Contained Events due to their character
and (ii),
furthermore, both electron-like events and muon-like events
give the single structure image free from arbitrary inter-
pretation with the proper electron/muon discrimination
procedure. SK treat neutrino events whose energies cover
from several hundred MeV to several GeV, if the neutrino
events occur inside the detector. In this energy region,
Quasi Elastic Scattering(QEL) [4] is dominant compared
with other physical processes, such as one-pion produc-
tion [5], coherent pion production [6] and deep inelastic
scattering [7]. Events due to other processes, except QEL,
are not free from ambiguities due to multi-ring structure
of the images.
Therefore, SK should have analyzed the muon-
like events and the electron-like events with the
single ring image in Fully Contained Events exculu-
sively where QEL is dominant, without utilizing
poorer quality events, if SK pursue to obtain the
clear cut conclusion on the neutrino oscillation 3.
2 It seems strange that the experimental data with different
qualities give similarly precise results, because Fully Contained
Events whose information is totally inside the detector are of
higher experimental qualities compared with those of both Up-
ward Through Going Events and Stopping Events.
3 In their analysis, they really add Partially Contained
Events to the experimental data as those with the same quality
under the assumption that they belong to muon-like events in
Fully Contained Events to raise the statistics higher. However,
such the assumption lacks in theoretical background. Further-
more, SK utilize to add multi-ring structure events which are
Therefore, it is essential for us to examine single ring
structure events among Fully Contained Events due to
QEL which have the least ambiguities among the neutrino
events concerned to obtain clear cut conclusion as for the
neutrino oscillation. Here, the main concern of the present
paper is devoted to the detailed analysis of the muon-like
events from QEL, focusing on the direction of the incident
neutrino. Situation around the corresponding electron-like
event is the same as in the muon-like event. The exami-
nation on the separation of Fully Contained Events from
Partially Contained Events will be discussed in subsequent
papers.
Here, it should be emphasized that the direction of
the incident neutrino is assumed to be the same as that
of the emitted charged lepton, i.e., the (anti-)muon or
(anti-)electron, in the SK analysis of both Fully Contained
Events and Partially Contained Events [8,9]. The SK De-
tector Simulation is to be constructed without any con-
tradiction with the SK assumption on the direction.
From the point of orthodoxical Monte Carlo Simula-
tion, it seems to be unnatural for SK to impose such the
assumption that the direction of the incident neutrino is
the same as that of the emitted lepton ( hereafter, we call
this assumption simply ”the SK assumption on the di-
rection”) upon their Detector Simulation. Obviously, one
need not any assumption on the relation between the di-
rection of the incident neutrino and that of the emitted
lepton in any sense, if we develop the Monte Carlo Simu-
lation in a rigorous manner, which will be shown later in
the present paper.
In order to avoid any misunderstanding toward the SK
assumption on the direction we reproduce this assumption
from the original SK paper:
”However, the direction of the neutrino must be estimated
from the reconstructed direction of the products of the neu-
trino interaction. In water Cherenkov detectors, the direc-
tion of an observed lepton is assumed to be the direction of
the neutrino. Fig.11 and Fig.12 show the estimated corre-
lation angle between neutrinos and leptons as a function of
lepton momentum. At energies below 400 MeV/c, the lep-
ton direction has little correlation with the neutrino direc-
tion. The correlation angle becomes smaller with increas-
ing lepton momentum. Therefore, the zenith angle depen-
dence of the flux as a consequence of neutrino oscillation
is largely washed out below 400 MeV/c lepton momentum.
With increasing momentum, the effect can be seen more
clearly. ” [8] 4.
On the other hand, Ishitsuka states in his Ph.D thesis
which is exclusively devoted into the L/E analysis of the
caused by one-pion roduction, coherent pion production and
deep inelastic scattering. However, the discrimination among
the multi-ring structures is not so easy, which may lead the
worse estimation of energies as well as directions of the events
concerned.
4 It could be understood from this statement that SK justify
the validity of this assumption above 400 MeV/c. However, it is
not correct, because SK put ”to be proved ” as the proposition.
See, page 101 in their paper [8].
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 3
atmospheric neutrino from Super Kamiokande as follows:
” 8.4 Reconstruction of Lν
Flight length of neutrino is determined from the neu-
trino incident zenith angle, although the energy and the
flavor are also involved. First, the direction of neutrino is
estimated for each sample by a different way. Then, the
neutrino flight lenght is calclulated from the zenith angle
of the reconstructed direction.
8.4.1 Reconstruction of Neutrino Direction
FC Single-ring Sample
The direction of neutrino for FC single-ring sample is
simply assumed to be the same as the reconstructed direc-
tion of muon. Zenith angle of neutrino is reconstructed as
follows:
cosΘrecν = cosΘµ (8.17)
,where cosΘrecν and cosΘµ are cosine of the reconstructed
zenith angle of muon and neutrino, respectively. ” [9] 5.
In our understanding, SK Monte Carlo Simulation is
named usually as the Detector Simulation. It is, however,
noticed that the effect of the azimuthal angles of the emit-
ted leptons in QEL could not be taken into account in their
Simulation. As will be shown in later (see Section 3), this
effect greatly influences over the final zenith angle distri-
bution of the emitted leptons. Also, the back scattering
due to QEL can not be neglected for the rigorous deter-
mination of the direction of the incident neutrino, but this
effect could not be treated in the SK Detector Simulation,
which is beyond the application limitaition 6.
On the other hand, we could take into account these
effects correctly in our Monte Carlo Simulation which is
named as Time Sequential Simulation.
In the present paper, we carry out the full Monte Carlo
Time Sequential Simulation as exactly as possible, with-
out the SK assumption on the direction to clarify the prob-
lematic issue raised by SK. We carry out simulation which
starts from the opposite side of the Earth to the SK de-
tector. A neutrino sampled from the atmospheric neutrino
energy spectrum at the opposite side of the Earth tra-
verses through the medium with different densities in the
interior of the Earth and penetrates finally into the SK
5 It should be noticed that the SK assumption on the di-
rection may hold on the following possible two cases: [1] The
scattering angle of the emitted lepton is so small that the ef-
fect of the scattering angle could be neglected really. However,
in the present case, it could not be true from Fig. 1 and Fig.
2 and Table 1. [2] One may assert that the assumption could
not hold on individual case, but it could hold statistically af-
ter accumulation of large amount of the data. However, such
assertion should be verified. We verify such assumption could
not hold. See, Fig. 11 and Fig. 12 in the present paper.
6 SK have never clarified not only the details, but also the
principle and its validity on their Monte Carlo Simulation. We
hope disclosure of their Detector Simulation for open and fair
scientific discussion.
detector where the neutrino interactions occur. The emit-
ted energy of the individual lepton thus produced and its
direction are simulated exactly based on the probability
function of the cross sections concerned.
We finally show the zenith angle distribution of the
emitted leptons as well as that of the incident neutrinos
are quite different from corresponding ones of the SK. This
indicates that the SK assumption on the direction coud
not be a reliable estimator as for the determination of the
direction of the incident neutrino (See, section 5).
2 Cross Sections of Quasi Elastic Scattering
in the Neutrino Reaction and the Scattering
Angle of Charged Leptons.
We examine the following reactions due to the charged
current interaction (c.c.) from QEL.
νe + n −→ p+ e−
νµ + n −→ p+ µ−
ν̄e + p −→ n+ e+ (1)
ν̄µ + p −→ n+ µ+
The differential cross section for QEL is given as fol-
lows [6].
dσℓ(ℓ̄)(Eν(ν̄))
G2F cos
ν(ν̄)
A(Q2)±B(Q2)
C(Q2)
where
A(Q2) =
+ f1f2
+ g21
B(Q2) = (f1 + f2)g1Q
C(Q2) =
f21 + f
+ g21
The signs + and − refer to νµ(e) and ν̄µ(e) for charged
current (c.c.) interactions, respectively. The Q2 denotes
the four momentum transfer between the incident neu-
trino and the charged lepton. Details of other symbols are
given in [4].
The relation among Q2, Eν(ν̄), the energy of the in-
cident neutrino, Eℓ, the energy of the emitted charged
lepton (muon or electron or their anti-particles) and θs,
the scattering angle of the emitted lepton, is given as
Q2 = 2Eν(ν̄)Eℓ(1− cosθs). (3)
4 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
Fig. 1. Relation between the energy of the muon and its
scattering angle for different incident muon neutrino energies,
0.5 GeV, 1 GeV, 2 GeV, 5 GeV, 10 GeV and 100 GeV.
Also, the energy of the emitted lepton is given by
Eℓ = Eν(ν̄) −
. (4)
Now, let us examine the magnitude of the scattering
angle of the emitted lepton in a quantitative way, as this
plays a decisive role in determining the accuracy of the
direction of the incident neutrino, which is directly related
to the reliability of the zenith angle distribution of both
Fully Contained Events and Partially Contained Events in
By using Eqs. (2) to (4), we obtain the distribution
function for the scattering angle of the emitted leptons
and the related quantities by a Monte Carlo method. The
procedure for determining the scattering angle for a given
energy of the incident neutrino is described in the Ap-
pendix A. Fig. 1 shows this relation for muon, from which
we can easily understand that the scattering angle θs of
the emitted lepton ( muon here ) cannot be neglected.
For a quantitative examination of the scattering angle,
we construct the distribution function for θs of the emit-
ted lepton from Eqs. (2) to (4) by using a Monte Carlo
method.
Fig. 2 gives the distribution function for θs of the muon
produced in the muon neutrino interaction. It can be seen
that the muons produced from lower energy neutrinos are
scattered over wider angles and that a considerable part
of them are scattered even in backward directions. Simi-
lar results are obtained for anti-muon neutrinos, electron
neutrinos and anti-electron neutrinos.
Also, in a similar manner, we obtain not only the dis-
tribution function for the scattering angle of the charged
leptons, but also their average values < θs > and their
standard deviations σs. Table 1 shows them for muon neu-
trinos, anti-muon neutrinos, electron neutrinos and anti-
electron neutrinos. In the SK analysis, it is assumed that
the scattering angle of the charged particle is zero [8,9].
Fig. 2. Distribution functions for the scattering angle of the
muon for muon-neutrino with incident energies, 0.5 GeV, 1.0
GeV and 2 GeV. Each curve is obtained by the Monte Carlo
method (one million sampling per each curve).
3 Influence of Azimuthal Angle of Quasi
Elastic Scattering over the Zenith Angle of
both the Fully Contained Events and
Partially Contained Events
In the present section, we examine the effect of the az-
imuthal angles of the emitted leptons over their own zenith
angles for given zenith angles of the incident neutrinos 7.
For three typical cases (vertical, horizontal and diag-
onal), Fig. 3 gives a schematic representation of the re-
lationship between, θν(ν̄), the zenith angle of the incident
neutrino, and (θs, φ) a pair of scattering angle of the emit-
ted lepton and its azimutal angle.
From Fig. 3(a), it can been seen that the zenith angle
θµ(µ̄) of the emitted lepton is not influenced by its φ in
the vertical incidence of the neutrinos (θν(ν̄) = 0
o), as it
must be. From Fig. 3(b), however, it is obvious that the
influence of φ of the emitted leptons on their own zenith
angle is the strongest in the case of horizontal incidence
of the neutrino (θν(ν̄) = 90
o). Namely, one half of the
emitted leptons are recognized as upward going, while the
other half is classified as downward going ones. The di-
agonal case ( θν(ν̄) = 43
o) is intermediate between the
vertical and the horizontal. In the following, we examine
the cases for vertical, horizontal and diagonal incidence of
the neutrino with different energies, say, Eν(ν̄) = 0.5 GeV,
Eν(ν̄) = 1 GeV and Eν(ν̄) = 5 GeV.
The detailed procedure for the Monte Carlo simulation
is described in the Appendix A.
7 Throughout this paper, we measure the zenith angles of
the emitted leptons from the upward vertical direction of the
incident neutrino. Consequently, notice that the sign of our
direction is oposite to that of the SK ( our cos θν(ν̄) = - cos θν(ν̄)
in SK)
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 5
Table 1. The average values < θs > for scattering angle of the emitted charged leptons and their standard deviations σs for
various primary neutrino energies Eν(ν̄).
Eν(ν̄) (GeV) angle νµ(µ̄) ν̄µ(µ̄) νe ν̄e
(degree)
0.2 < θs > 89.86 67.29 89.74 67.47
σs 38.63 36.39 38.65 36.45
0.5 < θs > 72.17 50.71 72.12 50.78
σs 37.08 32.79 37.08 32.82
1 < θs > 48.44 36.00 48.42 36.01
σs 32.07 27.05 32.06 27.05
2 < θs > 25.84 20.20 25.84 20.20
σs 21.40 17.04 21.40 17.04
5 < θs > 8.84 7.87 8.84 7.87
σs 8.01 7.33 8.01 7.33
10 < θs > 4.14 3.82 4.14 3.82
σs 3.71 3.22 3.71 3.22
100 < θs > 0.38 0.39 0.38 0.39
σs 0.23 0.24 0.23 0.24
Fig. 3. Schematic view of the zenith angles of the charged
muons for diffrent zenith angles of the incident neutrinos, fo-
cusing on their azimuthal angles.
3.1 Dependence of the spreads of the zenith angle for
the emitted leptons on the energies of the emitted
leptons for different incident directions with different
energies
We give the scatter plots between the fractional energies
of the emitted muons and their zenith angle for a definite
zenith angles of the incident neutrino with different ener-
gies in Figs. 4 to 6. In Fig. 4, we give the scatter plots
for vertically incident neutrino with different energies 0.5
GeV, 1 GeV and 5 GeV . In this case, the relations between
the emitted energies of the muon and and their zenith an-
gles are unique, which comes from the definition of the
zenith angle of the emitted lepton. However, the densities
(frequencies of event number) along each curve is differ-
ent in position to position and depend on the energies of
the incident neutrinos. Generally speaking,densities along
curves become higher toward cos θµ(µ̄) = 1. In this case,
cos θµ(µ̄) is never influenced by the azimuthal angel in the
scattering by the definition 8.
Fig. 5 tells us that the horizontally incident neutrinos
give the most widely spread of the zenith angle distribu-
tion of the emitted lepton influenced by the azimuthal an-
gle. The more lower incident neutrino energies, the more
wider spreads of the emitted leptons. The diagonally in-
cident neutrinos give the intermediate distribution of the
emitted leptons between those of vertically incident neu-
trinos and horizontally incident neutrinos.
8 The zenith angles of the particles concerned are measured
from the vertical direction.
6 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=1(θν=0°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=1(θν=0°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=1(θν=0°)
Fig. 4. The scatter plots between the fractional energies of the produced muons and their zenith angles for vertically incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=0(θν=90°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=0(θν=90°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=0(θν=90°)
Fig. 5. The scatter plots between the fractional energies of the produced muons and their zenith angles for horizontally incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
(a) (b) (c)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=0.5GeV
cosθν=0.731(θν=43°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=1GeV
cosθν=0.731(θν=43°)
0 0.2 0.4 0.6 0.8 1
Eµ / Eν
Eν=5GeV
cosθν=0.731(θν=43°)
Fig. 6. The scatter plots between the fractional energies of the produced muons and their zenith angles for diagonally incident
muon neutrinos with 0.5 GeV, 1 GeV and 5 GeV, respectively. The sampling number is 1000 for each case.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 7
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=1(θν=0°)
average=0.262
s.d.=0.547
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=1(θν=0°)
average=0.590
s.d.=0.439
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=1(θν=0°)
average=0.978
s.d.=0.067
Fig. 7. Zenith angle distribution of the muon for the vertically incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=0(θν=90°)
average=0.003
s.d.=0.564
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=0(θν=90°)
average=0.001
s.d.=0.480
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=0(θν=90°)
average=0.006
s.d.=0.141
Fig. 8. Zenith angle distribution of the muon for the horizontally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
(a) (b) (c)
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=0.5GeV
cosθν=0.731(θν=43°)
average=0.189
s.d.=0.556
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=1GeV
cosθν=0.731(θν=43°)
average=0.432
s.d.=0.463
−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
cosθµ
muon neutrino
Eν=5GeV
cosθν=0.731(θν=43°)
average=0.715
s.d.=0.103
Fig. 9. Zenith angle distribution of the muon for the diagonally incident muon neutrino with 0.5 GeV, 1 GeV and 5 GeV,
respectively. The sampling number is 10000 for each case. SK stand for the corresponding ones under the SK assumption.
8 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
3.2 Zenith angle distribution of the emitted lepton for
the different incidence of the neutrinos with different
energies
In Figs. 7 to 9, we give the zenith angle distributions of
the emitted muons for the given direction of the incident
neutrinos with different energies of the neutrino. These
figures are obtained through summation on the energies
of the emitted muons for their definite zenith angles in
Figs. 4 to 6.
In Figs. 7(a) to 7(c), we give the zenith angle distri-
bution of the emitted muon for the case of vertically inci-
dent neutrinos with different energies, say, Eν = 0.5 GeV,
Eν = 1 GeV and Eν = 5 GeV.
Comparing the case for 0.5 GeV with that for 5 GeV,
we understand the big contrast between them as for the
zenith angle distribution. The scattering angle of the emit-
ted muon for 5 GeV neutrino is relatively small (See, Table
1) that the emitted muons keep roughly the same direction
as their original neutrino. In this case, the effect of their
azimuthal angle on the zenith angle is also small. However,
in the case for 0.5 GeV which is the dominant energy for
Fully Contained Events in the Superkamiokande, there is
even a possibility for the emitted muon to be emitted in
the backward direction due to the large angle scattering,
the effect of which is enhanced by their azimuthal angle.
The most frequent occurrence in the backward scatter-
ing of the emitted muon appear in the horizontally inci-
dent neutrino as shown in Figs. 8(a) to 8(c). In this case,
the zenith angle distribution of the emitted muon should
be symmetrical to the horizontal direction. Comparing the
case for 5 GeV with those both for ∼0.5 GeV and for ∼1
GeV, even 1 GeV incident neutrinos lose almost the orig-
inal sense of the incidence if we measure it by the zenith
angle of the emitted muon. Figs. 9(a) to 9(c) for the di-
agonally incident neutrino tell us that the situation for
diagonal cases lies between the case for the vertically in-
cident neutrino and that for horizontally incident one.
4 Zenith Angle Distribution of Fully
Contained Events and Partially Contained
Events for a Given Zenith Angle of the
Incident Neutrino, Taking Their Energy
Spectrum into Account
In the previous sections, we discuss the relation between
the zenith angle distribution of the incident neutrino with
a single energy and that of the emited muons produced by
the neutrino for the different incident direction. In order to
apply our motivation around the uncertainty of the SK as-
sumption on the direction for Fully Contained Events and
Partially Contained Events, we must consider the effect of
the energy spectrum of the incident neutrino. The Monte
Carlo simulation procedure for this purpose are given in
the Appendix B.
In Fig. 10, we give the zenith angle distributions of the
sum of µ+(µ̄) and µ− for a given zenith angle of ν̄µ̄ and
νµ, taking into account primary neutrino energy spectrum
at Kamioka site.
In Table 2, the average values for cosθµ+µ̄ and their
standard deviation for different incidences of the incident
neutrinos with different energies are presented 9.
In the SK case, their average values are given by cosθν(ν̄)
themselves by definition and, consequently, the standard
deviations are zero under the assumption, because the SK
assumption is of the delta function for the incidence direc-
tion. They are shown in the bottom line of Table 2. In the
second line from the bottom in this table, we give the av-
erage values and their standard deviations for cos θν+ν̄ ob-
tained under the inclusion of the energy spectrum for pri-
mary neutrinos. Thus, we found these values correspond to
those for incident neutrino with the effective single energy
between 0.5 GeV and 1 GeV. If we compare the average
energies and the standard deviations for the inclusion of
incident neutrino energy spectrum with those under the
SK assumption, it is easily understood that SK assump-
tion does not represent real zenith angle distribution of
the emitted muon.
5 Relation between the Zenth angle
Distribution of the Incident Neutrinos and
that of the emitted leptons
Now, we extend the results for the definite zenith angle
obtained in the previous section to the case in which we
consider the zenih angle distribution of the incident neu-
trinos totally.
Here, we examine the real correlation between cos θν
and cos θµ, by peforming the exact Monte Carlo simula-
tion.
The detail for the simulation procedure is given the
Appendix C.
In Fig. 11 we classsify the correlation between cos θν
and cos θµ according to the different energy range of the
incident muon neutrinos. It should be noticed that the
SK assumption on cos θν = cos θµ is roughly hold only for
Eν ≥ 5 GeV, but the widths in cos θµ for the definite cos θν
near cos θν =0 (for horizontally incident neutrino ) are
much larger than those near cos θν =1 ( for the vertically
incident neutrino). Of course, this is due to the effect of
the azimutal angle in QEL which could not be derived by
the SK simulation (DETECTOR SIMULATION ). Such
tendencies become more remarkable in Eν ≤ 5 GeV and
in these energies the SK assumption on the direction does
not hold any more.
In Fig. 12, we classify the correlation between cos θν and
cos θµ according to the different energy range of Eµ. The
similar argument on Fig. 11 can be done on the case of
Fig.12
9 Notice that the difference in the corresponding quantites
between the case for single energy and the case for the energy
spectrum. The formers are given in the µ−,while th latter is
given in µ− and µ+. However, such the difference does not
change the essential recognition.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 9
Table 2. Average values and their standerd deviations in cosθµ+µ̄ for the zenith angle distributions of the muons with different
primary energies of the insident neutrinos.
Vertical Diagonal Horizontal
cosθν+ν̄ = 1 (0
◦) cosθν+ν̄ = 0.731 (43
◦) cosθν+ν̄ = 0 (90
Eν+ν̄(GeV) cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄ cos θµ+µ̄ σcos θµ+µ̄
0.5 0.262 0.547 0.189 0.556 -0.003 0.564
1.0 0.590 0.439 0.432 0.463 0.001 0.480
2.0 0.581 0.250 0.623 0.290 0.001 0.325
5.0 0.978 0.067 0.715 0.103 0.006 0.141
Spectrum∗ 0.468 0.531 0.339 0.519 -0.005 0.500
SK∗∗ 1.00 0.00 0.731 0.000 0.000 0.000
(a) (b) (c)
−1 −0.5 0 0.5 1
cosθµ+µ− 
µ+ and µ−
cosθν+ν − =1(θν+ν −=0°)
Avg.=0.81
S.D.=0.30
−1 −0.5 0 0.5 1
µ −  
cosθµ+µ −
µ+ and µ−
cosθν+ν −=0(θν+ν −=90°)
Avg.=0.00
S.D.=0.34
−1 −0.5 0 0.5 1
µ −  
cosθµ+µ −
µ+ and µ−
cosθν+ν −=0.73(θν+ν −=43°)
Avg.=0.60
S.D.=0.33
Fig. 10. Zenith angle distribution of µ− and µ+ for ν and ν̄ for the incident neutrinos with the vertical, horizontal and diagonal
directions, respectively. The overall neutrino spectrum at Kamioka site is taken into account. The sampling number is 10000
for each case. SK stand for the corresponding ones under the SK assumption.
0 0.2 0.4 0.6 0.8 1
cosθν
Eµ < 0.5 GeV
0.5 < Eµ < 1 GeV
Eµ > 1 GeV
0 0.2 0.4 0.6 0.8 1
cosθν
1 < Eµ < 2 GeV
2 < Eµ < 5 GeV
Eµ > 5 GeV
Fig. 12. Correlation diagrams between cos θν and cos θµ for different muon energy ranges.
10 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
0 0.2 0.4 0.6 0.8 1
cosθν
1 < Eν < 2 GeV
2 < Eν < 5 GeV
Eν > 5 GeV
Fig. 11. Correlation Diagram between cos θν and cos θµ for
different neutrino energy regions.
Thus, it could be surely concluded from Fig. 11 and
Fig. 12 that the SK assumption on the direction never
holds as a good estimator for the determination of the
directions of the incident neutrinos.
In order to obtain the zenith angle distribution of the
emitted leptons for that of the incident neutrinos, we di-
vide the cosine of the zenith angle of the incident neutrino
into twenty regular intervals from cos θν = 0 to cos θν = 1.
For the given interval of cos θν , we carry out the exact
Monte Carlo simulation, the detail of which is give in the
Appendix D and obtain the cosine of the zenith angle of
the emitted leptons, taking account of the geometry for
surronding the SK detector.
Thus, for each interval of cos θν , we obtain the corre-
sponding zenith angle distribution of the emitted leptons.
Then, we sum up these corresponding ones over all zenth
angles of the incident neutrinos and we finally obtain the
relation between the zenith angle distribution for the in-
cident neutrinos and that for the emitted leptons.
In a similar manner, we could obtain between cos θν̄
and cos θµ̄ for anti-neutrinos. The situation for anti-neutrinos
is essentially same as that for neutrinos.
Here, we examine the zenith angle distribution of the
muons from both upward neutrinos and downward ones
in the case that neutrino oscillation does not exist.
By performing the procedures described in Appendix
C, a pair of sampling ( cos θν+ν̄ , Eν+ν̄ ) gives a pair of (
cos θµ+µ̄, Eµ+µ̄ ). In Fig. 13, we give the zenith angle dis-
tribution of the upward neutrinos ( the sum of νµ and ν̄µ )
which is constructed from the energy spectra for different
cos θν+ν̄ . (see, Honda et. al. [10] and Appendix B)
Upward neutrinos may produce even downward lep-
tons due to both the backscattering effect and the effect
of azimuthal angle on larger forward scattering for the in-
teraction concerned (see, Figure 3 and Figures 4 to 6 in
−1 −0.5 0 0.5 1
cosθν+ν −
µ+ ,µ−  (pµ>0.4 GeV/c)
upward neutrinos
no oscillation
from upward neutrinos
cosθµ+µ −, 
Fig. 13. The relation between the zenith angle distribution
of the incident neutrino and corresponding ones of the emitted
lepton
the text). As the result of it, the zenith angle distribution
of the emitted muons for the upward neutrino may leak in
the downward direction. From Figure 13, it is very clear
that the shape of the zenith angle distribution for the in-
cident neutrinos is quite different from that of the emitted
muons produced by these neutrinos. If the SK assumption
on the direction statistically holds, the zenith angle dis-
tribution for the emitted muons should coincide totally
with that of the incident neutrinos. In other words, one
may say that the zenith angle distribution for the emitted
muons should be understood as that of the incident neu-
trino under the SK assumption on the direction. However,
the muon spectrum is distinctively different from the real
(computational) incident neutrino spectrum as shown in
the figure. Thus we conclude that SK assumption on the
direction leads to the wrong conclusion on the neutrino
oscillation. The further examination on the experimental
data obtained by SK will be carried out in the subsequent
papers.
It is, further, noticed that upward neutrino energy
spectrum in the figure biggest near cos θν+ν̄ = 0 and the
smallest near cos θν+ν̄ = 1, which reflects from the en-
hancement of the primary incident neutrino energy spec-
trum from the inclined direction and is independent on the
neutrino oscillation, while in SK opinion, such tendency
may be favor of the existence of neutrino oscillation.
6 Discussions and Conclusion
In order to extract the definite conclusion on the neutrino
oscillation from the experiment by cosmic ray neutrinos
whose intensity as well as interaction with the substance
are both very weak, first of all, one should analyze the
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 11
most clear cut and ambiguity free events. Among neutrino
events analyzed by SK, the most clear cut events are Single
Ring Events, such as electron-like events and muon-like
events in Fully Contained Events which are generated by
QEL. These events are provided with simplicity due to
single ring and all possible measureable physical quantities
are confined in the detector.
Furthermore, QEL is the most dominant source for
neutrino events which are generated in the SK detector.
This is the reason why we examine the QEL events excul-
sively in present and subsequent papers.
If the neutrino oscillation really exists, the most clear
cut evidence surely appears in the analysis of single ring
events due to QEL in Fully Contained Events and one does
not need the analysis of any other type of events, such as
single ring events in Partially Contained Events, multi-
ring events in either Fully Contained Events or Partially
Contained Evetns, all of which include inevitably ambigu-
ities for the interpretation and show merely sub-evidences
compared with that from the single ring events due to
QEL in Fully Contained Events.
SK analyze the zenith angle distribution of the inci-
dent neutrinos under the asssumption that the direction
of the incident neutrino is the same as that of the emitted
lepton. We conclude that this assumption is supplemented
by their Monte Carlo Simulation named as Detector Sim-
ulation 10.
In the present paper, we adopt Time Sequential Simu-
lation which starts from the incident neutrino energy spec-
trum on the opposite side of the Earth to the SK detec-
tor and simulate all posible physical processes which are
connected with the zenith angle distribution of the in-
cident neutrinos according to their probability functions
concerned for the examination on the validity of the SK
assumption on the direction.
Concretely speaking, we take the following treatment,
(i) the stochastic treatments of the scattering angle of the
emitted lepton in QEL, including the scattering on the
backward as well as the azimuthal angle, which could not
be treated in Detector Simulation, (ii) the stochastic treat-
ment on the zenith angle distribution of the emitted lep-
ton, considering the incident neutrino energy spectrum,
(iii) the stochastic treatment on the detection of the QEL
events inside the SK detector.
Furthermore, the discrimination between Fully Con-
tained Events and Partially Contained Events is only pos-
sible in the Time Sequential Simulation, because the events
concerned may be classified into different categories by
chance, Fully Contained Events and Partially Contained
Events due to different occurring points and different di-
rections.
10 The SK Detector Simulation for obtaing the zenith angle
distribtuion of the incident neutrino as for the neutrino oscilla-
tiom has never been disclosed in their papers, even in the Ph.D
thesis. Consequently, this is only our onjecture as for utiliza-
tion of the SK Detector Simulation . A clear thing is only that
SK impose the proposittion that the direction of the incident
neutrino is the same as that of the emitted lepton upon the
neutrino oscillation analysis.
The conclusions thus obtained are as follows:
(1) The zenith angle distributions of the emitted lepton
in QEL for the incident neutrino with both the def-
inite zenith angle and the definite energy are widely
spread, particularly, into even the backward region due
to partly pure backscattering and partly the combina-
tion of the azimuthal angle with the slant direction
of the incident neutrinos. However, for every incident
neutrino with a definte zenith angle, SK give the same
definite zenith angle to the emitted lepton. Already in
this stage, the SK assumption on the direction does
not hold.
(2) Taking account of the incident neutrino energy spec-
trum and simulating all physical processes concerned,
we obtain the zenith angle distribution of the emitted
leptons for the incident neutrino with a definite zenith
angle. It is proved that the SK assumption on the di-
rection does not hold again.
(3) The correlation diagrams between cos θν and cos θµ
show that SK assumption does noty hold well even
for higher energies of the incident neutrinos, and it
is shown that the correlation between them become
weaker in more inclined incident neutrinos due to the
effect of the azimuthal angle in QEL.
(4) Taking into account the detection efficiency for the
events concerned in the simulation for upward neu-
trinos and anti-neutrinos, we obtain the zenith angle
distribution of the leptons ( muons plus anti-muons
). According to the SK Assumption on the direction,
the zenith angle distribution is the same as that of the
incident neutrinos. However, the original zenith angle
distribution of incident neutrino is found to be quite
different from that derived from that of leptons. This is
the final conclusion that SK have not measured the di-
rection of the incident neutrinos reliably, which is quite
independent on either the existence or non-existence of
the neutrino oscillation.
(5) The SK assume that the Partially Contained Events
exclusively belong to the muon-like event. However,
such the assumption lacks in theoretical background.
Electron events can also contribute to the Partially
Contained Events under some geometrical condition,
for example, partly coming from the transformation
by Eq.(A.5). The quantitative examination on the Par-
tially Contained Events among the electron-like event
will be published elsewhere.
In subsequent papers, we will give the relation between
the zenith angle distributions of the incident neutrinos
and the corresponding muons in the cases with and with-
out neutrino oscillation, including downward neutrino and
will examine whether it is possible to or not to detect the
neutrino oscillation by using atmospheric neutrino.
12 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
In the following Appendices we give the concrete Monte
Carlo Simulations, namely, the details of our Time Se-
quential Simulation.
A Appendix: Monte Carlo Procedure for the
Decision of Emitted Energies of the Leptons
and Their Direction Cosines
Here, we give the Monte Carlo Simulation procedure for
obtaining the energy and its direction cosines, (lr,mr, nr),
of the emitted lepton in QEL for a given energy and its
direction cosines, (l,m, n), of the incident neutrino.
The relation among Q2, Eν+ν̄ , the energy of the inci-
dent neutrino, Eℓ, the energy of the emitted lepton (muon
or electron or their anti-particles) and θs, the scattering
angle of the emitted lepton, is given as
Q2 = 2Eν(ν̄)Eℓ(ℓ̄)(1− cosθs). (A·1)
Also, the energy of the emitted lepton is given by
Eℓ(ℓ̄) = Eν(ν̄) −
. (A·2)
Procedure 1
We decide Q2 from the probability function for the differ-
ential cross section with a given Eν(ν̄) (Eq. (2) in the text)
by using the uniform random number, ξ, between (0,1) in
the following
Pℓ(ℓ̄)(Eν(ν̄), Q
2)dQ2, (A·3)
where
Pℓ(ℓ̄)(Eν(ν̄), Q
dσℓ(ℓ̄)(Eν(ν̄), Q
dσℓ(ℓ̄)(Eν(ν̄), Q
(A·4)
From Eq. (A·1), we obtain Q2 in histograms together with
the corresponding theoretical curve in Fig. 14. The agree-
ment between the sampling data and the theoretical curve
is excellent, which shows the validity of the utlized proce-
dure in Eq. (A·3) is right.
Procedure 2
We obtain Eℓ(ℓ̄) from Eq. (A·2) for the given Eν(ν̄) and
Q2 thus decided in the Procedure 1.
Procedure 3
We obtain cos θs, cosine of the the scattering angle of the
emitted lepton, for Eℓ(ℓ̄) thus decided in the Procedure 2
from Eq. (A·1) .
Procedure 4
We decide φ, the azimuthal angle of the scattering lepton,
0 0.2 0.4 0.6 0.8 1
1 GeV
2 GeV
Eν=0.5 GeV
Fig. 14. The reappearance of the probability function for QEL
cross section. Histograms are sampling results, while the curves
concerned are theoretical ones for given incident energies.
which is obtained from
φ = 2πξ. (A·5)
Here, ξ is a uniform random number (0, 1).
As explained schematically in the text(see Fig. 3 in the
text), we must take account of the effect due to the az-
imuthal angle φ in the QEL to obtain the zenith angle
distribution of both Fully Contained Events and Partially
Contained Events correctly.
Procedure 5
The relation between direction cosines of the incident neu-
trinos, (ℓν(ν̄),mν(ν̄), nν(ν̄)), and those of the corresponding
emitted lepton, (ℓr,mr, nr), for a certain θs and φ is given
ℓ2 +m2
ℓ2 +m2
ℓν(ν̄)
ℓ2 +m2
ℓ2 +m2
mν(ν̄)
ℓ2 +m2 0 nν(ν̄)
sinθscosφ
sinθssinφ
cosθs,
(A·6)
where nν(ν̄) = cosθν(ν̄), and nr = cosθℓ. Here, θℓ is the
zenith angle of the emitted lepton.
The Monte Carlo procedure for the determination of θℓ
of the emitted lepton for the parent (anti-)neutrino with
given θν(ν̄) and Eν(ν̄) involves the following steps:
We obtain (ℓr,mr, nr) by using Eq. (A·6). The nr is
the cosine of the zenith angle of the emitted lepton which
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 13
Fig. 15. The relation between the direction cosine of the
incident neutrino and that of the emitted charged lepton.
should be contrasted to nν , that of the incident neutrino.
Repeating the procedures 1 to 5 just mentioned above, we
obtain the zenith angle distribution of the emitted leptons
for a given zenth angle of the incident neutrino with a def-
inite energy.
In the SK analysis, instead of Eq. (A·6), they assume
nr = nν(ν̄) uniquely for Eµ(µ̄) ≥ 400 MeV.
B Appendix: Monte Carlo Procedure to
Obtain the Zenith Angle of the Emitted
Lepton for a Given Zentith Angle of the
Incident Neutrino
The present simulation procedure for a given zenith an-
gle of the incident neutrino starts from the atmospheric
neutrino spectrum at the opposite site of the Earth to
the SK detector. We define, Nint(Eν(ν̄), t, cosθν(ν̄)), the in-
teraction neutrino spectrum at the depth t from the SK
detector in the following way
Nint(Eν(ν̄), t, cosθν(ν̄)) = Nsp(Eν(ν̄), cos θν(ν̄))×
λ1(Eν(ν̄), t1, ρ1)
× · · · ×
λn(Eν(ν̄), tn, ρn)
(B·1)
Here, Nsp(Eν(ν̄), cos θν(ν̄)) is the atmospheric (anti-)
neutrino spectrum for the zenith angle at the opposite
surface of the Earth.
Here λi(Eν(ν̄), ti, ρi) denotes the mean free path due
to the neutrino(anti neutrino) with the energy Eν(ν̄) from
QEL at the distance, ti, from the opposite surface of the
Earth inside whose density is ρi.
The procedures of the Monte Carlo Simulation for the
incident neutrino(anti neutrino) with a given energy,Eν(ν̄),
whose incident direction is expressde by (l,m, n) are as fol-
lows.
Procedure A
For the given zenith angle of the incident neutrino, θν(ν̄),
we formulate, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the produc-
tion function for the neutrino flux to produce leptons at
the Kamioka site in the following
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
= σℓ(ℓ̄)(Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄), (B·2)
where
σℓ(ℓ̄)(Eν(ν̄)) =
dσℓ(ℓ̄)(Eν(ν̄), Q
dQ2. (B·3)
Each differential cross section above is given in Eq. (2) in
the text.
Utilizing, ξ, the uniform random number between (0,1),
we determine Eν(ν̄), the energy of the incident neutrino in
the following sampling procedure
∫ Eν(ν̄)
Eν(ν̄),min
Pd(Eν(ν̄), t, cos θν(ν̄)(ν̄))dEν(ν̄), (B·4)
where
Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
. (B·5)
14 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
In our Monte Carlo procedure,
the reproduction of, Pd(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), the nor-
malized differential neutrino interaction probability func-
tion, is confirmed in the same way as in Eq. (A·4).
Procedure B
For the (anti-)neutrino concerned with the energy ofEν(ν̄),
we sample Q2 utlizing ξ3, the uniform random number be-
tween (0,1). The Procedure B is exactly the same as in the
Procedure 1 in the Appendix A.
Procedure C
We decide, θs, the scattering angle of the emitted lepton
for given Eν(ν̄) and Q
2. The procedure C is exactly the
same as in the combination of Procedures 2 and 3 in the
Appendix A.
Procedure D
We randomly sample the azimuthal angle of the charged
lepton concerned. The Procedure D is exactly the same as
in the Procedure 4 in the Appendix A.
Procedure E
We decide the direction cosine of the charged lepton con-
cerned. The Procedure E is exactly the same as in the
Procedure 5 in the Appendix A.
We repeat Procedures A to E until we reach the de-
sired trial number.
C Appendix: Correlation between the Zenith
Angles of the Incident Neutrinos and Those
of the Emitted Leptons
Procedure A
By using, Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄), which is defined
in Eq. (B·2),
we define the spectrum for cos θν(ν̄) in the following.
I(cos θν(ν̄))d(cos θν(ν̄)) =
d(cos θν(ν̄))
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄).(C·1)
By using Eq. (C·2) and ξ, a sampled uniform random num-
ber between (0,1), then we could determine cos θν(ν̄) from
the following equation
∫ cos θν(ν̄)
Pn(cos θν(ν̄))d(cos θν(ν̄)), (C·2)
where
Pn(cos θν(ν̄)) = I(cos θν(ν̄))
I(cos θν(ν̄))d(cos θν(ν̄)).
(C·3)
Procedure B
For the sampled d(cos θν(ν̄)) in the Procedure A, we sam-
ple Eν(ν̄) from Eq.(C·4) by using ξ, the uniform randum
number between (0,1)
∫ Eν(ν̄)
Eν(ν̄),min
Ppro(Eν(ν̄), cos θν(ν̄))dEν(ν̄), (C·4)
where
Ppro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄) =
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
∫ Eν(ν̄),max
Eν(ν̄),min
Npro(Eν(ν̄), t, cos θν(ν̄))dEν(ν̄)
. (C·5)
Procedure C
For the sampled Eν(ν̄) in the Procedure B, we sample
Eµ(µ̄) from Eqs. (A·2) and (A·3). For the sampled Eν(ν̄)
and Eµ(µ̄), we determine cos θs, the scattering angle of the
muon uniquely from Eq. (A·1).
Procedure D
We determine, φ, the azimuthal angle of the scattering lep-
ton from Eq. (A·5) by using ξ, an uniform randum number
between (0,1).
Procedure E
We obtain cos θµ(µ̄) from Eq. (A·6). As the result, we ob-
tain a pair of (cos θν(ν̄), cos θµ(µ̄)) through Procedures A
to E. Repeating the Procedures A to E, we finally the cor-
relation between the zenith angle of the incident neutrino
and that of the emitted muon.
E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK 15
D Appendix: Detection of the Neutrino
Events in the SK Detector and Their
Interaction Points
The plane ABCD is always directed vertically to the di-
rection of the incident neutrino with a given zenith an-
gle, which is shown in Fig. 16. The rectangular ABCDE-
FGH encloses the SK detector whose radius and height
is denoted by R and H, respectively. The width and the
height of the plane ABCD for a given zenith angle, θν(ν̄),
is given as, R and Rcos θν(ν̄) + H sin θν ,respectively, which
are shown in Fig. 16-c.
Now, let us estimate the ratio of the number of the neu-
trino events inside the SK detector to that in the rectan-
gular ABCDEFGH. As the number of the neutrino events
inside some material is proportional to the number of the
nucleons in the material concerned. The number of the
nucleons inside the SK detector (ρ = 1) is given as
Nsk =
NavogaR
2H, (D·1)
whereNavoga denotes the Avogadro number, and the num-
ber of the nucleons in the exterior of the SK detector inside
ABCDEFGH is given as
Fig 16-a Fig 16-b
Injection Point
(X,Y)
Neutrino
Injection Point
Fig 16-c Fig 16-d
Fig. 16. Sampling procedure for neutrino evens injected into
the detector
Nextr(cos θν(ν̄) ) = ρNavoga
R2H +
R(H2+R2) sin θν(ν̄) cos θν(ν̄)
, (D·2)
where ρ is the density of the rock which surrounds the SK
detector.
Then, the total number of the target in the rectangu-
larABCDEFGH is given as
Ntot(cos θν(ν̄)) = Nsk +Nextr(cos θν(ν̄)). (D·3)
Here, we take 2.65, as ρ (standard rock).
Then, Rtheor, the ratio of the number of the neutrino
events in the SK detector to that in the rectangular ABCDE-
FGH is given as
Rtheor(cos θν(ν̄)) = Nsk/Ntot(cos θν(ν̄)). (D·4)
We obtain Rtheor for different values of cos θν(ν̄) given in
the Table 3.
Here, we simulate neutrino events occured in the rectangu-
lar ABCDEFH, by using the atmospheric neutrino beam
which falls down on the plane ABCD. Thus, Nsmaple, the
sampling number of the (anti-)neutrino events inside the
rectangular ABCDEFG for a given cos θν(ν̄) is given as
Nsample(cos θν(ν̄)) = Ntot(cos θν(ν̄))×
∫ Eν(ν̄),max
Eν(ν̄),min
ℓ(ℓ̄)
(Eν(ν̄))Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄)
(D·5)
where σℓ(ℓ̄)(Eν(ν̄)) is the total cross section for (anti-)neutrino
due to QEL, and Nint(Eν(ν̄), t, cosθν(ν̄))dEν(ν̄) is the dif-
ferential nutrino energy spectrum for the definite zenith
angle, θν(ν̄), in the plane ABCD. The injection points of
the neutrinos in the plane ABCD are distributed over the
plane randomly and uniformely and the injection points
are determined from a pair of the uniform random num-
bers between (0,1). They penetrate into the rectangular
ABCDEFGH from the injection point in the plane ABCD
and some of them may penetrate into the SK detector or
may not, which depend on their injection point.
In the neutrino events which penetrate into the SK
detectorr, their geometrical total track length, Ttrack, are
devided into three parts
Ttrack = Tb + Tsk + Ta, (D·6)
where Tb denotes the track length from the plane ABCD
to the entrance point of the SK detector, Tsk denotes the
track length inside the SK detector, and Ta denotes the
track length from the escaping point of the SK detector
to the exit point of the rectangular ABCDEF, and thus
16 E.Konishi et. al.,: The Reliability on the Direction of Neutrino in SK
Table 3. Occurrence probabilities of the neutrino events in-
side the SK detector for different cos θν ’s. Comparison between
Rtheor and Rmonte. The sampling numbers for the Monte Carlo
Simulation are, 1000, 10000, 100000, respectively.
cos θν Rtheor Rmonte
Sampling Number
1000 10000 100000
0.000 0.58002 0.576 0.5750 0.57979
0.100 0.41717 0.425 0.4185 0.41742
0.200 0.32792 0.353 0.3252 0.32657
0.300 0.27324 0.282 0.2731 0.27163
0.400 0.23778 0.223 0.2329 0.23582
0.500 0.21491 0.206 0.2063 0.21203
0.600 0.20117 0.197 0.1946 0.19882
0.700 0.19587 0.193 0.1925 0.19428
0.800 0.20117 0.198 0.2002 0.20001
0.900 0.22843 0.230 0.2248 0.22803
1.00 0.58002 0.557 0.5744 0.57936
Ttrack denotes the geometrical length of the neutrino con-
cerned in the rectangular ABCDEFGH.
By the definition, the neutrinos concerned with Ttrack
interact surely somewhere along the Ttrack. Here, we are
interested only in the interaction point which ocuurs along
Tsk. We could determine the interaction point in the Tsk
in the following.
We define the following quantities for the purpose.
Tweight = Tsk + ρ(Tb + Ta), (D·7)
ρav = Tweight/Ttrack, (D·8)
ξρ = ρav/ρ, (D·9)
ξsk = Tsk/Tweight. (D·10)
The flow chart for the choice of the neutrino events in
the SK detector and the determination of the interaction
points inside the SK detector is given in Fig. 17. Thus, we
obtain neutrino events whose occurrence point is decided
in the SK detector in the following.
xf = x0, (D·11)
yf = y0 + ξTsk sin θν(ν̄) (D·12)
zf = z0 + ξTsk cos θν(ν̄). (D·13)
If we carry out the Monte Carlo Simulation, following
the flow chart in Fig. 17, then, we obtain Nevent, the num-
ber of the neutrino events generasted in the SK detector.
The ratio of the selected events to the total trial is given
Rmonte(cos θν(ν̄)) = Nevent(cos θν(ν̄))/Nsample(cos θν(ν̄)).
(D·14)
Comparison between Rtheor and Rmonte in Table 3 shows
that our Monte Carlo procedure is valid.
References
1. Kasuga, S. et al., Phys. Lett. B374 (1996) 238.
N Nsample
N=N+1
N=N+1
Entry
Determination of
Point in the plane ABCD
by using 1 and 2
Judgement on the
Event’s Entering
Determination of
Ta,Tb and Tsk
Determination of Tweight
av, and sk
Determination of e
interaction point of
e events inside SK
Fig. 17. Flow Chart for the determination of the interaction
points of the neutrino events inside the detector
2. Ashie,Y. et al., Phys. Rev. D 71 (2005) 112005.
3. Ashie,Y. et al., Phys. Rev. Lett.93 (2004) 101801.
4. Renton, P., Electro-weak Interaction, Cambridge University
Press (1990). See p. 405.
5. D.Rein and L.M.Sehgal, Ann. of Phys. 133 (1981) 1780.
6. D.Rein and L.M.Sehgal Nucl. Phys. B84 (1983) 29.
7. R.H.Gandhi et. al. Astropart. Phys. 5 (1996) 81.
8. Kajita, T. and Totsuka, Y. Rev. Mod. Phys., 73 (2001) 85.
See p. 101.
9. Ishitsuka, M., Ph.D thesis, University of Tokyo (2004). See
p. 138.
10. Honda, M., et al., Phys. Rev. D 52 (1996) 4985
	Introduction
	Cross Sections of Quasi Elastic Scattering in the Neutrino Reaction and the Scattering Angle of Charged Leptons.
	Influence of Azimuthal Angle of Quasi Elastic Scattering over the Zenith Angle of both the Fully Contained Events and Partially Contained Events
	Zenith Angle Distribution of Fully Contained Events and Partially Contained Events for a Given Zenith Angle of the Incident Neutrino, Taking Their Energy Spectrum into Account
	Relation between the Zenth angle Distribution of the Incident Neutrinos and that of the emitted leptons
	Discussions and Conclusion
	Appendix: Monte Carlo Procedure for the Decision of Emitted Energies of the Leptons and Their Direction Cosines 
	Appendix: Monte Carlo Procedure to Obtain the Zenith Angle of the Emitted Lepton for a Given Zentith Angle of the Incident Neutrino
	Appendix: Correlation between the Zenith Angles of the Incident Neutrinos and Those of the Emitted Leptons
	Appendix: Detection of the Neutrino Events in the SK Detector and Their Interaction Points
