Curiosity: This paper is mentioned in two different Physics Nobel P...
Atmospheric neutrinos are produced in hadronic showers that result ...
There are three flavours of neutrinos (electron, muon, tau): \begi...
It is very hard to detect neutrinos. Neutrinos are electrically neu...
Let us compute the oscillation probability. Consider a two Neutrino...
Figure 1 (bottom) shows a strong asymmetry for $\mu$-like events de...
Curiosity: In Particle Physics, there is a convention of a **five-s...
Phase space *("allowed solutions")* for $\Delta m^{2}$ and $\sin^2(...
Zenith angle distributions for $e$-like and $\mu$-like events. The ...
Figure 4 shows the ratio of data *"real"* events and Monte Carlo ex...
The conclusion states that the observed events are consistent with ...
VOLUME
81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
UGUST
1998
Evidence for Oscillation of Atmospheric Neutrinos
Y. Fukuda,
1
T. Hayakawa,
1
E. Ichihara,
1
K. Inoue,
1
K. Ishihara,
1
H. Ishino,
1
Y. Itow,
1
T. Kajita,
1
J. Kameda,
1
S. Kasuga,
1
K. Kobayashi,
1
Y. Kobayashi,
1
Y. Koshio,
1
M. Miura,
1
M. Nakahata,
1
S. Nakayama,
1
A. Okada,
1
K. Okumura,
1
N. Sakurai,
1
M. Shiozawa,
1
Y. Suzuki,
1
Y. Takeuchi,
1
Y. Totsuka,
1
S. Yamada,
1
M. Earl,
2
A. Habig,
2
E. Kearns,
2
M. D. Messier,
2
K. Scholberg,
2
J. L. Stone,
2
L. R. Sulak,
2
C. W. Walter,
2
M. Goldhaber,
3
T. Barszczxak,
4
D. Casper,
4
W. Gajewski,
4
P. G. Halverson,
4,
* J. Hsu,
4
W. R. Kropp,
4
L. R. Price,
4
F. Reines,
4
M. Smy,
4
H. W. Sobel,
4
M. R. Vagins,
4
K. S. Ganezer,
5
W. E. Keig,
5
R. W. Ellsworth,
6
S. Tasaka,
7
J. W. Flanagan,
8,
A. Kibayashi,
8
J. G. Learned,
8
S. Matsuno,
8
V. J. Stenger,
8
D. Takemori,
8
T. Ishii,
9
J. Kanzaki,
9
T. Kobayashi,
9
S. Mine,
9
K. Nakamura,
9
K. Nishikawa,
9
Y. Oyama,
9
A. Sakai,
9
M. Sakuda,
9
O. Sasaki,
9
S. Echigo,
10
M. Kohama,
10
A. T. Suzuki,
10
T.J. Haines,
11,4
E. Blaufuss,
12
B. K. Kim,
12
R. Sanford,
12
R. Svoboda,
12
M. L. Chen,
13
Z. Conner,
13,
J. A. Goodman,
13
G. W. Sullivan,
13
J. Hill,
14
C. K. Jung,
14
K. Martens,
14
C. Mauger,
14
C. McGrew,
14
E. Sharkey,
14
B. Viren,
14
C. Yanagisawa,
14
W. Doki,
15
K. Miyano,
15
H. Okazawa,
15
C. Saji,
15
M. Takahata,
15
Y. Nagashima,
16
M. Takita,
16
T. Yamaguchi,
16
M. Yoshida,
16
S. B. Kim,
17
M. Etoh,
18
K. Fujita,
18
A. Hasegawa,
18
T. Hasegawa,
18
S. Hatakeyama,
18
T. Iwamoto,
18
M. Koga,
18
T. Maruyama,
18
H. Ogawa,
18
J. Shirai,
18
A. Suzuki,
18
F. Tsushima,
18
M. Koshiba,
19
M. Nemoto,
20
K. Nishijima,
20
T. Futagami,
21
Y. Hayato,
21,
§
Y. Kanaya,
21
K. Kaneyuki,
21
Y. Watanabe,
21
D. Kielczewska,
22,4
R. A. Doyle,
23
J. S. George,
23
A. L. Stachyra,
23
L.L. Wai,
23,
R. J. Wilkes,
23
and K. K. Young
23
(Super-Kamiokande Collaboration)
1
Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo, 188-8502, Japan
2
Department of Physics, Boston University, Boston, Massachusetts 02215
3
Physics Department, Brookhaven National Laboratory, Upton, New York 11973
4
Department of Physics and Astronomy, University of California at Irvine, Irvine, California 92697-4575
5
Department of Physics, California State University, Dominguez Hills, Carson, California 90747
6
Department of Physics, George Mason University, Fairfax, Virginia 22030
7
Department of Physics, Gifu University, Gifu, Gifu 501-1193, Japan
8
Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822
9
Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK),
Tsukuba, Ibaraki 305-0801, Japan
10
Department of Physics, Kobe University, Kobe, Hyogo 657-8501, Japan
11
Physics Division, P-23, Los Alamos National Laboratory, Los Alamos, New Mexico 87544
12
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803
13
Department of Physics, University of Maryland, College Park, Maryland 20742
14
Department of Physics and Astronomy, State University of New York, Stony Brook, New York 11794-3800
15
Department of Physics, Niigata University, Niigata, Niigata 950-2181, Japan
16
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
17
Department of Physics, Seoul National University, Seoul 151-742, Korea
18
Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan
19
The University of Tokyo, Tokyo 113-0033, Japan
20
Department of Physics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
21
Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan
22
Institute of Experimental Physics, Warsaw University, 00-681 Warsaw, Poland
23
Department of Physics, University of Washington, Seattle, Washington 98195-1560
(
Received 6 July 1998)
We present an analysis of atmospheric neutrino data from a 33.0 kton yr (535-day) exposure of the
Super-Kamiokande detector. The data exhibit a zenith angle dependent deficit of muon neutrinos which
is inconsistent with expectations based on calculations of the atmospheric neutrino flux. Experimental
biases and uncertainties in the prediction of neutrino fluxes and cross sections are unable to explain our
observation. The data are consistent, however, with two-flavor n
m
$ n
t
oscillations with sin
2
2u.
0.82 and 5 3 10
24
,Dm
2
,6310
23
eV
2
at 90% confidence level. [S0031-9007(98)06975-0]
PACS numbers: 14.60.Pq, 96.40.Tv
Atmospheric neutrinos are produced as decay products
in hadronic showers resulting from collisions of cosmic
rays with nuclei in the upper atmosphere. Production
of electron and muon neutrinos is dominated by the pro-
cesses p
1
! m
1
1n
m
followed by m
1
! e
1
1 n
m
1
n
e
(and their charge conjugates) giving an expected ratio
1562 0031-9007y98y81(8)y1562(6)$15.00 © 1998 The American Physical Society
VOLUME
81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
UGUST
1998
s; n
m
yn
e
d of the flux of n
m
1 n
m
to the flux of n
e
1 n
e
of about 2. The n
m
yn
e
ratio has been calculated in detail
with an uncertainty of less than 5% over a broad range of
energies from 0.1 to 10 GeV [1,2].
The n
m
yn
e
flux ratio is measured in deep underground
experiments by observing final state leptons produced via
charged-current interactions of neutrinos on nuclei, n1
N!l1X. The flavor of the final state lepton is used to
identify the flavor of the incoming neutrino.
The measurements are reported as R ;smyed
DATA
y
smyed
MC
, where m and e are the number of muon-
like sm-liked and electronlike se-liked events observed
in the detector for both data and Monte Carlo simu-
lations. This ratio largely cancels experimental and theo-
retical uncertainties, especially the uncertainty in the
absolute flux. R 1 is expected if the physics in the
Monte Carlo simulation accurately models the data.
Measurements of significantly small values of R have
been reported by the deep underground water Cherenkov
detectors Kamiokande [3,4], IMB [5], and recently by
Super-Kamiokande [6,7]. Although measurements of R
by early iron-calorimeter experiments Fréjus [8] and NU-
SEX [9] with smaller data samples were consistent with
expectations, the Soudan-2 iron-calorimeter experiment
has reported observation of a small value of R [10].
Neutrino oscillations have been suggested to explain
measurements of small values of R. For a two-neutrino
oscillation hypothesis, the probability for a neutrino pro-
duced in flavor state a to be observed in flavor state b after
traveling a distance L through a vacuum is
P
a!b
sin
2
2u sin
2
µ
1.27Dm
2
seV
2
dLskmd
E
n
sGeVd
, (1)
where E
n
is the neutrino energy, u is the mixing angle
between the flavor eigenstates and the mass eigenstates,
and Dm
2
is the mass squared difference of the neutrino
mass eigenstates. For detectors near the surface of the
Earth, the neutrino flight distance, and thus the oscilla-
tion probability, is a function of the zenith angle of the
neutrino direction. Vertically downward-going neutrinos
travel about 15 km, while vertically upward-going neutri-
nos travel about 13 000 km before interacting in the detec-
tor. The broad energy spectrum and this range of neutrino
flight distances make measurements of atmospheric neu-
trinos sensitive to neutrino oscillations with Dm
2
down to
10
24
eV
2
. The zenith angle dependence of R measured
by the Kamiokande experiment at high energies has been
cited as evidence for neutrino oscillations [4].
We present our analysis of 33.0 kton yr (535 days) of
atmospheric neutrino data from Super-Kamiokande. In
addition to measurements of small values of R both above
and below ,1 GeV, we observed a significant zenith angle
dependent deficit of m-like events. While no combination
of known uncertainties in the experimental measurement
or predictions of atmospheric neutrino fluxes is able to
explain our data, a two-neutrino oscillation model of
n
m
$ n
x
, where n
x
may be n
t
or a new, noninteracting
“sterile” neutrino, is consistent with the observed flavor
ratios and zenith angle distributions over the entire energy
region.
Super-Kamiokande is a 50 kton water Cherenkov detec-
tor instrumented with 11146 photomultiplier tubes (PMTs)
facing an inner 22.5 kton fiducial volume of ultrapure wa-
ter. Interaction kinematics are reconstructed using the time
and charge of each PMT signal. The inner volume is sur-
rounded by a ,2mthick outer detector instrumented with
1885 outward-facing PMTs. The outer detector is used to
veto entering particles and to tag exiting tracks.
Super-Kamiokande has collected a total of 4353 fully
contained (FC) events and 301 partially contained (PC)
events in a 33.0 kton yr exposure. FC events deposit all
of their Cherenkov light in the inner detector while PC
events have exiting tracks which deposit some Cherenkov
light in the outer detector. For this analysis, the neutrino
interaction vertex was required to have been reconstructed
within the 22.5 kton fiducial volume, defined to be .2m
from the PMT wall.
FC events were separated into those with a single visible
Cherenkov ring and those with multiple Cherenkov rings.
For the analysis of FC events, only single-ring events were
used. Single-ring events were identified as c-like or m-like
based on a likelihood analysis of light detected around
the Cherenkov cone. The FC events were separated into
“sub-GEV” sE
vis
, 1330 MeVd and “multi-GeV” sE
vis
.
1330 MeVd samples, where E
vis
is defined to be the energy
of an electron that would produce the observed amount
of Cherenkov light. E
vis
1330 MeV corresponds to
p
m
, 1400 MeVyc.
In a full-detector Monte Carlo simulation, 88% (96%) of
the sub-GeV e-like sm-liked events were n
e
sn
m
d charged-
current interactions and 84% (99%) of the multi-GeV
e-like sm-liked events were n
e
sn
m
d charged-current (CC)
interactions. PC events were estimated to be 98% n
m
charged-current interactions; hence, all PC events were
classified as m-like, and no single-ring requirement was
made. Table I summarizes the number of observed events
for both data and Monte Carlo as well as the R values for
the sub-GeV and multi-GeV samples. Further details of
the detector, data selection, and event reconstruction used
in this analysis are given elsewhere [6,7].
We have measured significantly small values of R
in both the sub-GeV and multi-GeV samples. Several
sources of systematic uncertainties in these measurements
have been considered. Cosmic ray induced interactions in
the rock surrounding the detector have been suggested as a
source of e-like contamination from neutrons, which could
produce small R values [11], but these backgrounds have
been shown to be insignificant for large water Cherenkov
detectors [12]. In particular, Super-Kamiokande has 4.7 m
of water surrounding the fiducial volume; this distance
corresponds to roughly 5 hadronic interaction lengths
and 13 radiation lengths. Distributions of event vertices
1563
VOLUME
81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
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1998
TABLE I. Summary of the sub-GeV, multi-GeV, and PC
event samples compared with the Monte Carlo prediction based
on the neutrino flux calculation of Ref. [2].
Data Monte Carlo
Sub-GeV
Single-ring 2389 2622.6
e-like 1231 1049.1
m-like 1158 1573.6
Multi-ring 911 980.7
Total 3300 3603.3
R 0.63 6 0.03 sstat.d 6 0.05 ssyst.d
Multi-Gev
Single-ring 520 531.7
e-like 290 236.0
m-like 230 295.7
Multi-ring 533 560.1
Total 1053 1091.8
Partially contained 301 371.6
R
FC1PC
0.65 6 0.05 sstat.d 6 0.08 ssyst.d
exhibit no excess of e-like events close to the fiducial
boundary [6,7].
The prediction of the ratio of the n
m
flux to the n
e
flux is dominated by the well-understood decay chain of
mesons and contributes less than 5% of the uncertainty in
R. Different neutrino flux models vary by about 620% in
the prediction of absolute rates, but the ratio is robust [13].
Uncertainties in R due to a difference in cross sections
for n
e
and n
m
have been studied [14]; however, lepton
universality prevents any significant difference in cross
sections at energies much above the muon mass and thus
errors in cross sections could not produce a small value of
R in the multi-GeV energy range. Particle identification
was estimated to be * 98% efficient for both m-like and
e-like events based on Monte Carlo studies. Particle
identification was also tested in Super-Kamiokande on
Michel electrons and stopping cosmic-ray muons and the
m-like and e-like events used in this analysis are clearly
separated [6]. The particle identification programs in
use have also been tested using beams of electrons and
muons incident on a water Cherenkov detector at KEK
[15]. The data have been analyzed independently by
two groups, making the possibility of significant biases in
data selection or event reconstruction algorithms remote
[6,7]. Other explanations for the small value of R, such as
contributions from nucleon decays [16], can be discounted
as they would not contribute to the zenith angle effects
described below.
We estimate the probability that the observed m
ye ratios
could be due to statistical fluctuation is less than 0.001%
for sub-GeV R and less than 1% for multi-GeV R.
The m-like data exhibit a strong asymmetry in zenith
angle sQd while no significant asymmetry is observed in
the e-like data [7]. The asymmetry is defined as A
FIG. 1. The sU 2 DdysU 1 Dd asymmetry as a function
of momentum for FC e-like and m-like events and PC
events. While it is not possible to assign a momentum to
a PC event, the PC sample is estimated to have a mean
neutrino energy of 15 GeV. The Monte Carlo expecta-
tion without neutrino oscillations is shown in the hatched
region with statistical and systematic errors added in quadra-
ture. The dashed line for m-like is the expectation for
n
m
$ n
t
oscillations with ssin
2
2u 1.0, Dm
2
2.2 3
10
23
eV
2
d.
sU 2 DdysU 1 Dd where U is the number of upward-
going events s21 , cosQ,20.2d and D is the num-
ber of downward-going events s0.2 , cos Q,1d. The
asymmetry is expected to be near zero independent of the
flux model for E
n
. 1 GeV, above which effects due to
the Earth’s magnetic field on cosmic rays are small. Based
on a comparison of results from our full Monte Carlo simu-
lation using different flux models [1,2] as inputs, treat-
ment of geomagnetic effects results in an uncertainty of
roughly 60.02 in the expected asymmetry of e-like and
m-like sub-GeV events and less than 60.01 for multi-GeV
events. Studies of decay electrons from stopping muons
show at most a 60.6% up-down difference in Cherenkov
light detection [17].
Figure 1 shows A as a function of momentum for
both e-like and m-like events. In the present data, the
asymmetric as a function of momentum for e-like events is
consistent with expectations, while the m-like asymmetry
at low momentum is consistent with zero but significantly
deviates form expectations at higher momentum. The
average angle between the final state lepton direction and
the incoming neutrino direction is 55
±
at p 400 MeVyc
and 20
±
at 1.5 GeVyc. At the lower momenta in Fig. 1, the
possible asymmetry of the neutrino flux is largely washed
out. We have found no detector bias differentiating e-like
and m-like events that could explain an asymmetry in
m-like events but not in e-like events [7].
1564
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81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
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1998
Considering multi-GeV sFC 1 PCd muons alone, the
measured asymmetry, A 20.296 6 0.048 6 0.01 de-
viates from zero by more than 6 standard deviations.
We have examined the hypotheses of two-flavor n
m
$
n
e
and n
m
$ n
t
oscillation models using a x
2
com-
parison of data and Monte Carlo, allowing all important
Monte Carlo parameters to vary weighted by their expected
uncertainties.
The data were binned by particle type, momentum, and
cos Q.Ax
2
is defined as
x
2
X
cos Q,p
sN
DATA
2 N
MC
d
2
ys
2
1
X
j
e
2
j
ys
2
j
, (2)
where the sum is over five bins equally spaced in cos Q and
seven momentum bins for both e-like events and m-like
plus PC events (70 bins total). The statistical error, s,
accounts for both data statistics and the weighted Monte
Carlo statistics. N
DATA
is the measured number of events
in each bin. N
MC
is the weighted sum of Monte Carlo
events:
N
MC
L
DATA
L
MC
X
MC events
w . (3)
L
DATA
and L
MC
are the data and Monte Carlo live times.
For each Monte Carlo event, the weight w is given by
w s1 1adsE
i
n
yE
0
d
d
s1 1h
s,m
cos Qd
3 f
e,m
sss sin
2
2u, Dm
2
, s1 1ldLyE
n
ddd
3
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
s12b
s
y2d sub-GeV e-like ,
s1 1b
s
y2d sub-GeV m-like ,
s1 2b
m
y2d multi-GeV e-like ,
s1 1b
m
y2d
12
r
2
N
PC
N
m
¥
multi-GeV m-like ,
s1 1b
m
y2d
11
r
2
¥
PC .
(4)
E
i
n
is the average neutrino energy in the ith momentum bin;
E
0
is an arbitrary reference energy (taken to be 2 GeV);
h
s
sh
m
d is the up-down uncertainty of the event rate in
the sub-GeV (multi-GeV) energy range; N
PC
is the total
number of Monte Carlo PC events; and N
m
is the total
number of Monte Carlo FC multi-GeV muons. The factor
f
e,m
weights an event accounting for the initial neutrino
fluxes (in the case of n
m
$ n
e
), oscillation parameters,
and LyE
n
. The meaning of the Monte Carlo fit para-
meters, a and e
j
;sb
s
,b
m
,d,r,l,h
s
,h
m
dand their as-
signed uncertainties, s
j
, are summarized in Table II. The
overall normalization, a, was allowed to vary freely. The
uncertainty in the Monte Carlo LyE
n
ratio sld was con-
servatively estimated based on the uncertainty in an ab-
solute energy scale, uncertainty in neutrino-lepton angular
and energy correlations, and the uncertainty in production
height. The oscillation simulations used profiles of neu-
trino production heights calculated in Ref. [18], which ac-
count for the competing factors of production, propagation,
and decay of muons and mesons through the atmosphere.
TABLE II. Summary of Monte Carlo fit parameters. Best-
fit values for n
m
$ n
t
sDm
2
2.2 3 10
23
eV
2
, sin
2
2u
1.0d and estimated uncertainties are given. (*) The overall
normalization sad was estimated to have a 25% uncertainty but
was fitted as a free parameter.
Monte Carlo fit parameters Best fit Uncertainty
a Overall normalization 15.8% (*)
d E
n
spectral index 0.006 s
d
0.05
b
s
Sub-GeV mye ratio 26.3% s
s
8%
b
m
Multi-GeV mye ratio 211.8% s
m
12%
r Relative norm. of PC to FC 21.8% s
p
8%
l LyE
n
3.1% s
l
15%
h
s
Sub-GeV up-down 2.4% s
s
h
2.4%
h
m
Multi-GeV up-down 20.09% s
m
h
2.7%
For n
m
$ n
e
, effects of matter on neutrino propagation
through the Earth were included following Ref. [19,20].
Because of the small number of events expected from
t production, the effects of t appearance and decay were
neglected in simulations of n
m
$ n
t
. A global scan was
made on a ssin
2
2u, log Dm
2
d grid minimizing x
2
with re-
spect to a, b
s
, b
m
, d, r, l, h
s
, and h
m
at each point.
The best fit to n
m
$ n
t
oscillations, x
2
min
65.2y
67 DOF, was obtained at ssin
2
2u 1.0, Dm
2
2.2 3
10
23
eV
2
d inside the physical region s0 # sin
2
2u#1d.
The best-fit values of the Monte Carlo parameters (sum-
marized in Table II) were all within their expected errors.
The global minimum occurred slightly outside of the physi-
cal region at (sin
2
2u 1.05, Dm
2
2.2 3 10
23
eV
2
,
x
2
min
64.8y67 DOF). The contours of the 68%, 90%,
and 99% confidence intervals are located at x
2
min
1 2.6,
5.0, and 9.6 based on the minimum inside the physical re-
gion [21]. Thee contours are shown in Fig. 2. The region
near x
2
minimum is rather flat and has many local minima
so that inside the 68% interval the best-fit Dm
2
is not
well-constrained. Outside of the 99% allowed region the
x
2
increases rapidly. We obtained x
2
135y69 DOF,
when calculated at sin
2
2u 0, Dm
2
0 (i.e., assuming
no oscillations).
For the test of n
m
$ n
e
oscillations, we obtained a
relatively poor fit; x
2
min
87.8y67 DOF, at ssin
2
2u
0.93, Dm
2
3.2 3 10
23
eV
2
d. The expected asymmetry
of the multi-GeV e-like events for the best-fit n
m
$
n
e
oscillation hypothesis, A 0.205, differs from the
measured asymmetry, A 20.036 6 0.067 6 0.02,by
3.4 standard deviations. We conclude that the n
m
$ n
e
hypothesis is not favored.
The zenith angle distributions for the FC and PC samples
are shown in Fig. 3. The data are compared to the Monte
Carlo expectation (no oscillations, hatched region) and the
best-fit expectation for n
m
$ n
t
oscillations (bold line).
We also estimated the oscillation parameters consider-
ing the R measurement and the zenith angle shape sepa-
rately. The 90% confidence level allowed regions for each
1565
VOLUME
81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
UGUST
1998
FIG. 2. The 68%, 90%, and 99% confidence intervals are
shown for sin
2
2u and Dm
2
for n
m
$ n
t
two-neutrino oscil-
lations based on 33.0 kton yr of Super-Kamiokande data. The
90% confidence interval obtained by the Kamiokande experi-
ment is also shown.
case overlapped at 1 3 10
23
,Dm
2
,4310
23
eV
2
for sin
2
2u 1.
As a cross-check of the above analyses, we have re-
constructed the best estimate of the ratio LyE
n
for each
event. The neutrino energy is estimated by applying a
correction to the final state lepton momentum. Typi-
cally, final state leptons with p , 100 MeVyc carry 65%
of the incoming neutrino energy increasing to ,85% at
p 1 GeVyc. The neutrino flight distance L is esti-
mated following Ref. [18] using the estimated neutrino
energy and the reconstructed lepton direction and flavor.
Figure 4 shows the ratio of FC data to Monte Carlo for
e-like and m-like events with p . 400 MeV as a func-
tion of LyE
n
, compared to the expectation for n
m
$ n
t
oscillations with our best-fit parameters. The e-like data
show no significant variation in LyE
n
, while the m-like
events show a significant deficit at large LyE
n
. At large
LyE
n
, the n
m
have presumably undergone numerous os-
cillations and have averaged out to roughly half the
initial rate.
The asymmetry A of the e-like events in the present data
is consistent with expectations without neutrino oscilla-
tions and two-flavor n
e
$ n
m
oscillations are not favored.
This is in agreement with recent results from the CHOOZ
experiment [22]. The LSND experiment has reported the
appearance of n
e
in a beam of n
m
produced by stopped
pions [23]. The LSND results do not contradict the
present results if they are observing small mixing angles.
With the best-fit parameters for n
m
$ n
t
oscillations, we
expect a total of only 1520 events from n
t
charged-
current interactions in the data sample. Using the current
sample, oscillations between n
m
and n
t
are indistinguish-
able from oscillations between n
m
and a noninteracting
sterile neutrino.
Figure 2 shows the Super-Kamiokande results overlaid
with the allowed region obtained by the Kamiokande
FIG. 3. Zenith angle distributions of m-like and e-like events for sub-GeV and multi-GeV data sets. Upward-going particles
have cos Q,0and downward-going particles have cos Q.0. Sub-GeV data are shown separately for p , 400 MeVyc and
p . 400 MeVyc. Multi-GeV e-like distributions are shown for p , 2.5 and p . 2.5 GeVyc and the multi-GeV m-like are shown
separately for FC and PC events. The hatched region shows the Monte Carlo expectation for no oscillations normalized to the data
live time with statistical errors. The bold line is the best-fit expectation for n
m
$ n
t
oscillations with the overall flux normalization
fitted as a free parameter.
1566
VOLUME
81, NUMBER 8 PHYSICAL REVIEW LETTERS 24A
UGUST
1998
FIG. 4. The ratio of the number of FC data events to FC
Monte Carlo events versus reconstructed LyE
n
. The points
show the ratio of observed data to MC expectation in the
absence of oscillations. The dashed lines show the expected
shape for n
m
$ n
t
at Dm
2
2.2 3 10
23
eV
2
and sin
2
2u
1. The slight LyE
n
dependence for e-like events is due to
contamination (27%) of n
m
CC interactions.
experiment [4]. The Super-Kamiokande region favors
lower values of Dm
2
than allowed by the Kamiokande
experiment; however the 90% contours from both ex-
periments have a region of overlap. Preliminary stud-
ies of upward-going stopping and through-going muons
in Super-Kamiokande [24] give allowed regions consis-
tent with the FC and PC event analysis reported in this
paper.
Both the zenith angle distribution of m-like events
and the value of R observed in this experiment signifi-
cantly differ from the best predictions in the absence
of neutrino oscillations. While uncertainties in the flux
prediction, cross sections, and experimental biases are
ruled out as explanations of the observations, the present
data are in good agreement with two-flavor n
m
$ n
t
oscillations with sin
2
2u.0.82 and 5 3 10
24
,Dm
2
,
6310
23
eV
2
at a 90% confidence level. We con-
clude that the present data give evidence for neutrino
oscillations.
We gratefully acknowledge the cooperation of the
Kamioka Mining and Smelting Company. The Super-
Kamiokande experiment was built and has been operated
with funding from the Japanese Ministry of Education,
Science, Sports and Culture, and the United States De-
partment of Energy.
*Present address: NASA, JPL, Pasadena, CA 91109.
Present address: High Energy Accelerator Research
Organization (KEK), Accelerator Laboratory, Tsukuba,
Ibaraki 305-0801, Japan.
Present address: University of Chicago, Enrico Fermi
Institute, Chicago, IL 60637.
§
Present address: Institute of Particle and Nuclear
Studies, High Energy Accelerator Research Organization
(KEK), Tsukuba, Ibaraki 305-0801, Japan.
Present address: Stanford University, Department of
Physics, Stanford, CA 94305.
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1567

Discussion

![hadronic shower](http://www.mpi-hd.mpg.de/hfm/CosmicRay/shower.png "Hadronic Shower") Figure 4 shows the ratio of data *"real"* events and Monte Carlo expected events in the absence of oscillations for various $L/E_\nu$. The dashed lines correspond to the predictions of the best fit considering a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation. Once again we see a deficit of $\mu$-like events showing that the data collected by Super-Kamiokande is consistent with a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation scenario. The conclusion states that the observed events are consistent with a two flavour $\nu_\mu \leftrightarrow \nu_\tau$ oscillation scenario a gives new limits for $\sin^2(2\theta)$ and $\Delta m^2$. Figure 1 (bottom) shows a strong asymmetry for $\mu$-like events detected by Super-Kamiokande. The hatched region shows the Monte Carlo results considering no neutrino oscillations. We see that the data (black dots) deviate from these predictions. The dashed line corresponds to $$\nu_\mu \leftrightarrow \nu_\tau$$ oscillations and seems to be fairly consistent with the observations. Zenith angle distributions for $e$-like and $\mu$-like events. The hatched region shows the Monte Carlo expectation in the case of no oscillations. The bold line corresponds to the best fit considering a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation. We can see that the data (black dots) is consistent with an oscillation scenario for both types of events and various energy ranges. Curiosity: In Particle Physics, there is a convention of a **five-sigma effect (99.99994% confidence)** being required to qualify any detected event as a **"Discovery"**. Phase space *("allowed solutions")* for $\Delta m^{2}$ and $\sin^2(2\theta)$ for the results of the best fit to the data considering a two flavour oscillation scenarion from $\nu_\mu$ to $\nu_\tau$ for different confidence levels (68%, 90%, 99%). We see an overlap at the 90% confidence level between Super-Kamiokande and Kamiokande. Atmospheric neutrinos are produced in hadronic showers that result from the interaction of a cosmic ray at the top of the atmosphere. The cosmic rays start a cascade of reactions that produce pions that will decay and produce neutrinos. ![Leptons](http://thumbsnap.com/sc/7wsSurZQ.jpg "Leptons table") There are three flavours of neutrinos (electron, muon, tau): \begin{eqnarray*} e - \nu_e\\ \mu - \nu_\mu\\ \tau - \nu_\tau\\ \end{eqnarray*} (and the corresponding antineutrinos) Picture of the first neutrino observation in a Bubble chamer (1970) via it's charged current interaction with a proton. Once can see that the neutrino detection is made indirectly by studying the paths of the particles produced when the neutrino interaction with matter occurred. ![first neutrino](https://upload.wikimedia.org/wikipedia/commons/5/57/FirstNeutrinoEventAnnotated.jpg) It is very hard to detect neutrinos. Neutrinos are electrically neutral, so they are not affected by Electromagnetism; they are leptons so they are not affected by the Strong force. So they are only affected by the Weak force (short range interaction) and Gravity (extremly weak at the subatomic scale). Neutrinos are detected indirectly via their weak interactions with matter where they produce the charged leptons associated with them. Curiosity: This paper is mentioned in two different Physics Nobel Prize ceremonies 2002 and 2015. See here: [Physical Review Letters.](http://journals.aps.org/prl/50years/milestones#1998) Let us compute the oscillation probability. Consider a two Neutrino oscillation hypothesis from flavour a to flavour b, ($\nu_a\,,\nu_b$): \begin{eqnarray*} \left(\begin{array}{c} \nu_{a} \\ \nu_{b} \\ \end{array}\right) &=& \left(\begin{array}{cc} cos{\theta} & sin{\theta} \\ -sin{\theta} & cos{\theta} \\ \end{array}\right) \left(\begin{array}{c} \nu_{1} \\ \nu_{2} \\ \end{array}\right) \end{eqnarray*} In bra-ket notation: \begin{eqnarray*} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle #1|} \ket{\nu_a} &=& \cos\theta\ket{\nu_1} + \sin\theta\ket{\nu_2}\\ \ket{\nu_b} &=& -\sin\theta\ket{\nu_1} + \cos\theta\ket{\nu_2} \end{eqnarray*} Using the Schrödinger equation one can write the mass eigenstates of a particle as: \begin{eqnarray*} \ket{\nu_1(t)} &=& e^{-iE_1t}\ket{\nu_1} = e^{-i\sqrt{m_1^2+p_1^2}\,t}\ket{\nu_1} \nonumber\\ &\simeq& e^{-i(p+\frac{m_1^2}{2p})t}\ket{\nu_1}\\ \ket{\nu_2(t)} &\simeq& e^{-i(p+\frac{m_2^2}{2p})t}\ket{\nu_2} \end{eqnarray*} In natural units the energy of any given mass eigenstate $i$ can be written as: $E_i=\sqrt{m_i^2+p^2}\simeq p+\frac{m_i^2}{2p}$ where we consider that $m\ll p$. One can then rewrite the wave function for a flavour a neutrino ($\nu_a$) as: \begin{eqnarray*} \ket{\nu_a(t)} &=& c\ket{\nu_1(t)} + s\ket{\nu_2(t)}\\ &\simeq& e^{-ip}\left( ce^{-i\frac{m_1^2}{2p}t}\ket{\nu_1} + se^{-i\frac{m_2^2}{2p}t}\ket{\nu_2} \right) \end{eqnarray*} where $c \equiv \cos\theta$ and $s\equiv \sin\theta$. We do the same for flavour b ($\nu_b$) and then compute the probability function for the oscillation from flavour $\ket{\nu_a}$ to flavout $\ket{\nu_b}$ as: \begin{eqnarray*} P_{\nu_a\rightarrow \nu_b}(t) &=& \left| \left< \nu_b | \nu_a(t) \right> \right|^2 \nonumber \\ &=& \left| -sce^{-i\frac{m_1^2}{2p}t} + cse^{-i\frac{m_2^2}{2p}t} \right|^2 \nonumber \\ &=& 2s^2c^2 \left( 1-\cos{\frac{m_1^2-m_2^2}{2p}t} \right) \nonumber \\ &\simeq& \sin^2(2\theta)\sin^2(\delta_{12}) \end{eqnarray*} where:\begin{eqnarray*} \delta_{ij} = \frac{c^3}{\hbar}\frac{\Delta m_{ij}^2}{4E} L \simeq \frac{\Delta m_{ij}^2}{2p} L \simeq 1.27\, \frac {\Delta m_{ij}^2}{E} L \end{eqnarray*} with $\Delta m^2_{ij}=m^2_i-m^2_j$ and again we use natural units so that $t \simeq L$, $p \simeq E$, $L$ (km) is the distance travelled by the neutrino $E$ (GeV) the neutrino energy $\theta$ mixing angle. So finally one can write: \begin{eqnarray*} P_{\nu_a \rightarrow \nu_b} \left( L/E \right) &=& sin^2\left( 2\theta \right) \, sin^2\left( 1.27\,\Delta m^2 L/E \right).\\ \end{eqnarray*}