Ž . Nuclear Phys ics B 573 2000 3–26 www.elsevier.nlrlocaternpe Status of the MSW solutions of the solar neutrino problem M.C. Gonzalez-Garcia a , P.C. de Holanda a,b , C. Pena-Garay a , J.W.F. Valle a ˜ a Instituto de Fısica Corpuscular – C.S.I.C., UniÕersitat of Valencia, 46100 Burjassot, Valencia, Spain ´ ` ` b Instituto de Fısica Gleb Wataghin, UniÕersidade Estadua l de Campinas, UNICAMP, 13083- 970 –´ Campinas, Brazil Received 8 July 1999; received in revised form 6 October 1999; accepted 2 November 1999 Abstract We present an updated global analysis of two-flavour MSW solutions to the solar neutrino problem. We perform a fit to the full data set corresponding to the 825-day Super-Kamiokande data sample as well as to chlorine, GALLEX and SAGE experiments. In our analysis we use all measured total event rates as well as all Super-Kamiokande data on the zenith angle dependence, energy spectrum and seasonal variation of the events. We compare the quality of the solutions ofthe solar neutrino anomaly in terms of conversions ofninto activ e or sterile neutr inos. For the e case of conversions into active neutrinos we find that, although the data on the total event rates Ž . favour s the Small Mixin g Angle SMA soluti on, onc e the full dat a set is includ ed both SMA and Ž . Large Mix ing Ang le LMA solut ions gi ve an equall y good fi t to the data. We find t hat the best -fit poi nts for the combin ed anal ysi s are Dm 2 s3.6 =10 y5 eV 2 and si n 2 2 us0.79 wit h x2 s min 35.4r30 d.o.f. and Dm 2 s5.1 =10 y6 eV 2 and sin 2 2 us5.5 =10 y3 with x2 s37.4r30 d.o.f. min In contrast with the earlier 504-day study of Bahcall–Krastev–Smirnov our results indicate that the LMA solution is not only allowed, but slightly preferred. On the other hand, we show that seasonal effects, although small, may still reach 11% in the lower part of the LMA region, without Ž confli ct wi th the negati ve hint s of a day–night vari at ion 6% is due to the eccentricit y of the . Eart h’s orbit . In par ticular the bes t-f it LMA soluti on predicts a seas ona l effe ct of 8.5%. For conversions into sterile neutrinos only the SMA solution is possible with best-fit point Dm 2 s 5.0 =10 y6 eV 2 and sin 2 2 us3.=10 y3 and x2 s40.2r30 d.o.f. We also consider departures min Ž . Ž . of the Standar d Solar Mo del SSM of Bahcall and Pinsonn eaul t 1998 BP98 by all owin g arbitrary 8 B and hep fl uxes. These modi fi cat ions do not alter sig ni ficantl y the oscil lati on parameters. The best fit is obtained for 8 Br 8 B s0.61 and heprhep s12 for the SMA SSM SSM soluti on bo th for conversi on s into acti ve or st er il e neut rinos and 8 Br 8 B s1. 37 and SSM heprhep s38 for the LMA solution. q 2000 Elsevier Science B.V. All rights reserved. SSM 1. Introduction It has been already three decades since the first detection of solar neutrinos. It was w x realiz ed from the very beginni ng that the observed rate at the Homestak e experiment 1 w x was far lower than the theor eti cal expec tat ion based on the standard solar model 2,3 0550-3213r00r$ - see front matter q2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0550-3213 99 00709-9
www.elsevier.nlrlocaternpe
Status of the MSW solutions of the solar neutrino problem
M.C. Gonzalez-Garcia a, P.C. de Holanda a,b, C. Pena-Garay
a, J.W.F. Valle a˜ a Instituto de Fsica Corpuscular –
C.S.I.C., UniÕersitat of Valencia, 46100 Burjassot, Valencia,
Spain´ ` ` b Instituto de Fsica Gleb Wataghin, UniÕersidade
Estadual de Campinas, UNICAMP, 13083-970 – ´
Campinas, Brazil
Received 8 July 1999; received in revised form 6 October 1999;
accepted 2 November 1999
Abstract
We present an updated global analysis of two-flavour MSW solutions
to the solar neutrino
problem. We perform a fit to the full data set corresponding to the
825-day Super-Kamiokande
data sample as well as to chlorine, GALLEX and SAGE experiments. In
our analysis we use all
measured total event rates as well as all Super-Kamiokande data on
the zenith angle dependence,
energy spectrum and seasonal variation of the events. We compare
the quality of the solutions of
the solar neutrino anomaly in terms of conversions of
n into active or sterile neutrinos. For thee
case of conversions into active neutrinos we find that, although
the data on the total event rates .favours the Small Mixing Angle
SMA solution, once the full data set is included both SMA and
.Large Mixing Angle LMA solutions give an equally good fit to the
data. We find that the best-fit
points for the combined analysis are Dm2 s 3.6 = 10y5 eV 2
and sin2 2u s 0.79 with x 2 smin
35.4r30 d.o.f. and Dm2 s 5.1 = 10y6 eV 2 and sin2 2u s
5.5 = 10y3 with x 2 s 37.4r30 d.o.f.min
In contrast with the earlier 504-day study of
Bahcall–Krastev–Smirnov our results indicate that
the LMA solution is not only allowed, but slightly preferred. On
the other hand, we show that
seasonal effects, although small, may still reach 11% in the lower
part of the LMA region, without conflict with the negative hints of
a day–night variation 6% is due to the eccentricity of the
.Earth’s orbit . In particular the best-fit LMA solution predicts a
seasonal effect of 8.5%. For
conversions into sterile neutrinos only the SMA solution is
possible with best-fit point Dm2 s
5.0 = 10y6 eV 2 and sin2 2u s 3.= 10y3 and x 2 s
40.2r30 d.o.f. We also consider departuresmin
. .of the Standard Solar Model SSM of Bahcall and Pinsonneault 1998
BP98 by allowing
arbitrary 8
B and hep fluxes. These modifications do not alter
significantly the oscillation
parameters. The best fit is obtained for 8
Br 8
B s 0.61 and heprhep s 12 for the SMASSM
SSM
solution both for conversions into active or sterile neutrinos and
8Br8 B s 1.37 andSSM
heprhep s 38 for the LMA solution. q2000 Elsevier
Science B.V. All rights reserved.SSM
1. Introduction
It has been already three decades since the first detection of
solar neutrinos. It was w xrealized from the very beginning that
the observed rate at the Homestake experiment 1
w xwas far lower than the theoretical expectation based on the
standard solar model 2,3
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 4
with the implicit assumption that neutrinos created in the solar
interior reach the Earth
unchanged, i.e. they are massless and have only standard properties
and interactions. In
the first two decades of solar neutrino research, the problem
consisted only of the
discrepancy between theoretical expectations based upon solar model
calculations and
the observations of the capture rate in the chlorine solar neutrino
experiment. This
discrepancy led to a change in the original goal of using solar
neutrinos to probe the
properties of the solar interior towards the study of the
properties of the neutrino itself.
From the experimental point of view much progress has been done in
recent years.
We now have available the results of five experiments, the original
chlorine experiment w x w xat Homestake 4 , the radio chemical
gallium experiments on pp neutrinos, GALLEX 5
w x w xand SAGE 6 , and the water Cherenkov detectors Kamiokande 7
and Super-Kamio- w xkande 8,9 . The latter has been able not only
to confirm the original detection of solar
neutrinos at lower rates than predicted by standard solar models,
but also to demonstrate
directly that the neutrinos come from the Sun by showing that
recoil electrons are
scattered in the direction along the Sun–Earth axis. We now have
good information on
the time dependence of the event rates during the day and night, as
well as a
measurement of the recoil electron energy spectrum. After 825 days
of operation,
Super-Kamiokande has also presented preliminary results on the
seasonal variation of
the neutrino event rates, an issue which will become important in
discriminating the w xMSW scenario from the possibility of neutrino
oscillations in vacuum 10,11 .
On the other hand, there has been improvement on solar modelling
and nuclear cross
sections. For example, helioseismological observations have now
established that diffu-
sion is occurring and by now most solar models incorporate the
effects of helium and w xheavy element diffusion 12,13 . The
quality of the experiments themselves and the
robustness of the theory make us confident that in order to
describe the data one must .depart from the Standard Model SM of
particle physics interactions, by endowing
neutrinos with new properties. In theories beyond the Standard
Model of particle physics w xneutrinos may naturally have exotic
properties such as non-orthonormality 14 , flavour-
w x w xchanging interactions 15 , transition magnetic moments 16,17
and neutrino decays w x18–21 , the most generic is the existence of
mass. While many of these may play a role
w xin neutrino propagation and therefore in the explanation of the
data 22–30 it is
undeniable that the most generic and popular explanation of the
solar neutrino anomaly
is in terms of neutrino masses and mixing leading to neutrino
oscillations either in w x w xÕacuum 31–38 or via the
matter-enhanced MSW mechanism 39,40 .
In this paper we study the implications of the present data on
solar neutrinos in the
framework of the two-flavour MSW solutions to the solar neutrino
problem. We perform
a global fit to the full data set corresponding to 825 days of data
of the Super-Kamio-
kande experiment as well as to chlorine, GALLEX and SAGE. In our
analysis we use as
SSM the latest results from the most accurate calculation of
neutrino fluxes by Bahcall w xand Pinsonneault 41 which
incorporates the new normalization for the low energy
cross section S s 19q4 eV b indicated by the
recent studies at the Institute of Nuclear17 y2
w xTheory 42 . We have also considered the possibility of departing
from the SSM of
BP98 by allowing a free normalization of the 8
B flux and hep neutrino fluxes and we
present the results we obtain when we treat these normalization as
free parameters.
We combined the measured total event rates at chlorine, gallium and
Super-Kamio-
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 5
energy spectrum and seasonal variation of the events. The goal of
such analysis is not
only to compare the quality of the solutions to the solar neutrino
anomaly in terms of
flavour oscillations of n into active or
sterile neutrinos but also to study the weight of e
different observables on the determination of the underlying
neutrino physics parameters w xas emphasized in Ref. 43 . The
outline of the paper is as follows. In Section 2 we
present the basic elements that enter into our calculation of the
observables and the
definitions used in the statistical combination of the data.
Section 3 contains our results .of the allowed or excluded regions
of oscillation parameters from the analyses of the
different observables. The results on the allowed regions from the
combined analysis of
the total event rates is contained in Section 3.1. The constraints
arising from the
Super-Kamiokande searches for day–night variation of the event
rates are discussed in
Section 3.2. In Section 3.3 we discuss the information which can be
extracted from the
distortion of the recoil electron energy spectrum measured by
Super-Kamiokande. The
restrictions arising from the preliminary Super-Kamiokande data on
the seasonal varia-
tion of the event rates are studied in Section 3.4. In Section 3.5
we present our results
from the global fit to the full data set and we determine the
allowed range of oscillation
parameters which are consistent with all the data. Finally in
Section 4 we discuss the
possible implications of our results for future
investigations.
Our results show that for oscillation into active neutrinos
although the data on the
total event rates favours the SMA angle solution, once the full
data set is included both
SMA and LMA give an equally good fit to the data. We find that the
best-fit points for
the combined analysis are Dm2 s 3.6 = 10y5 eV 2 and sin2
2u s 0.79 with x 2 smin
35.4r30 d.o.f. and Dm2 s 5.1 = 10y6 eV 2 and sin2 2u s
5.5 = 10y3 with x 2 smin
w x37.4r30 d.o.f. We note that in contrast with the earlier 504-day
study of Ref. 43 our
results indicate that the LMA solution is not only allowed, but
actually slightly
preferred. The existence of hints that the LMA MSW solution could
be correct was also w xdiscussed in Ref. 44 . On the other hand, we
find good quantitative agreement with the
w x w xrecent results of Ref. 9 as well as qualitative agreement
with the old results of Ref. 45
based on a smaller sample. We show that seasonal effects may be
no-negligible in the
lower part of the LMA region, without conflict with the negative
hints of a day–night
variation. In particular the best-fit LMA solution predicts a
seasonal effect of 8.5%, 6%
of which is due to the eccentricity of the Earth’s orbit. For
conversions into sterile
neutrinos we find that only the SMA solution is possible with
best-fit point Dm2 s 5.0
= 10y6 eV 2 and sin2 2u s 3.0 = 10y3 and x 2 s
40.2r30 d.o.f. We also considermin
. .departures of the Standard Solar Model SSM of Bahcall and
Pinsonneault 1998 BP98
by allowing arbitrary 8
B and hep fluxes. These modifications do not affect
significantly
the oscillation parameters. We find that the best fit is obtained
for 8
Br 8
B s 0.61 andSSM
heprhep s 12 for the SMA solution both for conversions into
active or sterileSSM
neutrinos and 8
Br 8
B s 1.37 and heprhep s 38 for the LMA
solution.SSM SSM
2. Data and techniques
In order to study the possible values of neutrino masses and mixing
for the MSW
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 6
w xGALLEX and SAGE 5,6 and at the water Cerenkov detector
Super-Kamiokande. Apart
from total event rates we have in this case the zenith angle
distribution of the events, the
electron recoil energy spectrum and the seasonal distribution of
events, all measured w xwith their recent 825-day data sample 9
.
We first describe our calculation of the different observables. For
simplicity we
consider the two-neutrino mixing case
n s cosu n q sinu n ,
n s ysinu n q cosu n , 1 .e
1 2 x 1 2
where x can label either an active, x s
m,t , or sterile neutrino, x s s. In order to
account for Earth regeneration effects, we have determined the
solar neutrino survival
probability P in the usual way, assuming that the
neutrino state arriving at the Earth ise e
an incoherent mixture of the n and
n mass eigenstates.1 2
P s P Sun P Earth q P Sun P Earth , 2 .e e e1 1 e e 2 2
e
where P Sun is the probability that a solar neutrino, that
is created as n , leaves the Sune1 e
as a mass eigenstate n , and P Earth is
the probability that a neutrino which enters the1 1 e
Earth as n arrives at the detector as
n . Similar definitions apply to P Sun and
P Earth.1 e e 2 2 e
The quantity P Sun is given, after discarding the fast
oscillating terms, ase1
1 1Sun SunP s 1 y P s q y P cos 2u
t , 3 . . .e1 e 2 L Z m 02
2
w x .where P denotes the improved Landau–Zener
probability 46,47 and u t is
the L Z 0 m
mixing angle in matter at the neutrino production point. In our
calculations of the
expected event rates we have averaged this probability with respect
to the production
point. The electron and neutron number density in the Sun and the
production point w xdistribution were taken from Ref. 48 .
In order to obtain the conversion probabilities in the Earth,
P Earth, we integrate thei e
evolution equation in matter assuming a step function profile of
the Earth matter density the Earth as consisting of mantle and core
of constant densities equal to the correspond-
3 3.ing average densities, r , 4.5 grcm and
r , 11.5 grcm . To convert from them c
mass density to electron and neutron number density we use the
charge to nucleon ratio w x Z r A s 0.497 for the
mantle and Z r A s 0.467 for the core. In the
notation of Ref. 49 ,
we obtain for P Earth s 1 y P Earth 2 e 1 e
2 2EarthP F s Z sinu q
W cosu q W sinu , 4
. . . .2 e 1 3
where u is the mixing angle in vacuum and the
Earth matter effect is included in the w x Earthformulas for
Z ,W and W , which can
be found in Ref. 49 . P depends on the1 3 2 e
amount of Earth matter travelled by the neutrino on its way to the
detector, or, in other
words, on its arrival direction which is usually parametrized in
terms of the nadir angle, F , of the Sun at the detector site.
Due to this effect the survival probability is, in
general, time dependent. This Earth regeneration effect is
important in the study of the w xzenith angle distribution of
events as well as in their seasonal variation 10,50–62 .
2.1. Rates
w xHere we update previous analyses of solar neutrino data 43,45 by
including the
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 7
seasonal variation data and we anticipate its future role in
discriminating between
different solutions of the solar neutrino anomaly. Working in the
context of the BP98 w x 8
standard solar model of Ref. 41 we also allow for a free
normalization of the B flux
and of the hep flux. In our statistical treatment of the data we
follow closely the analysis w x w xof Refs. 63,64 with the updated
uncertainties given in Refs. 41,48 .
In our study we use the measured rates shown in Table 1. For the
combined fit we
adopt the x 2 definition
x 2 s R th y Rexp s y2 R th y Rexp
, 5 . . .Ý R i i i j j j
i , js1,3
where R th is the theoretical prediction of the event rate
in detector i and Rexp is thei i
measured rate. The error matrix s contains not
only the theoretical uncertainties buti j
also the experimental errors, both systematic and
statistical.
The general expression of the expected event rate in the presence
of oscillations in
experiment i is given by R th,i
R th s f dE l E
.Ý Hi k n k
n
k s1,8
= ² : ² :s E P E ,t q
s E 1 y P E ,t
, 6 . . . . . .e , i n e e
n x , i n e e n
where E is the neutrino energy, f
is the total neutrino flux and l is the
neutrino n k k
. w xenergy spectrum normalized to 1 from the solar nuclear
reaction k 65 with the w x .
.normalization given in Ref. 41 . Here s s is
the n n , x s m, t
interactione, i x, i e x
w xcross section in the Standard Model 66 with the target
corresponding to experiment i, ² .:and P
E ,t is the time-averaged n
survival probability.e e n e
.For the chlorine and gallium experiments we use improved cross
sections s E a , i
. w xa s e, x from Ref. 48 . For the
Super-Kamiokande experiment we calculate the
expected signal with the corrected cross section given in Section
2.3.
The expected signal in the absence of oscillations, RBP98,
can be obtained from Eq.i
.6 by substituting P s 1. In Table 1 we also give the
expected rates at the differente e
w xexperiments which we obtain using the fluxes of Ref. 41 .
2.2. Day–night Õariation
As already mentioned, in the MSW picture the expected event rates
can be different w xwhen the neutrinos travel through the Earth due
to the n regeneration effect 50–62 .e
As a result in certain regions of the oscillation parameters the
expected event rates
depend on the zenith angle of the Sun as observed from the
experiment site, since this
determines the amount of Earth matter crossed by the neutrino on
its way to the detector.
Table1
BP98Experiment Rate Ref. Units Ri
w xHomestake 2.56"0.23 4 SNU 7.8"1.1 w xGALLEX q SAGE 72.3"5.6 5,6
SNU 130"7
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 8
The Super-Kamiokande Collaboration has studied the dependence of
the event rates
with the period of time along the day and the night. They present
their results in the
form of a zenith angle distribution of events.
In our analysis we have used the experimental results from the
Super-Kamiokande
Collaboration on the zenith angle distribution of events taken on
five night periods and w xthe day averaged value, shown in Table 2.
which we graphically reduced from Ref. 9
We define x 2 for the zenith angle data as
2th BP98 expa R r R y R . z i i
i2x s , 7 .Ý Z 2s iis1
,6
where we have neglected the possible correlation between the errors
of the different
angular bins which could arise from systematical uncertainties. The
factor a is included z
in order to avoid over-counting the data on the total event rate
which is already included
in x 2. R
We compute the expected event rate in the period i in
the presence of oscillations as
1 .t cosF max,ith ² : R s
d t f dE l E
s E P E ,t . . .ÝH Hi k
n k n e , i
n e e n Dt .t
cosF i min,i k s1,8
² :qs E 1 y P E
,t , 8 . . . . x , i n e e
n
where t measures the yearly averaged length of
the period i normalized to 1, so . .Dt s
t cosF y t cosF s
0.500, 0.086, 0.091, 0.113, 0.111, 0.099 for the dayi max,i
min, i
and five night periods respectively.
Super-Kamiokande has also presented their results on the day–night
variation in the
form of a day–night asymmetry,
Day y Night A s 2 s y0.065 " 0.031 stat.
" 0.013 syst. . 9 . . .D r N Day q Night
Since the information included in the zenith angle dependence
already contains the
day–night asymmetry, we have not added the asymmetry as an
independent observable
in our fit. Notice also that, being a ratio of event rates, the
asymmetry is not a
Gaussian-distributed observable and therefore should not be
included in a x 2 analysis.
Table2 w xSuper-Kamiokande Collaboration zenith angle distribution
of events 9
Angular Range Data " s i i
Day 0 - cosu -1 0.463"0.0115
N1 y0.2 - cosu - 0 0.512"0.026
N2 y0.4 - cosu -y0.2 0.471"0.025
N3 y0.6 - cosu -y0.4 0.506"0.021
N4 y0.8 - cosu -y0.6 0.484"0.023
N5 y1- cosu -y0.8 0.478"0.023
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 9
Table3 w xRecoil energy spectrum of solar neutrinos from the
825-day Super-Kamiokande Collaboration data sample 9 .
Here s is the statistical error, s
is the error due to correlated experimental errors,
s is the errori,stat i,exp i,cal
due to the calculation of the expected spectrum, and s
is due to uncorrelated systematic errorsi,uncorr
. . .Energy bin Data " s s % s
% s %i i,stat i,exp i,cal
i,uncorr
5.5 MeV - E -6 MeV 0.472"0.037 1.3 0.3 4.0e
6 MeV - E -6.5 MeV 0.444"0.025 1.3 0.3 2.5e
6.5 MeV - E - 7 MeV 0.427"0.022 1.3 0.3 1.7e
7 MeV - E - 7.5 MeV 0.469"0.022 1.3 0.5 1.7e
7.5 MeV - E -8 MeV 0.516"0.022 1.5 0.7 1.7e
8 MeV - E -8.5 MeV 0.488"0.025 1.8 0.9 1.7e
8.5 MeV - E -9 MeV 0.444"0.025 2.2 1.1 1.7e
9 MeV - E -9.5 MeV 0.454"0.025 2.5 1.4 1.7e
9.5 MeV - E -10 MeV 0.516"0.029 2.9 1.7 1.7e
10 MeV - E -10.5 MeV 0.437"0.030 3.3 2.0
1.7e
10.5 MeV - E -11 MeV 0.439"0.032 3.8 2.3
1.7e
11 MeV - E -11.5 MeV 0.476"0.035 4.3 2.6
1.7e
11.5 MeV - E -12 MeV 0.481"0.039 4.8 3.0
1.7e
12. MeV - E -12.5 MeV 0.499"0.044 5.3 3.4
1.7e
12.5 MeV - E -13 MeV 0.538"0.054 6.0 3.8
1.7e
13 MeV - E -13.5 MeV 0.530"0.069 6.6 4.3
1.7e
13.5 MeV - E -14 MeV 0.689"0.092 7.3 4.7
1.7e
14 MeV - E - 20 MeV 0.612"0.077 9.2 5.8 1.7e
2.3. Recoil electron spectrum
The Super-Kamiokande Collaboration has also measured the recoil
electron energy w xspectrum. In their published analysis 67 after
504 days of operation they present their
.results for energies above 6.5 MeV using the Low Energy LE
analysis in which the
recoil energy spectrum is divided into 16 bins, 15 bins of 0.5 MeV
energy width and the
last bin containing all events with energy in the range 14 MeV to
20 MeV. Below 6.5
MeV the background of the LE analysis increases very fast as the
energy decreases. .Super-Kamiokande has designed a new Super Low
Energy SLE analysis in order to
reject this background more efficiently so as to be able to lower
their threshold down to w x5.5 MeV. In their 825-day data 9 they
have used the SLE method and they present
results for two additional bins with energies between 5.5 MeV and
6.5 MeV.
In our study we use the experimental results from the
Super-Kamiokande Collabora-
tion on the recoil electron spectrum on the 18 energy bins
including the results from the
LE analysis for the 16 bins above 6.5 MeV and the results from the
SLE analysis for the
two low energy bins below 6.5 MeV, shown in Table 3.
Notice that in Table 3 we have symmetrized the errors to be
included in our x 2
analysis. We have explicitly checked that the exclusion region is
very insensitive to this
symmetrization. We define x 2 for the spectrum as
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 10
where
s 2 s d s 2 q s 2 q s s
q s s . 11 . .i j i j i ,stat i ,uncorr
i ,exp j,exp i ,cal j,cal
Again, we introduce a normalization factor a in order
to avoid double-counting withs p
the data on the total event rate which is already included in
x 2. Notice that in our R
definition of x 2 we introduce the correlations
amongst the different systematic errors inS
the form of a non-diagonal error matrix in analogy to our previous
analysis of the total
rates. These correlations take into account the systematic
uncertainties related to the
absolute energy scale and energy resolution, which were not yet
available at the time the w xanalysis of Ref. 43 was performed.
Note that our procedure is different from that used
by the Super-Kamiokande collaboration. However, we will see in
Section 3.3 that both
methods give very similar results for the exclusion regions. This
provides a good test of
the robustness of the results of the fits, in the sense that they
do not depend on the
details of the statistical analysis.
The general expression of the expected rate in the presence of
oscillations R th in a .bin, is given from Eq. 6 but
integrating within the corresponding electron recoil energy
bin and taking into account that the finite energy resolution
implies that the measured
kinetic energy T of the scattered electron is
distributed around the true kinetic energy X X
. w xT according to a resolution function Res
T , T of the form 68
2X 1 T y T .
X Res T , T s exp y ,
12 . .2' 2 s2p s
where
X's s s T rMeV , 13 .0
w xand s s 0.47 MeV for Super-Kamiokande 8,69 . On
the other hand, the distribution of 0
the true kinetic energy T X
for an interacting neutrino of energy E is
dictated by the n
X . X w xdifferential cross section
d s E , T
rdT , that we take from 66 . The kinematic limits a
n
are
E n X X X
0 ( T ( T E , T E s
. 14 . . . n n
1 q m r2 E e n
. .For assigned values of s , T
, and T , the corrected cross section
s E a s e, x0 min max
a
is given as X
X d s E , T .T .
a n T E max X Xn s
E s dT dT Res T ,
T . 15 . . .H H Xa n dT T
0min
In Fig. 1 we show our results for the recoil electron spectrum in
the absence of
oscillations and compare it with the expectations from the
Super-Kamiokande Monte
Carlo. We see that the agreement is excellent. In this figure no
normalization has been
included.
2.4. Seasonal Õariation
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 11
. . .Fig.1. Theoretical recoil electron energy spectrum obtained by
our calculation in 6 and 15 solid histogram .compared with the
Super-Kamiokande MC predictions dotted histogram . Also shown are
the data points
w xfrom Super-Kamiokande data 9 .
seasonal variation of the data, especially for recoil electron
energies above 11.5 MeV . w xsee Table 4 . As discussed in
Refs. 10,11 , the expected MSW event rates do exhibit a
seasonal effect. In the LMA region such dependence can be expected
mainly due to the
different night duration throughout the year at the experimental
site and also due to the
different averaged Earth densities crossed by the neutrino during
the night periods,
which lead to a seasonal-dependent n
regeneration effect in the Earth. On the othere
hand, in the SMA region, Earth matter effects are due to resonant
conversion of
neutrinos in the Earth which is only possible when neutrinos travel
both through the w xmantle and the core 50– 62 . Thus we find that
in the SMA region the seasonal variation
is associated with the fact that at Super-Kamiokande site only in
October–March nights
there are neutrinos arriving at sufficiently low zenith angle to
satisfy the resonant
condition.
We define x 2 for the seasonal variation data as
2th BP98 expa R r R y R .sea
i i i2x s , 16 .ÝSea
2s iis1,8
where, as before, the normalization factor a is
introduced to avoid double-counting.sea
Table4 w xSeasonal distribution of events given by the
Super-Kamiokande Collaboration 9
. .Period month Data " s E )11.5i i
0.0 - t -1.5 0.588"0.057
1.5- t - 3.0 0.588"0.057
3.0 - t - 4.5 0.532"0.069
4.5- t -6.0 0.392"0.059
6.0 - t - 7.5 0.473"0.059
7.5- t -9.0 0.521"0.065
9.0 - t -10.5 0.548"0.065
10.5- t -12.0 0.522"0.058
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 12
Taking into account the relative position of the Super-Kamiokande
setup in each
period of the year, we calculate the distribution of the events
as
t qD t r2i thdt R t .Hth R t ,
Dt . t yD t r2i i i s . 17
.BP98 th R Dt DtR t . .i
th . .Here Dt s 1.5 months and R t
is obtained from Eq. 6 but using the time-dependent
survival probabilities and integrating the recoil electron energy
above 11.5 MeV. Notice
that, unlike in the 708 days data sample, in order to compare our
results with the recent
data on the seasonal dependence of the event rates from the
Super-Kamiokande
Collaboration for the 825 data sample, we have not included the
geometrical seasonal
neutrino flux variation due to the variation of the Sun–Earth
distance arising from the
Earth’s orbit eccentricity because the new Super-Kamiokande data is
already corrected
for this geometrical variation.
3. Fits: results
We now turn to the results of our fits with the observables
described above. We have
obtained the regions of allowed oscillation parameters
Dm2–sin2 2u by obtaining the 2 2 2 2 . 2
.minimum x and imposing the condition
x ( x q Dx 2,CL ,
where Dx 2,CLmin
. .s 4.61 9.21 for 90% 99% CL regions.
3.1. Rates
We first determine the allowed range of oscillation parameters
using only the total
event rates of the chlorine, gallium and Super-Kamiokande
experiments. The average
event rates for these experiments are summarized in Table 1. We
have not included in w xour analysis the Kamiokande data 7 as it is
well in agreement with the results from the
w xSuper-Kamiokande experiment and the precision of this last one
is much higher 9 . For
the gallium experiments we have used the weighted average of the
results from w x w xGALLEX 5 and SAGE 6 detectors.
Using the predicted fluxes from the BP98 model the x 2
for the total event rates is x 2 s 62.4 for 3 d.o.f. This
means that the SSM together with the SM of particleSSM
interactions can explain the observed data with a probability lower
than 10 y1 2!
In the case of active–active neutrino oscillations we find that the
best-fit point is
obtained for the SMA solution with
Dm2 s 5.6 = 10y6 eV 2 ,
sin2 2u s 6.3 = 10y3 ,
x 2 s 0.37, 18 .min
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 13
There are two more local minima of x 2. One is the
LMA solution with the best fit
for
sin2 2u s 0.67,
x 2 s 2.92, 19 .min
which is acceptable with at 91% CL. The other is the LOW solution
with best-fit point
Dm2 s 1.3 = 10y7 eV 2 ,
sin2 2u s 0.94,
which is only acceptable at 99% CL.
In the case of active–sterile neutrino oscillations the best-fit
point is obtained for the
SMA solution with
sin2 2u s 5.0 = 10y3 ,
x 2 s 2.6, 21 .min
acceptable at 90% CL. The LMA and LOW solutions are not acceptable
for oscillation
into sterile neutrinos. In those regions x 2 0 19.5
implying that they are excluded at themin
99.999% CL. Unlike active neutrinos which lead to events in the
Super-Kamiokande
detector by interacting via neutral current with the electrons,
sterile neutrinos do not
contribute to the Super-Kamiokande event rates. Therefore a larger
survival probability
for 8
B neutrinos is needed to accommodate the measured rate. As a
consequence a larger
contribution from 8
B neutrinos to the chlorine and gallium experiments is expected,
so
that the small measured rate in chlorine can only be accommodated
if no 7
Be neutrinos
are present in the flux. This is only possible in the SMA solution
region, since in the
LMA and LOW regions the suppression of 7
Be neutrinos is not enough.
In Fig. 2 we show the 90% and 99% CL allowed regions in the plane
Dm2–sin2 2u .
The best-fit points in each region are marked. We find that as far
as the analysis of the
2 2 .Fig.2. Allowed regions in Dm and sin 2u
from the measurements of the total event rates at chlorine,
. . .gallium and Super-Kamiokande 825-day data sample for a
active–active transitions and b active–sterile.
. .The darker lighter areas indicate the 99% 90% CL regions. Global
best-fit point is indicated by a star.
Local best-fit points are indicated by a dot.
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 14
Fig.3. Same as Fig. 2 but allowing a free 8
B flux normalization.
total rates is concerned, there is no substantial change in the
best-fit points in the three
regions as compared to the previous most recent analysis including
the 504 days of w xSuper-Kamiokande data 43 .
We have also considered the possibility of departing from the SSM
of BP98 by
allowing a free normalization of the 8
B flux and we treat this normalization as a free
parameter b in our analysis. Fig. 3 shows the
allowed regions in the MSW parameter
space when b is allowed to take arbitrary values.
The best fit SMA solution is obtained
for
sin2 2u s 5.0 = 10y3 ,
b s 0.82,
The best fit for the LMA solution occurs at
Dm2 s 1.6 = 10y5 eV 2 ,
sin2 2u s 0.63,
x 2 s 0.47, 23 .min
and the LOW solution has its best-fit point at b s 0.98
and therefore coincides with the .one obtained in Eq. 20 .
In the case of active–sterile neutrino oscillations the best-fit
point is obtained for the
SMA solution:
sin2 2u s 3.2 = 10y3 ,
b s 0.75,
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 15
2 2 .Fig.4. Allowed regions in Dm and sin 2u
from the measurements of the total event rates at chlorine,
.gallium and Super-Kamiokande 825-day data sample combined with the
zenith angle distribution observed in
. . .Super-Kamiokande for a active–active and b active–sterile
transitions. The darker lighter areas indicate .99% 90% CL regions.
Global best-fit point is indicated by a star. Local best-fit points
are indicated by a dot.
The shadowed area represents the region excluded by the zenith
angle distribution data at 99% CL.
For all solutions we find that the main effect of considering a
free normalization of
the 8
B flux is upon the quality of the fits, as measured by the depth of
the x 2. Next
comes the position of the best-fit points, mainly a reduction in
the value of the mixing
angle. The allowed regions are considerably enlarged as can be seen
by comparing Figs.
2 and 3.
3.2. Yearly aÕeraged zenith angle dependence
We now study the constraints on the oscillation parameters from the
Super-Kamio-
kande Collaboration measurement of the zenith angle distribution of
events. We use the
data taken on five night periods and the day averaged value shown
in Table 2 which we w xgraphically reduced from Ref. 9 .
Considering only the zenith angle data not including the
information from the total .rates , the best-fit point in the case
of active–active neutrino oscillations is obtained for
Dm2 s 4.5 = 10y5 eV 2 and sin2 2u s 1.0 with x 2 s
2.3 for 3 d.o.f. and in the case of min
active–sterile neutrino oscillations is obtained for Dm2 s
3.2 = 10y5 eV 2 and sin2 2u s
0.98 with x 2 s 2.2. With these values we calculate the
excluded region of parametersmin
at the 99% CL, shown in Fig. 4. Notice that the zenith angle data
favours the LMA
solution of the solar neutrino problem.
When combining the information from both total rates and zenith
angle data, we
obtain that in the case of active–active neutrino oscillations the
best-fit point is still
obtained for the SMA solution with
Dm2 s 5.0 = 10y6 eV 2 ,
sin2 2u s 6.3 = 10y3 ,
x 2 s 5.9, 25 .min
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 16
for 6 d.o.f. which is acceptable with a 56% CL. However, the LMA
solution becomes
relatively better with a local minimum at
Dm2 s 4.5 = 10y5 eV 2 ,
sin2 2u s 0.80,
x 2 s 7.2, 26 .min
valid at 70% CL. This difference arises from the fact that,
although small, some effect is
observed in the zenith angle dependence which points towards a
larger event rate during
the night than during the day, and that this difference is constant
during the night as w xexpected for the LMA solution 44 . In the
SMA solution, however, the enhancement is
w xexpected to occur mainly in the fifth night 50–62 .
The LOW solution is almost un-modified and presents the best-fit
point at
Dm2 s 1.0 = 10y7 eV 2 ,
sin2 2u s 0.94,
x 2 s 12.7, 27 .min
which is acceptable at 95% CL.
In the case of active–sterile neutrino oscillations the best-fit
point is obtained for the
SMA solution with
sin2 2u s 5.0 = 10y3 ,
x 2 s 8.1, 28 .min
valid at 77% CL.
Fig. 4 shows the regions excluded at 99% CL by the zenith angle
data alone, together
with the 90% and 99% CL allowed regions in the plane Dm2
–sin2 2u from the
combined analysis of rates plus zenith angle data. The best-fit
points in each region are
indicated. The main difference with Fig. 2 is observed in the LMA
allowed region which
is cut from below, as the expected day–night variation is too large
for smaller neutrino
mass differences.
3.3. Recoil electron spectrum
We now present the results of the study of the recoil electron
spectrum data observed
in Super-Kamiokande. Using the method described in Section 2.3 we
obtain that the x 2
.for the undistorted energy spectrum Standard Model case is 20.1
for 17 d.o.f. This
corresponds to an agreement of the measured with the expected
energy shape at the 27%
CL. This value depends on the degree of correlation allowed between
the different
errors. In this way, reducing the correlation in the error matrix
the agreement decreases
down to 13%. This suggests that the correlations amongst the
different systematic
uncertainties related to the energy resolution will play an
important role in the analysis
of the energy spectrum. Indeed in our definition of
x 2 we introduce the correlationsS
amongst the different systematic errors in the form of a
non-diagonal error matrix in
analogy to our previous analysis of the total rates. These
correlations take into account
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 17
2 2 .Fig.5. Allowed regions in Dm and sin 2u
from the measurements of the total event rates at the
chlorine, .gallium and Super-Kamiokande 825-day data sample
combined with the recoil electron spectrum data
. .observed in Super-Kamiokande for a active–active oscillations
and b active–sterile oscillations. The darker . .lighter area
indicate 99% 90% CL regions. Global best-fit point is indicated by
a star. Local best-fit points
are indicated by a dot. The shadowed area represents the region
excluded by the spectrum data at 99% CL.
constitute the main difference between our treatment of the
spectrum data and the earlier w xone presented in Ref. 43 , when
this experimental information was still unavailable.
We find that the best fit to the spectrum in the MSW plane for the
case of
active–active neutrino oscillations is obtained for Dm2 s 6.3
= 10y6 eV 2 and sin2 2u s
0.08 with x 2 s 17.9. For the case of active–sterile
neutrino oscillations we getmin
Dm2 s 6.3 = 10y6 eV 2 and sin2 2u s 0.08 with x 2
s 17. With these values we obtainmin
the excluded region of parameters at the 99% CL shown in Fig.
5.
We see that our results for the exclusion regions are
quantitatively very similar to w xthose of the Super-Kamiokande
collaboration 9 , even though our procedure is different
from theirs. This provides a good test of the robustness of the
results of the fit of the
spectrum, in the sense that they do not depend on the details of
the statistical analysis. In w xcontrast we note that our results
are different from those of Ref. 43 .
When combining the information from both total rates and the recoil
energy spectrum
data we obtain that in the case of active–active neutrino
oscillations both SMA and
LMA solutions lead to fits to the data if similar quality. In this
way, the best-fit for the
LMA solution is
sin2 2u s 0.67,
while for the SMA solution we find
Dm2 s 5.6 = 10y6 eV 2 ,
sin2 2u s 5.0 = 10y3 ,
x 2 s 23.4, 30 .min
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 18
for 18. d.o.f. which are acceptable at 83%. Finally for the LOW
solution we find
Dm2 s 1.0 = 10y7 eV 2 ,
sin2 2u s 0.94,
with agreement with data only at the 8.5% CL.
In the case of active–sterile neutrino oscillations the best-fit
point is obtained for the
SMA solution with
sin2 2u s 3.0 = 10y3 ,
x 2 s 26.3. 32 .min
acceptable at 90%.
In Fig. 5 we plot the excluded region at 99% CL by the energy
spectrum data
together with the 90% and 99% CL allowed regions in the plane
Dm2 –sin2 2u from the
combined analysis. The best-fit points in each region are
marked.
The main point here is that the oscillation hypothesis does not
improve considerably
the fit to the energy spectrum as compared to the no-oscillation
hypothesis. In this w xconnection it has been suggested 70,71 that
better description can be obtained by
allowing a larger flux of hep neutrinos as they contribute mainly
to the end part of the
spectrum. In order to account for this possibility we have also
analysed the data allowing
for a free normalization of the 8
B and hep fluxes, treating them as a free parameters
b
and g correspondingly. When doing so we find that
the no-oscillation hypothesis gives x 2 s 17.4 for 15
d.o.f. for b s 0.45 and g s 13.5.
When combining the information from both total rates and the recoil
energy spectrum
data in the case of active–active neutrino oscillations we obtain
that for the LMA
Dm2 s 1.6 = 10y5 eV 2 ,
sin2 2u s 0.63,
b s 1.3, g s 33,
x 2 s 17.5, 33 .min
for 16 d.o.f. which is acceptable at 66% while for the SMA
solution
Dm2 s 5.0 = 10y6 eV 2 ,
sin2 2u s 2.5 = 10y3 ,
b s 0.61, g s 13,
x 2 s 19.5, 34 .min
which is acceptable at 75% CL. Finally for the LOW solution we
find
Dm2 s 1.0 = 10y7 eV 2 ,
sin2 2u s 0.94,
acceptable at 93%.
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 19
In the case of active–sterile neutrino oscillations the best-fit
point is obtained for the
SMA solution:
sin2 2u s 2.0 = 10y3 ,
b s 0.61, g s 12,
x 2 s 22. 36 .min
which is acceptable at 89% CL.
In Fig. 6 we display the normalized expected energy spectra for
SMA, LMA
solutions for active–active oscillations and for no-oscillation
with non-standard 8
B and
hep fluxes.
Recently Super-Kamiokande has also presented preliminary results
which seem to
hint for a seasonal variation of the event rates, especially for
recoil electron energy
above 11.5 MeV. As explained in Section 2.4 the expected MSW event
rates do exhibit
a seasonal effect due to the seasonal-dependent n
regeneration effect in the Earth. Wee
explore here the constraints on the MSW oscillation parameters
which can be extracted
from the seasonal dependence data given in Table 4. Considering
only the seasonal variation data above 11.5 MeV allowing a
free
8 . 2normalization for the B flux we find that the SSM
yields a value of x s 8. for 7
d.o.f. This shows that the data is still not precise enough to
enable one to draw any
definitive conclusion. However, one can still obtain some
preliminary information on the
MSW parameters from the analysis of these data. In this way, when
allowing oscilla-
w xFig.6. Expected normalized recoil electron energy spectrum
compared with the experimental data 9 . The
solid line represents the prediction for the best-fit SMA solution
with free 8
B and hep normalizations .b s 0.61,
g s12 , while the dashed line gives the corresponding
prediction for the best-fit LMA solution .b s1.37,
g s 38 . Finally, the dotted line represents the
prediction for the best non-oscillation scheme with
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 20
tions into active flavours we obtain the best-fit point for
Dm2 s 3.2 = 10y6 eV 2 and
sin2 2u s 0.1 with x 2 s 1.8 for 5 d.o.f. and in
the case of active–sterile neutrinomin
oscillations best fit is obtained for Dm2 s 1.3 = 10y6 eV 2
and sin2 2u s 0.1 with
x 2 s 2.0.min
In Fig. 7 we show the exclusion regions at 95% and 99% CL. The
larger area
represents the allowed region at 95% CL. Being an
allowed region, it means that at 95%
CL, some effect is observed. The darker area shows the small
excluded region at 99%
CL.
Since the seasonal variation of the event rates in the MSW region
is due to neutrino
regeneration in the Earth one expects a correlation with the
day–night variation which
arises from the same origin. This correlation is observed as the
95% allowed regions in
Fig. 7 have a large overlap with the 99% excluded region from the
observed zenith angle
dependence in Fig. 4. Notice however that the upper part of the 95%
CL allowed region 2 .for the oscillation in active flavours
larger Dm and larger mixing angles is still in
agreement with the observed zenith angle dependence. Thus, should
the seasonal effect
be confirmed in the higher energy part of the spectrum, it will
favour the LMA solution
to the solar neutrino problem.
3.5. Combined analysis
We now present our results for the simultaneous fits to all the
available data and
observables. In the combination we define the global
x 2 as the sum of the different x 2
defined above. In principle such analysis should be taken with a
grain of salt as these
pieces of information are not fully independent; in fact, they are
just different projec-
tions of the double differential spectrum of events as a function
of time and energy.
Thus in our combination we are neglecting possible correlations
between the uncertain-
ties in the energy and time dependence of the event rates.
Fig.7. 95 and 99% CL regions obtained from the analysis of the
seasonal dependence of the event rates . .observed in
Super-Kamiokande from data with E )11.5 for a
active–active oscillations and b active–sterilee
oscillations. The larger area represents the allowed region at 95%
CL. The darker area is the excluded region
at 99% CL.
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 21
w xFrom the full data sample we obtain that the for the SSM of Ref.
41
x 2 SSM s 96. 37 . .min
for 32 d.o.f. So the probability of explaining the full data sample
as a statistical
fluctuation of the SSM together with the SM of particle
interactions is smaller than
10y7.
In the MSW oscillation region we obtain that for oscillations into
active flavours the
best fits are obtained for the LMA solution with
Dm2 s 3.65 = 10y5 eV 2 ,
sin2 2u s 0.79,
x 2 s 35.4, 38 .min
for 30 d.o.f. which implies that the solution is acceptable at 77%
CL., while in the SMA
region the local best-fit point is
Dm2 s 5.1 = 10y6 eV 2 ,
sin2 2u s 5.5 = 10y3 ,
x 2 s 37.4, 39 .min
valid at the 83% CL. The global analysis still presents a minimum
in the LOW region
with best-fit point:
sin2 2u s 0.94,
x 2 s 40., 40 .min
which is acceptable at 90% CL.
The results of the global analysis for the case of active–sterile
neutrino oscillations,
show the fit is slightly worse in this case than in the
active–active oscillation scenario.
The best-fit point is obtained for the SMA solution with
Dm2 s 5.0 = 10y6 eV 2 ,
sin2 2u s 3.0 = 10y3 ,
x 2 s 40.2, 41 .min
acceptable at 90% CL. . .In Fig. 8 we show the 90% lighter and 99%
darker CL allowed regions in the
plane Dm2–sin2 2u . Best-fit points in each regions are
also indicated.
By comparing Fig. 8 and Fig. 2 we see the effect of the inclusion
of the full data from
Super-Kamiokande on both time and energy dependence of the event
rates. For
oscillations into active neutrinos, the larger modification is in
the LMA solution region
which has its lower part cut by the day–night variation data while
the upper part is
suppressed by the data on the recoiled energy spectrum. The best
fit point is also shifted
towards a larger mass difference by a more than a factor 2. As for
the SMA solution
region the position of the best fit point is shifted towards a
slightly smaller mixing angle,
while the size of the region at the 90% CL is reduced. At 99% CL
the allowed SMA
region is very little modified. For oscillations into sterile
neutrinos the best fit point is
also shifted towards a smaller mixing angle.
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 22
2 2 .Fig.8. Allowed regions in Dm and sin 2u
from the measurements of the total event rates at the
chlorine, .gallium and Super-Kamiokande 825-day data sample
combined with the zenith angle distribution observed in
.Super-Kamiokande, the recoil energy spectrum and the seasonal
dependence of the event rates, for a . . .active–active
oscillations and b active–sterile oscillations. The darker lighter
areas indicate 99% 90% CL
regions. Global best-fit point is indicated by a star. Local
best-fit points are indicated by a dot.
We finally study the allowed parameter space with the normalization
of the 8
B and
hep fluxes are left free. Fig. 9 shows the allowed regions in
the MSW parameter space
when the normalizations are allowed to take arbitrary values. The
best fit for the LMA
solution occurs at
sin2 2u s 0.67, b s 1.37, g s 38,
x 2 s 30.7, 42 .min
which is acceptable at 64% CL. The best fit SMA solution is
obtained for
Dm2 s 5.0 = 10y6 eV 2 ,
sin2 2u s 2.5 = 10y3 , b s 0.61, g s
12,
x 2 s 34, 43 .min
which is acceptable at 80% CL., The best fit for the LOW solution
occurs at
Dm2 s 1.0 = 10y7 eV 2 ,
sin2 2u s 0.94, b s 0.97, g s 21,
x 2 s 38.1, 44 .min
which is acceptable at 90% CL.,
In the case of active–sterile neutrino oscillations
Dm2 s 5.0 = 10y6 eV 2 ,
sin2 2u s 2.0 = 10y3 , b s 0.61, g s
12,
x 2 s 35.8, 45 .min
which is acceptable at 85% CL.,
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 23
Fig.9. Same as previous figure allowing free 8
B and hep neutrino flux normalizations.
For all solutions we find that the main effect of considering free
normalization of the 8
B and hep fluxes is upon the position of the best-fit
points, mainly a reduction in the
value of the mixing angle. The allowed regions are also enlarged as
can be seen by
comparing Fig. 8 and Fig. 9.
4. Summary and discussion
We have presented an updated global analysis of two-flavour MSW
solutions to the
solar neutrino problem using the full data set corresponding to the
825-day Super-
Kamiokande sample plus chlorine, GALLEX and SAGE experiments. In
addition to all
measured total event rates we included all Super-Kamiokande data on
the zenith angle
dependence, energy spectrum and seasonal variation of the events.
We have given a
comparison of the quality of different solutions of the solar
neutrino anomaly in terms of
MSW conversions of n into active and
sterile neutrinos. For the case of conversionse
into active neutrinos we have found that once the full data set is
included both SMA and
LMA solutions give an equally good fit to the data. We find that
the best-fit points for
the combined analysis are Dm2 s 3.6 = 10y5 eV 2 and sin2
2u s 0.79 with x 2 smin
35.4r30 d.o.f. and Dm2 s 5.1 = 10y6 eV 2 and sin2 2u s
5.5 = 10y3 with x 2 smin
37.4r30 d.o.f. In contrast with the earlier 504-day study of
Bahcall–Krastev–Smirnov
our results indicate that the LMA solution is slightly preferred.
Although small, there is
a hint for seasonality in the data and this should therefore take
into account in future .studies, as we have indicated here.
Defining the seasonal variation in percent as
R th y R th max min
Var ' 2 , th th R q Rmax min
th .where R is the expected event rate during the
winter summer period, we findmaxmin .
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 24
6% comes from the variation expected from the geometric effect due
to the eccentricity .of the Earth’s orbit , without conflict with
the negative hints of a day–night variation.
Such seasonal dependence is correlated with the day–night effect
and this in turn can be
used in order to discriminate the MSW from the vacuum oscillation
solution to the solar w xneutrino anomaly. We have performed a
numerical study of this correlation 11 which
generalizes the estimate presented by Smirnov at the 1999 edition
of Moriond for a
constant Earth density. Our results show that due to the Earth
matter profile the seasonal
variation can be substantially enhanced or suppressed as compared
to the expected value
obtained with an average Earth density. One must notice, however,
that the exact values
of masses and mixing for which a significant enhancement is
possible may depend on
the precise model of the Earth density profile. The numerical
results presented above
were obtained using the step function profile and thus numerical
differences with the w xresults obtained with, for instance the
PREM model 44 can be expected for a given
point in the MSW plane.
With the results from the combined analysis one can also predict
the expected event
rates at future experiments. For example, we find that the average
oscillation probabili-
ties for 7
Be neutrinos in the different allowed regions for MSW oscillations
into active
neutrinos is
P 7
P 7
s 0.55q0.08 . 47 .Be LMA y0.08
These results imply that, at Borexino we expect a suppression on
the number of events
as compared with the predictions of the SSM of
RSMA r R BP98 s 0.22q0.11 , 48 .y0.004
R LMA r RBP98 s 0.65q0.06 . 49 .y0.06
For conversions into sterile neutrinos only the SMA solution is
possible with best-fit
point Dm2 s 5.0 = 10y6 eV 2 and sin2 2u s 3.2 = 10y3
and x 2 s 40.2r30 d.o.f.min
which leads to
and one expects a larger suppression of events at Borexino
Rsteriler R BP98 s 0.015q0.090 . 51 .y0.002
As a way to improve the description of the observed recoil electron
energy spectrum
we have also considered the effect of departing from the SSM of
Bahcall and
Pinsonneault 1998 by allowing arbitrary 8
B and hep fluxes. Our results show that this
additional freedom does not lead to a significant modification of
the oscillation
parameters. The best fit is obtained for 8
Br 8
B s 0.61 and heprhep s 12 for theSSM SSM
SMA solution both for conversions into active or sterile neutrinos
and 8 Br8 B s 1.37SSM
and heprhep s 38 for the LMA solution.SSM
( ) M.C. Gonzalez-Garcia et al.r Nuclear Physics B 573
2000 3– 26 25
Acknowledgements
M.C.G.-G. is thankful to the Instituto de Fisica Teorica of UNESP
and to the CERN
theory division for their kind hospitality during her visits. We
thank comments by John
Bahcall, Vernon Barger, Venya Berezinsky, Plamen Krastev and Alexei
Smirnov. This
work was supported by Spanish DGICYT under grant PB95-1077, by the
European
Union TMR network ERBFMRXCT960090 and by Brazilian funding agencies
CNPq,
FAPESP and the PRONEX program.
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www.elsevier.nlrlocaternpe
.Direct detection of Earth matter effects MSW in flavor
oscillations at neutrino beams from stored muon decays
A. Bueno a, M. Campanelli b, A. Rubbia a,b
a Institut fur Teilchenphysik, ETHZ, CH-8093 Zurich,
Switzerland ¨ ¨
b CERN, GeneÕa, Switzerland
Abstract
We explore the possibility of a neutrino oscillation experiment
with a very long baseline in the
range of 6500 km and a neutrino beam produced by the decays of
muons circulating in a storage
ring. The recent developments in view of muon colliders allow us to
envisage neutrino sources of
a sufficiently high intensity. We first consider n
l
n oscillations within a three flavor
oscillatione m
.framework. Evidence for this oscillation implies that the 1– 3
mixing is non-zero. We study the .effect of the neutrino
propagation through matter MSW effect . Given the density of the
Earth,
the existence of the MSW resonance can be experimentally proven by
comparing the oscillated
spectra of neutrinos obtained from the decays of muons of positive
and negative charges.
Moreover, a precision study of the oscillations of all three
flavors could be performed since .neutrinos are above the tau
production threshold appearance searches . q 2000 Elsevier
Science
B.V. All rights reserved.
1. Introduction
As is well known, neutrino flavor oscillations will take place if
the neutrino flavor . .eigenstates n ,
n and n do not coincide with
their mass eigenstates n , n
and n .e m t 1 2 3
Within the two flavor oscillation framework, the transition
probability in vacuum .between flavor n and
n a ,b s e,m,t , a/b is given
by
a b
L km . 2 2 2 2P n
™
n ssin 2u sin 1.27Dm eV , 1 . . .a
b i j / E GeV .
n
2 2 2 .where Dm s m y m i, j s
1,2,3,i/ j is the mass squared difference between thei
j i j
two neutrino mass eigenstates, n and
n , u is the mixing angle between
mass and weak i j
eigenstates, L is the baseline, and
E the neutrino energy. When the neutrinos propagate
n
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27–
3928
in matter, their equation of motion must include an interaction
with the medium. For a
given neutrino energy which depends on the matter electron density,
a ‘‘resonance’’ w x.condition will be reached the MSW effect
1–3 at which the flavor oscillation of
. .neutrinos antineutrinos will then be enhanced suppressed
compared to oscillations in
vacuum.
The results on the solar and atmospheric neutrino fluxes can be
naturally explained in
terms of neutrino oscillations. Since the energies and distances of
the solar neutrino
problem are widely different from those of the atmospheric neutrino
anomaly, the
neutrino oscillation solutions require very different values of
mass squared differences. w xThe solar neutrino deficit 4 can be
explained via the disappearance of n ’s due
toe
flavor oscillation enhanced by the MSW effect. A combined fit to
the experimental data
implies a Dm2 in the region 10y5 eV 2 and two possible
angular solutions: the small . .SMA and large LMA mixing angle
solutions. Another solution to these data would
require a Dm2 f 10y10 eV 2 if the oscillation took place in
vacuum.
The MSW solution of the solar neutrino problem is the more
attractive one because it
does not require the fine-tuning of the oscillation parameters. The
conversion in the Sun
is primarily a resonance phenomenon, which occurs at a specific
density that corre- 2 .sponds to a definite neutrino energy
for a given Dm . Because at night solar
neutrinos
will cross the Earth before reaching the detector, the MSW
mechanism may introduce a w xday-night asymmetry 18 by the
phenomenon of ‘‘regeneration’’. However, in the
relatively small density of the Earth, the resonance condition is
not met. Clearly the
search for the MSW effect on Earth will be of great importance. w
xThe observation of atmospheric neutrinos by SuperKamiokande 5 and
other detec-
w x .tors 6–11 Kamiokande, IMB, Soudan-II and MACRO has
shown evidence for
neutrino flavor oscillations, compatible with n
neutrinos converting to n with maximal m
t
mixing and 10y3 Q Dm2 Q 10y2 eV 2.
For the present generation of long baseline beams from CERN and
Fermilab, the w xbaseline of L s 730 km is too short to
provide a strong resonance signal 12 . Within
this context, we explore the possibility of an experiment using
neutrinos from stored
muon decays at a very long baseline of 6500 km, with neutrinos of
sufficient energy in
order to produce the resonance phenomena in their passage through
Earth for the set of
parameters suggested by the atmospheric neutrino results. Since
this phenomenology
implies oscillations into tau neutrinos, the energy should also be
large enough in order to
observe explicitly the charged current interactions of
n ’s. t
w xThe synergy with future muon colliders 13 is here twofold: Muon
colliders will
necessarily require very intense proton sources and it will be then
conceivable to obtain w xvery intense neutrino beams 14 to
compensate for the flux decrease with increasing
distance from the source. In addition, the well-defined flavor
composition of beams from
muon decays can be exploited using a detector with charge
identification capabilities to w xlook for flavor oscillations
15,16 .
We first consider n
l
n oscillations within a three flavor
oscillation framework, ate m
the Dm2 s m2 y m2 mass indicated by the atmospheric
neutrinos. Evidence for this32 3 2
.oscillation would imply that the 1– 3 mixing between the first and
the third family is
non-zero. We study the effect of the neutrino propagation through
the matter of the .Earth MSW effect . Given the density of the
Earth, the MSW resonance occur at high
energies accessible with accelerators.
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27– 39
29
With the use neutrino beams from stored muon decays, the matter
enhancement or
suppression of the oscillation can be experimentally tested by
comparing the event rates
and energy spectra obtained from decaying stored muons of positive
and negative
charges.
We then consider the study of oscillations of all three flavors.
Our high energy
neutrino beam is well above tau threshold, therefore opening the
channel of tau
appearance searches. These would be optimized for the Dm2
mass indicated by the
atmospheric neutrino anomaly since the baseline is such that the
relevant parameter
E r L is exactly in that range
of Dm2. The transition probabilities will therefore be
n
maximized and the flavor oscillation pattern will be visible as a
function of the incoming
neutrino energy. This is a main difference with respect to
presently planned long
baseline beams where given the baseline of 730 km the basic
oscillation can be
measured only in the upper part of the region indicated by the
atmospheric neutrinos.
Extending the baseline without a loss of rate will enlarge the
explored Dm2 region
correspondingly.
2. Neutrinos from decays of stored muons
w xThe neutrino beams from the decays of stored muon 14 provide an
ideal configura-
tion for the study of matter effects. First of all,
.1 they will contain neutrinos of the electron and muon flavors in
same quantity of
a well-defined helicity composition depending on the muon charge,
i.e. my
™
y q qe n n or m
™
e n n .e m e m
.2 the charge of the muon can be easily selected, ideally within
each fillings of the
storage ring.
These features distinguish them from traditional neutrino beams
where n dominates, m
where n comes from kaon decays since highly
suppressed
in p ™
en , and in which thee e
neutrino–antineutrino configurations are not symmetric. In
addition, the fluxes of neutrino beams from muon decays can
be easily predicted since no hadronic processes
.involved when the muon polarization is known and are flexible in
the choice of the
beam energy, since precisely determined by the muon storage ring
energy.
Secondly, the well-defined flavor composition can be exploited
using a detector with w x.charge identification capabilities
see Refs. 15,16 . For decays of negative stored
y y .muons m
™
e n n , the charged current neutrino
interactions will produce leadinge m q .electrons of
positive charge n
N ™
e q X and leading muons of negative
chargee
y .n
N ™
m q X . In case of neutrino
n
l
n oscillations, there will be appearance
of m e m
ynegative leading electrons n
™
n N
™
e q X and of positive leading muons
n
™
n N m e e m
™
mqq X . For decays of positive stored muons, the charge
conjugate configurations
will occur.
In the case of massive detectors, it will be easier to measure the
charge of leading
muons than that of electrons. The golden channel to look for
oscillations is therefore
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27–
3930
.Fig. 1. Possible very long baselines across the Earth seen from
above the North pole .
be discussed in Section 5. Some information will also be available
from the leading
electron sample without charge discrimination but with lesser
sensitivity.
3. Very long baselines and neutrino source requirements
Since muon colliders necessarily require very intense proton
sources, it will be
possible to obtain very intense neutrino beams to perform long and
very long baseline
oscillation experiments1.
We assume here for the sake of a concrete example that the neutrino
detector will be .located in Europe at the Laboratori Nazionali del
GranSasso LNGS . The neutrino
source could then be located in different laboratories around the
world, in the American . .BNL, FNAL or Asian KEK continent.
.The BNL-LNGS has a baseline of L s 6500 km see Fig. 1
. The beam goes to a
maximum depth of 900 km and arrives at GS with an angle with
respect to horizontal of o w xabout 30 . We estimate the
average density of the Earth 18 for this baseline to be
3 r s 3.6grcm . Other baselines are FNAL-GranSasso L s
7400 km, max. depth 1200 3 o . km, average density
r s 3.7grcm , beam angle 36 or KEK-GranSasso L s
8800
3 o .km, max. depth 1800 km, average density r s
4.0grcm , beam angle 44 . There is not
much differences in L between the various
baselines.
In a neutrino beam from stored decaying muons, the neutrino event
rate will grow as
E 3 where E is the energy of the
muon storage ring. We are interested in a neutrino m m
beam with energy in the range of 10–30 GeV. We list in Table 1 the
neutrino event rates
in case of no oscillation as a function of the muon storage ring
energy, for 10 21 muon
decays of a given charge and 10 kton target. Neutrino interactions
have been divided . .into charged current CC and neutral current NC
interactions. The CC events include
. .inelastic scattering DIS and quasi-elastic QE
interactions.
1 w xIn Ref. 17 , we have considered a n
disappearance experiment with a baseline of 730 km. Since the
m
neutrino beams contain both electron and muon flavors, the
disappearance of muon neutrino can be performed
by directly comparing electron and muon events. This method is self
normalizing, i.e. without the need of a
near detector to predict the original flux.
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27– 39
31
Table1 .The total number of neutrinos detected in a 10 kton
fiducial detector for a baseline Ls6500 km and a total
number of 1021 muons decays
21 y 21 q10 m decays 10 m
decays
. . . . . E GeV CC n CC
n NC CC n CC n
NC m e m e m
10 426 1152 196 1016 488 173
15 1414 3751 1129 3283 1624 1005
20 3313 8712 3542 7588 3820 3172
25 6412 16746 8221 14568 7401 7419
30 11010 28576 16008 24850 12710 14524
To reach 10 21 muon decays, we refer to recent simulations
performed for muon w xcolliders studies 13 which consider a scheme
in which the yield of accumulated muons
in the storage ring per 16 GeV proton impinging on the primary
target is about 10-15%.
Such a storage ring would be operated for four years, alternating
runs with positive
and negative muons. Given the shape of the muon storage ring, about
40% of the muons would decay in the direction towards the neutrino
detector the rest decays in the
.bending section and in the opposite direction . To achieve an
integrated intensity a total
of
N mq q N my
s N =0.15=0.4=4 , 2=10 21m’s . . p
requires a proton source of the order of N
, 10 22 protonsryear. An upgrade of the p
w xAGS accelerator in BNL 19 could yield an integrated intensity
close to this figure.
4. Matter enhanced neutrino oscillations
w xIn three-family scenarios 20 , the mixing between the neutrino
flavors will be
determined by a 3=3 unitary matrix describing the mixing between
flavor and mass
eigenstates. With three neutrino states, oscillations will be
determined by only two
independent mass square differences, say Dm2 s m2 y m2 and
Dm2 s m2 y m2.32 3 2 21 2 1
The current experimental results on solar and atmospheric neutrinos
allow us to
consider the approximation that only one mass scale is relevant,
since if we assign 2 2 . . 2 2Dm Dm to
the atmospheric solar oscillations, then Dm 4Dm
. This implies32 21 32 21
that the two oscillations driven by the mass differences Dm2
and Dm2 decouple and32 21
can be studied independently.
The three-family oscillation is then described by only three
parameters: the mass
difference Dm2 and the two mixing angles
u and f . The mass eigenstate m
is defined32 1
orthogonal to the electron flavor state. The mixing matrix takes
then the form
0 cosu sinu
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27–
3932
We assume that n is the heaviest state and the
atmospheric neutrino data implies3
2 2 .that the mixing is maximal between n
and n sin f s cos f s 0.5 .
The flavor m t
oscillation probabilities for neutrinos and antineutrinos are then
simply
P n
™
n s P n
™
n s P n
™
n s P n
™
n . . . .e m e t e
m e t
L km . 2 2 2 2s sin 2u sin 1.27Dm eV . 3 .
.32 / E GeV .n
w xThe negative result from CHOOZ 21 on electron disappearance and
the Su-
perKamiokande data themselves constrain the mixing angle
u . A value compatible with 2 w xobservation is sin
u s 0.025 12 .
In matter, the modification of the flavor transition is taken into
account by the mixing
angle in matter u , which ism
sin2 2u 2 .sin 2u x s , 4 .
.m 22sin 2u q x.cos2u .
where the minus sign applies to n ’s and the plus to
n ’s and
3'2 2 G n E r grcm E
GeV . .F e n n y4 x s f 0.76=10 , 5 .2
2 2Dm Dm eV .
where n is the electron density of the medium. The
transition probabilities are thene
ly km . 2 y 2 2 2P n
™
n s P n
™
n s sin 2u sin 1.27Dm eV ,
6 . . . . .e m e t m 32
/ E GeV .n
lq km . 2 q 2 2 2P n
™
n s P n
™
n s sin 2u sin 1.27Dm eV ,
7 . . . . .e m e t m
32 / E GeV .n
2" 2(where l s L= sin 2u q
x"cos2u . For neutrinos, the resonance
condition will . 2 .be met when x E , Dm
, r , cos2u and the oscillation
amplitude will reach a maxi-
n
mum. For the resonant neutrino energy, E res , this
reads n
1.32=10 4cos2uDm2 eV 2 . res 4 2 2 E f f 0.37=10
Dm eV , 8 . .n 3r grcm .
where we have assumed a constant Earth density r s 3.6
grcm3 and small mixing
angles. For the parameters indicated by the atmospheric neutrino
observation 10 y3 Q
Dm2 Q 10y2 eV 2, we obtain 3.7 Q E res Q 37 GeV, i.e. the
resonance energy lies in the n
region accessible to high energy accelerator neutrino beams.
The conversion from n
™
n flavor is independent of matter effects and
is given by m t
the probability
L km . 2 2 2P n
™
n s P n
™
n f sin 1.27Dm eV , 9 . . . .m t
m t 32 / E GeV .n
where we have made the approximation that cos 2u s 1 since
u is a small angle. Given 2 .our choice
of E r L f Dm , there will be large
oscillation to n n .
n 32 t t
( ) A. Bueno et al.r Nuclear Physics B 573 2000 27– 39
33
5. Detection of neutrino oscillations and matter effects
We consider in more detail neutrino oscillations for Dm2 s
3=10y3 eV 2. Our
conclusion remain unchanged for values of Dm2 in the
range 10y2 Q Dm2 Q 10y3 eV 2
w x.see Ref. 22 . We study the three-family oscillation of
the two neutrino flavors present
in the beam. In the case of a neutrino beam from negative muons, we
will observe n
™
n , n
m e m t e m e
t
.processes .
The oscillated neutrino fluxes as a function of energy for the
different flavors are yshown in Fig. 2 for a 30 GeV muon beam. The
difference between the n
™
n in me m
decays and the n
™
n in mq decays is clearly visible. We
observe a similar effect fore m y q . . .n n
appearance in m m decays. In both cases,
the n
™
n n
™ &n