





Neutrino Mixing and Future Solar Neutrino Experiments

S.M. Bilenky and C. Giunti
           





  





Joint Institute for Nuclear Research, Dubna, Russia
 





INFN Torino, Via P. Giuria 1, I10125 Torino, Italy
 





Dipartimento di Fisica Teorica, Universit`a di Torino


Possibilities of a model independent treatment of the data from future real-time solar neutrino experiments (SNO,
Super-Kamiokande and others) are discussed. It is shown that in the general case of transitions of the initial solar
's into and/or the total flux of initial neutrinos and the survival probability can be determined directly






from the experimental data. Lower bounds for the probability of transition of solar 's into all possible sterile
states are derived and expressed through measurable quantities.






Talk presented by S.M. Bilenky at TAUP93, LNGS (Assergi, Italy), September 1993. DFTT 66/93.



1. Introduction dent way whether there are sterile neutrinos in solar
neutrino flux on the earth.
As is well known [1], the existing solar neutrino
data can be described by the resonant MSW enhance- 2. Transitions of solar into and/or
U V U W U X





ment of the mixing between two neutrino flavours.
For the parameters The spectrum of the initial neutrinos is given by
$ and
! ( )




! " ! ! % & '





! #
( the expression
$ and are neutrino masses and is the vac-
)










!

uum mixing angle) a small mixing angle solution
Y ` a ` a

( and ) (4)
b c b c g





H d P e f H d P





( )

!



! 0 1 2 3 4 5 6 7 8 ! % & ' 0 9 2 3 4 5 @





and a large mixing angle solution ( ! 0 3 4 5 A 7 8 !
and ) were obtained. These solutions where is a known function (the phase space
g





H d P





! ( )




% & ' 0 4 B 9






F D G h D
were found, however, from relatively low statistics factor of the decay ) normalized
7





i p





data and under the assumption that the Standard Solar by the condition . The total flux
g





H d P q d e 3 r





Model (SSM) [2] correctly predicts the neutrino fluxes
`

a
from different reactions of the and CNO cycles. is determined by number of nuclei in the
b c





f





C C





We will discuss here [3, 4, 5] the possibilities of active zone of the sun and strongly depends on many
a model independent treatment of the data of the parameters (temperature, cross-sections of different
future solar neutrino experiments (SNO [6], Super- reactions and so on).
Kamiokande (S-K) [7], ICARUS [8], etc.). In these The flux of solar
's on the earth as a function
experiments high energy neutrinos mostly from of neutrino energy will be determined through the
d











observation of CC-induced events (1):
 26 Oct 1993 decay will be detected. The distinctive new feature
of the SNO experiment will be that in this experiment
Y





Q s s s s
solar neutrinos will be detected through the simultane- (5)
b c





c v
b





H d P e H d P t u H d P





ous observation of the CC, NC and neutrino-electron
elastic scattering (ES) processes where Q s s is the differential CC event rate and
H d P





s s is the total cross section of process (1). Fur-
c v
b

u H d P





thermore, through the observation of CC and NC or
D E F G D D (1)
5


C C H I I P CC and ES events the following ratios can be deter-
D E F D D Q (2)
C H R I P mined
D G F D G (3)
5 5



H S T P





p





Y

w x x x





s s s





b c





v
b



We will show here that in the general case of transi- e 3 u H d P H d P q d t y
#





tions of solar p
's into and/or the initial neu-






Y

w











trino flux can be determined independently on what b c b c
e 3 u H d P H d P q d t y





#





is going on with neutrinos on their way from the sun
to the earth. This will allow to determine the x
sur- where s and are the total NC and ES event
y y





vival probability directly from the experimental data. rates; x

s and are the cross sections of
c
b





v
b



u H d P u H d P





We will show also that the data of the SNO and S-K the
E F Q and G F G processes, respec-
C 5 5





w w
experiments will allow to check in a model indepen- tively. If x
s ( ) it will be a direct proof
4 4





1


Y

that there are
and/or in the solar neutrino flux where is the flux of sterile neutrinos on the
b





H d P





on the earth. earth. If it will occur that the left hand side of this
The total flux of neutrinos can be measured equation, in which only measurable quantities enter,






through the observation of NC events. In fact we is found to be not equal to zero, it will mean that in
have the flux of solar neutrinos on the earth besides active
neutrinos there are also sterile neutrinos. However,
` a

x x
s s (6)
g v
b c b
f e y t according to our model calculations the two terms in
p the right-hand side of Eq.(8) could cancel each other.
where x x
s s





v v
g b b g In ref.[5] we proposed other relations which will allow
" u H d P H d P q d e B 3 2





to reveal the presence of sterile neutrinos. In general,
. A comparison of the results of the mea-
5  @   !



3 4





for the average probability we have
surement of the total flux with the SSM will be a severe
test of the model.





x





For the s
survival probability we have y






c 
b b





e





`


a

x





s







x v
g b b c






f
s Q s s NC
v
g b




 H d P





c  c
b b





H d P e






x





s s s










v
b c g





u H d P H d P y














b c b






e





`

a


Thus the survival probability of solar
as a function
v
g b b c






f
ES
of neutrino energy can be determined directly from
d





the data of the SNO experiment without any assump- where
tion about the initial neutrino flux.












The total neutrino flux can be determined also






c 
from the observation of CC and ES events. We have b b













NC
`

a







p
(7)
c
b g b 





f e  t





x







s





c 
b b





v
b g





u H d P H d P H d P q d





where






p 





"





x





s

Y












v
g b





b c b b c






 " y u H d P u H d P  H d P q d













# #











p





c 
b b





and
















b 






g b  g (
" u H d P H d P q d e B 2 ES






p





. From Eq.(6) and Eq.(7) it follows that
5  A !

 



3 4













the quantities 
b b c b


x




g





u H d P H d P H d P q d
s and that will be measured in
y 

















the SNO and S-K experiments are not independent and
"





are connected by the relation
x
s , where
g

b 

y e 





x





s
. This relation can be
v
g b g b 

!



" t e 3 B 1 2 3 4











violated only if there are transitions of solar
's into Furthermore, taking into account that 
b c b c





H d P e





sterile states. ` `
Y a a Y

and hence  " ,
c c
b b





b c g b c g





H d P t f H d P f H t P !





for the average probability of transition of solar 's
3. Sterile Neutrinos into all possible sterile states we obtain the following
The problem of existence of sterile neutrinos is of lower bounds:
principal importance for the theory of neutrino mixing
#



x







(see ref.[9]). Future solar neutrino experiments could s
c  $
b b











3

x





s





# %





allow to reveal whether there are transitions of solar #







b c b $











3









into sterile states [4, 5]. Using only the unitarity # %





of the mixing matrix (conservation of probability) and where
excluding the initial flux, in the general case of active
and sterile neutrinos instead of the relation x
Y

x x x





s




s s s





 "


c
b



y e


g g

v
b



" y t H t P

!





we have %
Y



















 "




b c





g b  g





"  t H t P !





p





%





Y

x






s





b b






y  e u H d P H d P q d

are measurable quantities. If it will occur that x
s &





#



(8)
p








%
and/or & it will mean that there are transitions
3 3


Y

x





s






b

%



v
b
u H d P H d P q d of solar
's into sterile states. It is easy to see that
#





2


the ratios x
s and can be less than 1 only if the P.I. Krastev and S.T. Petcov, Phys. Lett. B 299
% %
transition probability of solar
's into all active states (1993) 99.
depends on neutrino energy . Thus, if x
s &





d 3





[2] J.N. Bahcall and R. Ulrich, Rev. Mod. Phys. 60
%
and/or & it will mean not only that sterile
3





(1988) 297; J.N. Bahcall, Neutrino Physics and
%
neutrinos exist but also that the transition probability
of solar Astrophysics, 1989; J.N. Bahcall and M.H. Pin-

's into sterile states depends on energy.
In S-K and other future solar neutrino experiments sonneault, Rev. Mod. Phys. 64 (1992) 885.
(ICARUS, etc.) a large number of solar neutrino in- [3] H.H. Chen, Phys. Rev. Lett. 55 (1985) 1534; R.S.
duced ES events will be observed. From these data Raghavan et al., Phys. Rev. Lett. 57 (1986) 1801;
the differential ES event rate
Q will be deter-
H d P J.N. Bahcall et al., Phys. Lett. B 178 (1986) 324;
mined and new possibilities for testing the existence S. Weinberg, Int. J. Mod. Phys. A 2 (1987) 301;
of sterile neutrinos will emerge. In fact, a measure- C. Rubbia, CERN-PPE/93-08.
ment of
Q and Q s s will allow to determine
H d P H d P





the differential flux of all types of active neutrinos on [4] S.M. Bilenky and C. Giunti, Phys. Lett. B 311
the earth: (1993) 179.







i












Q Q s s
b c [5] S.M. Bilenky and C. Giunti, DFTT 62/93.
Y

H d P u H d P H d P





D





b





H d P e 3 r











s s





b b
 




#





u H d P u H d P v
b c





u H d P

[6] SNO Coll., Phys. Lett. B 194 (1987) 321; H.H.
Chen, Nucl. Instr. Meth. A 264 (1988) 48.
From the conservation of probability we obtain
[7] Y. Totsuka, ICRR-report-227-90-20 (1990).
`

Y a

3 






b b c b






c
b
H d P e f 3 H d P  [8] ICARUS Coll., LNF-89/005(R) (1989).
g #




H d P

















[9] S.M. Bilenky and B. Pontecorvo, Phys. Rep. 41
If it will occur that the left-hand side of this equa- (1978) 225; S.M. Bilenky and S.T. Petcov, Rev.
tion, which contains only measurable quantities, de- Mod. Phys. 59 (1987) 671.
pends on energy, then it will mean that there are sterile
neutrinos in the flux of solar neutrinos on the earth.
Furthermore, in the case under consideration we can
obtain a lower bound for the transition probability of
solar 's into sterile states as a function of energy:






Y





b





H d P
























b c b












H d P 3











#





Y











b





g g  





H d P t
 

















 "





!





4. Conclusions

We have demonstrated that future solar neutrino ex-
periments in which neutrino will be detected through
the observation of different reactions have a great po-
tential for a model independent investigation of the
neutrino mixing problem. These experiments could
allow also to measure the flux of initial
's and to






test the SSM.


References

[1] GALLEX Coll., Phys. Lett. B 285 (1992) 390;
S.A. Bludman et al., Phys. Rev. D 47 (1993) 2220;
L.M. Krauss et al., Phys. Lett. B 299 (1993) 94;


3



