DFTT 75/93
November 1993





Future Solar Neutrino Experiments and
Neutrino Spin-Flavour Precession

 





S.M. Bilenky and C. Giunti

! #"





Joint Institute of Nuclear Research, Dubna, Russia
%$&"





INFN Torino, Via P. Giuria 1, I10125 Torino, Italy
%'"





Dipartimento di Fisica Teorica, Universit`a di Torino


Abstract

The main features of the observables in the SNO and Super-Kamiokande solar
neutrino experiments in the case of neutrino spin and/or spin-flavour precession
in the magnetic field of the sun are discussed. It is shown without any model
dependent assumption that in the case of Majorana transition magnetic moments the
NC event rate does not depend on time and a measurement of will allow
(0)21 (3)41





to determine the initial flux of neutrinos with a theoretical uncertainty of a few
576





%. In the case of Dirac magnetic moments the NC event rate will depend on time
 2 Dec 1993 and we obtain a model independent lower bound for the transition probability of
initial 's into right-handed sterile neutrinos.
8&9





@





BILENKY@TO.INFN.IT
A





GIUNTI@TO.INFN.IT


1 Introduction

It was shown in ref.[1, 2, 3] that future real-time solar neutrino experiments (SNO [4] and
others), in which high-energy neutrinos from decay will be detected through the observation
5

6





of different reactions, will allow to separate the investigation of neutrino properties (masses,
mixing, etc.) from the investigation of the central invisible region of the sun in which energy is
generated.
If only active neutrinos are present in the solar neutrino flux on the earth, the initial flux of
neutrinos and the survival probability can be determined directly from the data of future
5 6 8&9





solar neutrino experiments without any assumption about the mechanism of transition of into
89





and/or . In the general case of neutrino mixing, solar 's can transfer into active neutrinos






8 8 8 9





, , and into sterile left-handed (anti)neutrinos (see ref.[5]). Different model independent






8&9 8 8





relations and inequalities between observables, which could allow to reveal whether there are
sterile neutrinos in the flux of solar neutrinos on the earth, were derived in ref.[3].
In this paper we will discuss possibilities of a model independent treatment of the data from
future solar neutrino experiments in the case of spin and/or spin-flavour precession of solar 's
8 9





due to anomalously large neutrino magnetic moments. The effects of neutrino magnetic moments
[6, 7] were widely discussed in last years (see ref.[8]) in connection with a possible indication of
the existence of an anticorrelation between the flux of solar 's and the sunspot number [9].
8 9





The possible transitions of neutrinos in the magnetic field of the sun depend on the nature of
neutrinos. In the case of Dirac neutrinos, solar 's can be transformed into sterile right-handed
8 9





 






neutrinos ( ), quanta of right-handed fields. In the case of Majorana neutrinos,
8





 






transitions of solar 's into active right-handed antineutrinos ( ), quanta of left-
8&9 8






handed fields, could take place. Notice that direct transitions are forbidden by CPT
8 9
8&9





invariance. However, sizable transitions can occur under special conditions if both
8 9
8&9





spin-flavour precession and the MSW or vacuum oscillations mechanisms are operating [10] .
!





In the SNO experiment neutrinos (antineutrinos) from the sun will be detected through
observation of the following processes:


4353



('


(1)
8&9#"%$& "0)1"2)





98@3





(2)
8 8 "6$ 8 8 "0)1"67





9ACB2

' '



(3)
8 8 " 8 8 "





It is also planned [13] to search for from the sun with the help of the reaction
8 9





ED


(4)

8&9#"%$& "%7 "67





The threshold for neutrino detection in this experiment will be rather high ( for CC
FHGPIRQTSVUXW`Y





cbedfb

and ES, for NC and for reaction (4)). Thus SNO will detect
FCGaI UXW`Y FCGPIhgpiqUXWEY 5 6





r





Let us notice that the Mont Blanc collaboration has obtained the following upper bound for the flux of t(u in the
s





r




'4ikjmlon 'j





energy range : [11]. From the analysis of the background
4C(





vxwy4xwRy4 edfhg y





in the Kamiokande experiment the following upper bound for the flux of high-energy t(u with was
rq





s %pdf wy4





r




'4i@julon 'vj





obtained: [12].
#sq(






d#tdg y






1


's and neutrinos (and antineutrinos) originating from them. The energy spectrum of the initial
8&9





's is given by
5 9
6 8









 


(5)
 

F F











where is a known function (the phase space factor of the decay D ) and


F 5 6 576CW " " 8 9







 is the total initial flux (determined by the central temperature of the sun, the cross sections of







different reactions of the and CNO cycles, etc.). In the Super-Kamiokande (S-K) experiment
)o)





[14] high energy solar neutrinos ( ) will be detected through the observation of
F GaI2Q SVUXW`Y





process (3).
A measurement of the CC event rate as a function of neutrino energy will allow to obtain
F











the flux on the earth and to determine the survival probability up to a constant.






9 9
8 F 8





If it will occur that the CC event rate depends on time periodically, it will be an evidence that
neutrinos have large magnetic moments and their effects are important. In this paper we discuss
which additional informations can be extracted from a measurement of the NC and ES event rates
through the observation of the processes (2) and (3), respectively. In particular, we will show
that through the observation of the NC and ES reactions it will be possible to distinguish Dirac
from Majorana magnetic moments.



2 Neutral Current

Let us consider first the NC process (2). In the case of Majorana neutrino magnetic moments the
integral NC event rate is given by


&% 43 7% 43



 "! "!


5 (6)
21 21

5





)21 )21 )21

$# 6#

( F F F " F F F





 





('290) )  ('290) ) 









! !


where and are the cross sections of the processes and ,
5





$# $#



)41 F )41 F 8 $2 8o) 7 8
x$X 8
o) 7





% %






and 5 are the fluxes of all types of neutrinos and antineutrinos on the
21 21





F F





 





8'490) )  8'29) ) 





earth. Taking into account that

% %





9 @

5 (7)
 
1 1










F " F F





 





8'29) )  8'29) ) 





from Eq.(5) and Eq.(6) we obtain


)21

BA

( (8)
)21




"DC













)21













where

PI



A ! !





5




%

)41 )21 WI XYW3

HG $# 6#



F F
5 (9)
V XV
 21  21





)41FE



b Q

C F F F F





)21









('29) ) SRUT T







2


% %

  





Here 5 is the transition probability of solar 's into neutrinos
V V
X  X 
1 1





F F 8 9





 





T T

('29) )  ('29) ) 





(antineutrinos) of all types and


5





)41 )21

$# 6#

"
(10)
)21





E



b





43




!

(11)
5 5





)41 E )41

$#  # $# 6#



F F F





The cross sections of the processes and where calculated by several groups
8 $& 8 7e) 8 $ 8 7 )





and reviewed in ref.[15]. Using the results presented in ref.[15], we obtained


!



 d(b Am'


(12)
)41
$#

i





!

'

 d A" A

(13)
5





)41
6#



i S





It is easy to see that the value of the integral is very small. In fact, for the absolute value of
C )41





this integral we have the following upper bound:


I





# #





5





)41 )41

$# 6# '
b A


# # (14)
)41

# #%$

C





5





)41 )21


$# 6#'&

"





The upper bound of the integral is so small because the cross sections of the processes
C )41





and are very close to each other in the energy region near the threshold.
8 $ 8 7e) 8
x$ 8 7 )





The argument in favour of this fact is rather general: from symmetry considerations it follows
that near the threshold (if only the s-state of the final nucleons is taken into account) the vector
current does not contribute to the matrix elements of the processes and
8 $ 8 7 ) 8 $ 8 7 )





! !

and the cross sections and are equal. The corrections due to higher states are small in
5





 #  #



)21 )41





the relevant energy region (see ref.[15] and references therein). Thus, the term in Eq.(8) can
)21
C





be safely neglected and, in the case of Majorana magnetic moments, we come to the following
conclusions:


1. The NC event rate does not depend on time (within less than 2%).


2. The flux of the initial 's is given by
5 6 8&9





)41



( (15)







)41

&











and therefore can be determined directly from the experimental data .

l "





We used the values of the function given in ref.[16].
(





)





Notice that the expression (15) for coincides with that obtained in ref.[2] for the case of transitions of solar







1032





t u 's only into t!4 and/or t65 due to usual neutrino mixing.


3


3. It is possible to obtain the survival probability directly from measurable quantities:
8 9





% %

I WI



 A





5





V V V
  X   
1 1





F F F





 





T 8' ) "T 8'29) )  T





(16)










)41





X









F











)21



F (












where is the flux of on the earth, which can be determined from the CC event







F 8&9





rate.

Consider now the case of Dirac neutrino magnetic moments. In this case


7% 43



 "!


(17)
21





)21 )21

$#

( F F F











('290) ) 





%

I



9


where and is the flux of sterile right-handed neutrinos
  
1









F F F F











8'29) ) 





on the earth. It is clear that in the Dirac case will depend on time. Hence, a time dependence
( )41





of the CC and NC event rates will be a signal that neutrinos have large Dirac magnetic moments.
For the average transition probability of solar 's into sterile states we obtain the following
8 9





model independent lower bound:







A





% %

 43




!





V V
   
1 1





E )41

$#



F F F F F





)41

$#

 
8'490) )  T 8'29) )  T (18)
)41





I

)41

A

(






g














)41

$#














where indicates the maximum value of , which can be obtained from
  







F F





the data on the CC event rate. Let us stress that, unlike the case considered in ref.[3], in the case
of spin and/or spin-flavour transitions due to Dirac magnetic moments the lower bound (18) will
depend on time.


3 Elastic Scattering

Let us consider now the ES process (3). Using Eq.(5), in the case of Majorana neutrinos we have







!





BA


! (19)
"DC





"





















where







I WI & 43



$# ! !



" !





 % X





E





( 9 F 9 F F F





(20)
I WI & 43 



$# ! !






5 5 5





 % X





9 F 9 F F F





4






! !


" is the integral ES event rate, ( 5 ) is the total cross section of the process
21 21





( 9 F 9 F









 p

( ) with , and
   
8 8 8 8





WI



! !

A





5





% %
 





&% WI XYW3

G

9 F 9 F



"
5 (21)
V V
 81  81





E



Q
b

C F F F F




!











R T T

8'29) ) 





where

! "





5





"

% %



"





9 9


(22)
E



b





3




!

"
5 (23)
% %
5    





E

% %
   



9 9 F F F





9 9





f



! ! t


The values of the cross sections and 5 ( ) with a 5 MeV threshold kinetic
 
1 1





9 F 9 F





energy for electron detection are depicted in Fig.1 in the energy range relevant for solar neutrino
I I



 ed   d


experiments. We used and [17]. For " and " we obtained the
5





% %



  S S





9 9





following values:

!



Tb d A" A '

! (24)
%






9





!

'

Tb d A A

! (25)
5






 %



i





9





Using these values, we have the following upper bound for the absolute value of the integral " :
C





I





" "





# #





5





% %





ed AEb

9 9
"
# # (26)
# #$

C





" !





5





% %
  &



"





9 9





# #





Let us notice that the real value of "
# # can be significantly smaller than the upper bound given
# #

C





in Eq.(26). We calculated the integral ! in the simplest model with two non-mixed massive
C











Majorana neutrinos and a large transition magnetic moment . In ref.[18] it was shown that the






9









A


existing solar neutrino data can be described by this model with ' under specific








9 !4!





&
assumptions for the magnetic field of the sun. For the average value of found in ref.[18]






I



 A md A d A


( ' ) we obtained that ' '
! , where the lower and
$ $

5 WEY C






upper bounds correspond to high and low solar activity, respectively.







The time dependence of the quantity ! is determined by the integral ! , which is less
C





A

than . Neglecting " in Eq.(19), we obtain the following approximate expression for the
C





&
initial flux of neutrinos:
5 6













"









(27)







"





&











Thus, in the case of Majorana magnetic moments, the initial flux of neutrinos can be determined
5 6





in two independent ways: from the NC event rate (see Eq.(15)) and from the ES and CC event
rates (see Eq.(27)). Therefore, independently from the value of the initial neutrino flux, we have
the following approximate relation between measurable quantities:




)41







! (28)
)41




(





!





&











5


This relation is a generalization of an analogous relation that was obtained in ref.[2] for the case
in which only active neutrinos , , are present in the flux of solar neutrinos on the earth (in






9 
8 8 8





that case the relation is exact).
Let us notice that, in the case of Majorana magnetic moments, the ES event rate will depend
on time. This time dependence is determined by the time dependence of the CC event rate.
Let us consider now the case of Dirac neutrino magnetic moments. In this case we have

A

"





%

I 43



A !


(29)
V
%   1







9 F F F F





"








"





%







 %



9

('290) )  T





9





where

I WI  3



 0! !


! " (30)
 % 
  





E





( 9 F 9 F F F






In the case under consideration the quantity " will depend on time. From Eq.(29), we obtain
the following lower bound for the average transition probability of solar 's into all possible
8 9





right-handed sterile states:







A





% %

43




!





V V
X  %  
1 1





E





F 9 F F F F





!





 %

 





9

('29) ) "T ('29) )  T
! (31)


!





I



A





g 





!











 %







9





In the case of Dirac magnetic moments, instead of relation (28) we have


)21

 #



! (32)
)41



( "





"





%







9





where


)21 3 I 43

 #


 ! !

(33)
%  





)41

 #



9
 F F F F F F




"





%








9





Let us assume that is not observed. If the relation (28) is violated and depends on time, we

8&9 





will have an additional argument in favour of Dirac magnetic moments. Notice, however, that,
according to our model calculations, the two terms in the right-hand side of Eq.(33) could cancel
each other.


4 On the transition probability of into and/or
 
 






Let us consider the case of Majorana magnetic moments (the CC event rate depends on time
but the NC event rate is constant). As we have seen in Sec.2, from the data on the CC and NC
reactions (and the process (4)) it will be possible to determine the sum of the probabilities of


6


the transitions of initial 's into , , , (see Eq.(16)). In this section we will discuss
 





8 9 8 8 8 8






possibilities to obtain from the SNO and S-K data an information about the transition probability
%







5 . With the help of Eq.(15) we obtain
V
X  1





F











T

8' ) 









!







)41





I


A

" (34)
C




"





)41

% &


(





9





where

I WI & 43



$# ! !


! "





X % 





E





( 9 F 9 F F F





(35)
I WI & 43



$# ! !






5 5





X % 





9 F 9 F F F





and

A











& %
WI 3


$#
! !



!
5 5 (36)
V
% %   1





E





9 9
C F F F F F





!





%









9

T

8' ) 





It is easy to see that

I





! "





5





% %
 







9 9  dfb A

! (37)
$ $

C




!





%







9





%





Hence, in order to obtain an information about the transition probability 5 , it is
V
X  1





F











T

8' ) 







" "
necessary to know the measurable quantity with an accuracy better than
)41











)21

%



(





9





b 

. If this accuracy will be achieved, the average probability of the transition of solar 's
9
8





&
into and/or can be determined directly from the experimental data:







8 8











A





% WI & % 43




# ! !






5 5 5





V V
  % % X 
1 1





E





F I 9 F 9 F F F F





! !





5





%  %



 





9 9

T T

(' )  8' ) 





"







"





)21




%








 I 



9 A (38)
I





! ! "





5





)41

% % %
&   

(





9 9 9





5 Conclusions

In SNO, Super-Kamiokande and other future solar neutrino experiments high energy solar neutri-



nos will be detected through the observation of CC, NC and ES ( - elastic scattering) reactions.
8





If neutrinos have large (Majorana or Dirac) magnetic moments, solar neutrinos can undergo spin
and/or spin-flavour precessions in the magnetic field of the sun. In this case the CC event rate
will manifest a periodical time dependence.
We have shown that in the case of Majorana neutrino magnetic moments the NC event rate
in the SNO experiment ( ) will not depend on time and a measurement of will allow to
( )21 ( )41





7


determine in a model independent way the total flux of initial neutrinos (see Eq.(15)) and the
5 6





survival probability (see Eq.(16)). In the case of Majorana neutrino magnetic moments the
9
8





ES event rates will depend on time. We have shown, however, that a combination of ES and CC







event rates ( ! , see Eq.(20)) can have only a minor time dependence. We have also shown that,
in the case of Majorana neutrino magnetic moments, there is a relation between the NC, CC and
ES event rates (Eq.(28)).
In the case of Dirac neutrino magnetic moments the NC event rate in the SNO experiment,
as well as the CC and ES event rates, will depend on time. We have shown that a measurement
of the NC (or ES) and CC event rates will allow to determine a (time dependent) lower bound
for the average transition probability of solar 's into all possible right-handed sterile states (see
8 9





Eq.(18) and Eq.(31)).
Finally, for the case of Majorana magnetic moments we have derived an expression that
allows to obtain the average probability of the transition of initial 's into and/or from






8 9 8 8






measurable quantities (see Eq.(38)). The determination of this average probability will require a
measurement of the event rates with rather high accuracy.



Acknowledgments

It is a pleasure for us to express our gratitude to Wanda Alberico for very useful discussions.


References

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8


[8] P.B. Pal, Int. J. Mod. Phys. A 7 (1992) 5387.

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9


100

)
eV
/M
2
cm 10
6-4 0(1
-e
-e
/ E 1 -e
e
-e
--e
0.1
6 7 8 9 10 11 12 13 14 15
E (MeV)






! ! ! !


Figure 1: Values of the cross sections , , 5 , 5 in the energy range
 %  %





9 F 9 F 9 F 9 F





relevant for solar neutrino experiments, with a 5 MeV threshold kinetic energy for electron
I I



 ed   d


detection. We used and [17].
  S S





10



