Solar neutrino interactions in the detector
Solar neutrinos in Super-Kamiokande are measured through the following reaction;
|
(1) |
,The advantage of an experiment using this interaction is that we are able to obtain information on
- (1)
- the direction of the incident neutrinos because the recoil electrons in Eq((1)) are scattered to almost same direction as the neutrinos,
- (2)
- the exact arrival time of the incident neutrinos and
- (3)
- the energy distribution of the recoil electrons which reflects the energy spectrum of the incident neutrinos.
The cross section for this scattering can be calculated using standard electroweak theory, and the differential cross section is
 , |
(2) |
where 
are
the Fermi coupling constant, the electron mass ,the incoming neutrino energy and the
kinetic energy of the recoil electron, respectively. The parameters 
and 
are defined by
 , |
 , |
||
 , |
where 
is the Weinberg angle. From Eq(2), the overall factor in the differential cross
section is
 , |
(3) |
We also consider radiative corrections for the one-loop electroweak and QCD in the
interaction Eq(1). A QED radiative correction is also considered. These corrections reduce
the relative probability of observing recoil electrons by about 4% in our energy region
for 
.
The cross section as a function of neutrino energy including radiative corrections is
shown in Fig 1. This cross section can be calculated by integrating Eq(2) between 0
and the maximum value,
, of the electron kinetic energy. Here 
for given 
is limited by
kinematics as follows,
|
(4) |
As shown in the figure, when is 10MeV, which is a
typical energy of 
solar neutrinos, the cross section of neutrino
scattering on electron is:
 , |
(5) |
|
 , |
(4) |
The differences of the cross section between 
and 
is because the scattering of on an
electron can take place only through the neutral current interaction, while in case of ,
both neutral and charged current interactions take place. The 
cross section is
approximately six times less than 
cross section.
|
|
Fig 1: |
The cross section of the interaction for  , with electoron as a function of
neutrino energy. |
The energy distribution of recoil electrons scattered by solar neutrinos is given by
 , |
(7) |
where 
is the solar neutrino spectrum at earth,
and 
is the maximum energy of solar neutrinos. Fig 2 shows the energy
distribution of recoil electrons with only 
and hep solar neutrinos considered.
|
|
| Fig 2: | The energy spectrum of recoil electrons scattered off by and hep solar neutrinos. |
The angle, 
, between the direction of a recoil electron and an incoming
neutrino is given by
 |
(8) |
The angular distribution of recoil electrons is given by
 . |
(9) |
Fig 3 shows the angular distribution of recoil electrons with total energy greater
than 0, 5, 7 and 10MeV. From this figure, the scattering angle is less than 20
if the
threshold energy is greater than 5MeV. For a larger threshold energy, the angular
distribution becomes much more strongly forward peaked. The angular dispersion defined by
the value which includes 68% of the distribution, is 12.3, 9.1
and 5.7for the electrons with total energy greater than 5, 7 and 10MeV,
respectively.
|
|
| Fig 3: | The angular distribution of recoil electrons from incident solar
neutrinos. Here,"ALL" means over the and hep SSM spectra. |
