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* Integration over neutrino energy spectrum. To
compute the event rate one has to integrate Eq. (3)
over neutrino energies. For sufficiently large Am2
the last term averages out to zero, leading effec-
tively to the loss of coherence between the two mass
eigenstates.
As Am2 increases, coherence is first lost for reac-
tions with broad energy spectra, such as pp and 8B,
and persist the longest for neutrinos produced in two-
body final states. The most important such reaction is
7Be+e ->7Li +ve, which produces the 7Be neutrinos.
The 7Be neutrinos have an energy spread of only a few
keV, arising from the Doppler shift due to the motion
of the 7Be nucleus and the thermal kinetic energy of the
electron. A detailed discussion of this phenomenon can
be found in [7,5].
In order to properly take these effects into account, in
our codes we numerically integrate over the exact 7Be
line profile, computed in [23]. As Fig. 3 shows, the neu-
trino survival probability becomes independent of L for
Am2 > 6 x 10-9 eV2. For this reason, we present our
fits for Am2 ranging from 10-1 eV2 to 10-8 eV2. Un-
fortunately, in the literature vacuum oscillations are usu-
ally studied in the range from 10-1 eV2 to 10-9 eV2
[24,25,4], although the allowed regions in all these pa-
pers seem to extend above 10-9 eV2.
5. In summary, the preceding examples clearly illus-
trate the importance of including the solar matter ef-
fects when studying vacuum oscillation of solar neutri-
nos with Am2 ? 10-10 eV2. Because to describe such
effects one has to use the full range of the mixing angle
0 < 0 < r/2, future fits to the data should be extended
to 0 > r/4. This seems especially important in light
of the latest analyses [25], [4], which in addition to the
total rates also use the information on the neutrino spec-
trum and time variations at Super-Kamiokande. In this
case the allowed vacuum oscillation regions are mostly
located in the Am2 > 4 x 10-10 eV2 region [4], pre-
cisely where the matter effects are relevant. (The best
fit to the Super-Kamiokande electron recoil spectrum is
achieved for Am2 6.3 x 10-10 eV2, sin2 20 1 [25].)
It would be very desirable to repeat these analyses with
the solar matter effects included.
Additionally, since the 7Be neutrinos remain (par-
tially) coherent for Am2 > 10-9 eV2, it is desirable to
present the results of the fits in the range 10-1 eV2 <
Am2 < 10-8 eV2, as was done in [2].
ACKNOWLEDGMENTS
I am very grateful to Hitoshi Murayama and John Bah-
call for their support. I would like to thank John Bahcall
for including in the BP2000 solar model the data for the
outer regions of the Sun. I would also like to thank JamesPantaleone, Plamen Krastev, M.C. Gonzalez-Garcia, and
Yosef Nir for their valuable input. This work was in part
supported by the U.S. Department of Energy under Con-
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[21] Notice that this is different from the conventional ap-
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Friedland, Alexander. MSW Effects in Vacuum Oscillations, article, February 6, 2000; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc927089/m1/5/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.