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MSW Effects in Vacuum Oscillations
Department of Physics, University of California, Berkeley, CA 94720, USA;
Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
(February 1, 2008)
We point out that for solar neutrino oscillations with the mass squared difference of Am2 t
1010 - 10-9 eV2, traditionally known as "vacuum oscillation" range, the solar matter effects are
non-negligible, particularly for the low energy pp neutrinos. One consequence of this is that the
values of the mixing angle a and </2 - a are not equivalent, leading to the need to consider the
entire physical range of the mixing angle 0 < a
neutrino oscillation parameters.
1. The field of solar neutrino physics is currently un-
dergoing a remarkable change. For 30 years the goal
was simply to confirm the deficit of solar neutrinos. The
latest experiments, however, such as Super-Kamiokande,
SNO, Borexino, KamLAND, etc, aim to accomplish more
than that. By collecting high statistics real time data
sets on different components of the solar neutrino spec-
trum, they hope to obtain unequivocal proof of neu-
trino oscillations and measure the oscillation parameters.
With the physics of solar neutrinos quickly becoming a
precision science, it is more important then ever to ensure
that all relevant physical effects are taken into account
and the right parameter set is used.
It has been a long standing tradition in solar neutrino
physics to present experimental results in the Am2 -
sin2 20 space and to treat separately the "vacuum os-
cillation" (Am2 ~ 10- - 10-9 eV2) and the MSW
(Am2 i-8 - 10-s eV2) regions. In the vacuum oscil-
lation region the neutrino survival probability (i.e. the
probability to be detected as ve) was always computed
according to the canonical formula,
P 1 -sin2 20 sin2 (1.27A 2L) , (1)
where the neutrino energy E is in GeV, the distance L in
km, and the mass squared splitting Am2 in eV2. Eq. (1)
makes sin2 20 seem like a natural parameter choice. As
sin2 20 runs from 0 to 1, the corresponding range of the
mixing angle is 0 < 0 < r/4. There is no need to
treat separately the case of Am2 < 0 (or equivalently
r/4 < 0 < r/2), since Eq. (1) is invariant with respect
to Am2 -> -Am2 (0 -> /2 -0).
The situation is different in the MSW region, since
neutrino interactions with matter are manifestly flavor-
dependent. It is well known that for Am2 > 10-8 eV2
matter effects in the Sun and Earth can be quite large.
In this case, if one still chooses to limit the range of
the mixing angle to 0 < 0 < </4, one must consider
both signs of Am2 to describe all physically inequivalent
situations. As was argued in , to exhibit the continuity
of physics around the maximal mixing, it is more natural
to keep the same sign of Am2 and to vary the mixing
angle in the range 0 < 0 < r/2.
< </2 when determining the allowed values of the
Historically, a possible argument in favor of not consid-
ering 0 > </4 in the MSW region might have been that
this half of the parameter space is "uninteresting", since
for 0 > r/4 there is no level-crossing in the Sun and the
neutrino survival probability is always greater than 1/2.
However, a detailed analysis reveals that allowed MSW
regions can extend to maximal mixing and beyond, as
was explored in  (see also  and  for a treatment of
3- and 4- neutrino mixing schemes).
In this letter we point out that for solar neutrinos with
low energies, particularly the pp neutrinos, the solar mat-
ter effects can be relevant even for neutrino oscillations
with Am2 ~ 10--1 - 10-9 eV2. These effects break the
symmetry between 0 and /2 - 0 making it necessary
to consider the full physical range of the mixing angle
0 < 0 r/2 even in the "vacuum oscillation" case.
2. For simplicity, we will only consider here the two-
generation mixing. If neutrino masses are nonzero then,
in general, the mass eigenstates vi,2) are different from
the flavor eigenstates ve, ). The relationship between
the two bases is given in terms of the mixing angle 0:
vi) cos0 ve) - sin06v ),
v22) sin6ve) + cos06v ).
In our convention v2) is always the heavier of the two
eigenstates, i.e. Am2 m - m ;> 0. Then, as already
mentioned, 0 < 0 < r/2 encompasses all physically dif-
Neutrinos are created in the Sun's core and exit the
Sun in the superposition of vi) and v2). For Am2 in
the vacuum oscillation region, the neutrino is produced
almost completely in the heavy Hamiltonian eigenstate
v+). In this case, if the evolution inside the Sun is adia-
batic, the exit state is purely v2). In the case of a nona-
diabatic transition there is also a nonzero probability Pc
to find the neutrino in the vi) state (a "level crossing"
probability). For a given value of Pc, the survival prob-
ability for neutrinos arriving at the Earth is determined
by simple 2-state quantum mechanics [5,7,8]:
P = zPccos20+ (1 -Pc)sin20
. ( ~Am 2L (3
+ 2 Pc(1 -P) sin B cos B cos 2.54 E. + , . (3)
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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/m1/2/: accessed June 17, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.