The above formalism applies to neutrinos passing through vacuum. In the
presence of matter the neutrinos acquire effective masses from coherent
scattering processes. In particular, coherent 
scattering via the charged current amplitude differentiates the 
from the other neutrinos. The MSW propagation equation [15] is
where 
GeV
is the Fermi constant and
is the density of electrons (neutrons). The extra term appears
for sterile neutrinos because of the difference in neutral current
amplitudes. In the absence of matter, (17) reproduces
the vacuum oscillation equation. However, the matter term can be extremely
important. The MSW resonance occurs for
Then the diagonal components become equal and even a small mixing term can be amplified to maximal mixing.
If one had a definite density of matter then the resonance would require a
fine-tuning. However, the neutrinos are born in the center of the sun
where the density is high and as they move outward the density decreases.
For a range of energies (for a given 
and 
) a neutrino
will encounter a layer of just the right density for the MSW resonance. As
it passes through the two energy eigenvalues will ``cross''. If the density
varies slowly enough (adiabatically) there will be an almost certainty of
conversion. If the crossing is non-adiabatic, the conversion probability
is smaller.
The conversion probability as the neutrino emerges from the sun depends on
the energy as well as the parameters 
and 
.
The survival probability is constant (for a fixed energy) within a triangular
region of the 
plane, as shown for the
Kamiokande energies in Figure 7. The upper branch of the triangle
is the adiabatic solution [53]. On this branch the
density varies adiabatically, and all neutrinos are converted if they
encounter a resonance density. This occurs for the high energy but not the
low energy neutrinos, so one expects suppression of the high energy part of
the spectrum. The diagonal, or non-adiabatic, branch (NA) is the one in
which the adiabatic approximation is breaking down [54]. In
this case the dominant suppression will be in the middle of the spectrum.
Finally, the vertical or large angle branch (LA) is an extension of vacuum
oscillations. In the regime 
it yields roughly
equal suppression for all energies.
Figure: Contours of constant survival probability for the Kamiokande
experiment. The Earth effect is not included. The contours average over
the relevant neutrino energies.
Typical probabilities of survival as a function of the neutrino energy for realistic MSW parameters on the NA and LA branch are indicated in Figure 8.
Figure: Survival probabilities for the non-adiabatic and large angle
solutions.
For several years my collaborators and I have been carrying out the best
MSW analysis of the data that we could [16,17]. We generally use the
Bahcall-Pinsonneault [26] standard solar model predictions for the
initial fluxes and also for the distributions 
, 
, and
, which are respectively the radial distributions of the production
locations of each neutrino flux component and of the electron and neutron
number densities. We also use other ansatzes for the initial fluxes. In
analyzing the data one must take into account the energy resolution and
threshold effects for the Kamiokande experiment. In addition, if the
converted neutrino is a 
or a 
the neutral current cross
section 
, which is about 
as strong
as the 
charged current cross section, must be included for
Kamiokande. This effectively lowers the number of surviving 
observed by Kamiokande.
It is important to properly incorporate the theoretical uncertainties
in the initial neutrino fluxes. These can be due to the core temperature
, as well as the production and detector cross sections. One must
also include the correlations
of those uncertainties between different flux
components and between different experiments [16]. For example, if
the core temperature is higher than in the SSM, it is higher for all of the
flux components and all of the experiments. To allow for comparison with
updated SSMs and with alternate SSMs we have generally worked with
error matrices parameterized by the temperature and cross
section uncertainties [17,16]. These are calibrated from
specific Monte Carlos [46], and the agreement between the two methods
is excellent, both for the uncertainties and their correlations.
Altogether, the theoretical errors are important but not
dominant. In analyzing the data it is important to do a joint 
analysis of the data to find the allowed regions. Simply overlapping
allowed regions between different experiments necessarily neglects
correlations and tends to overestimate the allowed regions. There are also
complications in the analysis due to the multiple solutions [16].
The Earth effect [55], i.e., the
regeneration of 
in the Earth at night, is significant for a
small but important region of the MSW parameters, and not only affects the
time-average rate but can lead to day/night asymmetries. The Kamiokande
group has looked for such asymmetries and has not observed them
[56], therefore excluding a particular region of the MSW
parameters in a way independent of astrophysical uncertainties. We fold
both the time-averaged and the day/night data into the overall fits
[57,16].
Figure: Allowed regions at 95% CL from individual experiments and from
the global fit. The Earth effect is included for both time-averaged and
day/night asymmetry data, full astrophysical and nuclear physics
uncertainties and their correlations are accounted for, and a joint
statistical analysis is carried out. The region excluded by the Kamiokande
absence of the day/night effect is also indicated. From
[16,50].
The allowed regions from the overall fit for normal oscillations
or 
are shown in Figure 9. There
are two solutions at 95% C.L., one in the NA branch for the Homestake and
Kamiokande experiments (and the adiabatic branch for the gallium
experiments), and one on the LA branch. The former gives a much better
fit, as can be seen in Table 5. There is a second large angle
solution with smaller 
, which only occurs at 99% C.L.
Table: MSW parameters for the non-adiabatic and large angle solutions as
well as the overall 
(7df). There is also a second large angle
solution, which is allowed at 99% C.L. but
gives a much poorer fit. The last
two rows are the probability in each case of obtaining a larger 
,
and the relative probabilities of the various solutions. From
[16,50].
MSW fits can also be performed using other solar models as inputs, as a way of getting a feeling for the uncertainties. Figure 10 shows the MSW fit assuming the TCL SSM [27]. One sees that the allowed regions are qualitatively similar, but differ in detail.
Figure: Allowed regions assuming the TCL SSM. From
[16,50].
One can also consider transitions 
into
sterile neutrinos. These are different in part because the MSW formulas
contain a small contribution from the neutral current scattering from
neutrons. Much more important is the lack of the neutral current
scattering of the 
in the Kamiokande experiment. There is a
non-adiabatic solution similar to the one for active neutrinos, though the
fit is poorer. However, there is no acceptable large angle solution
because of the lack of a neutral current, which makes that case similar to
astrophysical solutions. Oscillations into a sterile neutrino in that
region are also disfavored by Big Bang nucleosynthesis [58].
It is interesting to go a step further and consider nonstandard solar
models and MSW simultaneously [17,16,10]. There is now
sufficient data to determine both the MSW parameters and the core
temperature in a simultaneous fit. One obtains [10,50], 
, in remarkable agreement with the standard solar model
prediction 
. The allowed MSW parameters are shown in
Figure 11. The regions are larger than when one accepts the
SSM, but still constrained.
Figure: Allowed regions of the MSW parameters when 
is allowed to be
free. From [10,50].
Figure: Allowed MSW parameters when the 
flux is free. From
[10,50].
Alternatively, one can allow the 
flux 
to be free, as
would be expected in models with lower 
, for example. The data are
consistent with the SSM value with large errors, but favor a slightly
higher value 
. The allowed regions
of the MSW parameters are shown in Figure 12.
Although the MSW mechanism gives a perfect description of
existing data, there is one
alternative, vacuum oscillations [37]--[42]. There are fine-tuned
solutions with the earth-sun distance being at a node of the oscillations,
corresponding to parameter ranges 
and
.
