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  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 . 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 . 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  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 . 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 , 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 .
The Earth effect , 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 , 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 . 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 .
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 --. There are fine-tuned solutions with the earth-sun distance being at a node of the oscillations, corresponding to parameter ranges and .