The Mikheyev–Smirnov–Wolfenstein effect (often referred to as the matter effect) is a particle physics process which modifies neutrino oscillations in matter of varying density. The MSW effect is broadly analogous to the differential retardation of sound waves in density-variable media, however it also involves the propagation dynamics of three separate quantum fields which experience distortion. In free space, the separate rates of neutrino eigenstates lead to standard neutrino flavor oscillation. Within matter – such as within the Sun – the analysis is more complicated, as shown by Mikheyev, Smirnov and Wolfenstein. It leads to a wide admixture of emanating neutrino flavors, which provides a compelling solution to the solar neutrino problem. Works in 1978 and 1979 by American physicist Lincoln Wolfenstein led to understanding that the oscillation parameters of neutrinos are changed in matter. In 1985, the Soviet physicists Stanislav Mikheyev and Alexei Smirnov predicted that a slow decrease of the density of matter can resonantly enhance the neutrino mixing. Later in 1986, Stephen Parke of Fermilab, Hans Bethe of Cornell University, and S. Peter Rosen and James Gelb of Los Alamos National Laboratory provided analytic treatments of this effect. Mikheyev, Smirnov and Wolfenstein were nominated in 2015 for the Nobel Prize in Physics, but did not win, the prize instead going to Takaaki Kajita and Arthur B. McDonald for their discovery of neutrino oscillations. Summary The presence of electrons in matter changes the instantaneous Hamiltonian eigenstates (mass eigenstates) of neutrinos due to the charged current's elastic forward scattering of the electron neutrinos (i.e., weak interactions). This coherent forward scattering is analogous to the electromagnetic process leading to the refractive index of light in a medium and can be described either as the classical refractive index, or the electric potential, . The difference of potentials for different neutrinos and : induces the evolution of mixed neutrino flavors (either electron, muon, or tau). In the presence of matter, the Hamiltonian of the system changes with respect to the potential: , where is the Hamiltonian in vacuum. Correspondingly, the mass eigenstates and eigenvalues of change, which means that the neutrinos in matter now have a different effective mass than they did in vacuum: . Since neutrino oscillations depend upon the squared mass difference of the neutrinos, neutrino oscillations experience different dynamics than they did in vacuum. Similar to the vacuum case, the mixing angle describes the change of flavors of the eigenstates.