Simple and Accurate Oscillation Probabilities for Three Coupled   Neutrinos Propagating in Matter
  Within a conventional Hamiltonian description, we find accurate closed-form expressions for the oscillation probabilities of three coupled neutrinos propagating in matter. Subtle cancelations that occur in coefficients of our formulation are avoided for all transitions $\nu_a \to \nu_b$ by transforming to a different set of coefficients presented in an appendix of this paper. The neutrino mass eigenvalues are easily obtained numerically as the solution of a cubic equation. Our methods are illustrated for flavor-changing transitions in the $(\nu_e,\nu_\mu)$ sector. The resulting analytic expressions oscillation probabilities, which are particularly simple, are also accurate to a few percent over all regions of interest at present and the envisioned future neutrino facilities. While somewhat less accurate than numerical simulations, our approximate expressions are sufficiently accuracy to obviate the need for exact computer simulations in many circumstances. 
Mikkel B. Johnson and Leonard S. Kisslinger
hep-ph