  We present neutrino bulb model simulations of Majorana neutrino coherent spin
transformation (i.e., neutrino-antineutrino transformation), coupled to
neutrino flavor evolution, for conditions corresponding to the neutronization
burst epoch of an Oxygen-Neon-Magnesium (O-Ne-Mg) core collapse supernova.
Significant neutrino spin transformation in, for example, the neutronization
burst, could alter the fluences of neutrinos and antineutrinos in a way which
is potentially detectable for a Galactic core collapse supernova. Our
calculations for the first time incorporate geometric dilution in the spin
evolution of the neutrinos and combine two-flavor and three-flavor evolution
with spin mixing physics. We find that significant spin transformations can
occur, but only with a large neutrino luminosity and an electron fraction
($Y_e$) profile which facilitates adiabatic conditions for the spin-channel
resonance. Using our adopted parameters of neutrino energy spectra, luminosity,
density and $Y_e$ profiles, our calculations require an unrealistically large
neutrino rest mass to sustain the spin transformation. It is an open question
whether examining different density profiles or incorporating other sources of
nonlinear feedback, such as $Y_e$ feedback, could mitigate this need. We find
that spin transformations are not sensitive to the flavor structure of
neutrinos, i.e., the spin transformations occur regardless of whether we
simulate two- or three-flavor transformations. In the two-flavor case, spin
transformations were insensitive to the choice of solar or atmospheric
mass-squared splitting as well as the choice of the Majorana phase.
Importantly, our three-flavor simulations, as well as our two-flavor
simulations done with the atmospheric mass-squared splitting, show that the
inclusion of spin degrees of freedom can significantly and qualitatively alter
neutrino flavor evolution.
