Neutrino Oscillations in the Early Universe: A Real Time Formulation

Neutrino oscillations in the early Universe prior to the epoch of primordial nucleosynthesis is studied by implementing real time non-equilibrium field theory methods. We focus on two flavors of Dirac neutrinos, however, the formulation is general. We obtain the equations of motion for neutrino wavepackets of either chirality and helicity in the plasma allowing for CP asymmetry. Contributions non-local in space-time to the self-energy dominate over the asymmetry for $T \gtrsim 3-5 \mathrm{MeV}$ if the lepton and neutrino asymmetries are of the same order as the baryon asymmetry. We find a new contribution which cannot be interpreted as the usual effective potential. The mixing angles and dispersion relations in the medium depend on \emph{helicity}. We find that resonant transitions are possible in the temperature range $10 \lesssim T \ll 100 \mathrm{MeV}$. Near a resonance in the mixing angle, the oscillation time scale in the medium as compared to the vacuum is \emph{slowed-down} substantially for small vacuum mixing angle. The time scale of oscillations \emph{speeds-up} for off resonance high energy neutrinos for which the mixing angle becomes vanishingly small. The equations of motion reduce to the familiar oscillation formulae for negative helicity ultrarelativistic neutrinos, but include consistently both the \emph{mixing angle and the oscillation frequencies in the medium}. These equations of motion also allow to study the dynamics of right handed as well as positive helicity neutrinos.