Gravitational Correction in Neutrino Oscillations

Recently, the neutrino oscillation phase due to the presence of gravity is much discussed [1], [2], [3]. If the neutrinos are massive particles and mixed, the neutrino oscillation between different flavor states occurs. Suppose that neutrinos are created as weak flavor eigenstate, say, at ~rA, and propagate to ~rB . The state is a linear superposition of mass eigenstates and each phase of mass eigenstate evolves in different way. As a result, the mixing phase angle for the relativistic neutrinos propagating in a flat space is given by φ0 = ∆m L/(4h̄E), where E is energy, L = |~rB − ~rA|, and ∆m = m21 −m2. (See e.g., [4].) Gravitational effect was not seriously studied so far. Gravity can be eliminated by choosing appropriate inertial frame locally. The propagating distance of the neutrinos is so long in some case, that gravitational effect may become important. Ahluwalia and Burgard [1] considered the gravitational effect on the neutrino oscillation. They showed that the external weak gravitational field of a star with mass M adds a new contribution to the phase difference, denoted by φG = −∆m2L〈φ〉/(4h̄E), where 〈φ〉 is defined by the average of gravitational potential over the semi-classical path, i.e., 〈φ〉 = −( ∫ ~rB ~rA dLGM/r)/L. This gravitationally induced phase can be estimated as φG = −〈φ〉φ0. The phase becomes to the extent of roughly 20% of φ0, near neutron stars. They suggested that the new oscillation phase may be significant effect on the supernova explosions, since the extremely large fluxes of neutrinos are produced with different energies corresponding to the flavor states. The gravitationally induced oscillation phase may have the important astrophysical consequences. However, their derivation and even the definitions such as energy were not clear in their original paper. Bhattacharya, Habib and Mottola [2] critically re-examined the quantum mechanical phase mixing. They calculated the phase difference for radially propagating particles, and found that the term of ∆m2L〈φ〉/(h̄E) ∼ GM∆m log(rB/rA)/(h̄E) is canceled out. They showed that the possible gravitational effect appears at the higher order, ∆m/E, and that the phase difference for radially propagating particles is GM∆m log(rB/rA) /(4h̄E ). Numerically its magnitude is equal to ∼ 10 (M/M⊙) (∆m /eV) (E/MeV) log(rB/rA), which is completely negligible in typical astrophysical applications. Therefore, the conclusion of Ahluwalia and Burgard [1] seems to be incorrect for radially propagating case. Natural question is what happens in more general case. Does the term of GM∆m/E always disappear?