Neutrinos in general relativity: Four(?) levels of approach

The three subsequent levels of approach to the problem of the neutrino in general relativity which have been exploited till now, are:

1. ‘classical particle’ approach, i.e. a study on the neutrino as a classical particle in a classical, given gravitational field;
2. ‘quantum particle’ approach, i.e. considering the Dirac equation for the neutrino in a given gravitational field;
3. ‘classical field’ approach comprising the study of combined neutrino-gravitational fields.

Many results obtained along these lines are presented, with emphasis upon the geometrical theory of two-component neutrino-gravitational fields. A synthesis of the particle and fields aspects of the neutrino could provide a possible fourth, till now non-existing, ‘quantum field’ level of approach. This should deal with a guantized neutrino field in a curved space-time (which might be also quantized, but perhaps this should belong already to a next, fifth level of approach). Studies on the neutrino physics in gravitational fields reveal not only a series of results which are of interest in se, and which may be used as the basis to a unified theory of neutrino and gravitational fields (the Rainich problem for the neutrino). They provide in addition the necessary material for astrophysical and cosmological applications; some of these are outlined in relation to the results presented.     

Interference Phase of Neutrino Oscillation in Schwarzschild-de Sitter Space-Time

We study the mass neutrino interference phase in Schwarzschild-de Sitter space time along the null trajectory and the geodesic line and obtain the effects of cosmological constant A on the neutrino oscillation. Firstly, in the high energy limit, we find that the phase along the geodesic keeps the double of that along the null. Secondly, we calculate the phase on the condition that the cosmological constant, A, is a small quantity. The correction of the phase due to A is given. Finally, we calculate the proper oscillation length in Schwarzschild-de Sitter space-time, which increases because of the existence of A, compared with the result in Schwarzschild space-time. All of our results can be reduced to those in Schwarzschild space-time as A approaches to zero.     

Neutrino Oscillations in the de Sitter and the Anti-de Sitter Space-Time

General expressions of the neutrino oscillation phase in the generally static space-time with spherical symmetry are given. The effect of the gravitational field on the oscillation length is embodied in the gravitational red shift factor. We find that a blue shift of the oscillation length takes place when the neutrino travels out of the gravitational field. Then, we discuss the variation of the oscillation length influenced by the cosmological constant. In the de Sitter space-time, the positive cosmological constant prolongs the oscillation length. And, in the anti-de Sitter space-time, the negative cosmological constant shortens it as expected.     

Gravitational effects on the neutrino oscillation in vacuum

The propagation of neutrinos in a gravitational field is studied by developing a method of calculating a covariant quantum-mechanical phase in a curved space-time. The result is applied to neutrino propagation in the Schwarzschild metric.     

Does CP Phase Exist Above the GUT Scale

We consider non-reormalizable interaction term as a perturbation of the conventional neutrino mass matrix. Quantum gravitational (Planck scale )effects lead to an effective SU(2) _L ×U(1) invariant dimension-5 Lagrangian involving neutrino and Higgs fields, which gives rise to additional terms in neutrino mass matrix. There additional term can be considered to be perturbation of the GUT scale bi-maximal neutrino mass matrix. In particular, for the $\theta_{13}'$ range 0.00005–0.28, indicates the existence of CP violating phase above the GUT scale. We assume that the gravitational interaction is flavor blind. In this paper, we further investigate the possibility of CP phase exist from Quantum gravity.     

Using Hamilton-Jacobi Equation to Study the Neutrino Oscillations in the Stationary Space-Time

We study the mass neutrino oscillation by solving Hamilton-Jacobi equation in the Kerr-Newman-Kasuya space-time, as an important example of the stationary space-time, and give the general expression of the oscillation phase. A special case, the geodesic with L=aE is considered. Then, the proper oscillation length is studied carefully. The effects of the gravitational field, the rotating parameter a, the electric charge and magnetic charge on the oscillation length are given. It is worth noting that a blue shift of the oscillation length rather than a red shift takes place as the neutrino travels out of the gravitational field.     

Neutrino wave packet propagation in gravitational fields

We discuss the propagation of neutrino wave packets in a Lense–Thirring metric using a gravitational phase approach. We show that the neutrino oscillation length is altered by gravitational corrections and that neutrinos are subject to helicity flip induced by stellar rotation. For the case of a rapidly rotating neutron star, we show that absolute neutrino masses can be derived, in principle, from rotational contributions to the mass-induced energy shift, without recourse to mass generation models presently discussed in the literature.     

Can gravity distinguish between Dirac and Majorana neutrinos?

We show that spin-gravity interaction can distinguish between Dirac and Majorana neutrino wave packets propagating in a Lense-Thirring background. Using time-independent perturbation theory and the gravitational phase to generate a perturbation Hamiltonian with spin-gravity coupling, we show that the associated matrix element for the Majorana neutrino differs significantly from its Dirac counterpart. This difference can be demonstrated through significant gravitational corrections to the neutrino oscillation length for a two-flavor system, as shown explicitly for SN 1987A.     

Interference phase of mass neutrino in CM space--time

In the gravitational field of central mass with electric and magnetic charges and magnetic moment (CM space-time), this paper calculates the interference phase of mass neutrino along geodesic in the radial direction, and discusses the contribution of the electric and magnetic charges and magnetic moment of the central mass to the phase.     

Neutrino oscillations in the gravitational field

We calculate the gravitational correction to the phase difference between neutrino mass eigenstates for the spherically symmetric gravitational field described by the Schwarzschild metric. This correction was calculated in a number of works, but the results of these works differ from each other. Our result does not coincide with the results ever published. In this work, we make calculations in the simplest way and verify our result by several tests.     

Neutrino radiation in spherically-symmetric gravitational fields II. The structure of the radiation field

It is shown that the neutrino radiation field emitted by a star may be described by Vaidya's radiating Schwarzschild metric. The gravitational energy shift of the neutrino field is also considered, both in terms of an exact solution and in the weak field approximation.     

Neutrino oscillations in the general spherically symmetric gravitational field

The results for neutrino oscillations in the gravitational field described by the Schwarzschild metric are generalized to the general spherically symmetric gravitational field.     

Effect of Majorana Phases in Neutrino Oscillation

Quantum gravity (Planck scale effects) lead to an effective SU(2) _L ×U(1) invariant dimension-5 Lagrangian involving neutrino and Higgs fields. On symmetry breaking, this operator gives rise to correction to the above masses and mixing. The gravitational interaction M _X =M _pl , we find that for degenerate neutrino mass spectrum, it is shown that the Majorana phase of the neutrino mixing matrix can effects in neutrino oscillation probability.     

The General Treatment of the High and Low Energy Particle Interference Phase in a Gravitational Field

The interference phase of the high energy mass neutrinos and the low energy thermal neutrons in a gravitational field are studied. For the mass neutrinos, we obtain that the phase calculated along the null is equivalent to the half phase along the geodesic in the high energy limit, which means that the correct relative phase of the mass neutrinos is either the null phase or the half geodesic phase. Further we point out the importance of the energy condition in calculating the mass neutrino interference phase. Moreover, we apply the covariant phase to the calculation of the thermal neutron interference phase, and obtain the consistent result with that exploited in COW experiment.     

Einstein-Weyl space-times with geodesic and shear-free neutrino rays: Asymptotic behaviour

We consider a neutrino field with goodesic and shear-free rays, in interaction with a gravitational field according to the Einstein-Weyl field equations. Furthermore we suppose that there exists a Killing vector r^μ whose magnitude is almost everywhere bounded at the future and past endpoints of the neutrino rays. The implications of the asymptotic behavior of r^μ on the structure of space-time are investigated and a useful set of reduced equations is obtained. It is found that under these hypothesis the space-time cannot be asymptotically flat if the neutrino field is nonvanishing. All the Demianski-Kerr-NUT-like space-times as well as the space-times which admit a covariantly constant null vector are explicitly obtained.     

Quantum phase transition for the BEC-BCS crossover in condensed matter physics and CPT violation in elementary particle physics

We discuss the quantum phase transition that separates a vacuum state with fully gapped fermion spectrum from a vacuum state with topologically protected Fermi points (gap nodes). In the context of condensed-matter physics, such a quantum phase transition with Fermi point splitting may occur for a system of ultracold fermionic atoms in the region of BEC-BCS crossover, provided Cooper pairing occurs in the non-s-wave channel. For elementary particle physics, the splitting of Fermi points may lead to CPT violation, neutrino oscillations, and other phenomena.     

Vaidya's spherically symmetric radiating solution

Conclusions An analysis of Vaidya's solution has shown that it cannot describe electromagnetic radiation in a gravitational field: the Einstein-Maxwell equations for this solution are incompatible.The neutrino treatment of this solution is somewhat dubious.An investigation using the Rodichev energy tensor and Petrov's classification indicates that there is also no gravitational radiation in this solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 130–132, November, 1971.In conclusion the authors wish to thank V. I. Rodichev for valuable discussions of this work.     

Atomic Interferometry in Gravitational Fields: Influence of Gravitation on the Beam Splitter

The laser induced splitting of atomic beams inthe presence of a gravitational field is analyzed. Inthe frame of a quasiclassical approximation, the motionof the atomic beam through a laser region with rectangular profile is calculated. Beside theusual beam splitting due to the atomlaser interaction,an additional splitting occurs due to the anomalouseffective interaction with the gravitational field. In a first order approximation in thegravitational acceleration, the outcome of an atominterferometry experiment is given, which includes thevarious corrections owing to the gravitationalmodification of the beam splitting process.     

Nonconservation of lepton current and asymmetry of relic neutrinos

The neutrino asymmetry, \({n_v} - {n_{\bar v}}\), in the plasma of the early Universe generated both before and after the electroweak phase transition (EWPT) is calculated. It is well known that in the Standard Model the leptogenesis before the EWPT, in particular, for neutrinos, owes to the Abelian anomaly in a massless hypercharge field. At the same time, the generation of neutrino asymmetry in the Higgs phase after the EWPT has not been considered previously due to the absence of any quantum anomaly in an external electromagnetic field for such electroneutral particles as neutrinos, in contrast to the Adler anomaly for charged left- and right-handed massless electrons in the same electromagnetic field. Using the Boltzmann equation for neutrinos modified to include the Berry curvature term in momentum space, we establish a violation of the macroscopic neutrino current in the plasma after the EWPT and exactly reproduce the non-conservation of the lepton current in the symmetric phase before the EWPT that owes to the contribution of the triangle anomaly in an external hypercharge field but already without computing the corresponding Feynman diagrams. We apply the new kinetic equation to calculate the neutrino asymmetry by taking into account the Berry curvature and the electroweak interaction with plasma particles in the Higgs phase, including that after the neutrino decoupling in the absence of their collisions in the plasma. We find that this asymmetry is too small for observations. Thus, a difference between the relic neutrino and antineutrino densities, if it exists, must appear already in the symmetric phase of the early Universe before the EWPT.     

Is the size of <Emphasis Type="Italic">θ</Emphasis> <Subscript>13</Subscript> related to the smallness of the solar mass splitting?

θ₁₃ is small compared to the other neutrino mixing angles. The solar mass splitting is about two orders smaller than the atmospheric splitting. We indicate how both could arise from a perturbation of a more symmetric structure. The perturbation also affects the solar mixing angle and can tweak alternate mixing patterns such as tribimaximal, bimaximal, or other variants to viability. For real perturbations only normal mass ordering with the lightest neutrino mass less than 10^?2 eV can accomplish this goal. Both mass orderings can be accommodated by going over to complex perturbations if the lightest neutrino is heavier. The CP-phase in the lepton sector, fixed by θ₁₃ and the lightest neutrino mass, distinguishes different options.