Gravity induced geometric phases and entanglement in spinors and neutrinos: Gravitational Zeeman effect

We show Zeeman-like splitting in the energy of spinors propagating in the background gravitational field, analogous to the spinors in electromagnetic field, otherwise termed as Gravitational Zeeman Effect. These spinors are also found to acquire a geometric phase, in a similar way as they do in the presence of magnetic fields. Based on this result, we investigate geometric phases acquired by neutrinos propagating in a strong gravitational field. We also explore entanglement of neutrino states due to gravity which could induce neutrino-antineutrino oscillation in the first place. We show that entangled states also acquire geometric phases which are determined by the relative strength between gravitational field and neutrino masses.


I. INTRODUCTION
It is well known that if the time dependence in the Hamiltonian arises through certain parameters, namely adiabatic parameters, then the system develops a nondynamic phase, called the Berry phase [1]. Spinors propagating in the magnetic fields are known to acquire such a Berry phase. Interestingly a neutrino propagating through a medium also develops such a system, while the varying matter density corresponds to the adiabatic parameter.
Several authors have studied the geometric phases in neutrino oscillations. Although it was argued in an earlier work that the Berry phase plays no role in two flavor neutrino oscillations in matter [2], the work was restricted to a limited region in the parameter space. However, it was shown by exploiting the spin degree of freedom that the interaction of neutrinos with the transverse magnetic field can lead to a geometric effect [3]. Later on, it was argued [4] that the Berry phase can only appear in the presence of nonstandard (e.g. R-parity violating supersymmetry) neutrino-matter interactions for the particular case of two flavor oscillations in matter. Essentially all the above papers argued that geometric phases do not arise in the two flavor neutrino oscillation probabilities with CP conservation in vacuum or in matter, in the absence of any nonstandard neutrino-matter interactions. It was, however, furthermore argued [5] that even in the absence of CP violation, neutrinos in two flavor oscillation in vacuum in a period can acquire an overall phase consisting of a dynamical phase and a phase depended on mixing angle only. The second part of the phase, which is of geometric origin, was called Berry phase. Note that this phase does not arise due to slowly varying parameters leading to adiabatic evolution, rather due to Schrödinger evolution of the system giving a closed loop in the Hilbert space. As the phase is a global phase at the amplitude level, it does not appear in measurable quantities like probabilities of appearance or survival of neutrinos. These cyclic geometric phases were furthermore extended by the later authors [6] to obtain noncyclic phases for two and three flavor neutrinos in vacuum, which remain unobservable because of the same reason as before. Also the geometric phases for neutrinos propagating in varying magnetic fields have been reported [7].
It is interesting to note that [8] the Berry phase has a connection to the phase discovered by Pancharatnam [9]. In fact, both the phases can be described under the same platform [10]. Unlike the Berry phases obtained in above work, recently it has been established [11] that Pancharatnam phase can appear in detection probabilities and hence can be observed directly even in an effective two flavor approximation. However, none of the work considered the effects of gravity in the calculations; whether the interaction of spinors and then neutrinos with gravitational field brings any effect in it or not. This issue particularly arises due to the fact that neutrinos interacting with background gravity may not preserve CPT [12,13], which may be shown as a natural candidate for governing the Berry phase even in the evolution of neutrinos due to the split of dispersion energy between neutrino and antineutrino. Indeed within the pure standard model of particle physics, the neutrino oscillations can not be understood and hence relaxing the CPT conservation through gravitational interaction is one of the natural steps forward to beyond standard model. While the Berry phase arises in the presence of nonstandard matter-neutrino interactions, neutrino spin and magnetic field interactions, it is a natural question if the coupling between spin of neutrino and in general spinor and spin connection to the background gravity generates any geometric effect.
Two flavor neutrino oscillation in the background gravity has been discussed in various astrophysical contexts. One of the current authors explored possible Lorentz and CPT violations in the neutrino sector in the presence of background gravity and its astrophysical consequences [12,13,[15][16][17]. Earlier, the analogy of solar neutrino oscillations with the precession of electron spin in a time-dependent magnetic field was discussed [18]. Then based on the evolution of a statistical ensemble, oscillations for neutrinos from supernovae or in the early universe in the presence of mixing and matter interactions in a thermal environment were shown to be viewed in terms of precession [19]. It was also observed [20] that spin flavor resonant transitions of neutrinos emanating from active galactic nuclei may occur in the vicinity of black hole due to gravitational effects and due to the presence of a large magnetic field. Interestingly, the matter effects therein become negligible in comparison to gravitational effects.
In the present paper, we start by recapitulating the origin of Berry phase in spinors in the presence of external magnetic fields in §II. Then we show the analogous effects in the presence of background gravitational fields, namely gravitational geometric phase in spinors in the same section. Subsequent plan is to apply this result in the neutrino sector. To do so, we first recapitulate the basic solutions of previous work discussing neutrino oscillations in curved spacetime [12,13] in §III, which are used in subsequent sections. Based on these neutrino states evolving in the gravitational background, we explore any geometric (as well as dynamic) effect/phase arisen due to gravity in §IV. Subsequently, our aim is to explore the possible entanglement of neutrino states coupled with background gravitational field and to compute the geometric phase arisen in their evolution in §V. Finally we discuss how the geometric phases actually vary with gravitational field in §VI and summarize results in §VII.

GRAVITATIONAL FIELDS
A. In electromagnetic field Let us recall the Dirac equation describing dynamics of spinors in the presence of electromagnetic field given by where the various components of γ µ , where µ = 0, 1, 2, 3, are Dirac matrices with their usual meaning, e is the electric charge, m is the mass of the spinor and A µ is the electromagnetic where A 0 is the temporal component of A µ which is basically the Coulomb potential, p is the momentum vector and σ is the Pauli spin matrix. In the nonrelativistic limit, when m is much larger than the rest of the terms in the R.H.S. of Eq. (2), it reduces to Apart from the split due to the positive and negative energy solutions, clearly there is an additional split in the respective energy levels. This is basically Zeeman-splitting governed by the term with Pauli's spin matrix, in the up and down spinors for positive and negative energy spinors induced by magnetic fields, whether we choose relativistic or nonrelativistic regimes. The same governing term involved with σ is also responsible for the Berry phase if B is varying, which in spherical polar coordinates with R ≡ (r,θ,φ) is given by Note that in above calculation for Berry phase, B is assumed to be B = | B| r sinθ cosφ +θ sinθ sinφ +φ cosθ . Hence, when R is constant, Φ g = 0. Figure 1 represents the energy splitting given by Eq. (3). While the primary splitting corresponds to positive and negative energy solutions and the secondary splitting corresponds to the interaction between the spin and magnetic fields.

B. In gravitational field
Dirac equation in the presence of background gravitational fields has already been shown to have many consequences (see, e.g., the work by one of the present authors [12,13,16,21]) and is known to have the form (see, e.g., [13,22,23]) where B g µ is the gravitational covariant 4-vector potential (gravitational coupling with the spinor) and γ 5 = γ 5 = iγ 0 γ 1 γ 2 γ 3 as usual. Here we do not repeat the calculation to obtain the reduced form of the Dirac equation given by Eq. (5), which is available in the existing literature, see, e.g., [16,17,24] for details. In brief, while expanding the various terms of the Dirac Lagrangian (and equation) in curved spacetime, one obtains a hermitian-like and an another anti-hermitian-like parts (assuming gravitational coupling does not change in sign under PT transformation, e.g. constant locally), apart from the part already there in Minkowski spacetime. Hence on adding the Lagrangian with its hermitian conjugate, giving rise to the total Lagrangian (and corresponding equation); the anti-hermitian part drops out and one obtains Eq. (5) given above. Nevertheless, the appearance anti-hermitian-like part (which need not always be anti-hermitian, depending on the underlying spacetime) is independent of the hermitian-like term [21] that alone could lead to the axial-vector term given by Eq. (5), which is the basic building block of the following discussion. Hence, for simplicity, here we do not consider the apparent anti-hermitian term.
where B g 0 is the temporal component of B g µ . In the regime of weak gravity and when m is much larger than the rest of the terms in the R.H.S. of Eq. (6), it reduces to Here, there are two-fold split in dispersion energy, governed by two terms associated with the Pauli spin matrix, between up and down spinors for positive and negative energy spinors induced by gravitational fields, whether the field is weak or strong. The same governing terms are also responsible for Berry phase, as is for electromagnetic fields, which in spherical polar coordinates with R ≡ (r,θ,φ) is given by Interestingly, even if B g µ is constant (at a fixed local inertial frame) but nonzero, as p is varying -at least expected to change direction due to the propagation of spinor, Φ g survives, as seen from the last term in Eq. (7). Hence, while in electromagnetic fields the magnetic potential and hence field has to be varying, in gravitational field even the constant (but nonzero) gravitational potential still would produce Berry phase.   7). Here the splittings are different than those in electromagnetic case. Both the splittings are involved with the interaction between the spin and background gravitational fields. Hence, the gravitational "Zeeman-effect" appears to be different than the conventional electromagnetic Zeeman-effect. Nevertheless, the total energy of the system of particles remains conserved in electromagnetic and gravitational cases both (which indeed should be in the time-independent spacetime). Also, in the local inertial frame, at a given epoch if the process is considered in expanding universe, gravitational potential B g µ appears to be constant acting as a background effect. Note that B g µ can be computed for various spacetime metrics, as given by previous work [12,13,[15][16][17]21]. In order to have nonzero B g µ , spherical symmetry has to be broken and hence in Schwarzschild geometry (and hence for the spacetime around a nonrotating black hole), it vanishes. On the other hand, in Kerr geometry (and hence for the spacetime around a rotating black hole), it survives independent of the choice of coordinates: in Boyer-Lindquist as well as Kerr-Schild [12,15]. Also it survives in other natural spacetimes breaking spherical symmetry, e.g. in early universe under gravity wave perturbation, Bianchi II, VIII and IX anisotropic universe, in Fermi-normal coordinate upto second order correction [12,13,17,21]. In Kerr-Schild coordinate, the temporal part of gravitational potential reads as [12,13] r is the radial coordinate of the system and M and a are respectively mass and angular momentum per unit mass of the black hole. Naturally, B g 0 survives (and is varying with space coordinates) for any nonspinning black hole leading to gravitational Zeeman effect and Berry phase independent of spatial part B g . Similarly nonzero B g leads to gravitational Zeeman effect and Berry phase (when B g has to be varying as well), independent of the value of B g 0 . In Bianchi II spacetime with, e.g., even equal scalefactors in all directions, B g 0 survives as [12,17] leading to gravitational Zeeman splitting and Berry phase both even though B g = 0 (provided p for latter to emerge).
(5) and the related term involving with σ in Eq. (7) contributes as long as the spacetime naturally has some handedness, independent of the choice of coordinates.

III. NEUTRINO STATES IN THE PRESENCE OF GRAVITATIONAL FIELD
A. Neutrino-antineutrino mixing Recalling the work by Sinha & Mukhopadhyay [13] describing the mixing of neutrino (ψ) and antineutrino (ψ c ) in the presence of gravitational coupling, based on the formalism discussed above, let us write down the mass eigenstates ν 1 and ν 2 for a particular flavor at where B 0 is the gravitational scalar coupling potential and m the Majorana mass of the neutrino. Henceforth, by B µ we will mean B g µ itself, defined in the previous section, in order keep the same notation as of previous papers. However, at an arbitrary time t the mass eigenstates are when E ψ and E ψ c , are dispersion energies of neutrino and anti-neutrino respectively, are given by where B is the gravitational vector coupling potential and p the momentum of the neutrinos.
In the absence of gravitational field, neutrino and antineutrino mix in the same angle.
The oscillation length can also be recalled, for ultra-relativistic neutrinos, as and at an arbitrary time t as |ψ c (t) = cos θ e −iE 1 t |ν 1 (0) − e iφ sin θ e −iE 2 t |ν 2 (0) (17a) where in ultra-relativistic limit the energies of the mass eigenstates are when the corresponding masses in the presence of lepton number conserving mass (m n ) and violating mass (m). Here | p| ∼ E, the mean energy of the neutrinos, and θ and φ are same as in Eq. (12). The oscillation length can be recalled as which indicates that only for B 0 m, the gravitational field could affect the oscillation.

IV. DYNAMIC AND GEOMETRIC PHASES
Let us consider the wavefunction Ψ(t) of a system evolving over a time interval t ∈ [0, τ ], where Ψ(0) is its initial value and Ψ(τ ) being the final. The total phase accumulated over the entire evolution is given by Φ t = arg( Ψ(0)|Ψ(τ ) ) and the corresponding dynamic phase is given by Φ d = − τ 0 Ψ(t)|i∂ t |Ψ(t) dt. The difference between the two phases is defined as the geometric phase [14], given by In a situation when the system could oscillate back-and-forth between Ψ(t) andΨ(t) (which is antiparticle of Ψ(t)), e.g. the case of neutrino oscillation, we define new total and dynamic phases respectively given by Φ to = arg( Ψ (0)|Ψ(τ ) ) and Φ do = − τ 0 Ψ (t)|i∂ t |Ψ(t) dt. We term them as respective oscillation phases, when the geometric oscillation phase is given by Below we use these definitions to evaluate various phases in the neutrino sector. More precisely, we evaluate Φ t , Φ d , Φ g and Φ to , Φ do , Φ go for various neutrino states recalled in the previous section.
The phases, as we show below, depend on B µ , which furthermore is determined by the nature of underlying spacetime and the corresponding parameter values. For the explicit computations of B µ , see previous papers, e.g. [12,13,[15][16][17]21]. Nevertheless, for the present purpose we do not consider the contribution due to the spatial variation of neutrino states at t = 0, which is obvious from section II. Our interest rather is the contribution to the geometric phases due to mixing and oscillation of states, which arise due to the effect of spacetime curvature on to the time evolution of neutrino states.
The other phase is given by for |ν 1 as and for |ν 2 as when θ is independent of time. Even if θ is not constant, the part outside the integral in either of the Eqs. (32) and (33) always contributes to Φ g1 and Φ g2 respectively, revealing a τ -independent phase, as long as φ in the neutrino states is not constant. For θ = 0 which corresponds to B 0 >> m, the total geometric phases for ν 1 and ν 2 turn out to be nπ and nπ − φ respectively with n = 0, 1, 2, 3.... For θ = π/4 which corresponds to B 0 << m, they are nπ − φ/2. However, generally speaking neutrino mass does not vary with time and hence φ remains fixed throughout the propagation. Thus all the terms associated with φ actually vanish and any geometric contribution to the phase arises from other terms in, e.g., For the oscillation between mass eigenstates, total phase and the other phase is given by As before, the term outside the integral in Φ do1 survives only if φ varies with time, which generally may not be the case as the neutrino mass is fixed.
From previous work [12,16], B 0 can be computed for the spacetime around black holes when M be the mass of the black hole and M the mass of Sun. Therefore, for a black hole in an X-ray binary with M = 10M , B 0 << m, when the Majorana mass of a neutrino m ∼ 10 −2 eV. In this case, there is apparently no effect of gravity to the geometric and dynamic phases and Φ do1 turns out to be τ -independent and arisen due to Majorana nature of the neutrino because E ψ = E c ψ . This is purely the consequence of the mixing of neutrino and antineutrino, which occurs due to the presence of Majorana mass. The same is true for black holes at the center of AGNs.
For primordial black holes with M ≤ 10 24 gm, on the other hand, B 0 ≥ 1eV so that B 0 >> m. Therefore, θ → 0 and hence the part outside the integral of Φ d1 → 0 and and that of Φ d2 → −φ. Moreover, Φ do1 → 0. In this case gravitational field removes any possibility of mixing and then oscillation, which however affects geometric and dynamic phases.
When the mass of primordial black hole increases to M = 10 26 gm, B 0 ∼ m which alters the mixing angle compared to that in the absence of gravitational effect, and hence affects the phases. Important point to note is that larger is the mass of black hole, larger its radius, and hence smaller is the density in the surrounding disk. Therefore, in order to affect geometric and dynamic phases due to gravitational effect, the gravitational mass should not be more than ∼ 10 −6 M .

B. Mixing of mass eigenstates
In this case, for |ψ c the total phase and the dynamical phase containing a term which does not explicitly depend on τ due to nonzero neutrino phase φ, given by Similarly, for |ψ For neutrino-antineutrino oscillation, the total phase and the other phase Here θ is assumed to be independent of time. If, in general, θ is not a constant, then other terms will contribute to Φ d1,2 and Φ do1 . For B 0 >> m, Φ do1 → 0, while for m >> B 0 , all parts of the phases survive. Oscillation is also possible for m >> B 0 , as long as m n = 0.
Simultaneously oscillation and modified geometric and dynamic phases due to gravity are revealed, only when B 0 ∼ m, m n . Note that θ and φ are the same as that for the cases of neutrino-antineutrino mixing for the various parameters of spacetime, e.g. the mass of black hole. More so, as mentioned before, any term in the phase associated with φ does not survive if φ is not a time-varying function which generally is the case for neutrinos whose mass is assumed to be fixed.

METRIC PHASES
We begin by showing that neutrino (ψ) and antineutrino (ψ c ) combined system, as given by Eq. (16), forms entanglement. As ψ c = −iσ 2 ψ * , and if ψ c is purely spin-down with only one component nonzero then ψ is purely spin-up, hence  where a is the nonzero component of ψ and we choose Weyl representation for the convenience. As it stands, the first and second terms cannot be decomposed into direct-product of two independent states, hence they entangle. Similarly the combined mass eigenstates in the presence of gravitational field and Majorana mass, given by Eq. (11), can be shown to exhibit entangled states. Now in the presence of flavor mixing, as given by Eqs. (23) and (24), the states ν e1 and ν µ1 are orthogonal to each other and ν e2 and ν µ2 do so. Also without mixing term, ν e1 and ν e2 form two orthogonal mass eigenstates for neutrino-antineutrino mixing in the electron sector and ν µ1 and ν µ2 in the muon sector (when we consider only two flavors for simplicity).
Interestingly it is clear from Eqs. (13) and (14) that gravitational field converts ν e1 (and ν µ1 ) to ν e2 (and ν µ2 ) by oscillation, leading both of them to be present at an arbitrary time. Hence, Eqs. (23) and (24) show that gravitational effect brings out two independent sets of flavor neutrinos: {ν e1 , ν µ1 } and {ν e2 , ν µ2 }, satisfying respective orthogonality conditions between electron and muon neutrinos in the respective Hilbert spaces H 1 and H 2 independently. Hence the neutrino states in H 1 should entangle with those in H 2 which are noninteracting. Therefore, following the conventional approach (e.g. [30]) we can construct the entangled states at t = 0 when |ν e1 and |ν e2 (and |ν µ1 and |ν µ2 ) in Eq. (50a) are two points on the Poincare sphere and so on for others equations. The angle α determines the degree of entanglement. As is the case in the Poincare sphere of a single spin-1/2 particle, above equation suggests that α and β parameterize a two-sphere called Schmidt sphere.

VI. VARIATION OF MIXING ANGLES WITH GRAVITATIONAL FIELD
The phases independent of τ are associated with mixing angles and phases of neutrinos, e.g. θ 1 , θ 2 , α (also θ) and φ 1 , φ 2 , β (also φ). Therefore, depending on the values of θ-s, which are determined by gravitational field and the physical nature of spacetime geometry, the τ -independent parts of phases vary. In the absence of gravitational field and in the presence of lepton number violating interaction and hence Majorana mass, neutrino and antineutrino mix with θ = π/4. Figure 3 shows that how the mixing angle of the basic neutrino-antineutrino states changes with gravitational coupling, which furthermore controls the geometric Berry like phases associated with Φ t1,2 − Φ d1,2 given in §IV.A,B.
In the presence of very strong gravitational effect (B 0 >> m e , m µ ) (see, e.g., [12,13]), θ 1,2 → π/4 and for entangled states Φ do1,2 → β cos α sin α. Similarly, the τ -independent part of Φ d1,2,3,4 of entangled states survives even at a very strong gravitational field for arbitrary α. Figure 4 shows that θ 1 and θ 2 decrease with the increase of |B 0 |, which furthermore controls geometric Berry like phases associated with entangled states given in §V. Figure   5 shows, how the corresponding τ -independent part of Φ d1 varies with the change of the strength of gravitational coupling.

VII. SUMMARY
Spinors interacting with background gravity of arbitrary strength in an arbitrary spacetime are known to be divided into states of positive and negative energies. Only requirement is that the spacetime should not be spherical symmetric. It has been shown that such spinors acquire a geometric phase due to background gravitational field in the same way as they do in a magnetic field. The necessary condition for so is either the spacetime curvature coupling to the spinor (gravitational 4-vector potential) is not constant or the momentum of spinor is not constant along with non-zero (even constant) temporal part of curvature coupling.
Neutrinos as a class of spinors in nature are shown to acquire geometric as well as dynamic phases during their propagation under background gravitational field. To have a nontrivial phase induced by the gravitational field of compact objects (e.g. black hole), the mass of the object producing gravitational fields must not be more than a millionth of a solar mass. In the flavor sector, when the background gravity is much stronger than the lepton number violating (Majorana) masses, the mixing parameters θ 1,2 become a constant equal to π/4, which could lead to φ-dependent geometric phases if φ varies. However for a weak background gravity, θ 1,2 are found to depend on the specific values of the neutrino masses.