Quantum Field Theory of neutrino mixing in spacetimes with torsion

In the framework of quantum field theory, we analyze the neutrino oscillations in the presence of a torsion background. We consider the Einstein-Cartan theory and we study the cases of constant torsion and of linearly time dependent torsion. We derive new neutrino oscillation formulae which are depending on the spin orientation. Indeed the energy splitting induced by the torsion influences oscillation amplitudes and frequencies. This effect is maximal for values of torsion of the same order of the neutrino masses and for very low momenta, and disappears for large values of torsion. Moreover, neutrino oscillation is inhibited for intensities of torsion term much larger than neutrino masses and momentum. The modifications induced by torsion on the $CP$-asymmetry has been also presented. Future experiments, such as PTOLEMY, could provide insights into the effect shown here.


I. INTRODUCTION
Theories of gravity beyond General Relativity (GR) have a long and complex history [1].Stimulated by the need of dealing with the shortcomings of GR, providing an explanation for the dark components of the universe, and possibly to set a viable framework for the quantization of gravity, there is by now a plethora of such theories.Some, as the early attempt to incorporate Mach's principle by Brans and Dicke [2], involve additional fields other than the metric [3,4].Other theories generalize the Einstein-Hilbert action, eventually including higher order curvature invariants [5].Quite a natural generalization of GR emerges when one considers a non symmetric connection, allowing for the possibility of torsion [6,7].Gravitational theories including torsion might be able to account for dark matter and dark energy [8].Torsion couples naturally to the spin density of matter, inducing a spin-dependent splitting of the energy levels [9] and spin oscillations [10].
In this paper we analyze the propagation of neutrinos on a torsion background and study its impact on the flavor oscillations.Neutrino oscillations in presence of torsion have been studied in the quantum mechanical framework [36,37].We here approach the subject from the point of view of quantum field theory and quantise the neutrino fields on a torsion background.We focus on the simplest generalization of GR including torsion, the Einstein-Cartan theory.We consider the cases of constant torsion and of torsion linearly depending on time, and we assume that spacetime curvature is absent.We show that the energy splitting induced by the torsion term leads to spin-dependent neutrino oscillation formulae.Indeed, the spin orientation affects the frequencies, as expected also in QM framework, and the oscillation amplitudes which in QFT are ruled by the Bogoliubov coefficients.This last effect is a pure consequence of the non-trivial condensate structure induced by neutrino mixing in QFT.
The spin dependence of the oscillation formulae is maximal for intensities of torsion comparable to the neutrino masses.On the other hand, much larger values of torsion carry out to flavor oscillations which are identical for the two spins, since they become essentially independent of the spin.Another effect is that a torsion large enough can effectively inhibit the flavor oscillations, since in this case the energy differences due to the various masses become irrelevant with respect to the common torsional energy term.The presence of torsion is more relevant on neutrino oscillations in non-relativistic regimes, for which the QFT effects are also more emphasized.Some phenomenological consequences of the theoretical results presented here could then be provided, in the future, by experiments that analyze non-relativistic neutrinos, such as PTOLEMY [33,38].We additionally discuss the modifications induced by torsion on the CP -asymmetry, which is a byproduct of the Dirac CP -violating phase in the mixing matrix.We show that also the CP asymmetry depends on the spin orientations in presence of the torsion background.
The paper is structured as follows.In section II we introduce the concept of spacetime torsion and we quantize a Dirac field on torsional background.In section III, we analyze the three flavor neutrino mixing in the presence of constant and time dependent torsion, and in section IV, we derive new oscillation formulae depending on the orientation of the spin and in section V, new expressions of CP violation are shown.The last section is devoted to the conclusions, while in the appendix we report the analysis of currents and charges for flavor mixing in the presence of torsion.

II. SPACETIME TORSION AND DIRAC FIELD QUANTIZATION
Here, we briefly recall the notion of spacetime torsion, then we quantize the Dirac field minimally coupled to the torsion in the framework of the Einstein-Cartan theory.We study the cases of constant and time-dependent torsion.

A. Spacetime Torsion
In general relativity, the requirements of metricity of the covariant derivative and of symmetry uniquely determine the connection coefficients (Christoffel symbols) in terms of the metric as follows: A more general theory, the so called Einstein-Cartan (or Riemann-Cartan geometry), is obtained if the assumption of symmetry is relaxed, keeping only metricity.In this case, the connection coefficients acquire an antysimmetric part given by where the tensors T ρ µν and K ρµν = 1 2 (T ρµν + T µνρ − T νρµ ) are respectively known as torsion and contorsion.It is also convenient to introduce [7] the trace vector V µ = T ρ µρ , the axial vector T µ = ϵ αβγµ T αβγ and the tensor q ρ µν , in terms of which the torsion is expressed as and the scalar curvature reads as Here R, is the general relativistic Ricci scalar given in terms of the metric.Notice that the covariant derivatives in this context are the usual ones involving only the Christoffel symbols.The vacuum action for Einstein-Cartan is given by the natural generalization of the Einstein-Hilbert action.It is written as with κ = 8πG c 4 .The torsion-related terms in Eq. ( 2) form a total derivative, not contributing to the field equations.As a consequence the vacuum theory is equivalent to general relativity.On the other hand, the situation changes in presence of matter, where a coupling of the form appears.The spin tensor, here denoted with Σ µν ρ , is constructed out of matter fields.We point out that, the field equations obtained by varying the total action with respect to contorsion simply lead to the algebraic constraint K ρµν ∝ Σ ρµν , expressing the proportionality of torsion and spin angular momentum.In the following we will be interested in Dirac spinors minimally coupled to torsion.The spin covariant derivatives, in presence of torsion, get modified as follows [9] where D µ is the general relativistic spin covariant derivative and the Lorentz indices on the contorsion tensor result from contraction with the tetrads K ABµ = e ρ A e σ B K ρσµ .Then, the spinor action is simply given by where S D is the Dirac action in general relativity and S T D = 3 d 4 x √ −gT µ S µ is the action term due to the Dirac -torsion coupling.Moreover, S µ = 1 2 ψγ µ γ 5 ψ is the Dirac spin vector.We remark that in all the above expressions the spacetime dependence of the curved gamma matrices is kept implicit γ µ = γ µ (x) = e µ A (x)γ A .

B. Dirac field quantization on constant torsional background
From now on we shall assume that some astrophysical source other than the Dirac field itself generates a background torsion.As far as minimally coupled Dirac fields are concerned, the information about torsion is stored in the axial vector field T µ (x).Since we are specifically interested in the effects of torsion on Dirac fields, we will assume that spacetime curvature is absent (although the most general case can be treated in a similar fashion, see e.g.[12,[39][40][41][42][43]), so that the covariant derivatives in (5) are replaced with standard derivatives and the gamma matrices reduce to the flat ones.Under these assumptions the Dirac equation becomes Canonical quantization proceeds as in flat spacetime, and the Dirac field may be expanded on any complete set of solutions of Eq. ( 6).We shall see that the expansion closely resembles that of flat spacetime when a constant torsion background is considered.
It is important to remark that the lepton charge Q = d 3 x ψγ 0 ψ is conserved as a consequence of the U (1) gauge invariance of the action (5).
In this subsection, we deal with the simplest possible torsion background.We consider a constant axial torsion directed along the third spatial axis.The study of time dependent torsion background will be carried out below.The Dirac equation for constant torsion reads and is solved [9] in momentum space by the spinors These solutions are formally the same as in flat space, except for a spin-dependent mass term m ± = m ± 3 2 T 3 .The torsion has indeed the effect of lifting the degeneracy in energy between the two spin orientations ⃗ k (t) = e iE r t v r ⃗ k , the Dirac field is expanded as with the coefficients obeying the canonical anticommutation relations.Since the solutions to eq.( 6) are similar to those obtained in flat space time, to derive the neutrino oscillation formulas in the presence of torsion, we can follow a procedure analogous to the one presented in ref. [20] where the oscillation formulas for neutrinos in quantum field theory in flat space were found.Here, we obtain new oscillation formulae, showing a behavior different with respect to the ones of ref. [20].The differences are all contained in the Bogoliubov coefficients which characterize the amplitudes of the oscillation formulae and which are depending on the spin orientation.

C. Dirac field quantization with time-dependent torsion
We now quantize the Dirac field coupled to a certain class of time-dependent of torsional backgrounds, namely with T 0 spacetime constant and the spatial components T i (t) , i = 1, 2, 3 having an arbitrary time dependency (yet retaining constancy with respect to the spatial variables).This class of backgrounds allows for a simple non-trivial generalization of the constant torsion treatment presented above.For concreteness we treat in some more detail the case of a linearly time-dependent torsion, i.e.T i = α i t for some constants α i .The Dirac equation is formally equivalent to (6) except for the explicit dependency of the torsion on time.In order to derive the solution of the Dirac equation with torsion, we write the spinor in the following form We use the ansatz for negative energy.Here ξ λ (p) denote the helicity eigenspinors, satisfying Then, the solution of the Dirac equation is determined by solving the following system: The eigenvalues of the matrix in eq.( 10) are and the eigenvectors are , then the system of eqs.( 10) can be solved by means of a simple exponentiation: It is here that the requirement of constancy of T 0 becomes relevant, since the condition [A (t), A (t ′ )] = 0 is fulfilled for T 0 independent of time (i.e.T 0 = α 0 ).The solutions can be explicitly written as for some constant C p p p,λ and ω p p p,λ = m 2 + p + ηλ T 0 2 .In the specific case of By imposing the normalisation condition (ω p,λ +m) 2 +(p+ηλ T 0 ) 2 .

III. FLAVOR MIXING WITH TORSION
In this section, we analyze the three-flavor neutrino mixing in the presence of torsion, in particular we consider the cases of constant and time dependent torsion.In both the two cases, the neutrino fields with definite masses Ψ T m ≡ (ν 1 , ν 2 , ν 3 ) satisfy the equation with The fields with definite masses shall be expanded as in eq.( 9), except for acquiring an additional label j = 1, 2, 3 distinguishing the mass (u r ⃗ k,j , α r ⃗ k,j , ...).The flavor fields are obtained by performing the appropriate SU (3) rotation on the mass triplet.We choose the CKM parametrization of the PNMS matrix to link the the triplet of flavor fields ψ T f = (ν e , ν µ , ν τ ) to the fields with definite masses Ψ T m .As shown in ref. [20] the rotation to flavor fields can be recast in terms of the mixing generator I θ as ν α σ = I −1 θ (t)ν α i (x)I θ (t) , where (σ, i) = (e, 1), (µ, 2), (τ, 3), and I θ (t) = I 23 (t)I 13 (t)I 12 (t) .For reader convenience, we report in the appendix A the explicit form of the formulae used in the computations.
We note that the generator I −1 θ (t) here introduced, is formally identical to the generator G −1 θ (t) presented in ref. [20], where the mixing of three families of neutrinos in flat space-time has been studied.The difference consists in the fact that while G −1 θ (t) of ref. [20] is expressed in terms of the Dirac fields in flat space-time, I −1 θ (t) contains Dirac fields which are the solution of the Dirac equations for fields in the presence of torsions (constant and time depentent).As we will see below, this leads to two new set of Bogoliubov coefficients, one for constant torsion and one for time depending torsio, which are dependent on the spin.At the operational level, I −1 θ (t) shares the same properties as G −1 θ (t).However, it is essential to underline that, despite the formal analogy, the result here obtained presents completely new behaviors, since the new neutrino oscillation formulas, which will be derived in the following, have amplitudes and frequencies depending on the spin orientation.This effect, due to the torsion, is also depending on the form of the torsion and can in principle affect neutrinos produced in the nuclei of spiral galaxies or in rotating black holes.
In the following, adopting the procedure used in ref. [20], and taking into account the presence of torsion, we show the intermediate steps to derive the new oscillation formulae and we show the different behaviors of the oscillation formulae for the adopted torsions.We start by recalling some properties of the mixing generator I −1 θ (t) shared with G −1 θ (t).I −1 θ (t) is a map between the Hilbert space of free fields H 1,2,3 and that of interacting fields H e,µ,τ : → H e,µ,τ .At finite volume, the vacuum |0⟩ 1,2,3 , relative to the space H 1,2,3 , is connected to the vacuum |0⟩ e,µ,τ , relative to the space H e,µ,τ , in the following way: , where |0⟩ e,µ,τ is the vacuum for the flavour fields.The explicit form of I −1 θ (t) is reported in the appendix A. The action of the mixing generator defines the plane wave expansion of the flavor fields where the flavor annihilators are given by α r ⃗ k,νσ (t .
The Bogoliubov coefficients Γ rr ij; ⃗ k and Σ rr ij; ⃗ k , appearing in the expressions of the flavor annihilators, are given by the inner product of the solutions of Dirac equations with different masses.In order to distinguish the case of constant torsion from that of time-dependent torsion, we use the following notation: Γ rr ij; ⃗ k = Ξ rr ij; ⃗ k and Σ rr ij; ⃗ k = χ rr ij; ⃗ k , for constant torsion, and Γ rr ij; ⃗ k = Π rr ij; ⃗ k and Σ rr ij; ⃗ k = Υ rr ij; ⃗ k , for time dependent torsion.The explicit form of the Bogoliubov coefficients in the two cases analyzed are reported in the following.

A. Bogoliubov coefficients with constant torsion
For constant torsion, the modules of the Bogoliubov coefficients are given by Ξ r,s i,j; ⃗ k Notice that, in the reference frame ⃗ k = (0, 0, ⃗ k ), Ξ r,s i,j; ⃗ k and χ r,s i,j; ⃗ k vanish for r ̸ = s.Explicitly one has: with the spin-dependent masses and the normalisation coefficients given explicitly by , respectively.The sign factor is defined as

B. Bogoliubov coefficients with time dependent torsion
In this case, the Bogoliubov coefficients are denoted with . The mixed coefficients are zero and explicitly we have: , where i, j = 1, 2, 3 and j > i. 1 The canonicity of the Bogoliubov transformations is satisfied by the following relations 1 In the ultrarelativistic case (p ≫ m j ), one has: t) −→ 0 for any t.Moreover, in the absence of torsion (i.e.T µ = 0) these coefficients coincide with those presented in the Minkowski metric.

IV. NEUTRINO OSCILLATIONS WITH BACKGROUND TORSION
In this section, we derive the neutrino oscillation formulae in the presence of torsion and we study, in particular, the two cases of constant and linear time dependent torsion.By analyzing flavor currents and charges in a way similar to what was done in the ref.[20], and as shown in appendix A, we can define the flavor charges in the presence of torsion as :: , with σ = e, µ, τ and, :: • • • :: , denoting the normal ordering with respect to the flavor vacuum state |0⟩ f .
The oscillation formulas are obtained by computing, in the Heisenberg picture, the expectation values of the above charges on the (flavor) neutrino state, defined at t = 0, as ν r † ⃗ k,σ (0) = α r † ⃗ k,νσ (0) |0⟩ f .At a fixed momentum ⃗ k they are given by: where , and J CP denotes the Jarlskog invariant J CP ≡ Im u iα u jβ u * iβ u * jα .In the parameterization under consideration, J CP is given by: J CP = 1 8 sin δ sin(2θ 12 ) sin(2θ 13 ) cos(θ 13 ) sin(2θ It is also easy to check that the above oscillation formulae reduce to the Pontecorvo formulae in absence of torsion in the ultrarelativistic limit | ⃗ k| ≫ m 1 , m 2 , m 3 .Then, the oscillation formulae are highly spin-dependent, , since in QFT framework, the oscillation amplitudes and the frequencies are spin depending.Notice that, in QM mixing treatment, the spin orientation affects only the frequencies ∆ ij , being in this case: In the following, we analyze the behaviour of the oscillation formulae for constant and for time dependent torsions.

A. Neutrino oscillation with constant torsion
We report the transition formulas for sample values of torsion and momentum.We consider values of neutrino masses m 1 ≈ 10 −3 eV, m 2 ≈ 9×10 −3 eV, and m 3 ≈ 2×10 −2 eV, in order that ∆m 2  12 ≈ 7.56×10 −5 eV 2 and ∆m 2 23 ≈ 2.5×10 −3 eV 2 , and of mixing angles such that sin 2 (2θ 13 ) = 0.10, sin 2 (2θ 23 ) = 0.97, and sin 2 (2θ 12 ) = 0.861, which are compatible with the experimental data.We also consider δ = π/4, and a fixed value of the momentum k ≃ 2 × 10 −2 eV and of the torsion , with σ = e, µ, τ , as a function of time, and we compare such formulae with the corresponding quantum mechanics ones.Such formulae can be obtained from eqs.( 17), ( 18), (19), by setting Q e→e (t) Figure 1.Color on line.Plots of the oscillation formulae in a constant torsion background: in the left-hand panel Q ↑ ⃗ k νe→νe (t) (blue line) and Q ↓ ⃗ k νe→νe (t) (red line) as a function of time.Torsion was picked to be comparable to the momentum as T 3 = 2 × 10 −4 eV.In the right panel is reported the detail of the same formulae and the comparison with the corresponding quantum mechanics oscillation formulae (dashed line).The plots of the neutrino oscillation formulae for the constant torsion background displayed in figs.(1), ( 2) and (3) show the strong dependence of them on the spin orientation.The difference is maximal when the torsion is comparable with the neutrino momentum and neutrino masses.On the other hand, for very big values of torsion, T 3 ≫ m i , | ⃗ k|, the energy terms are dominated by the torsion, indeed , so that E + ≃ E − .This implies that both the Bogoliubov coefficients Ξ rr , χ rr and the phase factors ∆ r , Ω r become essentially independent of the spin, and the flavor oscillations become independent on the spin orientation.We also note that a torsion large enough can effectively inhibit the flavor oscillations, since for T 3 ≫ m i , the energy differences due to the mass differences (e.g.∆m 12 , ∆m 13 and ∆m 23 ) become irrelevant with respect to the common torsional energy term.

B. Neutrino oscillations with time dependent torsion
The neutrino oscillation formulae, in the case of linearly time-dependent torsion for fixed momentum ⃗ k and spin (↑), are given by Eqs.(17,18,19) with the Bogoliubov coefficients expressed in Eqs.(15), (16).By utilizing the same values of the masses, of the angles and of the momentum used in fig.4), (5), and (6) the oscillation formulae for time dependent torsion.We assume η T 0 ≃ 5 × 10 −3 eV.We observe that, also in this case, the formulas strongly depend on the orientation of the spin.In the computations here presented, we neglected the spin-flip transition due to the torsion term.This analysis will be carried out in a forthcoming work.Q e→τ (t) Figure 6.Color online.In the left-hand panel plot of Q ↑ ⃗ k νe→ντ (t) (blue line) and Q ↓ ⃗ k νe→ντ (t) (red line) as a function of time.In the right panel is reported the detail of the same formulae and the comparison with the corresponding QM formulae (dashed line).

V. CP VIOLATION AND FLAVOR VACUUM
We now study the impact of torsion on the CP violation in neutrino oscillation due to the presence of Dirac phase in the mixing matrix.For fixed spin orientation, say ↑, the CP asymmetry ∆ ρσ ↑;CP can be defined in QFT, in a similar way to what was done in the ref.[20], and then: , where ρ, σ = e, µ, τ .Notice that a + sign appears in front of the probabilities for the antineutrinos, in place of −, because the antineutrino states already carry a negative flavor charge Q σ .For the ν e → ν µ transition, with r =↑, ↓, the CP asymmetry is explicitly where one has to consider Γ ++ Furthermore, we make some observations on the condensate structure of the flavor vacuum in the presence of torsion.In this case, |0 f (t)⟩ breaks the spin symmetry, resulting in a different condensation density for particles of spin up and down.Such densities are evaluated by computing the expectation values of the number operators for free fields N r αj , ⃗ k = α r † ⃗ k,j α r ⃗ k,j and   4), ( 5) and ( 6).In the right panel) Plots of N ↓ i; ⃗ k as a function of ⃗ k for the same choice of parameters.
It is worth noting that the well shape appearing in the right panel of Fig. (10) is due to the proportionality of Υ −− i,j,⃗ p to (p−η T 0 ) (see Eq. ( 16)), so that it vanishes for p = η T 0 , bringing also the condensation density to zero.

VI. CONCLUSIONS
We analyzed the Einstein-Cartan theory and by studying the neutrino propagation on a torsion background in the framework QFT, we derived new oscillation formulae which are depending on the spin orientations of the neutrino fields.Indeed, we have shown that the energy splitting induced by the torsion term affects the oscillation frequencies and the Bogoliubov coefficients which represent the amplitudes of the oscillation formulae.We considered flat space-time and two different kind of of torsion terms, the constant and the linearly time dependent torsion.
The two analyzed cases share the following behavior: the spin dependence of the oscillation is maximal for values of torsion comparable to the neutrino momentum and masses, while much larger values of torsion lead to flavor oscillations which are almost independent of the spin.Moreover, a torsion large enough can effectively inhibit the flavor oscillations.Such behaviours characterize also the CP -asymmetry.
The torsion effects are relevant on neutrino oscillations in non-relativistic regimes.Therefore, experiments studying neutrinos with very low momenta, such as PTOLEMY, could provide verification of such results in the future.

VII. APPENDIX A: USEFUL FORMULAE
For reader convenience, we report formulas useful for the computations.We consider the PNMS matrix matrix.Then, denoting with ψ T f = (ν e , ν µ , ν τ ), the flavor fields and with Ψ T m = (ν 1 , ν 2 , ν 3 ) the fields with definite masses, the mixing relations are: Here, c ij = cos θ ij , s ij = sin θ ij and δ is the Dirac CP -violating phase.

. 2 + χ ±r ij; ⃗ k 2 = 1
The canonicity of the Bogoliubov transformations is ensured by the relations r Ξ ±r ij; ⃗ k where i, j = 1, 2, 3 and j > i.Moreover, the time dependence of Ξ ±r ij; ⃗ k and χ ±r ij; ⃗ k is expressed by

Figure 2 .
Figure 2. Color on line.In the left-hand panel, plot of Q ↑ ⃗ k νe→νµ (t) (blue line) and Q ↓ ⃗ k νe→νµ (t) (red line) as a function of time.In the right panel, detail of the same formulae and comparison with the corresponding QM oscillation formulae (dashed line).

Figure 3 .
Figure 3. Color on line.In the left-hand panel, plot of Q ↑ ⃗ k νe→ντ (t) (blue line) and Q ↓ ⃗ k νe→ντ (t) (red line) as a function of time.In the right panel, detail of the same formulae and comparison with the corresponding QM formulae (dashed line).

Figure 4 .Figure 5 .
Figure 4. Color on line.Plots of the oscillation formulae in a time-dependent torsion: in the left-hand panel are plotted Q ↑ ⃗ k νe→νe (t) (blue line) and Q ↓ ⃗ k νe→νe (t) (red line) as a function of time.In the right panel is reported the detail of the same formulae and the comparison with the corresponding quantum mechanics oscillation formulae (dashed line).We consider η T 0 = 5 × 10 −3 eV.
CHARGES FOR THREE FLAVOR MIXING WITH TORSIONCharges are introduced, by using the symmetries of the Lagrangian for free field operators:L = ψ m (x)(iγ µ ∂ µ − M )ψ m (x).The Lagrangian is invariant under global transformation of a phase factor U (1) of the type Ψ ′ m = e iα Ψ m .Then a charge is introduced via Noether's theorem:Q = d 3 xΨ m (x)γ 0 Ψ m ; itrepresents the total charge of the system.Considering a field transformation Ψ m under global transformation SU (3), we obtain Noether charges Q m,j of the form: Q m,j (t) ≡ d 3 xJ 0 m,j (x) , with j = 1, 2, • • • , 8 and J 0 m,j (x), time component of the SU (3) currents.The charges satisfy the SU (3) algebra: [Q m,j (t), Q m,k (t)] = if jkl Q m,l(t) .Note that only charges Q m,3 and Q m,8 are not time-dependent.Appropriate combinations of these charges allow to define the quantities:
12s 23 + e iδ s 12 c 23 s 13 2 Σ ±± with i = 1, 2, 3, as a function of ⃗ k .We use the same values of the parameters adopted in the plots of the oscillation formulae.