Decoherence in neutrino oscillations, neutrino nature and CPT violation

We analyze many aspects of the phenomenon of the decoherence for neutrinos propagating in long baseline experiments. We show that, in the presence of an off-diagonal term in the dissipative matrix, the Majorana neutrino can violate the CP T symmetry, which, on the contrary, is preserved for Dirac neutrinos. We show that oscillation formulas for Majorana neutrinos depend on the choice of the mixing matrix U. Indeed, different choices of U lead to different oscillation formulas. Moreover, we study the possibility to reveal the differences between Dirac and Majorana neutrinos in the oscillations. We use the present values of the experimental parameters in order to relate our theoretical proposal with experiments.


INTRODUCTION
The phenomena of neutrino mixing and oscillations, induced by the non-zero neutrino mass, represent an hint of physics beyond the Standard Model of particles. It has been confirmed by many experiments [1]- [6]. At the present, the main issues of the neutrino physics are the determination of the absolute neutrino mass and its nature. As a matter of fact, since neutrino is electrically neutral, two possibilities exist, either neutrino is distinct from its antiparticle and hence is of the Dirac type, or it is equal to its antiparticle and it is of the Majorana type.
To reveal the neutrino nature, many experiments, based on the detection of the neutrinoless double beta decay, have been proposed [7]. Recently, it has been shown that quantities such as the Leggett-Garg K 3 quantity [8] and the geometric phase for neutrinos [9], can, in principle, discriminates between Dirac and Majorana neutrinos. Moreover, it has been shown that in the presence of decoherence, the neutrino oscillation formulas can depend on the Majorana phase [10]. However, at the moment the nature of the neutrino remains an open question.
On the other hand, particle mixing phenomenon, in particular the B 0 − B 0 mixing is used to test the CP T symmetry. The CP T theorem, affirms that the simultaneous transformations of charge conjugation C, parity transformation P , and time reversal T , is an exact symmetry of nature at the fundamental level [11].
In this paper, we show that, if quantum decoherence appears in neutrino oscillations, then long baseline experiments might allow to investigate the nature of neutrinos and the CP T symmetry. Moreover, if the neutrinos are Majorana particles, the decoherence could allow the right choice of the matrix mixing U .
The phenomena of dissipation and decoherence are consequences of interaction with the environment, which, in neutrino case, could be originated by quantum grav-ity effects, or strings and branes. A significant research effort had been undertaken in the study of dissipation in neutrino oscillations [10,12,13]. It has been shown that such phenomena can modify the oscillation frequencies and the oscillation formulas [10,12,13]. Moreover, it has been noted that the dissipation can generate oscillation formulas for Majorana neutrinos different with respect to the ones for Dirac neutrinos [10]. Still, other theoretical results can be obtained which are extremely relevant.
Here, by considering the neutrino as an open quantum system interacting with its environment, we analyze many aspects of the decoherence effect in flavor mixing. We study the time evolution of the density matrix representing the neutrino state in the flavor basis and we analyze the case in which the matrix describing the dissipator has off-diagonal terms. Specifically, we reveal the possible CP T symmetry breaking in the Majorana neutrino oscillation, and we study the differences between Majorana and Dirac neutrinos. We prove that, the presence of off-diagonal terms in decoherence matrix, leads to probability of transitions depending on the representation of the Majorana mixing matrix. Thus, if the decoherence exists in neutrino propagation, the oscillation formulas could provide a tool to determine the choice of the mixing matrix U. Moreover, by considering the data of IceCube and DUNE experiments and the recent constraints on decoherence parameters [24,25], we show that long baseline experiments on atmospheric neutrinos, like IceCube experiment, could reveal the nature of neutrino and could allow to test the CP T symmetry.
We analyze the neutrinos propagation in the vacuum and through a medium. The matter effects, are taken into account by replacing in the oscillation formulas in vacuum, ∆m 2 with ∆m 2 m = ∆m 2 R ± , and sin 2θ with sin 2θ m = sin 2θ/R ± . The coefficients R ± describing the Mikhaev-Smirnow-Wolfenstein (MSW) effect [14,15] are given by, with + for oscillation of antineutrinos and − for oscillations of neutrinos. Here, n e is the electron density in the matter, G F is the Fermi constant, and E is the neutrino energy. Notice that the MSW effect can be relevant in the ν e ↔ ν µ oscillations, since the ν e and ν µ indices of refraction are different in media like the Earth and the Sun, κ(ν e ) = κ(ν µ ). In particular, κ(ν e ) − κ(ν µ ) = − √ 2G F n e /p. On the contrary, in the case of the ν µ ↔ ν τ mixing, the ν µ and ν τ indices of refraction are different only in very dense matter, like the core of supernovae, but they are almost identical in the matter of Earth and the Sun. Therefore, in such media, R ∼ 1 and the ν µ ↔ ν τ oscillations are almost identical to the ones in vacuum [16]. In the following we consider the propagation through Earth in which the MSW effect is relevant only for ν e ↔ ν µ oscillations.
The paper is organized as follows. In Sec.II we report the main differences between Majorana and Dirac neutrinos. In Sec.III we analyze quantum decoherence in neutrino propagation and we show the effects induced on neutrino by an off-diagonal term in the dissipation matrix. Numerical analysis, for neutrino propagation in vacuum and through matter, are reported in Sec.IV. Sec.V is devoted to the conclusions.

MAJORANA AND DIRAC NEUTRINOS
A very important difference between Dirac and Majorana neutrinos consists in the fact that, while the Dirac Lagrangian is invariant under U (1) global transformation, and hence the charges associated (electric, leptonic, etc.) with the transformations are conserved, the Majorana Lagrangian breaks the U (1) symmetry [17]. This fact implies that processes violating the lepton number, such as neutrinoless double beta decay, are allowed for Majorana neutrinos and forbidden for Dirac ones. In the case of neutrino mixing, the breaking of the U (1) global symmetry of the Majorana Lagrangian implies that the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix with dimension n × n, contains a total number of physical phases for Majorana neutrinos different with respect to the one of Dirac neutrinos. Indeed, in the case of the mixing of n Dirac fields, one has N D physical phases given by N D = (n−1)(n−2) 2 , and in the case of the mixing of n Majorana fields, one has N M phases given by N M = n(n−1) where φ i , with i = 1, ..., n − 1, are the Majorana phases.
Other representations of Majorana mixing matrix can be obtained by the rephasing the lepton charge fields in the charged current weak-interaction Lagrangian, (for details see Ref. [18]). For example, for two mixed Majorana fields, one can consider the following mixing relations or where θ is the mixing angle and φ is the Majorana phase. Such a phase can be removed for Dirac neutrinos by rephasing the mass term of the Dirac Lagrangian. For example, in Eq.
(3), φ is eliminated by means of the replacements, ν 1 → ν 1 and ν 2 → ν 2 e iφ . Notice that, in the case of absence of decoherence (i.e. in standard treatment of neutrino mixing) and in the case of dissipation with diagonal decoherence matrix, the probabilities of neutrino transitions are invariant under the rephasing U αk → e iφ k U αk (α = e, µ; k = 1, 2). Then, in such cases, the presence of the Majorana phases φ i do not affect the oscillation formulas for neutrino propagating in the vacuum, through matter and through a magnetic field, being such formulas equivalent for Majorana and for Dirac neutrinos [19]. On the contrary, in the presence of off-diagonal terms in decoherence matrix, the oscillation formulas for Majorana neutrinos, depend on the phases φ i [10].
Moreover, we will reveal other two important aspects: a) Majorana neutrinos can violate CP T symmetry; b) different choices of the mixing matrix for such neutrinos can lead to different probability of transitions, (for example, the oscillation formulas obtained by using Eq.(2) can be different with respect to the ones obtained by means of Eq.(3), see below).

DECOHERENCE AND NEUTRINO OSCILLATIONS
The evolution of the neutrino considered as an open system, can be expressed by the Lidbland-Kossakowski master equation [20] ∂ρ(t) ∂t Here, H ef f = H † ef f is the effective hamiltonian, and D[ρ(t)] is the dissipator defined as The coefficients a ij of the Kossakowski matrix, could be derived by the properties of the environment [10] and F i , with i = N 2 − 1, are a set of operators such that T r(F k ) = 0 for any k and T r(F † i F j ) = δ ij . In the three flavor neutrino mixing case, F i are represented by the Gell-Mann matrices λ i . In the two flavor neutrino mixing F i , are the Pauli matrices σ i .
For the sake of simplicity, in the following we consider the mixing between two flavors (our results can be extended to the three flavor neutrino mixing case). Expanding Eqs. (4) and (5) in the bases of the SU (2), Eq. (4) can be written as where ρ µ = Tr(ρ σ µ ), with µ ∈ [0, 3] and D λµ is a 4 × 4 matrix. The conservation of the probability implies D λ0 = D 0µ = 0, then All the parameters of Eq. (7) are reals and the diagonal elements are positive in order to satisfy the condition, T r(ρ(t)) = 1 for any time t.
In order to study the effects of a non-diagonal form of the decoherence matrix on neutrino physics, we consider, for simplicity D λµ given by Such a dissipator is obtained by Eq. (7), by setting, γ 1 = γ 2 = γ , and β = δ = 0. The condition of complete positivity of the density matrix ρ(t), implies ∀t, the following condition |α| ≤ γ 3 /2 ≤ γ. By setting, ∆ = ∆m 2 2E , and by taking into account Eq.(8), one hasρ 0 (t) = 0, which for two neutrino families implies ρ 0 (t) = 1/2. Then the master equation (6) can be written as By solving Eq.(9), one gets where Ξ ± = α ± ∆ and Ω α = √ α 2 − ∆ 2 . Hence, the matrix density, at any time, t reads By using the mixing relations for Majorana neutrinos, given in Eq.(3), the matrix density of the electron neutrino, at time t = 0, is and similar for muon neutrino. Then at time t, one has where, Λ ± = 1 2 [1 ± cos 2θe −γ3t ] , and The probabilities of transition P νσ →ν̺ (t), with σ and ̺ neutrino flavors, are given by P νσ In a similar way, for anti-neutrino, one has Eqs. (14) and (15) show an asymmetry between the transitions ν σ → ν ̺ and ν σ → ν ̺ , i.e. P νσ →ν̺ (t) = P νσ →ν̺ (t). Such an asymmetry, is due to Majorana phase φ and appears also in the probability of an electron, muon or tau neutrino preserving its flavor σ, (σ = e, µ, τ ), i.e. P νσ →νσ (t) = P νσ →νσ (t). The CP violation, induced by the oscillation formulas Eqs. (14) and (15) is explicitly given by, The definition of the T-violating quantity in the case of dissipative matter is more delicate. Indeed, the decoherence and the dissipation induce an explicit violation of the T symmetry, which is independent on the nature of the particle. Here we are interested to the study of T symmetry in neutrino oscillation.
In QM mixing treatment, the T violating asymmetry can be obtained by means of two equivalent definitions, as follows However, in the presence of decoherence, the definition ∆ M T (t) = P νσ →ν̺ (t) − P νσ→ν̺ (−t) , cannot be used, since the complete positivity of the matrix density is not satisfied for any time. Indeed, one has The presence of hyperbolic functions in Eq. (18)  On the contrary, the relation, ∆ M T (t) = P νσ →ν̺ (t) − P ν̺→νσ (t), is defined properly at any time t. By using such a relation, we have, i.e., the two flavor Majorana neutrino oscillation, in the presence of decoherence, does not violate the T symmetry.
The CP T invariance imposes the relationship ∆ CP = ∆ T . However, by comparing Eq.(16) with Eq.(19), we have ∆ M CP = ∆ M T , which implies the violation of the CP T symmetry for Majorana neutrinos, ∆ M CP T = 0. Let us consider now Dirac neutrinos. The phase φ can be set equal to zero, then the oscillation formulas P νσ →ν̺ (t) and P νσ→ν̺ (t) are equivalent, P νσ →ν̺ (t) = P ν σ →ν̺ (t) and reduce to In such a case, the neutrino oscillation preserves the CP and T symmetries, ∆ D CP = ∆ D T = 0. The above results show that the decoherence could produce another difference between Majorana and Dirac neutrinos, i.e. the CP T symmetry is violated by Majorana neutrinos, but it is preserved by Dirac neutrinos.
The kind of CP T violation here presented is due to the mixing in the presence of the decoherence. It could represent an effect induced by the quantum gravity [21]. We emphasize that such a violation is different from an explicit CP T symmetry breaking in the Hamiltonian dynamics such that [CP T, H] = 0. In such a case, a possible cause of the CP T breaking can be represented by the Lorentz violation due to a propagation in a curved space violating the Lorentz invariance. In the present framework, the decoherence may lead to an effectively ill-defined CP T quantum mechanical operator [22].
Another result of the present paper, is the discovery of the fact that the oscillation formulas for Majorana neutrinos depend on the choice of the mixing matrix U . Different choices of U lead to different oscillation formulas. Indeed, Eqs. (14) and (15) are obtained by using the mixing relations given in Eq.(3). On the other hand, by using the mixing matrix of Eq.(2), the oscillation formula for neutrinos Eq. (14) is replaced by the one for antineutrinos Eq.(15) and viceversa. The dependence of the oscillation formulas by the choice of the representation of the mixing matrix characterizes the Majorana neutrinos. Therefore, if the neutrinos are Majorana particles, the study of the oscillation formulas in long baseline experiments could also allow the determination of the right mixing matrix.
Notice that similar effects are produced by the following dissipator On the other hand, the dissipator generates oscillation formulas depending on the phase φ, but there is no CP and CP T violations, being, in such a case, P νσ →ν̺ (t) = P νσ →ν̺ (t).
In order to emphasize the role of off-diagonal elements in the dissipator, we compare the above results with the one obtained in the case of diagonal dissipator, i.e. α = 0, and D µν = −diag(0, γ, γ, γ 3 ). In such a case, we have In such a case, for two flavor neutrino oscillation, one has, ∆ CP = ∆ T = ∆ CP T = 0. The Pontecorvo formulas [19] are recovered by setting in Eq.(23), γ = γ 3 = 0. The results here presented hold for neutrino propagation in vacuum and also for ν µ ↔ ν τ oscillation through the matter of the Earth. Indeed, the ν µ → ν τ oscillations and the CP violation in the Earth are practically identical to the ones in vacuum.
In the case of mixing between ν e and ν µ through media, the Earth is not charge-symmetric, indeed it contains electrons, protons and neutrons, but not contains their antiparticles. Then, the behavior of neutrinos is different with respect to the one of antineutrinos, also without decoherence (we have to consider R − in ∆m 2 m and sin 2θ m for neutrinos and R + for antineutrinos). This fact implies that, the ν e ↔ ν µ oscillations in matter break the CP and CP T symmetry also in absence of decoherence. Therefore, the ν e ↔ ν µ oscillations in matter cannot be used to study the CP T violation induced by the decoherence. Such an analysis, together with the one on the neutrino nature can be better done by studying the ν µ ↔ ν τ oscillation in vacuum or through the Earth, and the ν e ↔ ν µ oscillation in vacuum.

NUMERICAL ANALYSIS
We now present a numerical analysis of Eqs. (14), (15), (20) in order to study the nature of neutrinos. Moreover, we consider Eq. (16) to study the CP and CP T violations in Majorana neutrinos.
We consider the characteristic parameters of the Ice-Cube DeepCore experiment, which detects neutrino oscillations from atmospheric cosmic rays, over a baseline across the Earth [23]. Such an experiment is sensitive at neutrino energies in the range (6 − 100)GeV and it is mainly sensitive to muon neutrinos. Therefore, we analyze the mixing between ν µ and ν τ and we compute the probability of transition P νµ→ντ (x) and the corresponding oscillation formula for the anti-neutrinos P ν µ →ντ (x). We also study the mixing between ν e and ν µ and the oscillation formulas P νe→νµ (x) and P νe→νµ (x). The range of neutrino energy analyzed for ν e ↔ ν µ oscillation is (0.3 − 5)GeV , which is characteristic of DUNE experiment.
We use, in natural units, the approximation, x ≈ t, where x is the distance travelled by neutrinos. We analyze the neutrino propagation in the vacuum and through the matter.
By analyzing the plots in Figs.1 and 2, one can see that the differences between Majorana and Dirac neutrinos, the CP and CP T violations are, in principle, detectable. Indeed, considering the CP violation, which, in the case of two flavor neutrino mixing, is different from zero only for Majorana neutrinos, one finds: a) for ν µ ↔ ν τ neutrino oscillation in vacuum and through matter, in particular ranges of the energy, ∆ In Fig.3, we include the matter effect for the oscillation ν e ↔ ν µ and we plot the CP asymmetry ∆ CP = P νe→νµ (t) − P νe→νµ (t), for Majorana and for Dirac neutrinos in the presence of decoherence with off-diagonal term. Moreover, we plot ∆ CP in absence of decoherence. In our computations, we consider electron number density n e = 2.36cm −3 N A and the range of energy [0.3 − 1]GeV . The value of n e represents the weighted arithmetic mean of the mean electron densities in the Earth mantle n m e and in the Earth core n c e . (The Earth mantle has a radius of R m = 2885km and an estimated mean electron number density n m e ≃ 2.2cm −3 N A , the Earth core has a radius of R c = 3846km and an estimated mean electron number density n c e ≃ 5.4cm −3 N A ). The plots show different behaviors of ∆ CP for Majorana and for Dirac neutrinos in the presence of decoherence with off-diagonal term. Such behaviors are different with respect to the one of ∆ CP obtained by considering the two flavor neutrino mixing without decoherence.

CONCLUSIONS
We have studied different features of the phenomenon of the decoherence in neutrino oscillations. We have shown the possible CP T symmetry breaking in the Majorana neutrino oscillation, we have shown that the probability of transitions for Majorana neutrinos depend on the representation of the mixing matrix, and we have study the phenomenological differences between Majorana and Dirac neutrinos in their oscillations.
By using the data of IceCube DeepCore and DUNE experiments, and by considering the constraints on decoherence parameters [24,25], we have analyzed the oscillation formulas for atmospheric neutrinos, P νµ→ντ and P νµ→τ µ , and for neutrinos produced in accelerator, P νe→νµ and P νe→νµ . We have studied the behaviors of CP and CP T violations in neutrino oscillation and we have shown that, the differences between Majorana and Dirac neutrinos, together with the CP T violation could be detected if the phenomenon of decoherence is taken into account during the neutrino propagation in long baseline experiments. Moreover, the oscillation formulas could provide a tool to determine the choice of the mixing matrix for Majorana neutrinos, if the neutrino is a Majorana fermion. Since, at the Earth densities, the MSW effect affects only the ν e ↔ ν µ oscillations and such oscillations in matter are neither CP , nor CP T invariant, thus, ν e ↔ ν µ oscillations in matter are not suitable to study the possibility of CP T breaking induced by the decoherence. For such an analysis, one has to consider long baseline experiments analyzing atmospheric neutrinos, such as the IceCube experiment, or one has to study the ν e ↔ ν µ oscillations in vacuum. We point out that the CP T violation, the difference between the oscillation formulas of Majorana and Dirac neutrinos, and the dependence of such formulas on the representation of the mixing matrix appear only in the cases of a not diagonal form of the dissipator, similar to the one presented in Eq. (8). In the case of diagonal dissipator, such effects disappear. Then experiments like IceCube, could allow the determination of the correct form of the matrix describing the decoherence, if such phenomenon is relevant in neutrino oscillation.
Neutrino decoherence and CPT violation could be signals of quantum gravity. Therefore, our analysis could open new interesting scenarios not only in the study of neutrinos, but also in other fields of fundamental physics.
Notice also that, non-perturbative field theoretical effects of particle mixing [26], [27], can be neglected in the our treatment.