Revealing neutrino nature and $CPT$ violation with decoherence effects

We study decoherence effects on mixing among three generations of neutrinos. We show that in presence of a non--diagonal dissipation matrix, both Dirac and Majorana neutrinos can violate the $CPT$ symmetry and the oscillation formulae depend on the parametrization of the mixing matrix. We reveal the $CP$ violation in the transitions preserving the flavor, for a certain form of the dissipator. In particular, the $CP$ violation affects all the transitions in the case of Majorana neutrinos, unlike Dirac neutrinos which still preserve the $CP$ symmetry in one of the transitions flavor preserving. This theoretical result shows that decoherence effects, if exist for neutrinos, could allow to determine the neutrino nature and to test fundamental symmetries of physics. Next long baseline experiments could allow such an analysis. We relate our study with experiments by using the characteristic parameters and the constraints on the elements of the dissipation matrix of current experiments.

phases (the Dirac phase and the two Majorana phases) in the mixing matrix. On the contrary, Dirac neutrinos can break CP symmetry only in two of the three flavor preserving transitions. A difference between Dirac and Majorana neutrinos can be revealed also in the case of diagonal dissipator with γ 1 = γ 2 , or γ 4 = γ 5 , or γ 6 = γ 7 . Moreover, we show that the oscillation formulae for Majorana neutrinos depend on the parametrization of the Majorana mixing matrix. Therefore, if the decoherence affects neutrino evolution, the oscillation formulae could reveal the neutrino nature and, if the neutrinos are Majorana fermions, one could determine the right parametrization of the Majorana mixing matrix. Our theoretical studies suggest that the neutrino nature and the violation of fundamental symmetries could be analyzed with future long baseline experiments [25,26]. We consider the neutrino oscillations in vacuum since in this case the violation of CP and CP T symmetries due to the decoherence are not affected by other phenomena. In fact, for neutrinos travelling, for example, through Earth, the MSW effect already introduces an additional degree of CP T violation [27]. Therefore, one has to be careful to identify the right contribution responsible for violations purely induced by decoherence. Since we are mainly interested in highlighting the difference between Dirac and Majorana neutrinos, for simplicity we compare them in the vacuum. In a forthcoming paper we will extend our treatment in the presence of matter.
The paper is organized as follows. In Section II we briefly review the concepts of Dirac and Majorana neutrinos and introduce the mathematical tools of the density matrix needed to compute all the oscillation formulae for three flavors neutrinos in presence of decoherence. In Section III, we consider a diagonal dissipator and show that in this case the oscillation formulae are independent of the neutrino nature. In Section IV, we show the effects of an off-diagonal dissipator on the oscillation formulae and on the violation of CP and CP T symmetries. Moreover, we show the dependence of these quantities on the representation of the Majorana mixing matrix. In Section V, we present a numerical analysis by using the available data of the characteristic parameters involved in long baseline experiments. In Section VI we summarize the contents of this paper by emphasizing the relevance of the main results, and draw our conclusions.

II. NEUTRINO MIXING AND DECOHERENCE
The main distinction between Dirac and Majorana neutrinos relies on the fact that Dirac Lagrangian is invariant under the global transformation of U (1) so that all the associated charges (like electric, leptonic, etc.) turn out to be conserved, while Majorana Lagrangian breaks the U (1) symmetry. A process in which the lepton number is violated and therefore would be allowed only for Majorana neutrinos and not for Dirac is the neutrinoless double beta decay.
The breaking of the U (1) global symmetry has also consequences on the form of the mixing matrix [16] which contains a different number of physical phases for the two kind of neutrinos. Indeed, in the general case of the mixing with n Dirac fields, there exist N D = (n−1)(n−2) 2 physical phases, while for n Majorana fields, one has additional N M = n(n−1) 2 phases. The n − 1 extra phases are called Majorana phases and their detection would allow to identify the nature of neutrinos.
Let us recall that the Lagrangian density for Dirac neutrinos in flavor basis is given by where Ψ T f = (ν e , ν µ , ν τ ) and M † = M is the mixed mass term. The mixing relations are [16,17]: where U D is the Dirac mixing matrix, δ is the Dirac phase, c ij = cos(θ ij ) and s ij = sin(θ ij ) , with θ ij being the mixing angles between the fields with definite masses ν i , ν j with i, j = e, µ, τ , and Ψ T m = (ν 1 , ν 2 , ν 3 ) . Eq.(1) is diagonalized by using Eq.(2), so that we obtain the Lagrangian for free Dirac fermions with masses m 1 , m 2 and m 3 : where M d = diag(m 1 , m 2 , m 3 ) . For Majorana neutrinos, different parametrizations of the mixing matrix U M , exist. In fact, when decoherence is negligible, and even in the case of a diagonal dissipator, all the transition probabilities turn out to be invariant under the rephasing U αk → e iφ k U αk (α = e, µ; k = 1, 2). This means that the Majorana phases φ i do not affect the oscillation formulae which are the same as for Dirac neutrinos [18]. For instance, one can write where φ 1 and φ 2 are the two Majorana phases. Another possible parametrization is the following: and other choices leading to the same oscillation formulae are presented in Ref. [19]. This fact is no longer true when there are off-diagonal elements in dissipation matrix and also in the case of diagonal dissipator with γ 1 = γ 2 , or γ 4 = γ 5 , or γ 6 = γ 7 . Indeed, one can obtain oscillation formulae for Majorana neutrinos depending on the phases φ i , and on the parametrization of the mixing matrix, as shown in Ref. [11] for two flavor mixing and non-diagonal dissipator. In the following we will consider the case of three flavor neutrino mixing and reveal new aspects of neutrino oscillations which are absent in the case of mixing between two neutrinos. In the rest of the paper, we mainly focus on the matrix given in (5), which will be very useful to highlight the main features in presence of decoherence.
By treating the neutrino as an open quantum system, we analyze the physical implications of decoherence in flavor mixing. In particular, we study the time evolution of the density matrix corresponding to the neutrino state in the flavor basis and compute several transition probabilities for both diagonal and non-diagonal dissipation matrix.
The state evolution of neutrinos seen as an open system, can be described by the Lindblad-Kossakowski master equation [20]: where H = H † is the total Hamiltonian of the system and D[ρ(t)] is the dissipator defined as with a ij Kossakowski coefficients whose form is related to the characteristics of the environment [10]. The operators F i , with i = 1, . . . , N 2 − 1 , satisfy the relations Tr(F i ) = 0 and Tr F † i F j = δ ij , and in the case of three flavor neutrinos they are the Gell-Mann matrices λ i which satisfy the following properties: Here the non-vanishing f ijk are given by 2 . Let us now expand Eqs. (6) and (7) in the basis of SU (3) : where ρ µ = Tr (ρλ µ ) , with µ = 0, . . . , 8 . Given the mass differences ∆m 2 21 = m 2 2 − m 2 1 and ∆m 2 31 = m 2 3 − m 2 1 , the Hamiltonian reads where ∆ 21 = 1 2E ∆m 2 21 and ∆ 31 = 1 2E ∆m 2 31 . One can show that the only non-vanishing components H µ are The dissipator in Eq.(9) is given by where we considered the probability conservation which implies D µ0 = D 0ν = 0. All the elements in the matrix (12) are real and the ones on the diagonal are positive in order to satisfy the relation Tr (ρ(t)) = 1 . Hence, from Eq. (9) it is now clear that we have nine equations among which the µ = 0 component is trivial. Indeed, since f ij0 = 0 and D 0ν = 0 we obtainρ 0 (t) = 0 ⇒ ρ 0 (t) = 1. The density matrix written in terms of the components ρ µ in the basis λ µ reads With this expression of the density matrix, the neutrino oscillation formulae reads Notice that, the CP symmetry violation is defined as ∆CP ab ≡ P νa→ν b − Pν a →ν b = 0 and the T violation is given by The CP T symmetry is violated when ∆CP = ∆T .
In similar way, ∆T aa = 0. These result are the same of those obtained in the absence of decoherence. Moreover, like in the standard case, CP and T symmetries are violated because of the presence of the Dirac phase δ, while the presence of diagonal elements in the dissipation matrix only introduces a damping factor which is physically expected. For instance, the three channels responsible for CP violations read ∆CP eµ = P νe→νµ − Pν e→νµ = sin δ cos 2 θ 13 sin(θ 12 )sin(2θ 23 )sin θ 13 sin(∆ 32 t)e −γ67t + sin(∆ 21 t)e −γ12t − sin(∆ 31 t)e −γ45t , Note that the sum of CP violations for fixed family is vanishing, as expected, i.e. we have ∆CP eµ + ∆CP eτ = 0 , ∆CP µe + ∆CP µτ = 0 , ∆CP τ e + ∆CP τ µ = 0 .
Hence, in presence of a diagonal dissipation matrix, CP and T are violated, but CP T is still preserved as in the standard case where no decoherence effects are present, i.e. ∆CP ab = ∆T ab . Furthermore, it is clear that in such a case, the Majorana phases φ 1 and φ 2 do not play any role, indeed all oscillation formula are independent of them. The violation of CP and T is related only to the presence of the Dirac phase, indeed if we set δ = 0 we recover CP and T invariance also in presence of a diagonal dissipator. Different results are obtained for diagonal dissipators with γ 1 = γ 2 , or γ 4 = γ 5 , or γ 6 = γ 7 . In these cases, one can show that the oscillation formulae and the CP and T violations depend on the Majorana phases.

IV. NON-DIAGONAL DISSIPATOR
We now study the scenario with a non-diagonal dissipator. We consider the cases for which only two symmetric off-diagonal elements are non-zero. In particular, we mainly focus on following form for the dissipator: and then we will also comment on what happens if other off-diagonal elements are switched on. In the case described by Eq.(25) the system of differential equation in Eq.(16) will differ for the componentsρ 1 andρ 2 which now satisfy the two differential equationsρ respectively, and whose solutions read while the other components are the same the ones in (17). We have defined the quantities Ω ≡ α 2 1 − ∆ 2 21 and Ξ ± ≡ α 1 ± ∆ 21 . The initial conditions ρ i (0) are the same as in Eqs. (18), (19) and (20) for electronic, muon and tau neutrinos, respectively.
Let us now distinguish two cases: (A) first, we consider a mixing matrix with zero Majorana phases to show the role played by the Dirac phase in the violation of CP and CP T symmetries; (B) subsequently, we compute the oscillation probabilities considering non-zero Majorana phases and analyze the effects on CP and CP T violations.
In Eqs. (28) it is shown that the violation of CP appears in the transitions ν µ → ν µ and ν τ → ν τ . On the contrary, the transition ν e → ν e preserves such a symmetry. Notice that ∆CP µµ and ∆CP τ τ does not appear either in absence of decoherence or in presence of a diagonal dissipator. As we will see in the next subsection, for Majorana neutrinos ∆CP ee = 0. Therefore, the analysis of such a violation could be crucial in order to discriminate between Dirac and Majorana neutrinos in presence of an off-diagonal dissipation matrix. Moreover, the CP violating channels for different neutrinos are modified as follows: The T violations also differ from the diagonal case and are given by Therefore, unlike the case of a diagonal dissipator, when α 1 = 0, not only CP and T are violated, but also CP T symmetry is not preserved: Such violations are related to the presence of the Dirac phase, indeed by setting δ = 0, all the three symmetries are preserved even if α 1 = 0. Let us point out that such an effect is not present in the two flavors case analyzed in [11] since in that case no Dirac phase is present and one can not find any relation between the phase δ and CP T violation. The CP T violation induced by Dirac phase is a new feature in presence of decoherence and dissipation. If we set α 1 = 0 we recover the case of diagonal dissipator where CP T symmetry is preserved. Furthermore, in presence of an off-diagonal dissipator, the oscillation formula depends on the choice of the mixing matrix, indeed one can straightforwardly check that different parametrizations of the mixing matrix for Dirac neutrinos give different physical results. In this paper we focus on the Pontecorvo-Maki-Nakagawa (PMNS) parametrization in Eq. (2) for Dirac neutrinos. Our results show that, if the decoherence characterizes neutrino oscillations, next long baseline experiments could reveal which matrix elements contain the δ-phase. Let us emphasize that so far we have only considered one possible case of non-diagonal dissipator, in which only α 1 is non-zero. Of course, also other kinds of dissipation matrices can be studied in which other off-diagonal elements are non-zero. By making computations similar to those presented above, one can show that all the possible choices of the dissipator (12) lead to CP and T violations, as it also happens in the diagonal case. On the other hand, CP T is violated in most of the cases; however, there are some off-diagonal choices which still preserve it. Indeed, CP T symmetry is respected when the only non-zero off-diagonal element is one of the following: β 1 , α 3 , δ 3 , ξ 3 , η 1 , ζ 2 , χ 4 , δ 5 , β 6 , α 7 , γ 8 .

B. Non-zero Majorana phases
In this Subsection we repeat the previous analysis for the mixing matrix in Eq. (5) where the Majorana phases φ 1 and φ 2 are non-zero. We show that in presence of an off-diagonal dissipator, the oscillation formulae, the CP and T violations can depend on the Majorana phases, thus providing a new framework in which the real nature of neutrino can be challenged.
Here, with the letter M we mean the transition probabilities for Majorana neutrinos. By comparing Eqs. (32), (33) and (34) with the analogue in Eq. (28), we can immediately note that the presence of non-zero Majorana phases introduces new terms in the formulae, and in particular, generate a CP violation also in the transition ν e → ν e . This violation is absent for Dirac neutrinos, and depends on φ 1 for the dissipator considered. The transition (32) has a very peculiar meaning: unlike the case of zero Majorana phases, here ∆CP ee turns out to be non-vanishing, and becomes zero only when φ 1 = 0 . Such a feature is crucial in order to discriminate between Dirac and Majorana neutrinos and provides a completely new way to test the real nature of neutrinos in future experiments. Indeed, by considering the mixing matrix in Eq.(5) and the dissipator in Eq.(25), we have ∆CP ee = 0 for Dirac neutrinos and ∆CP M ee = 0 for Majorana neutrinos. Let us also clarify that such a difference in the CP violation for ν e → ν e transition, with respect to the other two, depends on the form of the dissipation matrix and on the representation of the mixing matrix for Majorana neutrinos.
The possibility to violate the CP symmetry in the transitions flavor preserving, here revealed, is a new result which can indicate the presence of decoherence and allow us to fix the form of the mixing matrix, besides the neutrino nature.
The T violating channels are not affected by the Majorana phases for our choice of the dissipator, indeed they are the same as in Eq. (30): This fact induces an extra violation of the CP T symmetry since we have In presence of an off-diagonal dissipator, Dirac and Majorana phases induce two independent CP T violations. The results here presented are obtained by considering the non-diagonal dissipator in which only α 1 is non-zero; see Eq. (25). Other kinds of dissipation matrices can be studied with other off-diagonal elements switched on. Like for the mixing matrix in Eq. (2), also for the matrix (5), CP and T are always violated, while CP T can be still preserved for some non-zero off-diagonal elements. Indeed, CP T is respected if the only non-zero off-diagonal element is one of among these: β 1 , α 3 , δ 3 , ξ 3 , η 1 , ζ 2 , χ 4 , δ 5 , β 6 , α 7 , γ 8 . Notice also that other choices of the Majorana matrix would give different results. For instance the mixing matrix U M in Eq. (4) give different expressions for the oscillation formula as compared to Eq. (5). This implies that the physical results depend on the chosen parametrization of the Majorana mixing matrix.
Summarizing, in presence of an off-diagonal dissipator, the neutrino oscillation formula depend on the parametrization of the mixing matrix. A physical implication is that Dirac and Majorana neutrinos are two totally distinct entities and their nature, together with CP T violation, can be tested with future experiments.

V. COMPARISON BETWEEN DIRAC AND MAJORANA NEUTRINOS
In this Section we relate our theoretical analysis to the parameters of neutrino experiments. We compare the behavior of Dirac and Majorana neutrinos considering some specific transition probabilities. In order to connect our results with existing long baseline experiments such as IceCube and DUNE, one should consider neutrino propagation in the matter and to adopt the formalism presented in Ref. [21], which generalize the Mikheyev-Smirnov-Wolfenstein (MSW) effect [22][23][24] to the case of decoherence. However, since the Earth is not charge-symmetric (it contains electrons, protons and neutrons, but it does not contain their antiparticles), then the oscillations in matter involving electron neutrino already induce the CP and CP T violations also in absence of decoherence. Therefore, one has to be careful to identify the right contribution responsible for violations purely induced by decoherence. Since we are mainly interested in highlighting the effects of the decoherence, we consider the neutrino oscillations in vacuum. In the following we approximate x ≈ t in Natural units.
In Fig. 1, panel (a), we plot the ν µ → ν τ oscillations in vacuum and ∆CP µτ as functions of the neutrino energy, by using the range of energy of the IceCube DeepCore experiment E ∈ (6 − 120)GeV [25,28] and a distance equal to Earth diameter x = 1.3 × 10 4 km, corresponding to t = 6.58 × 10 22 GeV −1 . We draw the oscillation formula P νµ→ντ and the quantity ∆CP µτ obtained by using the diagonal and the off-diagonal dissipators with zero and non-zero Majorana phases, respectively.
cases of a diagonal and an off-diagonal dissipation matrix. By analyzing Dirac neutrinos, we have shown that in presence of a diagonal dissipator, the oscillation formula do not depend on the parametrization of the mixing matrix and CP T symmetry is still preserved. Subsequently, we have switched on an off-diagonal elements in the dissipation matrix, and shown that for Dirac neutrinos the oscillation formula can depend on the parametrization of the mixing matrix. Moreover, we have revealed the possibility of a CP violation in the neutrino transitions preserving the flavor and the existence of a CP T violation due to the Dirac phase δ . By performing analogue computations for Majorana neutrinos, we have shown that in presence of an off-diagonal dissipation matrix, the oscillation formulae can depend on the Majorana phases φ i . These formulae depend on the choices of the parametrization of the Majorana mixing matrix. Indeed, different parametrizations lead to different formulae. We have also revealed a CP T violation term purely induced by the Majorana phases, which generalize the result in [11] obtained for two flavors neutrinos. For a specific form of the dissipator whose non-zero off-diagonal element is α 1 , we have shown that ∆CP ee = P νe→νe − Pν e →νe is zero for Dirac neutrinos, while it is non-vanishing for Majorana ones. ∆CP ee could be analyzed in next experiments to discriminate between Dirac and Majorana neutrinos.
The CP T violation induced by Dirac and Majorana phases, together with the distinction in the oscillation formula for Dirac and Majorana neutrinos, can be really tested in long baseline experiments if the phenomenon of decoherence is not negligible. Very interestingly, such a phenomenon might be even more accessible than the neutrinoless double beta decay, and represent a totally new scenario where to test the real nature of neutrinos. By using the parameters of IceCube DeepCore and DUNE experiments, and the constraints on dissipation matrix [28,29], we have analyzed the transitions ν µ → ν τ and ν e → ν e , and made a comparison between Dirac and Majorana. A detection of CP T violation induced by decoherence effects could be attributed to fluctuations of the space-time [30,31], thus such a detection might represent a signature of quantum gravity. Moreover, the studies on neutrino mixing in curved space [32,33] could be also generalized by including in them the decoherence and dissipation effects here presented. Therefore, our study might open new windows of opportunity to address several open questions in fundamental physics. It is worthwhile note that, non-perturbative field theoretical effects of particle mixing [34], [35] can be neglected in the our treatment.