The Leggett–Garg K_{3} quantity discriminates Dirac and Majorana neutrinos

The Leggett–Garg quantity discriminates Dirac and Majorana neutrinos

Monika Richter Institute of Physics, University of Silesia in Katowice, Katowice, Poland    Bartosz Dziewit Institute of Physics, University of Silesia in Katowice, Katowice, Poland    Jerzy Dajka Institute of Physics, University of Silesia in Katowice, Katowice, Poland and
Silesian Center for Education and Interdisciplinary Research, University of Silesia in Katowice, Chorzów, Poland

The quantity, introduced in a context of the Leggett–Garg inequality violation, is studied for the neutrino oscillations in matter with phenomenologically modelled dissipative environment. It is shown that the function acquires different values depending on whether neutrino is Dirac or Majorana particle, provided that there is a dissipative interaction between matter and neutrinos. The difference occurs for various matter densities and can serve as a potential quantifier verifying the neutrino nature. Moreover, working within phenomenological model one can suggest the values of the matter density and dissipation for which the difference is the most visible. There exist also special conditions in which the violation of the Leggett–Garg inequality, to a different extent for both kinds of neutrino, is observed.

neutrino oscillations, decoherence, Leggett–Garg inequality
14.60.Pq, 03.65.Yz, 03.65.Ta

I Introduction

For a long time there were two central controversies in neutrino physics. The first was the (already solved) problem of neutrino masses. Neutrinos are massive white; data. The second (still alive) concerns neutrino nature which can be either Dirac, with particles and antiparticles being different, or Majorana, with particles and antiparticles being the same as it is in the case of photons. Measuring properties of such fragile and weakly interacting particles as neutrinos is a challenge for experimentalists. One can invoke here a short but embarrassing history of faster-than-light neutrino anomaly. That is why the proposal of a potentially helpful quantifier credibly indicating desired neutrino’s property is of interest for both theoretical and experimental physicists. There are proposals of solving the Dirac–Majorana neutrino controversy propp. In particular, interferometric pattern expp1; gutier or geometric phases of various type bea1; Dajka can serve as a quantifiers of Majorana and Dirac neutrino’s difference.

In 1964 Bell proposed a way to experimentally exclude deterministic local (hidden variable theories) as a possible interpretation of quantum mechanics epr. His celebrated inequality has been later verified in the experiments performed on the pairs of entangled photons (clauser,aspect) which provided then quite convincing evidence against the hidden local-variable theories. Quantum entanglement has also been studied also in a context of oscillating neutrino systems blas. In 1984 another test of the quantum mechanics principles has been developed podstawa: while the Bell inequality questioned the correlations between spatially separated systems, the Leggett-Garg inequalities (LGI) put the limits on the time correlations. They were derived basing on the assumptions of macroscopic realism (MR) and non-invasive measurability (NIM), the first of which presupposes that the system can be only in one of the available set of states, whereas the second one claims that it is possible to perform a measurement without disturbing a system review; fil. Just as the Bell inequalities, Leggett-Garg inequalities have been reported to be violated in the variety of systems (see for instance experiment2,experiment1). Especially interesting in terms of a study of the Leggett-Garg inequalities seem to be the neutrinos oscillations, for which the coherence length (the distance beyond which, the interference of different massive neutrinos is not observable Giunti) is much bigger than for the other systems. Such research has been already made both from the theoretical (leggettneutrinos, leggettneutrinos2) and experimental points of view – cf. Ref. kaiser reporting violation of the LGI in neutrino systems based upon the MINOS experiment’s data.

Our aim in this work is essentially more modest: instead of searching for violations of neutrino’s macro–realism our primary aim is to present a quantifier which, being (at least in principle) measurable, can discriminate between Dirac and Majorana neutrinos.

In this work, complementary to recent studies of neutrinos conducted from the perspective of quantum information bea2, we propose such a quantifier. We show that the correlator, used in the simplest class of LGIs calculated for neutrinos oscillating in matter in a presence of decohering interaction, has different value for both kinds of neutrinos. A possible mechanism causing decoherence in neutrino oscillations originate from quantum gravity effects as suggested in Refs. gr1; gr2; gr3. Instead of complex modelling in this work, following ben1, we adopt Kossakowski–Lindblad–Gorini–type phenonological equations. This type of describing open systems is particularly useful in quantum optics breu but nevertheless its is very general as it can be applied to any completely positive dynamics of quantum systems alicki. Although there certain problems of open quantum dynamics where one can directly relate parameters of of the Lindblad equation to the microscopic system–plus–environment description of a considered model alicki, in a case of neutrino oscillation our modelling is purely phenonemological and the effect of decoherence is encoded in a Kossakowski matrix describing non–unitary character of quantum dynamics.

Furthermore, we show that certain form of Kossakowski matrix describing decoherence is a sine qua non condition for usefulness of the function in determining the neutrino’s nature. We also theoretically investigate the behaviour of the LGI violation for oscillating neutrinos treated as an open quantum system undergoing the dissipation and decoherence while interacting with an environment.

The paper is organized as follows: first (in Sec. II) for self–consistency of the presentation we review dissipative dynamics of the neutrino oscillations following Benatti; Dajka, then in Sec. III, we introduce the notion of the Leggett–Garg inequalities review simultaneously providing the conditions for which the correlator allows to discriminate between Dirac and Majorana neutrinos. Finally, we summarize our work.

Ii Dissipative neutrino oscillations in matter

As the neutrino of the given flavour is produced at some point, it propagates further as the coherence mixture of 3 massive states. Mathematically, it means that neutrino flavour states can be considered the linear combinations of the massive states . The correspondence between them is expressed via unitary transformation sometimes referred to as the lepton mixing matrix Giunti:


In this paper, we limit ourselves only to the two neutrino oscillations case (between and ). Since the third neutrino effectively decouples it seems to be a quite reasonable assumption Smirnov. Then matrix can be parametrized by one mixing angle and in the most general case by one CP violating phase :


At this point, it is necessary to emphasize the importance of the phase . The neutrino is the only massive and electrically neutral elementary fermion. Its nature is therefore ambiguous: neutrino can be Dirac or Majorana (natureNeutrino,Petcov). In the Dirac case, the phase is not physical and can be easily removed by rephasing the neutrino fields occurring in the weak lepton charged current. On the other hand, in the Majorana case remains and is considered as real and measurable quantity.

Let us now consider Hamiltonian describing neutrino propagating in matter, which is divided into kinetic () and interaction () parts:


In the ultrarelativistic limit energy of an individual neutrino can be expressed as (Giunti):


where is an average energy of a neutrino. Then can be, in the mass basis, expressed as function of Smirnov:


The potential part arises due to the coherent forward scattering of the electron neutrinos with the electrons (charged current) and all kinds of neutrinos with the electrons, neutrons and protons (neutral current). Since the contribution from the neutral current is the same for all flavours, the real influence on the neutrino oscillation comes only from the charged current :


where is the mixing matrix given in Eq.(2).

It can be easily seen that Hamiltonian given in Eq.(3) does not allow for any discrimination between Dirac and Majorana neutrino: its transformation to the flavour basis makes the phase disappear. However, the treatment of the neutrinos propagating in matter as an open quantum system can dramatically change the situation.

In general, the Markovian evolution of a system interacting with an external environment can be described by the Lindblad-Kossakowski master equation, which in the Heisenberg picture reads as (kossakowski; Lindblad; breu; alicki):


where stand for an observable and denotes a set of the matrices creating a basis in the -dimensional space.

Note that the right-hand side of Eq.(7) is split into two parts. In the first part one recognizes the ordinary Heisenberg equation, whereas the second part is responsible for additional effects connected with an interaction of a system with an environment. As for latter part, in order to assure the trace- and hermicity-preserving evolution, the coefficients must obey the following conditions Romano; breu:


Neutrino during its propagation interacts weakly with matter. It is therefore subjected to the decoherence and dissipation effects, which justifies the application of Eq.(7). Since our system is two-dimensional, the master equation is presented as ():


where denotes one of the set of matrices (,,) for , respectively.

Assuming the Hamiltonian given in Eq.(3) one can easily simplify the form of Eq.(9) by rearranging it to the Schrödinger-like form:


where is a vector represented in the basis of the sigma matrices. Let us notice that Eq.(10) is nothing more than a convenient way of rewritting the Lindblad Master equation Eq.(7).

As for the matrix , it consists of two parts. The dissipative part can be parametrized with six real independent parameters Benatti:


where elements of has got the following correspondence with the matrix elements:


Subsequently, the direct calculation of the matrix elements reveals that:


The remaining elements in the and matrices are all equal to .

It is now easy to solve Eq.(10), for which the formal solution has an exponential form:


Let us notice that in principle it is possible to derive the coefficients starting from microscopic description of a system including its environment alicki. However, for the systems, which requires description based on the relativistic field theory this task is often very challenging calz.

Iii Leggett–Garg function for the neutrino

The Leggett-Garg inequalities are usually constructed for the dichotomic (i.e. whose eigenvalues are either +1 or -1) observable for which quantity defined as:


constitutes a benchmark of the correlations between its measurements performed in the different times review. Then, the Leggett-Garg parameter :


defined for 3 times is lower and upper-bounded under the MR and NIM assumptions review:


which is known as the simplest from the family of the Leggett-Garg inequalities.

It is shown that for –level system (the kind of which is considered in this paper) the correlation function can be expressed as the expectation value of the symmetrised product fryc; review; emary:


where states for the anticommutator of the operators and . Then, representing an observable in the basis of sigma matrices as , one arrives to the simplified formula for the coefficient kaiser:


It is important to point out that in the above-introduced formulas we made an assumption of the stationarity of the system (no explicit time dependence in the Lindblad equation governing time evolution of the neutrino system Eq.(10)), according to which the values of the correlation functions are not the functions of the specific times and but rather the time interval . Such a requirement accounts to the formal fulfillment of the NIM condition huelga.

In order to observe the behaviour of the LGI for the neutrinos propagating in the interactive environment, it is convenient to choose as a dichotomic observable. Note that, since


it measures the neutrino flavour as projected on the z-axis kaiser.

Making use of the general solution (Eq.(14)) to the Lindblad-Kossakowski master equation (Eq.(9)) for we are able to examine the variability of the parameter (Eq.(16)) as a function of different quantities. In particular, we focus on a difference


between the function calculated for the Dirac neutrino, corresponding to vanishing CP–violating phase , and the Majorana with . Non–vanishing is a potential quantifier, a hallmark, indicating different behaviour of these two types of neutrinos.

An influence of decoherence for the properties of for equal time intervals and for different values of the off–diagonal elements , with remaining parameters (such as and ) fixed, is presented in Fig. (1). As we set all other parameters are expressed in suitable powers of . In particular, in our calculations we set and assume the experimental values of the solar mixing angle and olive. Furthermore, the values of the Kossakowski matrix, rather than derived from any fundamental properties of the system, such as the coupling with the well-described thermal environment lobejko; alicki, result from a phenomenological modeling Benatti; Dajka. Our choice of the non–vanishing , yet the simplest possible, has no qualitative influence on the presented results. Increasing either the number or the value (provided that Eq.(8) is satisfied) of non–vanishing matrix elements simply lowers the amplitude of .

Let us explicitly state here the central result of our work: non–vanishing of any of the off–diagonal elements of the Kossakowski matrix is necessary for . In other words, any ’off–diagonal decoherence’ is a sine qua non condition for to be able to discriminate Dirac and Majorana nature of oscillating neutrinos.

In Fig.(2) we present the parameter as function of influenced by different values of quantifying interaction of neutrinos with the normal matter. The results of the numerical tests for Dirac and Majorana neutrinos bring us to a conclusion that there is an optimal value of (in the case of parameters chosen here ) for which the difference is maximal and allows for the most efficient discrimination between Dirac and Majorana neutrino’s nature. For our parameters the difference is the most apparent for , whereas is hardly visible for and . As the potential describes interaction of neutrinos with matter one can conjecture that this type of interaction and, in particular, a presence of matter, can enhance the possible implications of the neutrino’s nature at least on certain measurable properties.

Figure 1: The difference as a function of time for Dirac and Majorana neutrinos versus for different values of , fixed potential and with .

In studies of the correlator, it is natural to investigate possibility of violation of the LGI Eq.(17) in dissipative neutrino systems. We investigate whether the condition may be satisfied more likely either for Dirac or for Majorana neutrinos. The first observation is that for both non-dissipative environments (with ) and the diagonal Kossakowski matrix, there is no possibility to distinct from Dirac and Majorana neutrinos solely upon studies of LGIs’ violation.

Figure 2: The difference as a function of time for Dirac and Majorana neutrinos for different values of potential , given energy and elements of the Kossakowski matrix with .

On the other hand, from Fig.(3) one infers that for off–diagonal decoherence there exists a certain value of the CP-violating phase for which the value of is maximized. Similarly, there is also an optimal value of , in our case , such that the violation of the LGI is most significant. Let us notice, however, that the value of which leads to a maximal violation of the LGI is different comparing to that which results in a maximal value of discussed previously. According to results of our numerical analysis not reproduced here this property is valid universally i.e. for any choice of parameters of the system.

Figure 3: The dependence of the coefficient on the potential in the dissipative () environment with .

Let us notice that both the LG–correlator Eq.(16) and the difference in Eq.(21) are formed by a very special combination of correlation functions defined in Eq. (18). It is worth to mention that all the properties of the LG–correlators which seem to be useful for distinguishing Dirac and Majorana neutrinos essentially originate form the properties of and as presented in Fig. (4) and Fig. (5) respectively. However, the additional physical content present in the LG–correlator supported by recent experimental efforts on studies of a LGI violation in neutrino systems kaiser is encouraging to relate this phenomenon – quantified by rather than – and the investigation of the neutrino’s nature.

Eventually, let us notice that, contrary to , an energy in Eq.(5) – one of these parameters which are crucial for the properties of oscillating neutrinos – affects neither nor . The chosen value of in Fig. (1), Fig. (2) and Fig.(3) is completely arbitrary. In other words, one cannot either make Dirac and Majorana neutrinos distinguishable or improve their distinguishability solely via setting a value of in a similar manner as it can be done with .

Figure 4: The dependence of the correlation function Eq.(18) on the potential in the dissipative () environment with .
Figure 5: The difference of the correlation function for Dirac and Majorana neutrinos for different values of potential , given energy and elements of the Kossakowski matrix with .

Iv Conclusions

Determination of the neutrino nature, if it is Dirac or Majorana particle, is one of the most important, still unsolved, problems in studies of these fascinating particles. There are various problems arising in this context. First of all, the measurement of any of the neutrinos’ properties is a hard task due to a well known weakness of the neutrino’s interactions. Secondly, one needs to find and measure an observable, which would allow for the establishment of the neutrino nature. In this theoretical work we attempt to contribute to this second aspect. We show that the correlator studied recently in a context of violation of the Leggett–Garg inequalities takes different values for Dirac and Majorana neutrinos under certain conditions which were identified in our work. We studied neutrino’s oscillation in a presence of decoherence and matter using phenomenological approach based on the Kossakowski–Lindblad–Gorini completely positive dynamics alicki.

The main conclusion of our modelling is that the difference in occurs provided that (i) there is a certain type of decoherence represented by an off–diagonal Kossakowski matrix and (ii) the neutrino interacts with a matter i.e. in Eq.(6). The two conditions are necessary for making the difference non–vanishing and potentially useful candidate for a neutrino’s nature quantifier.

We also studied the difference in a possible violation of the LGI for Dirac and Majorana neutrinos. We concluded that the same two conditions as formulated above are valid also for the Leggett–Garg condition. However, there is a crucial difference: the value of necessary for violation of the LGI () is one order of magnitude larger then the corresponding value necessary for . As the value of is related to density of matter affecting neutrino’s oscillations, this observation may be potentially interesting and useful for experimentalists.

Let us emphasize here that credibility of any quantitative results obtained via any phenomenological model are questionable as long as the model is not confronted with a huge amount of experimental data. As neutrino physics suffers from a shortage of experimental data quantitative predictions presented in our work require considerable caution in interpretation. One can not exclude a very pessimistic scenario such that the proposed quantifier discriminating neutrinos nature works in a tailored range of parameters which is essentially inaccessible in any direct experiment and that our predictions are of purely theoretical character.

Mysterious neutrinos, more than any other ’fundamental’ particles, attract physicists outside the Particle Physics Community. We hope that our results are not only a modest contribution to physics of neutrinos per se but also indicate some unusual properties of neutrinos shedding light on even more unusual properties of the Quantum World.


The work has been supported by the NCN project UMO-2013/09/B/ST2/03382 (B.D. and M.R.) and the NCN grant 2015/19/B/ST2/02856 (J.D) .


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