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Advances in High Energy Physics

Volume 2013 (2013), Article ID 852987, 20 pages

http://dx.doi.org/10.1155/2013/852987

## The Nature of Massive Neutrinos

^{1}SISSA, INFN, Via Bonomea 265, 34136 Trieste, Italy^{2}Kavli IPMU, University of Tokyo (WPI), Kashiwa 277-8583, Japan

Received 6 September 2012; Accepted 25 October 2012

Academic Editor: Jose Bernabeu

Copyright © 2013 S. T. Petcov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The compelling experimental evidences for oscillations of solar, reactor, atmospheric, and accelerator neutrinos imply the existence of 3-neutrino mixing in the weak charged lepton current. The current data on the 3-neutrino mixing parameters are summarised and the phenomenology of 3- mixing is reviewed. The properties of massive Majorana neutrinos and of their various possible couplings are discussed in detail. Two models of neutrino mass generation with massive Majorana neutrinos—the type I see-saw and the Higgs triplet model—are briefly reviewed. The problem of determining the nature, Dirac or Majorana, of massive neutrinos is considered. The predictions for the effective Majorana mass in neutrinoless double-beta-(-) decay in the case of 3-neutrino mixing and massive Majorana neutrinos are summarised. The physics potential of the experiments, searching for -decay for providing information on the type of the neutrino mass spectrum, on the absolute scale of neutrino masses, and on the Majorana CP-violation phases in the PMNS neutrino mixing matrix, is also briefly discussed. The opened questions and the main goals of future research in the field of neutrino physics are outlined.

#### 1. Introduction: The Three Neutrino Mixing—An Overview

It is a well-established experimental fact that the neutrinos and antineutrinos which take part in the standard charged current (CC) and neutral current (NC) weak interaction are of three varieties (types) or flavours: electron, and , muon, and , and tauon, and . The notion of neutrino type or flavour is dynamical: is the neutrino which is produced with or produces an in CC weak interaction processes; is the neutrino which is produced with or produces , and so forth. The flavour of a given neutrino is Lorentz invariant. Among the three different flavour neutrinos and antineutrinos, no two are identical. Correspondingly, the states which describe different flavour neutrinos must be orthogonal (within the precision of the current data): , , .

It is also well known from the existing data (all neutrino experiments were done so far with relativistic neutrinos or antineutrinos) that the flavour neutrinos (antineutrinos ) are always produced in weak interaction processes in a state that is predominantly left handed (LH) (right handed (RH)). To account for this fact, and are described in the Standard Model (SM) by a chiral LH flavour neutrino field , . For massless , the state of (), which the field annihilates (creates), is with helicity (−1/2) (helicity +1/2). If has a nonzero mass , the state of () is a linear superposition of the helicity (−1/2) and (+1/2) states, but the helicity +1/2 state (helicity (−1/2) state) enters into the superposition with a coefficient , being the neutrino energy, and thus is strongly suppressed. Together with the LH charged lepton field , forms an SU doublet in the Standard Model. In the absence of neutrino mixing and zero neutrino masses, and can be assigned one unit of the additive lepton charge and the three charges , , are conserved by the weak interaction.

At present there is no compelling evidence for the existence of states of relativistic neutrinos (antineutrinos), which are predominantly right handed, (left handed, ). If RH neutrinos and LH antineutrinos exist, their interaction with matter should be much weaker than the weak interaction of the flavour LH neutrinos and RH antineutrinos ; that is, () should be “sterile” or “inert” neutrinos (antineutrinos) [1]. In the formalism of the Standard Model, the sterile and can be described by singlet RH neutrino fields . In this case, and will have no gauge interactions, that is, will not couple to the weak and bosons. If present in an extension of the Standard Model (even in the minimal one), the RH neutrinos can play a crucial role (i) in the generation of neutrino masses and mixing, (ii) in understanding the remarkable disparity between the magnitudes of neutrino masses and the masses of the charged leptons and quarks, and (iii) in the generation of the observed matter-antimatter asymmetry of the Universe (via the leptogenesis mechanism [2, 3]; see also, e.g., [4, 5]). In this scenario which is based on the see-saw theory [6–9], there is a link between the generation of neutrino masses and the generation of the baryon asymmetry of the Universe. The simplest hypothesis (based on symmetry considerations) is that to each LH flavour neutrino field there corresponds an RH neutrino field , , although schemes with less (more) than three RH neutrinos are also being considered (see, e.g., [10]).

The experiments with solar, atmospheric, reactor, and accelerator neutrinos (see [11] and the references quoted therein) have provided compelling evidences for flavour neutrino oscillations [1, 12–14]—transitions in flight between the different flavour neutrinos , , (antineutrinos , , ), caused by nonzero neutrino masses and neutrino mixing. As a consequence of the results of these experiments, the existence of oscillations of the solar , atmospheric and , accelerator (at , , and , with being the distance traveled by the neutrinos), and reactor (at and ), was firmly established. The data imply the presence of neutrino mixing in the weak charged lepton current:
where are the flavour neutrino fields, is the left-handed (LH) component of the field of the neutrino having a mass , and is a unitary matrix—the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [1, 12–14], . All compelling neutrino oscillation data can be described assuming 3-neutrino mixing in vacuum, . The number of massive neutrinos can, in general, be bigger than 3 if, for example, there exist right-handed (RH) sterile neutrinos [1] and they mix with the LH flavour neutrinos. It follows from the current data that at least 3 of the neutrinos , say , , , must be light, , and must have different masses, . At present there is no compelling experimental evidence for the existence of more than 3 light neutrinos. Certain neutrino oscillation data exhibit anomalies that could be interpreted as being due to the existence of one or two additional (sterile) neutrinos with mass in the eV range, which have a relatively small mixing *~*0.1 with the active flavour neutrinos (see, e.g., [15] and the references quoted therein).

In the case of 3 light neutrinos on which we will concentrate on in this review, the neutrino mixing matrix can be parametrised by 3 angles and, depending on whether the massive neutrinos are Dirac or Majorana [16] particles, by 1 or 3 CP violation (CPV) phases [17–20]: where and are the two Majorana CPV phases and is a CKM-like matrix containing the Dirac CPV phase : In (3), , , the angles and the Dirac phase lie in the intervals and , and, in general, , [21, 22]. If CP invariance holds, we have , and [20, 23–25], , .

Thus, in the case of massive Dirac neutrinos, the neutrino mixing matrix is similar, in what concerns the number of mixing angles and CPV phases, to the CKM quark mixing matrix. The presence of two additional physical CPV phases in if are Majorana particles is a consequence of the special properties of the latter (see, e.g., [17, 26]). On the basis of the existing neutrino data it is impossible to determine whether the massive neutrinos are Dirac or Majorana fermions.

The neutrino oscillation probabilities depend, in general, on the neutrino energy , the source-detector distance, , on the elements of , and, for relativistic neutrinos used in all neutrino experiments performed so far, on the neutrino mass squared differences , (see, e.g., [11, 26]). In the case of 3-neutrino mixing there are only two independent neutrino mass squared differences, say and . The numbering of massive neutrinos is arbitrary. We will employ here the widely used convention of numbering of which allows to associate with the smallest mixing angle in the PMNS matrix , and , , and , , with the parameters which drive, respectively, the solar (), and the dominant atmospheric (and ) (and accelerator ) oscillations. Under the assumption of CPT invariance, which we will suppose to hold throughout this article, and drive also the reactor oscillations at km (see, e.g., [11]). In this convention , , and, depending on , we have either or (see further). In the case of (), the neutrino mass squared difference , as it follows from the data to be discussed below, is much smaller than , . This implies that in each of the two cases and we have . The angles and are sometimes called “solar” and “atmospheric” neutrino mixing angles and are often denoted as and , while and are sometimes referred to as the “solar” and “atmospheric” neutrino mass squared differences and correspondingly are denoted as , .

Before continuing we would like to note that the preceding discussion is to a large extent based on parts of the text of the review article [11].

The neutrino oscillation data, accumulated over many years, allowed to determine the parameters which drive the solar, reactor, atmospheric, and accelerator neutrino oscillations, , , and , with a rather high precision. Furthermore, there were spectacular developments in the period since June 2011 in what concerns the angle (see, e.g., [11]). They culminated in March of 2012 in a high precision determination of in the Daya Bay experiment with reactor [27, 28]: where we have quoted the latest result of the Daya Bay experiment published in [28]. Subsequently the RENO [29], Double Chooz [30], and T2K [31] (see also [32]) experiments reported, respectively, , , and evidences for a nonzero value of , compatible with the Day Bay result.

A global analysis of the latest neutrino oscillation data presented at the Neutrino 2012 International Conference, held in June of 2012 in Kyoto, Japan, was performed in [33]. We give below the best fit values of , , , , and , obtained in [33]: where the values (the values in brackets) correspond to (). The uncertainties and the ranges of the neutrino oscillation parameters found in [33] are given in Table 1 (note that we have quoted the value of in (5), while the mass squared difference determined in [33] is ()).

A few comments are in order. We have , as was indicated earlier. The existing data do not allow to determine the sign of . As we will discuss further, the two possible signs correspond to two different basic types of neutrino mass spectrum. Maximal solar neutrino mixing, that is, , is ruled out at more than 6 by the data. Correspondingly, one has at . The results quoted in (6) imply that is close to (but can be different from) , and that . Thus, the pattern of neutrino mixing is drastically different from the pattern of quark mixing. As we have noticed earlier, the neutrino oscillations experiments are sensitive only to neutrino mass squared differences , , and cannot give information on the absolute values of the neutrino masses, that is, on the absolute neutrino mass scale. They are insensitive also to the nature-Dirac or Majorana, of massive neutrinos and, correspondingly, to the Majorana CPV phases present in the PMNS matrix [17, 34].

After the successful measurement of , the determination of the absolute neutrino mass scale, of the type of the neutrino mass spectrum, of the nature-Dirac or Majorana, of massive neutrinos, and getting information about the status of CP violation in the lepton sector, are the most pressing and challenging problems and the highest priority goals of the research in the field of neutrino physics.

As was already indicated above, the presently available data do not permit to determine the sign of . In the case of 3-neutrino mixing, the two possible signs of correspond to two types of neutrino mass spectrum. In the widely used convention of numbering the neutrinos with definite mass employed by us, the two spectra read:(i)*spectrum with normal ordering (NO)*: , , , ;(ii)*spectrum with inverted ordering (IO)*: , , , , .

Depending on the value of the lightest neutrino mass, , the neutrino mass spectrum can be(a)*normal hierarchical (NH)*: , eV, eV; or(b)*inverted hierarchical (IH)*: , with eV; or(c)*quasidegenerate (QD)*: , , .

The type of neutrino mass spectrum (hierarchy), that is, the sign of , can be determined (i) using data from neutrino oscillation experiments at accelerators (NOA, T2K, etc.) (see, e.g., [35]), (ii) in the experiments studying the oscillations of atmospheric neutrinos (see, e.g., [36–39]), and (iii) in experiments with reactor antineutrinos [40–47]. The relatively large value of is a favorable factor for the determination in these experiments. If neutrinos with definite mass are Majorana particles, information about the can be obtained also by measuring the effective neutrino Majorana mass in neutrinoless double -decay experiments [48, 49].

More specifically, in the cases (i) and (ii), the can be determined by studying oscillations of neutrinos and antineutrinos, say, and , in which matter effects are sufficiently large. This can be done in long base-line oscillation experiments (see, e.g., [35]). For and , information on might be obtained in atmospheric neutrino experiments by investigating the effects of the subdominant transitions and of atmospheric neutrinos which traverse the Earth (for a detailed discussion see, e.g., [36–39]). For (*or *) crossing the Earth core, new type of resonance-like enhancement of the indicated transitions takes place due to the (*Earth*)* mantle-core constructive interference effect *(*neutrino oscillation length resonance *(*NOLR*)) [50] (see also [51]). As a consequence of this effect, the corresponding (*or *) transition probabilities can be maximal [52–54] (for the precise conditions of the mantle-core (NOLR) enhancement see [50, 52–54]). It should be noted that the Earth mantle-core (NOLR) enhancement of neutrino transitions differs [50] from the MSW one. It also differs [50, 52–54] from the mechanisms of enhancement discussed, for example, in the articles [55, 56]: the conditions of enhancement considered in [55, 56] cannot be realised for the or transitions of the Earth core crossing neutrinos. For , the neutrino transitions are enhanced, while for the enhancement of antineutrino transitions takes place [50] (see also [51–54, 57]), which might allow to determine . Determining the type of neutrino mass spectrum is crucial for understanding the origin of neutrino masses and mixing as well.

All possible types of neutrino mass spectrum we have discussed above are compatible with the existing constraints on the absolute scale of neutrino masses . Information about the absolute neutrino mass scale can be obtained by measuring the spectrum of electrons near the end point in H -decay experiments [58–60] and from cosmological and astrophysical data (see, e.g., [61]). The most stringent upper bound on the mass was obtained in the Troitzk [62] experiment (see also [63]):
We have in the case of quasidegenerate (QD) spectrum. The KATRIN experiment [63], which is under preparation, is planned to reach sensitivity of ; that is, it will probe the region of the QD spectrum. Information on the type of neutrino mass spectrum can also be obtained in -decay experiments having a sensitivity to neutrino masses *~* eV [64] (i.e., by a factor of *~*4 better sensitivity than that of the KATRIN experiment [63]). Reaching the indicated sensitivity in electromagnetic spectrometer -decay experiments of the type of KATRIN does not seem feasible at present. The cosmic microwave background (CMB) data of the WMAP experiment, combined with supernovae data and data on galaxy clustering can be used to derive an upper limit on the sum of neutrinos masses (see, e.g., [61]). Depending on the model complexity and the input data used one obtains [65] eV, 95% C.L. Data on weak lensing of galaxies, combined with data from the WMAP and PLANCK experiments, may allow to be determined with an uncertainty of eV [66, 67].

Thus, the data on the absolute scale of neutrino masses imply that neutrino masses are much smaller than the masses of the charged leptons and quarks. If we take as an indicative upper limit , , we have It is natural to suppose that the remarkable smallness of neutrino masses is related to the existence of a new fundamental mass scale in particle physics, and thus to new physics beyond that predicted by the Standard Model. A comprehensive theory of the neutrino masses and mixing should be able to explain the indicated enormous disparity between the neutrino masses and the masses of the charged leptons and quarks.

At present no experimental information on the Dirac and Majorana CPV phases in the neutrino mixing matrix is available. Therefore the status of the CP symmetry in the lepton sector is unknown. The importance of getting information about the Dirac and Majorana CPV phases in the neutrino mixing matrix stems, in particular, from the possibility that these phases play a fundamental role in the generation of the observed baryon asymmetry of the Universe. More specifically, the CP violation necessary for the generation of the baryon asymmetry within the “flavoured” leptogenesis scenario [68–70] can be due exclusively to the Dirac and/or Majorana CPV phases in the PMNS matrix [71, 72] and thus can be directly related to the low energy CP-violation in the lepton sector. If the requisite CP violation is due to the Dirac phases , a necessary condition for a successful (flavoured) leptogenesis is that [72], which is comfortably compatible with the Daya Bay result, (4).

With , the Dirac phase can generate CP violating effects in neutrino oscillations [73] (see also [17, 74]), that is, a difference between the probabilities of and oscillations in vacuum: , . The magnitude of the CP violating effects of interest is determined [75] by the rephasing invariant associated with the Dirac CPV phase in . It is analogous to the rephasing invariant associated with the Dirac CPV phase in the CKM quark mixing matrix [76, 77]. In the “standard” parametrisation of the PMNS neutrino mixing matrix, (2)-(3), we have Thus, given the fact that , , and have been determined experimentally with a relatively good precision, the size of CP violation effects in neutrino oscillations depends essentially only on the magnitude of the currently unknown value of the Dirac phase . The current data imply , where we have used (9) and the ranges of , , and given in Table 1. Data on the Dirac phase will be obtained in the long baseline neutrino oscillation experiments T2K, NOA, and other (see, e.g., [78]). Testing the possibility of Dirac CP violation in the lepton sector is one of the major goals of the next generation of neutrino oscillation experiments (see, e.g., [35, 78]). Measuring the magnitude of CP violation effects in neutrino oscillations is at present also the only known feasible method of determining the value of the phase (see, e.g., [79]).

If are Majorana fermions, getting experimental information about the Majorana CPV phases in the neutrino mixing matrix will be remarkably difficult [80–86]. As we will discuss further, the Majorana phases of the PMNS matrix play important role in the phenomenology of neutrinoless double-beta- decay—the process whose existence is related to the Majorana nature of massive neutrinos [87]: . The phases can affect significantly the predictions for the rates of the (LFV) decays , , and so forth, in a large class of supersymmetric theories incorporating the see-saw mechanism [88, 89]. As was mentioned earlier, the Majorana phase(s) in the PMNS matrix can be the leptogenesis CP violating parameter(s) at the origin of the baryon asymmetry of the Universe [21, 22, 71, 72].

Establishing whether the neutrinos with definite mass are Dirac fermions possessing distinct antiparticles, or Majorana fermions, that is, spin particles that are identical with their antiparticles, is of fundamental importance for understanding the origin of neutrino masses and mixing and the underlying symmetries of particle interactions. Let us recall that the neutrinos with definite mass will be Dirac fermions if particle interactions conserve some additive lepton number, for example, the total lepton charge . If no lepton charge is conserved, the neutrinos will be Majorana fermions (see, e.g., [26]). The massive neutrinos are predicted to be of Majorana nature by the see-saw mechanism of neutrino mass generation [6–9], which also provides an attractive explanation of the smallness of neutrino masses and, through the leptogenesis theory [2, 3], of the observed baryon asymmetry of the Universe. The observed patterns of neutrino mixing and of neutrino mass squared differences driving the solar and the dominant atmospheric neutrino oscillations can be related to Majorana massive neutrinos and the existence of an *approximate* symmetry in the lepton sector corresponding to the conservation of the *nonstandard* lepton charge [90]. They can also be associated with the existence of *approximate* discrete symmetry (or symmetries) of the particle interactions (see, e.g., [91–94]). Determining the nature (Dirac or Majorana) of massive neutrinos is one of the fundamental and most challenging problems in the future studies of neutrino mixing [11].

#### 2. The Nature of Massive Neutrinos

##### 2.1. Majorana versus Dirac Massive Neutrinos (Particles)

The properties of Majorana particles (fields) are very different from those of Dirac particles (fields). A massive Majorana neutrino (or Majorana spin 1/2 particle) with mass can be described in local quantum field theory which is used to construct, for example, the Standard Model, by 4-component complex spin 1/2 field which satisfies the Dirac equation and the Majorana condition:
where is the charge conjugation matrix, (, ), and is, in general, an unphysical phase. The Majorana condition is invariant under *proper* Lorentz transformations. It reduces by a factor of 2 the number of independent components in .

The condition (10) is invariant with respect to global gauge transformations of the field carrying a charge , , only if . As a result, (i) cannot carry nonzero additive quantum numbers (lepton charge, etc.), and (ii) the field cannot “absorb” phases. Thus, describes 2 spin states of a spin 1/2, *absolutely neutral particle*, which is identical with its antiparticle, . As is well known, spin Dirac particles can carry nonzero charges: the charged leptons and quarks, for instance, carry nonzero electric charges.

Owing to the fact that the Majorana (neutrino) fields cannot absorb phases, the neutrino mixing matrix contains in the general case of charged leptons and mixing of massive Majorana neutrinos , altogether CPV phases [17]. In the case of mixing of massive Dirac neutrinos, the number of CPV phases in , as is well known, is Thus, if are Majorana particles, contains the following number of additional Majorana CP violation phases: . In the case of charged leptons and massive Majorana neutrinos, the PMNS matrix can be cast in the form [17] where the matrix contains the Dirac CP violation phases, while is a diagonal matrix with the additional Majorana CP violation phases , As will be discussed further, the Majorana phases will conserve CP if [23–25] , . In this case and have a simple physical interpretation: these are the relative CP-parities of the Majorana neutrinos and and of and , respectively.

It follows from the preceding discussion that the mixing of massive Majorana neutrinos differs, in what concerns the number of CPV phases, from the mixing of massive Dirac neutrinos. For of interest, we have one Dirac and two Majorana CPV phases in , which is consistent with the expression of given in (2). If , there is one Majorana CPV phase and no Dirac CPV phases in . Correspondingly, in contrast to the Dirac case, there can exist CP violating effects even in the system of two mixed massive Majorana neutrinos (particles).

The Majorana phases do not enter into the expressions of the probabilities of oscillations involving the flavour neutrinos and antineutrinos [17, 34], and . Indeed, the probability to find neutrino (antineutrino ) at time if a neutrino (antineutrino ) has been produced at time and it had traveled a distance in vacuum is given by (see, e.g., [11, 26]) where and are the energy and momentum of the neutrino . It is easy to show, using the expression for in (13), that and do not depend on the Majorana phases present in since The same result holds when the neutrino oscillations take place in matter [34].

If -invariance holds, Majorana neutrinos (particles) have definite -parity : where , and is the unitary CP-transformation operator. In contrast, Dirac particles do not have a definite -parity—a Dirac field transforms as follows under the CP-symmetry operation: being an unphysical phase factor. In the case of invariance, the -parities of massive Majorana fermions (neutrinos) can play important role in processes involving real of virtual Majorana particles (see, e.g., [26, 95]).

Using (18) and (17) and the transformation of the boson field under the CP-symmetry operation, where is an unphysical phase, one can derive the constraints on the neutrino mixing matrix following from the requirement of CP-invariance of the leptonic CC weak interaction Lagrangian, (1). In the case of massive Dirac neutrinos we obtain , , . Setting the product of unphysical phases , one obtains the well-known result: In the case of massive Majorana neutrinos we obtain using (10), (17), (18), and (19): . It is convenient now to set , , and . In this (commonly used by us) convention we get [26] Thus, in the convention used the elements of the PMNS matrix can be either real or purely imaginary if are Majorana fermions. Applying the above conditions to, for example, , , and elements of the PMNS matrix (2) we obtain the CP conserving values of the phases , , and , respectively: , , , .

One can obtain in a similar way the CP-invariance constraint on the matrix of neutrino Yukawa couplings, , which plays a fundamental role in the leptogenesis scenario of baryon asymmetry generation, based on the (type I) see-saw mechanism of generation of neutrino masses [2–5, 79]: Here is the field of a heavy right-handed (RH) sterile Majorana neutrino with mass , denotes the Standard Model left-handed (LH) lepton doublet field of flavour , and is the Standard Model Higgs doublet field whose neutral component has a vacuum expectation value GeV. The term includes all the necessary ingredients of the see-saw mechanism. Assuming the existence of two heavy Majorana neutrinos, that is, taking in (22), and adding the term to the Standard Model Lagrangian, we obtain the minimal extension of the Standard Model in which the neutrinos have masses and mix and the leptogenesis can be realised. In the leptogenesis formalism it is often convenient to use the so-called orthogonal parametrisation of the matrix of neutrino Yukawa couplings [96]: where is, in general, a complex orthogonal matrix, . The CP violation necessary for the generation of the baryon asymmetry of the Universe is provided in the leptogenesis scenario of interest by the matrix of neutrino Yukawa couplings (see, e.g., [4, 5, 79]). It follows from (23) that it can be provided either by the neutrino mixing matrix , or by the matrix , or else by both the matrices and . It is therefore important to derive the conditions under which , , and respect the CP symmetry. For the PMNS matrix these conditions are given in (21). For the matrices and in the convention in which (i) satisfy the Majorana condition with a phase equal to 1 (i.e., ), (ii) and , and being the unphysical phase factors which appear in the CP-transformations of the LH lepton doublet and Higgs doublet fields and , respectively (this convention is similar to, and consistent with, the convention about the unphysical phases we have used to derive the CP-invariance constraints on the elements of the PMNS matrix ), they read [72] where is the CP-parity of . Thus, in the case of CP invariance also the elements of and can be real or purely imaginary. Note that, as it follows from (21) and (24), given which elements are real and which are purely imaginary of any two of the three matrices , and , determines (in the convention we are using and if CP invariance holds), which elements are real or purely imaginary in the third matrix. If, for instance, is purely imaginary () and is real (), then must be purely imaginary. Thus, in the example we are considering, a real would signal that the CP symmetry is broken [72].

The currents formed by Majorana fields have special properties, which make them also quite different from the currents formed by Dirac fields. In particular, it follows from the Majorana condition that the following currents of the Majorana field are identically equal to zero (see, e.g., [26]):
Equations (25) and (26) imply that Majorana fermions (neutrinos) cannot have nonzero charges and intrinsic magnetic and electric dipole moments, respectively. A Dirac spin particle can have nontrivial charges, as we have already discussed, and nonzero intrinsic magnetic moment (the electron and the muon, e.g., have it). If CP invariance does not hold, Dirac fermions can have also nonzero electric dipole moments. Equations (26) imply also that the Majorana particles (neutrinos) cannot couple to a real photon. The axial current of a Majorana fermion, . Correspondingly, can have an effective coupling to a *virtual* photon via the *anapole momentum* term, which has the following form in momentum space:
where is the momentum of the virtual photon and is the anapole form factor of . The fact that the vector current of is zero while the axial current is nonzero has important implications in the calculations of the relic density of the lightest and stable neutralino, which is a Majorana particle and the dark matter candidate in many supersymmetric (SUSY) extensions of the Standard Model [97].

In certain cases (e.g., in theories with a keV mass Majorana neutrino (see, e.g., [15]), in the TeV scale type I see-saw model (see, e.g., [98]), in SUSY extensions of the Standard Model) one can have effective interactions involving two different massive Majorana fermions (neutrinos), say and . We will consider two examples. The first is an effective interaction with the photon field, which can be written as where and are, in general, complex constants, , being the 4-vector potential of the photon field. Using the Majorana conditions for and in the convention in which the phases , it is not difficult to show that the constants and enter into the expression for in the form: , , that is, is purely imaginary and is real. In the case of , the current has to be Hermitian, which implies that should be real while should be purely imaginary. Combined with constraints on and we have just obtained, this leads to , which is consistent with (26). In the case of CP invariance of , the constants () and () should satisfy Thus, if , that is, if and possess the same CP-parity, and (and ) can be different from zero. If , that is, if and have opposite CP-parities, and (and ) can be different from zero. If CP invariance does not hold, we can have both and ( and ).

As a second example we will consider effective interaction of and with a vector field (current), which for concreteness will be assumed to be the -boson field of the Standard Model: Here and are, in general, complex constants. Using the Majorana conditions for and with , one can easily show that has to be purely imaginary, while has to be real. In the case of , the hermiticity of the current implies that both and have to be real. This, together with constraints on and just derived, leads to , which is consistent with the result given in (25). The requirement of CP invariance of , as can be shown, leads to (): Thus, we find, similarly to the case considered above, that if and posses the same CP-parity (), and can be different from zero; if and have opposite CP-parities (), while can be different from zero. If CP invariance does not hold, we can have both and .

These results have important implications, in particular, for the phenomenology of the heavy Majorana neutrinos in the TeV scale (type I) see-saw models, for the neutralino phenomenology in SUSY extensions of the Standard Model, in which the neutralinos are Majorana particles, and more specifically for the processes , (), , where and are, for instance, two neutralinos of, for example, the minimal SUSY extension of the Standard Model (see, e.g., [95, 99]).

Finally, if is a Dirac field and we define the standard propagator of as
one has
In contrast, a Majorana neutrino field has, in addition to the standard propagator
two nontrivial *nonstandard* (*Majorana*) propagators
This result implies that if in (1) are massive Majorana neutrinos, -decay can proceed by exchange of virtual neutrinos since . The Majorana propagators play a crucial role in the calculation of the baryon asymmetry of the Universe in the leptogenesis scenario of the asymmetry generation (see, e.g., [4, 5, 79]).

##### 2.2. Generating Dirac and Majorana Massive Neutrinos

The type of massive neutrinos in a given theory is determined by the type of the (effective) mass term neutrinos have, more precisely, by the symmetries of and of the total Lagrangian of the theory. A fermion mass term is bilinear in the fermion fields which is invariant under the proper Lorentz transformations.

Massive Dirac neutrinos arise in theories in which the neutrino mass term conserves some additive quantum number that could be, for example, the (total) lepton charge , which is conserved also by the total Lagrangian of the theory. A well-known example is the Dirac mass term, which can arise in the minimally extended Standard Model to include three RH neutrino fields , as singlets: where is a , in general complex, matrix. The term can be generated after the spontaneous breaking of the Standard Model gauge symmetry by an invariant Yukawa coupling of the lepton doublet, Higgs doublet, and the RH neutrino fields [100]: If the nondiagonal elements of are different from zero, , , the individual lepton charges , , will not be conserved. Nevertheless, the total lepton charge is conserved by . As in the case of the charged lepton and quark mass matrices generated via the spontaneous electroweak symmetry breaking by Yukawa type terms in the SM Lagrangian, is diagonalised by a biunitary transformation: , where and are unitary matrices. If the mass term in (36) is written in the basis in which the charged lepton mass matrix is diagonal, coincides with the PMNS matrix, . The neutrinos with definite mass are Dirac particles: their fields do not satisfy the Majorana condition, . Although the scheme we are considering is phenomenologically viable (it does not contain a candidate for a dark matter particle though), it does not provide an insight of why the neutrino masses are much smaller than the charged fermion masses. The only observable “new physics” is that related to the neutrino masses and mixing: apart from the neutrino masses and mixing themselves, this is the phenomenon of neutrino oscillations [100].

Indeed, given the fact that the lepton charges , , are not conserved, processes like decay, decay, decay, and so forth are allowed. However, the rates of these processes are suppressed by the factor [100] , , GeV being the -mass and , for the decay, and so forth, and are unobservably small. For instance, for the decay branching ratio we have [100] where we have used the best fit values of the neutrino oscillation parameters given in (5) and (6) and the two values correspond to and 0. The current experimental upper limit reads [101] . Thus, although the predicted branching ratio , its value is approximately by 43 orders of magnitude smaller than the sensitivity reached in the experiments searching for the decay, which renders it unobservable in practice.

As was emphasised already, massive Majorana neutrinos appear in theories with no conserved additive quantum number, and more specifically, in which the total lepton charge is not conserved and changes by two units. In the absence of RH singlet neutrino fields in the theory, the flavour neutrinos and antineutrinos and , can have a mass term of the so-called Majorana type:
where is a , in general complex matrix. In the case when all elements of are nonzero, , neither the individual lepton charges nor the total lepton charge is conserved: ., . As it is possible to show, owing to the fact that are fermion (anticommuting) fields, the matrix has to be symmetric (see, e.g., [26]): . A complex symmetric matrix is diagonalised by the *congruent transformation*:
where is a unitary matrix. If is written in the basis in which the charged lepton mass matrix is diagonal, coincides with the PMNS matrix: . The fields of neutrinos with definite mass are expressed in terms of and :
They satisfy the Majorana condition with , as (43) shows.

The Majorana mass term (40) for the LH flavour neutrino fields can be generated(i)effectively after the electroweak symmetry (EWS) breaking in the type I see-saw models [6–9],(ii)effectively after the EWS breaking in the type III see-saw models [102],(iii)directly as a result of the EWS breaking by an triplet Higgs field which carries two units of the weak hypercharge and couples in an invariant manner to two lepton doublets [18, 103, 104] (the Higgs triplet model (HTM) sometimes called also “type II see-saw model”),(iv)as a one-loop correction to a Lagrangian which does not contain a neutrino mass term [105, 106] (see also [107]),(v)as a two-loop correction in a theory where the neutrino masses are zero at tree and one-loop levels [108, 109] (see also [107]),(vi)as a three-loop correction in a theory in which the neutrino masses are zero at tree, one-loop and two-loop levels [107].

In all three types of see-saw models, for instance, the neutrino masses can be generated at the EWS breaking scale and in this case the models predict rich beyond the Standard Model physics at the TeV scale, some of which can be probed at the LHC (see, e.g., [110] and further). We will consider briefly below the neutrino mass generation in the type I see-saw and the Higgs triplet models.

In a theory in which the singlet RH neutrino fields , , are present (we consider in the present paper the case of three RH sterile neutrinos, but schemes with less than 3 and more than 3 sterile neutrinos are also discussed in the literature, see, e.g., [10, 15]), the most general neutrino mass Lagrangian contains the Dirac mass term (36), the Majorana mass term for the LH flavour neutrino fields (40), and a Majorana mass term for the RH neutrino fields [111]: where and , , and are , in general complex matrices. By a simple rearrangement of the neutrino fields this mass term can be cast in the form of a Majorana mass term which is then diagonalised with the help of the congruent transformation [26]. In this case there are six Majorana mass eigenstate neutrinos; that is, the flavour neutrino fields are linear combinations of the fields of six Majorana neutrinos with definite mass. The neutrino mixing matrix in (1) is a block of a unitary matrix.

The Dirac-Majorana mass term is at the basis of the type I see-saw mechanism of generation of the neutrino masses and appears in many grand unified theories (GUTs) (see, e.g., [26] for further details). In the see-saw models, some of the six massive Majorana neutrinos typically are too heavy to be produced in the weak processes in which the initial states of the flavour neutrinos and antineutrinos and , used in the neutrino oscillation experiments, are being formed. As a consequence, the states of and will be coherent superpositions only of the states of the light massive neutrinos , and the elements of the neutrino mixing matrix , which are determined in experiments studying the oscillations of and , will exhibit deviations from unitarity. These deviations can be relatively large and can have observable effects in the TeV scale see-saw models, in which the heavy Majorana neutrinos have masses in the *~* GeV range (see, e.g., [112]).

If after the diagonalisation of more than three neutrinos will turn out to be light, that is, to have masses *~*1 eV or smaller, active-sterile neutrino oscillations can take place (see, e.g., [15, 26]): an LH (RH) flavour neutrino (antineutrino ) can undergo transitions into LH sterile antineutrino(s) (RH sterile neutrino(s) ). As a consequence of this type of oscillations, one would observe a “disappearance” of, for example, and/or ( and/or ) on the way from the source to the detector.

We would like to discuss next the implications of CP invariance for the neutrino Majorana mass matrix, (40). In the convention we have used to derive (24), in which the unphysical phase factor in the CP transformation of the lepton doublet field , and thus of , , the requirement of CP invariance leads to the reality condition for : Thus, is real and symmetric and therefore is diagonalised by an orthogonal transformation; that is, if CP invariance holds, the matrix in (41) is an orthogonal matrix. The nonzero eigenvalues of a real symmetric matrix can be positive or negative (the absolute value of the difference between the number of positive and number of negative eigenvalues of a real symmetric matrix is an invariant of the matrix with respect to transformations , where is a real matrix which has an inverse). Consequently, in (41) in general has the form Let us denote the neutrino field which has a mass by . According to (43), the field satisfies the Majorana condition: . One can work with the fields remembering that some of them have a negative mass. It is not difficult to show that the CP-parity of the fields is , . The physical meaning of the signs of the masses of the Majorana neutrinos becomes clear if we change to a “basis” of neutrino fields which have positive masses . This can be done, for example, by introducing the fields [26]: As it is not difficult to show, if has a mass , CP-parity and satisfies the Majorana condition , the field possesses a mass , CP-parity and satisfies the Majorana condition : Thus, in the case of CP invariance, the signs of the nonzero eigenvalues of the neutrino Majorana mass matrix determine the CP-parities of the corresponding positive mass Majorana (mass eigenstate) neutrinos (for further discussion of the properties of massive Majorana neutrinos (fermions) and their couplings, see, e.g., [26]).

##### 2.3. A Brief Historical Detour

It is interesting to note that Pontecorvo in his seminal article on neutrino oscillations [13], which was published in 1958 when only one type of neutrino and antineutrino was known, assumed that the state of the neutrino , emitted in weak interaction processes, is a linear superposition of the states of two Majorana neutrinos and which have different masses, , opposite CP-parities, and are maximally mixed, while the state of the corresponding antineutrino is just the orthogonal superposition of the states of and : Thus, the oscillations are between the neutrino and the antineutrino , in full analogy with the oscillations. From contemporary point of view, Pontecorvo proposed active-sterile neutrino oscillations with maximal mixing and massive Majorana neutrinos. To our knowledge, the article [13] was also the first in which fermion mixing in the weak interaction Lagrangian was introduced.

The article of Maki et al. [14] was inspired, in part, by the discovery of the second type of neutrino—the muon neutrino, in 1962 at Brookhaven. These authors considered a composite model of elementary particles in which the electron and muon neutrino states are superpositions of the states of *composite Dirac neutrinos * and which have different masses, :
where is the neutrino mixing angle. The model proposed in [14] has lepton-hadron symmetry built in and as consequence of this symmetry the neutrino mixing angle coincides with what we call today the Cabibbo angle (the article by Maki et al. [14] appeared before the article by Cabibbo [113] in which the ‘‘Cabibbo angle’’ was introduced and the hadron phenomenology related to this angle was discussed, but after the article by Gell-Mann and Lévy [114] in which was also introduced (by the way, in a footnote)). The authors of [14] discuss the possibility of oscillations, which they called “virtual transmutations.”

In an article [115] by Katayama et al., published in 1962 somewhat earlier than [14], the authors also introduce two-neutrino mixing. However, this is done purely for model construction purposes and does not have any physical consequences since the neutrinos in the model constructed in [115] are massless particles.

In 1967 Pontecorvo independently considered the possibility of oscillations in the article [1], in which the notion of a “sterile” or “inert” neutrino was introduced. Later in 1969, Gribov and Pontecorvo [116] introduced for the first time a Majorana mass term for the LH flavour neutrinos and , the diagonalisation of which leads to two Majorana neutrinos with definite but different masses , , and two-neutrino mixing with an arbitrary mixing angle : This was the first modern treatment of the problem of neutrino mixing which anticipated the way this problem is addressed in gauge theories of electroweak interactions and in grand unified theories (GUTs). In the same article for the first time the analytic expression for the probability of oscillations was also derived.

##### 2.4. Models of Neutrino Mass Generation: Two Examples

*Type I See-Saw Model. *A natural explanation of the smallness of neutrino masses is provided by the type I see-saw mechanism of neutrino mass generation [6–9]. Integral part of this rather simple mechanism are the RH neutrinos (RH neutrino fields ). The latter are assumed to possess a Majorana mass term as well as Yukawa type coupling with the Standard Model lepton and Higgs doublets, and , given in (37). In the basis in which the Majorana mass matrix of RH neutrinos is diagonal, we have
where we have combined the expressions given in (22). When the electroweak symmetry is broken spontaneously, the neutrino Yukawa coupling generates a Dirac mass term: , with , GeV being the Higgs doublet v.e.v. In the case when the elements of are much smaller than , , , , the interplay between the Dirac mass term and the mass term of the heavy (RH) Majorana neutrinos generates an effective Majorana mass (term) for the LH flavour neutrinos [6–9]:
In grand unified theories, is typically of the order of the charged fermion masses. In theories, for instance, coincides with the up-quark mass matrix. Taking indicatively , GeV, one obtains GeV, which is close to the scale of unification of the electroweak and strong interactions, GeV. In GUT theories with RH neutrinos one finds that indeed the heavy Majorana neutrinos naturally obtain masses which are by few to several orders of magnitude smaller than (see, e.g., [7, 8]). Thus, the enormous disparity between the neutrino and charged fermion masses is explained effectively in this approach by the huge difference between the electroweak symmetry breaking scale and .

An additional attractive feature of the see-saw scenario under discussion is that the generation and smallness of neutrino masses are related via the leptogenesis mechanism [2, 3] (see also, e.g., [4, 5, 68–70, 79]) to the generation of the baryon asymmetry of the Universe. Indeed, the Yukawa coupling in (52), in general, is not CP conserving. Due to this CP-nonconserving coupling, the heavy Majorana neutrinos undergo, for example, the decays , , which have different rates: . When these decays occur in the Early Universe at temperatures somewhat below the mass of, say, , so that the latter are out of equilibrium with the rest of the particles present at that epoch, CP violating asymmetries in the individual lepton charges and in the total lepton charge of the Universe are generated. These lepton asymmetries are converted into a baryon asymmetry by conserving, but violating, sphaleron processes, which exist in the Standard Model and are effective at temperatures GeV [117]. If the heavy neutrinos have hierarchical spectrum, , the observed baryon asymmetry can be reproduced provided the mass of the lightest one satisfies GeV [118] (in specific type I see-saw models this bound can be lower by a few orders of magnitude, see, e.g., [119]). Thus, in this scenario, the neutrino masses and mixing and the baryon asymmetry have the same origin—the neutrino Yukawa couplings and the existence of (at least two) heavy Majorana neutrinos. Moreover, quantitative studies based on advances in leptogenesis theory [68–70], in which the importance of the flavour effects in the generation of the baryon asymmetry was understood, have shown that the Dirac and/or Majorana phases in the neutrino mixing matrix can provide the CP violation, necessary in leptogenesis for the generation of the observed baryon asymmetry of the Universe [71, 72]. This implies, in particular, that if the CP symmetry is established not to hold in the lepton sector due to the PMNS matrix , at least some fraction (if not all) of the observed baryon asymmetry might be due to the Dirac and/or Majorana CP violation present in the neutrino mixing.

In the see-saw scenario considered, the scale at which the new physics manifests itself, which is set by the scale of masses of the RH neutrinos, can, in principle, have an arbitrary large value, up to the GUT scale of GeV and even beyond, up to the Planck mass. An interesting possibility, which can also be theoretically well motivated (see, e.g., [120, 121]), is to have the new physics at the TeV scale, that is, GeV. Low scale see-saw scenarios usually predict a rich phenomenology at the TeV scale and are constrained by different sets of data, such as, the data on neutrino oscillations, from EW precision tests and on the lepton flavour violating (LFV) processes , , conversion in nuclei. In the case of the TeV scale type I see-saw scenario of interest, the flavour structure of the couplings of the heavy Majorana neutrinos to the charged leptons and the bosons, and to the LH flavour neutrinos and the boson, are essentially determined by the requirement of reproducing the data on the neutrino oscillation parameters [98]. All present experimental constraints on this scenario still allow (i) for the predicted rates of the decay, decay, and conversion in the nuclei to be [122] within the sensitivity range of the currently running MEG experiment on decay [101] planned to probe values of BR, and of the future planned experiments on decay and conversion [123–127], (ii) for an enhancement of the rate of neutrinoless double-beta--) decay [98], which thus can be in the range of sensitivity of the -decay experiments which are taking data or are under preparation (see, e.g., [128]) even when the light Majorana neutrinos possess a normal hierarchical mass spectrum (see further), and (iii) for the possibility of an exotic Higgs decay channel into a light neutrino and a heavy Majorana neutrino with a sizable branching ratio, which can lead to observables effects at the LHC [110] (for further details concerning the low energy phenomenology of the TeV scale type I see-saw model, see, e.g., [98, 120–122]).

Let us add that the role of the experiments searching for lepton flavour violation to test and possibly constrain low scale see-saw models, and more generally, extensions of the Standard Model predicting “new” (lepton flavour violating) physics at the TeV scale, will be significantly strengthened in the next years. Searches for conversion at the planned COMET experiment at KEK [124] and Mu2e experiment at Fermilab [125] aim to reach sensitivity to conversion rates , while, in the longer run, the PRISM/PRIME experiment in KEK [126] and the project-X experiment in Fermilab [127] are being designed to probe values of the conversion rate on , which are smaller by 2 orders of magnitude, [126]. The current upper limit on the conversion rate is [129]. There are also plans to perform a new search for the decay [123], which will probe values of the corresponding branching ratio down to BR, that is, by 3 orders of magnitude smaller than the best current upper limit [130]. Furthermore, searches for tau lepton flavour violation at superB factories aim to reach a sensitivity to BR (see, e.g., [131]).

*The Higgs Triplet Model (HTM). *In its minimal formulation this model includes one additional triplet Higgs field , which has weak hypercharge [18, 103, 104]:
The Lagrangian of the Higgs triplet model which is sometimes called also the “type II see-saw model,” reads (we do not give here, for simplicity, all the quadratic and quartic terms present in the scalar potential (see, e.g., [132])):
where , being the charge conjugation matrix, is the SM Higgs doublet, and is a real parameter characterising the soft explicit breaking of the total lepton charge conservation. We will discuss briefly the low energy version of HTM, where the new physics scale associated with the mass of takes values GeV TeV, which, in principle, can be probed by LHC (see [132, 133] and references quoted therein).

The flavour structure of the Yukawa coupling matrix and the size of the lepton charge soft breaking parameter are related to the light neutrino Majorana mass matrix , which is generated when the neutral component of develops a “small” vev . Indeed, setting and with GeV, from Lagrangian (55) one obtains The matrix of Yukawa couplings is directly related to the PMNS neutrino mixing matrix , which is unitary in this case: An upper limit on can be obtained from considering its effect on the parameter . In the SM, at tree-level, while in the HTM one has The measurement leads to the bound , or GeV (see, e.g., [132]).

For GeV, the model predicts a plethora of beyond the SM physics phenomena (see, e.g., [132, 134–139]), most of which can be probed at the LHC and in the experiments on charged lepton flavour violation, if the Higgs triplet vacuum expectation value is relatively small, roughly eV. As can be shown (see, e.g., [132]), the parameters and are related: for GeV we have , while if , then . Thus, a relatively small value of in the TeV scale HTM implies that has also to be small, and vice versa. A nonzero but relatively small value of can be generated, for example, at higher orders in perturbation theory [140]. The smallness of the neutrino masses is therefore related to the smallness of the vacuum expectation value , which in turn is related to the smallness of the parameter .

Under the conditions specified above one can have testable predictions of the model in low energy experiments, and in particular, in the ongoing MEG and the planned future experiments on the lepton flavour violating processes and (see, e.g., [122]). The HTM has also an extended Higgs sector including neutral, singly charged and doubly charged Higgs particles. The physical singly charged Higgs scalar field (particle) practically coincides with the triplet scalar field , the admixture of the doublet charged scalar field being suppressed by the factor . The singly and doubly charged Higgs scalars and have, in general, different masses [140]: . Both cases and are possible. The TeV scale HTM predicts the existence of rich new physics at LHC as well, associated with the presence of the singly and doubly charged Higgs particles and in the theory (see, e.g., [132, 137–139]).

#### 3. Determining the Nature of Massive Neutrinos

The Majorana nature of massive neutrinos typically manifests itself in the existence of processes in which the total lepton charge changes by two units: , , and so forth. Extensive studies have shown that the only feasible experiments having the potential of establishing the Majorana nature of massive neutrinos at present are the-decay experiments searching for the process (for reviews see, e.g., [26, 128, 141, 142]). The observation of-decay and the measurement of the corresponding half-life with sufficient accuracy not only would be a proof that the total lepton charge is not conserved, but might provide also information (i) on the type of neutrino mass spectrum [48, 49], and (ii) on the absolute scale of neutrino masses (see, e.g., [81]).

The observation of -decay and the measurement of the corresponding half-life with sufficient accuracy, combined with data on the absolute neutrino mass scale, might provide also information on the Majorana phases in [80, 82, 83, 143, 144]. If the neutrino mass spectrum is inverted hierarchical or quasidegenerate, for instance, it would be possible to get information about the phase . However, establishing even in this case that has a CP violating value would be a remarkably challenging problem [83] (see also [84]). Determining experimentally the values of both the Majorana phases and is an exceptionally difficult problem. It requires the knowledge of the type of neutrino mass spectrum and high precision determination of both the absolute neutrino mass scale and of the -decay effective Majorana mass, (see, e.g., [80, 83]).

##### 3.1. Majorana Neutrinos and -Decay

Under the assumptions of 3- mixing, for which we have compelling evidence, of massive neutrinos being Majorana particles and of-decay generated *only by the *(V-A)* charged current weak interaction via the exchange of the three Majorana neutrinos * having masses few MeV, the-decay amplitude of interest has the form (see, e.g., [80, 141, 142]): , where is the corresponding nuclear matrix element (NME) which does not depend on the neutrino mixing parameters, and
is the effective Majorana mass in-decay, , , . In the case of CP-invariance one has or and,
being the relative CP-parity of the Majorana neutrinos and .

It proves convenient to express [145] the three neutrino masses in terms of and , measured in neutrino oscillation experiments, and the absolute neutrino mass scale determined by (for a detailed discussion of the relevant formalism, see, e.g., [26, 80, 141, 142]). In both cases of neutrino mass spectrum with normal and inverted ordering one has (in the convention we use): , . For normal ordering,