Can one ever prove that neutrinos are Dirac particles?

According to the"Black Box"theorem the experimental confirmation of neutrinoless double beta decay ($0 \nu 2 \beta$) would imply that at least one of the neutrinos is a Majorana particle. However, a null $0 \nu 2 \beta$ signal cannot decide the nature of neutrinos, as it can be suppressed even for Majorana neutrinos. In this letter we argue that if the null $0 \nu 2 \beta$ decay signal is accompanied by a $0 \nu 4 \beta$ quadruple beta decay signal, then at least one neutrino should be a Dirac particle. This argument holds irrespective of the underlying processes leading to such decays.

Majorana mass term, since such term would necessarily break the SU (3) C ⊗ U (1) EM gauge symmetry. This implies that, although two-component spinors are indeed fundamental, the requirement that color and electromagnetic charges remain conserved, forces all the quarks and charged leptons to be Dirac particles. On the basis of this argument it has been argued in [5] that, thanks to their complete charge neutrality, only neutrinos can be -and should be -Majorana fermions. However, nature need not follow our theoretical prejudices, so that only experiments can settle whether neutrinos are Dirac or Majorana particles. Thanks to the small neutrino mass m ν and the V-A nature of the weak interaction, discerning the nature of neutrinos from experiments is a formidable task. A basic difference between Dirac and Majorana fermions resides in the CP phases present in their mixing matrices [5]. Indeed the sensitivity to the physical Majorana phases present in neutrino to anti-neutrino oscillations [6] is well below any conceivable test. Likewise, electromagnetic properties of neutrinos [7][8][9] have a hidden dependence on m ν . Indeed, all observables sensitive to the Majorana nature of neutrinos end up being suppressed by a power of m ν . The small scale of the active neutrino masses makes such differences very tiny.
However, there is a potentially feasible process which may settle the issue, namely the neutrinoless double beta decay, which has long been hailed as the ultimate test concerning the nature of neutrinos 1 . Indeed, if 0ν2β decay is ever observed, its amplitude can always 1 In SM extensions there may be feasible complementary probes of lepton number violation at collider energies [10][11][12][13][14]. arXiv:1711.06181v1 [hep-ph] 16 Nov 2017 be "dressed" so as to induce a Majorana mass, ensuring that at least one of the neutrinos is of Majorana type [15], as illustrated in Fig. 1. See Ref. [16,17] Figure 1. The "Black Box" theorem states that a 0ν2β signal ensures that at least one neutrino is Majorana in nature [15].
However, the non-observation of 0ν2β decay so far [18][19][20][21] has raised the intriguing possibility that neutrinos might well be Dirac particles. Several well motivated high-energy completions of the SM do lead to naturally light Dirac-type neutrinos [22][23][24][25]. Alternatively, the absence of a 0ν2β signal is not inconsistent with the Majorana nature of neutrinos, since the decay amplitude may be suppressed as a result of a destructive interference amongst the three active neutrinos, even if they are Majorana type [26,27]. Thus, although the observation of 0ν2β decay would necessarily imply that at least one neutrino species is Majorana in nature, the converse is not true: a negative 0ν2β decay signal does not tell us anything about the nature of neutrinos.
This prompts us to search for processes beyond the simplest 0ν2β decay which can also shed light upon the nature of neutrinos. 2 We will specifically focus on the two lowest 0ν2nβ processes characterized by n = 1, 2, namely, the neutrinoless double beta decay 0ν2β and the neutrinoless quadruple beta decay.
An experimental search for the 0ν4β process has been recently performed by the NEMO-3 collaboration, using 150 Nd [28]. The possible existence of 0ν4β decays has been first suggested in [29], and it is expected to arise in a number of models with family symmetries leading to Dirac neutrinos [30][31][32]. Here we argue that the combination of the 0ν2β and 0ν4β processes may be enough to settle the nature of neutrinos within a very broad class of models.
In order to proceed let us first look at the 0ν2β process and the neutrino mass generation from the symmetry point of view. In the Standard Model the neutrinos are massless and there is an accidental global "classically conserved" U (1) L symmetry in the lepton sector associated to Lepton number for all the leptons in SM 3 . By just adding right handed neutrinos ν iR sequentially to the SM particle content one can give mass to neutrinos without breaking the lepton number symmetry. In such a case neutrinos will necessarily be Dirac particles and the 0ν2nβ; n ≥ 1 decays will all be absent.
We now turn to the cases when this lepton number is broken down to a discrete Z m subgroup (m ≥ 2) which remains conserved. Notice that a U (1) symmetry only admits Z m subgroups, where Z m is a cyclic group of m elements, characterized by the property that if x is a nonidentity group element, then x m+1 ≡ x. The Z m groups only admit one-dimensional irreducible representations, conveniently represented by using the n-th roots of unity, ω = e 2πI m , where ω m = 1. If lepton number is broken to a Z m subgroup (with neutrinos transforming non-trivially under Z m ) by the new physics responsible for neutrino mass generation, then we have two possible cases: If the U (1) L is broken to a Z 2n subgroup, then one can make a further broad classification depending on the charges of neutrinos under the unbroken Z 2n symmetry. For neutrinos transforming nontrivially under any unbroken Z 2n+1 symmetry, they must be Dirac particles. For neutrinos transforming nontrivially under the Z 2n symmetry, they can be Majorana if and only if ν ∼ ω n . For any other transformation neutrinos will be Dirac particles. Thus, from a symmetry point of view, in contrast to popular belief, the Majorana neutrinos are the special ones, emerging only for certain transformation properties under the unbroken residual Z 2n symmetry. Now the simplest Z m group to which the U (1) L can break is Z 2 . This case is special, as it only offers two possibilities for neutrino transformation i.e. ν ∼ +1 or − 1, both of which satisfy Eq. (2) and only allows for Majorana neutrinos. Breaking U (1) L to Z 2 is quite simple, through a Majorana mass term νν arising effectively from new physics, as is the case of Weinberg's dimension 5 operatorL c ΦΦL [33]. Most popular in the literature, this case covers a big chunk of model setups, which typically involve breaking of lepton number to a residual Z 2 symmetry. This also induces a nonzero 0ν2β decay amplitude, as this decay is now allowed by the symmetry. The converse is also true, namely, if the 0ν2β decay process is allowed, it always implies that lepton number is broken and the associated new physics is bound to generate Majorana mass terms 4 . Notice that, since the higher 0ν2nβ beta process are also allowed by the residual Z 2 symmetry, they all will also occur through 'multiples of n" 0ν2β amplitudes as illustrated in Fig.2, for the simplest case of n=2. These higher processes can be intuitively thought of as "multiples" of the basic 0ν2β process, 0ν2nβ ≡ n(0ν2β) and thus we have Γ 0ν2nβ Γ 0ν2β . We now turn to the case of U (1) L broken to higher symmetries, with neutrinos transforming non-trivially under the residual Z m symmetry. Clearly if U (1) L breaks to an Z 2n+1 symmetry, the lowest possible allowed neutrinoless beta decay process will be 0ν(2n + 1)β, where 4 Notice that the Majorana mass term might be generated at the loop level, and need not be the dominant source of 0ν2β decay.
n is a positive integer. But such processes are forbidden, as can be easily seen. Consider, for simplicity 0ν3β. This process would require us to write down a 9-fermion operator, which is of course not possible. 5 . Hence in such cases no neutrinoless beta decay of any order below 0ν2(2n+1)β are possible and neutrinos can only be Dirac particles [34,35].
The more interesting case is when U (1) L breaks to even residual Z 2n symmetries, with n > 1. As already mentioned, in such cases both Dirac and Majorana neutrinos are possible, depending on how they transform under the Z 2n symmetry. Also, irrespective of the Dirac or Majorana nature of neutrinos, if U (1) L breaks to an even residual Z 2n symmetry, there is an associated 0ν2nβ processes allowed by the residual symmetry. However, an important distinction comes for the case of Dirac or Majorana neutrinos. As mentioned above, if neutrinos transform as ω n under the Z 2n symmetry, they must be Majorana particles. Moreover, in this case not only the 0ν2nβ process is allowed, but all other lower dimensional 0ν2n 1 β processes, where n 1 < n is a positive integer, are also allowed. However, if neutrinos are Dirac particles, then for the case of a Z 2n symmetry, it follows that ν ω n . This implies that the lowest process allowed by Z 2n symmetry is 0ν2nβ decays, all other lower dimensional processes being forbidden by the unbroken residual Z 2n symmetry. This is better illustrated by the simple example of U (1) L breaking to a Z 4 residual symmetry, called quarticity. Such a breaking has been accomplished within concrete realistic gauge models [29][30][31][32]. As already mentioned, in this case both Dirac as well as Majorana neu-trinos are possible. Neutrinos will be Dirac if they transform as ω or ω 3 ; with ω 4 = 1. They will be Majorana otherwise, if they transform trivially or transform as ω 2 under Z 4 . However, the quadruple beta decay 0ν4β illustrated in Fig. 3 is always allowed, irrespective of the nature of neutrinos.
Notice that, if neutrinos are Majorana particles transforming as 1, ω 2 under the Z 4 symmetry, the lower dimensional 0ν2β diagram of Fig. 1 is also allowed by the Z 4 symmetry. By dimensional power counting one sees that the 0ν2β decay is induced by a dim-9 operator, whereas 0ν4β decay is induced by dim-18 operator. Barring extremely fine tuned cancellations, one naively expects that Γ 0ν2β Γ 0ν4β . We estimate R = Γ 0ν2β /Γ 0ν4β for two simple cases: (a) 0ν2β and 0ν4β induced by effective d = 9 and d = 18 operators, respectively. And, (b) 0ν2β induced by a Majorana neutrino mass, while 0ν4β is induced by the "lepton quarticity" d = 6 operator. We find for case (a): while case (b) gives: Here Q ββ and Q 4β are the Q-values of the decays, both of order MeV. G F is the Fermi constant, q 0.1 GeV is the typical momentum transfer in the nucleus and Λ is the scale characterizing the new physics. The numbers correspond to Λ ∼ 1 TeV and for the mass mechanism we have included that the resulting neutrino mass should be at the level of the current experimental bound. Thus for Majorana neutrinos one naively expects to first see neutrinoless double beta decay, if at all. In contrast, for Dirac neutrinos, the dim-18 neutrinoless quadruple beta decay process is still allowed by Z 4 symmetry, while "conventional" neutrinoless double beta decay process is forbidden. Therefore, barring exceptional cases, if future experiments were to observe neutrinoless quadruple beta decay [28] without a positive neutrinoless double beta decay signal, then neutrinos should be Dirac particles. This conclusion can be easily generalized to higher Z m symmetries and higher 0ν2nβ decays.
Another important conclusion is that, for neutrinos to be Majorana particles, lepton number U (1) L must be broken to an even Z 2n subgroup under which neutrinos must transform in a very special way. Such possible "spe-cial nature" of Majorana neutrinos following from symmetry considerations is at odds with popular prejudices. In such a case all 0ν2nβ neutrinoless beta decay processes can be potentially induced, as they are all allowed by the Z 2n symmetry. If neutrinos do not transform appropriately then they must be Dirac particles. In such a case the lowest possible neutrinoless beta decay process allowed by symmetry is 0ν2nβ instead of the "conventional" 0ν2β decay. All lower dimensional neutrinoless beta decay process are forbidden by Z 2n symmetry. If, by contrast, U (1) L is broken to an odd Z 2n+1 symmetry with neutrinos transforming non-trivially under it, then neutrinos are necessarily Dirac particles and all neutrinoless beta decay process of any dimension lower than 0ν2(2n + 1)β decay are forbidden. It may also happen that U (1) L is either completely broken with no residual subgroup or is broken in such a way that the neutrinos transform trivially under the residual discrete lepton number Z m symmetry. In either of these cases, neutrinos can again be Majorana and all 0ν2nβ processes, including the 0ν2β decay are allowed by symmetry. In such a scenario also, on dimensional grounds, one should expect to first observe 0ν2β decay before observing any higher 0ν2nβ process.
The whole discussion above leads to an important conclusion concerning the nature of neutrinos, namely, if we first observe a higher 0ν2nβ decay, unaccompanied by a lower dimensional neutrinoless beta decay signal, such as 0ν2β decay, that would be a strong indication in favour of the Dirac nature of at least one neutrino. This statement holds in general, with the possible exception of very special cancellations as present say, in the quasi-Dirac neutrino situation [26,27], which would require fine tuning at the ∼ 10 −30 level. In contrast, if neutrinos are Majorana particles, then we should first observe 0ν2β decay, if at all, well before observing any higher dimensional 0ν2nβ decays. In short, we have argued that, should a 0ν4β decay signal ever be established, unaccompanied by 0ν2β decays, then one would rule out Majorana neutrinos.