One of the unrevealed properties of the neutrinos is whether they are Dirac type or Majorana type. Some important physics occur only for the Majorana type: the leptogenesis that can explain the baryon asymmetry of the universe (BAU) and the seesaw mechanism that can explain the smallness of the neutrino masses. Experiments such as neutrinoless double beta decay can expect signals only for the Majorana neutrinos.

The true nature of the neutrinos may not be simple enough to identify them as either Dirac or Majorana type, though. It is especially so in view that the properties of the dark matter, which is another mystery in particle physics, are also unrevealed. It is quite possible that neutrino and dark matter are tightly linked, affecting each other.

In this paper, we propose a new scenario in which the neutrino type oscillates due to the wave dark matter. We show that the amplitude of the oscillating dark matter can be large enough to change the neutrino type back and forth between the Dirac and the Majorana while satisfying all the constraints for dark matter. It can be tested, for instance, by the modulation search in the neutrinoless double beta decay. The neutrino type oscillation may not have occurred in the early universe though, allowing the widely adopted leptogenesis for the BAU.

Coupling an oscillating scalar field to a particle to vary its mass is not new, including the neutrino masses [1–14]. Our study differs from the existing works in the sense that it is the first scenario of the neutrino type oscillation.¹

We note that it was pointed out in Ref. [9], in which the active-sterile neutrino oscillations in the quasi-Dirac limit were mainly studied, the neutrino might be out of the quasi-Dirac type in the early universe because the Majorana mass term undergoes the suppression over the cosmic time originating from the scaling of the ultralight dark matter.

The Dirac neutrino νD=νL+νR is constructed by left and right-handed Weyl spinors. In contrast, the Majorana neutrino νM=νR+νRc is defined using only one Weyl spinor, and satisfies νM=νMc condition. In general, Lagrangian for the neutrino mass for a single flavor can be written as L=-mDν¯DνD-12mRν¯MνM=-12(ν¯Lν¯Rc)(0mDmDmR)(νLcνR)+H.c.(1)

The light and heavy mass eigenvalues are ml,h=12(mR∓mR2+4mD2). The mass matrix is diagonalized in the mass eigenstates νl (light one), νh (heavy one).²

For convenience, we use the following notations interchangeably: N≡νh, M≡mh.

The ratio of the Majorana and Dirac masses (mR/mD) determines the left-right (LR) mixing angle θLR and the composition of each neutrino mass eigenstate. The mR=0 case gives a pure Dirac spinor of the degenerate mass eigenvalues mD and the maximal mixing (θLR=π/4).

For mD≫mR case, the mass eigenvalues are nearly degenerate at mD, and mixing is almost maximal (θLR≃π/4). |ml,h|≃mD(1∓2δ),(3)νl,h≃±1±δ2(νL+νLc)+1∓δ2(νR+νRc),(4)where δ=mR/4mD. This case is called the quasi-Dirac neutrino (or pseudo-Dirac neutrino) [15–19]. Even though the mass eigenstates are still Majorana spinors, they act similarly to Dirac spinors. The Dirac mass term becomes more dominant than the Majorana mass term, so the lepton number violation (LNV) process is suppressed in this limit.

In contrast, for the mD≪mR case, the mass eigenvalues are given by the seesaw mechanism [20–23]. ml≃-mD2mR,mh≃mR,(5)νl≃νL+νLc,νh≃νR+νRc,(6)and the mixing angle θLR≃mD/mR is very small. The mass eigenstates behave almost like Majorana spinors.

The mass eigenvalues and the LR mixing angle are depicted in Fig. 1 as a function of mR/mD.

The oscillation between Dirac and Majorana states requires a time variation of θLR and thus the variation of mR/mD. Such a scenario can be constructed by introducing coupling between neutrino and oscillating scalar field ϕ such as a wave dark matter [24], which has periodic behavior for 10-22  eV<mϕ<30  eV mass range with corresponding oscillation period is T=2π/mϕ=O(ps)∼O(year). The lower bound originates from the Lyman-α forest data. The black hole superradiance search may further constrain the mass ranges [25,26].

Assuming the spatial homogeneity of the scalar field, the equation of motion in the expanding universe is ϕ¨+3Hϕ˙+∂V(ϕ)∂ϕ=0.(7)H is the Hubble parameter and the scalar field potential is V(ϕ)=12mϕ2ϕ2. We assume the mass mϕ remains constant over the cosmic evolution. The Hubble friction is ignored in the current universe (H0≪mϕ), so the scalar field has the periodic solution. ϕ(t)=ϕ0 cos mϕt,(8)where ϕ0=2ρϕ/mϕ. The energy density of the oscillating scalar, ρϕ=12ϕ˙2+12mϕ2ϕ2, has the local density of total dark matter ρDM=0.3  GeV/cm3 [27] as its upper limit.

There are several ways to introduce interactions between neutrinos and the oscillating scalar field [3–10,13,14]. In general, one can vary both Dirac and Majorana masses by making them functions of an oscillating scalar field, thus periodically changing mR/mD. We will consider the case only the Majorana mass is a function of an oscillating scalar field while the Dirac mass mD is fixed [4–6,8–10,13]. Especially, for the definiteness, we take the simplest case where the Majorana mass is given by [9] L⊃-12gϕν¯RcνR+H.c.(9)This term can serve as an addition to V(ϕ) that depends on the density of the right-handed neutrino. However, since the heavy right-handed neutrinos decay promptly in the early universe, its effect is negligible. Besides, this coupling generates the effective ϕ4 term by the loop with the neutrino [9,13], but the desired mass variation behavior of changing mR/mD is still guaranteed despite the existence of such a nonlinear term [28].

This interaction induces time-varying Majorana mass mR(t)=gϕ(t)≃mR,0cosmϕt,(10)where mR,0=gϕ0 is the current oscillating amplitude of right-handed Majorana mass.³

In general, there may exist a bare Majorana mass term -12mν¯RcνR+H.c., which makes the center of mR(t) shifted from zero to m. In this case, the oscillation amplitude mR,0 should be similar to or larger than m for the neutrino type oscillation to occur, but we assume m=0 here.

In our scenario, the present-time Majorana mass amplitude mR,0 is larger than mD, making mR(t) cross between quasi-Dirac and Majorana regions. We call this phenomenon the Dirac-Majorana neutrino type oscillation. The right-handed Majorana mass term mR in Eq. (1) is now replaced with mR(t) in Eq. (10). Thus, the mR/mD in Fig. 1 may oscillate between the (quasi-)Dirac and Majorana limit. The value of the ratio touches the pure Dirac neutrino case instantly when mR=0.

The required coupling strength g to give sufficient amplitude mR,0 is given by the following equation. mR,0≃1019  eV(g1)(ρϕρDM)1/2(10-22  eVmϕ).(11)The current dark matter density naturally provides such a large scale of the oscillating Majorana mass with respect to the typical Dirac mass scale of the standard model (SM) fermions, making the oscillating scalar dark matter a good candidate for our scenario (See Fig. 2).

The variation of the neutrino mass may affect neutrino-related physics including the following. (i) The neutrino flavor oscillation parameters (Δmij2 and sin2θij) and the apparent unitarity of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix for three-flavor mixing of the active neutrinos [33,34].⁴

See Refs. [3,5–7,9,10,13,14] for the effect of the mass variation on the neutrino flavor oscillation.

Although it is pending the scrutiny of the data with dedicated analysis, we assume it is okay if the time-averaged quantities satisfy the known bounds. For instance, we impose the active neutrino mass bound [27,35–38] on the time-averaged neutrino mass: ⟨|ml|⟩=1T∫0T|ml(t)|dt<0.1  eV. The allowed parameter space of Dirac mass mD and Majorana mass amplitude mR,0 is shown in Fig. 3. This time-average analysis can also be applied to the effect of the mass variation on the neutrino flavor oscillation parameters, but it is out of the scope of this paper.

For the convenient analysis, we call mD>mR(t) part the quasi-Dirac region. The time interval τ for this region is given by mDmR,0=sin(π2τT)for  mD<mR,0,(12)and the “quasi-Dirac time ratio” τ/T is 1 for mD>mR,0 case, meaning the neutrino is continually in the quasi-Dirac region.

The neutrinoless double beta decay (0νββ) 2n→2p+2e- as a LNV process [39] is one of the best examples that give a distinction for quasi-Dirac and Majorana neutrinos.

The 0νββ decay rate is proportional to the square of the following amplitude A. A=|∑iUei2miM(mi)|,(13)where U is the extended PMNS matrix including LR mixing, and M is the nuclear matrix element [40–44].

For a simple case of a single flavor, the amplitude is A=|mlcos2θLRM(ml)+mhsin2θLRM(mh)|.(14)For pure Dirac neutrinos, the amplitude is exactly zero because the two terms in A cancel each other precisely. In the quasi-Dirac region, the amplitude A is still highly suppressed [16,40]. In the Majorana region, A is relatively large and the 0νββ process is activated. If the Dirac-Majorana neutrino type oscillation occurs, the 0νββ process turns on and off periodically as mR(t) oscillates. The three-flavor case shows similar behavior.

The time-averaged amplitude ⟨A⟩ can be constrained by the current bound from the 0νββ experiments [35,38]. The Majorana region occurs most of the time according to Fig. 3, in which the LR mixing angle is negligibly small. Thus, only the first term in Eq. (14) remains, and the 0νββ bound is reduced to ⟨|ml|⟩ bound.

The neutrino type oscillation can be tested by analyzing 0νββ data reflecting time modulation. The event number will show a sharp decrease when the state enters the quasi-Dirac region. The time length of the 0νββ rate reduction depends on the dark matter mass mϕ and the quasi-Dirac time ratio τ/T.

Such a modulation search can apply to others, too. For instance, the cosmic neutrino background signal rate is larger for the Majorana neutrino compared to the Dirac neutrino by a factor of two [45–47]. A modulation search could be potentially much more sensitive than a typical signal-over-background search for these extremely weak signals.

The SM does not provide an explanation for the observed BAU. Leptogenesis is a possible model for baryogenesis by generating nonzero B-L via the LNV process, in the presence of the Majorana mass term. A decay of a heavy Majorana neutrino generates lepton asymmetry and this nonzero lepton number leads to baryogenesis via the sphaleron process.

The LNV comes from the CP violating decay of heavy Majorana neutrinos Ni. The CP asymmetry εi for this process is defined as εi=Γ(Ni→ℓΦ)-Γ(Ni→ℓ¯Φ†)Γ(Ni→ℓΦ)+Γ(Ni→ℓ¯Φ†),(15)where ℓ is the lepton doublet and Φ is the Higgs doublet. To produce enough lepton number to explain the observed BAU, at least εi≈10-6 is needed [48].

The CP asymmetry εi is proportional to εi∝∑k≠iIm(y†y)ik2(y†y)ii(y†y)kk(Mi2-Mk2)MiΓk(Mi2-Mk2)2+Mi2Γk2,(16)where y is the Yukawa matrix for the Dirac mass term and Γk is the decay rate of kth heavy neutrino. In the minimal leptogenesis scenario, where very heavy neutrinos are assumed (with a hierarchy M1≪M2,3), it reduces to ε1∝M1. Here, the lower bound M1>109  GeV is required [48,49]. In another scenario, called the resonant leptogenesis, nearly degenerate Mi’s are assumed, and the mass bound can be as low as TeV scale [50–52].

Since the amplitude of the oscillating scalar field decays as ϕ0∝a-3/2, the Majorana masses of neutrinos are expected to be far larger in the early universe than the present values when they are functions of the oscillating scalar dark matter [4–10,13].⁵

The scalar field becomes radiationlike and ϕ0∝a-1 if the ϕ4 term is dominant [9,53], but we set the small ϕ4 term here.

The Majorana mass could be considered as a constant before the scalar field oscillation began when the Hubble parameter H≃T2/MPl is about the dark matter mass mϕ, where T is the temperature of the universe and MPl is the Planck mass. Since the scale factor value a* at H≈mϕ can be determined when mϕ is given, the temperature T* at the dark matter production stage is given by T*≃103  eV(mϕ10-22  eV)1/2.(17)Using T∝1/a, mR∝a-3/2, and T0≃2.73  K, the Majorana mass M* at that stage can be obtained. M*M0≃1011(mϕ10-22  eV)3/4,(18)where M0 is the current Majorana mass amplitude.

The grand unification scale is typically taken for the heavy Majorana neutrino mass in the minimal leptogenesis, but it can decrease drastically over time. For mϕ=10-22  eV as a benchmark point, M*=1015  GeV decreases to M0∼10  TeV at the present time. This small M0 suppresses y∝M0 according to the Casas-Ibarra parametrization [54], which is widely adopted to ensure consistency with the neutrino oscillation data. Nevertheless, sufficient CP asymmetry ε can be achieved if the masses are degenerate.

We note that large M* and small y∝M0 may not provide the thermal production of the heavy Majorana neutrinos in the early universe, but the leptogenesis can be still achieved if, for instance, the thermal equilibrium is obtained with a new gauge boson that couples to the right-handed neutrinos and other particles. Also, nonthermal production of the Majorana neutrinos from inflaton decays may occur.

The scale of M* for the leptogenesis links with M0, which determines the Dirac-Majorana neutrino type oscillation in the present universe. This relation is schematically illustrated in Fig. 4.

We studied the scenario in which the neutrino masses are given as a function of the wave scalar dark matter in the way it results in Dirac-Majorana neutrino type oscillations. We showed this scenario is consistent with the various constraints on dark matter, and even if the oscillating scalar field is a fraction of the relic dark matter, the scenario works well.

The amplitude of the oscillating dark matter decreases over time, but the leptogenesis may still occur with the frozen Majorana mass in the early universe, while the suppressed oscillating mass amplitude at the present time may result in unique periodic conversion signatures in various neutrino experiments including the neutrinoless double beta decays. Rich physics and cosmology with neutrino type oscillation are warranted.