Lepton number violation and ‘Diracness’ of massive neutrinos composed of Majorana states

Majorana neutrinos naturally lead to a lepton number violation (LNV). A superposition of Majorana states can mimic Dirac-type neutrinos, leading to lepton number conservation. Using the example of specific observables related to high- and low-energy processes, we demonstrate how the strength of LNV correlates with neutrino parameters, such as $CP$ phases, flavor mixings, and mass ratios. We stress the coaction of low- and high-energy studies for quantitatively testing phenomenological models. Second, we conclude that in order to fully study the role of heavy neutrinos in the search for new physics signals, a departure from trivial scenarios assuming degeneracy in mass and no flavor mixing or $CP$ phases becomes necessary for a proper physical analysis.

Figure 1

Feynman diagram responsible for the “golden” $pp\to lljj$ signal. For Majorana neutrinos two signals are possible with same-sign leptons $pp\to W_2^{\pm }\to l_i^{\pm }{N}_a\to l_i^{\pm }{l}_j^{\pm }jj$ and opposite-sign leptons $pp\to W_2^{\pm }\to l_i^{\pm }{N}_a\to l_i^{\pm }{l}_j^{\mp }jj$. In the internal frame two related LNV processes can be identified: $ll\to W_2{W}_2$ (possible at the future lepton colliders) and $W_2^{-}{W}_2^{-}\to e^{-}{e}^{-}$ [a part of the low-energy neutrinoless nuclear double beta decay $(\beta \beta {)}_{0\nu }$].

Figure 2

“Christmas Tree” exclusion plot for $m^N$ dominating the $(\beta \beta {)}_{0\nu }$ half-life $T_{1/2}^{0\nu }[{}^{76}\mathrm{Ge}\to {}^{76}\mathrm{Se}]$ as a function of ${\theta }_{13}$. ${\eta }_{CP}(N_1)=-{\eta }_{CP}(N_3)=i$, ${\varphi }_3=\pi /2$, and $M_{1,2,3}=M_{W_2}/2=1.1\mathrm{TeV}$. The star represents the infinite half-life Dirac scenario (maximal ${\theta }_{13}$ mixing). The bands correspond to different evaluations of the nuclear matrix elements [36]. The excluded region (below the solid horizontal line) comes from Ref. [53]. The dashed horizontal line represents the expected future bound from the $\mathrm{Majorana}+\mathrm{GERDA}$ experiment [25].

Figure 3

The CMS vs $m^N$ dominant $(\beta \beta {)}_{0\nu }$ exclusion limits on the masses of $W_2$ and $N_a$ in the case when $(K_R{)}_{aj}={\delta }_{aj}$ as in the (A) scenario. The shaded region is excluded by the CMS data related to $pp\to eejj$ at the LHC Run 1 [47]. Present $(\beta \beta {)}_{0\nu }$ experiments exclude the region under the blue solid curve. The dotted orange curve corresponds to a future bound on $T_{1/2}^{0\nu }$ [25]. For comparison, when the mixing matrix $K_R$ is of the form (5) with ${\theta }_{13}=0.9\times \pi /4$ and ${\varphi }_3=\pi /2$, only the region under the dashed red curve is excluded. There are no available LHC data exclusion analyses for such “almost” Dirac neutrinos.

Figure 4

Dependence of $r_0$ and $r_1$ on ${\theta }_{13}$ for case (B) for two neutrinos with opposite $CP$ parities, $\eta (N_1)=-\eta (N_3)=i$ [${\varphi }_3=\pi /2$ in Eq. (7)]. The subscript 0(1) of $r_{0(1)}$ means that there is a 0(1) GeV mass splitting between the $N_1$ and $N_3$ states. The star represents a scenario with ${\theta }_{13}=\pi /4$ equivalent to one Dirac heavy neutrino. Black squares correspond to pure Majorana states (no interferences). The shaded region is already excluded by the $m^N$ dominant $(\beta \beta {)}_{0\nu }$ experimental data given in Fig. 2.

Figure 5

Interference effects in the $e^{-}{e}^{-}\to W_2^{-}{W}_2^{-}$ LNV process for three different splittings $\mathrm{\Delta }M=0$, 100, and 500 GeV between masses of heavy neutrinos $N_{1,3}$. $M_{W_2}=1.4\mathrm{TeV}$, $\sqrt{s}=3\mathrm{TeV}$, $M_{1,3}=1\mathrm{TeV}\pm \mathrm{\Delta }M/2$, $M_2=10\mathrm{TeV}$, $K_R$ is given as in Eq. (6), and ${\varphi }_3=\pi /2$.

Figure 6

$\tau \to e\gamma$ branching ratio as a function of mass splitting $\mathrm{\Delta }M=M_3-M_1$. The mixing between the first and third generations is assumed to be maximal. We chose $M_{W_2}=2.2\mathrm{TeV}$ (dotted line), $M_{W_2}=3\mathrm{TeV}$ (dashed line), and $M_{W_2}=5\mathrm{TeV}$ (solid line). Also, $M_1=M_{W_2}/2$.