Scotogenic neutrino masses and dark matter stability
from residual gauge symmetry

The lack of a viable dark matter (DM) candidate within the Standard Model is one of the most pressing issues requiring new physics. In addition to new particles, the existence of DM requires new symmetries to stabilise the corresponding candidate on cosmological scales, e.g. R-parity symmetry in supersymmetric schemes Jungman:1995df . Another important open question in need of new physics is that of neutrino masses, which are necessary to account for neutrino oscillation data deSalas:2017kay .

In order to deal simultaneously with the DM and neutrino mass issues, “low-scale” realisations where dark matter appears as a radiative mediator of neutrino mass generation, known as scotogenic models, are appealing alternatives. In its original version Ma:2006km , the symmetry stabilising dark matter is also responsible for the radiative origin of neutrino masses in a very elegant way. Yet, in this case as well as in other proposals Hirsch:2013ola ; Merle:2016scw , the stabilisation symmetry is introduced in an ad hoc manner.

Extending the Standard Model gauge symmetry can provide a natural setting for a theory of dark matter where stabilisation is automatic Alves:2016fqe ; Dong:2017zxo ; Kang:2019sab . Such electroweak extensions involve the SU(3)_L gauge symmetry, which remarkably explains the observed number of fermion families from the anomaly cancellation requirement Singer:1980sw ; Pisano:1991ee ; Frampton:1992wt . In the extended models discussed in Alves:2016fqe ; Dong:2017zxo ; Kang:2019sab , DM stability follows naturally from the spontaneous breaking of the extended gauge symmetry into the residual matter-parity symmetry, $M_{P}$, a non-supersymmetric version of R-parity.

We show here how scotogenic neutrino masses and automatic DM stabilisation are intrinsically linked in a 3-3-1-1 model Leite:2019grf . The present construction has the special feature of predicting that one of the light neutrinos is massless, which, in turn, implies that there is a lower bound for neutrinoless double beta decay rate. This feature arises in a novel way when compared to other schemes in the literature, such as the “incomplete” seesaw mechanism Schechter:1980gr or similar radiative mechanisms Reig:2018ztc .

In our model Leite:2019grf , the electric charge and $B-L$ generators are given by $Q=T_{3}-(1/\sqrt{3})T_{8}+X~{}$ and $B-L=-(2/\sqrt{3})T_{8}+N~{}$, respectively, where $T_{3}$ and $T_{8}$ are the diagonal $\mathrm{SU(3)_{L}}$ generators, while $X$ and $N$ are the $\mathrm{U(1)_{X}}$ and $\mathrm{U(1)_{N}}$ generators, respectively. Notice that, due to the extra $U(1)_{N}$ symmetry, the $B-L$ symmetry is fully gauged. The lepton and scalar fields and their respective transformations are summarised in Table 1. In the lepton sector, the left-handed fields appear as $SU(3)_{L}$ triplets, while the right-handed are $SU(3)_{L}$ singlets. The scalar sector contains five triplets and one singlet.

Symmetry breaking takes place when scalar fields acquire vacuum expectation values (vevs), according to the pattern below,

$\displaystyle\left\langle\sigma\right\rangle=\frac{v_{\sigma}}{\sqrt{2}},~{}\langle\chi\rangle=\frac{1}{\sqrt{2}}(0,0,w)^{T},\left\langle\eta\right\rangle=\frac{1}{\sqrt{2}}(v_{1},0,0)^{T},~{}\langle\rho\rangle=\frac{1}{\sqrt{2}}(0,v_{2},0)^{T},~{}\left\langle\zeta\right\rangle=\frac{1}{\sqrt{2}}(0,v_{2}^{\prime},0)^{T}.$

(1)

The spontaneous symmetry breaking process can be divided into two main steps

	$\displaystyle SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$		$\displaystyle\xrightarrow{v_{\sigma},w}SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes M_{P}$
			$\displaystyle\xrightarrow{v_{1},v_{2},v_{2}^{\prime}}~{}SU(3)_{c}\otimes U(1)_{Q}\otimes M_{P}~{},$

for $v_{\sigma},w\gg v_{EW}$ and $v_{EW}=(v_{1}^{2}+v_{2}^{2}+v_{2}^{\prime 2})^{1/2}=246$ GeV. After the breaking a residual matter-parity symmetry, $M_{P}$, is present

$\displaystyle M_{P}$

$\displaystyle=$

$\displaystyle(-1)^{3(B-L)+2s}~{},$

(3)

where $s$ is the particle’s spin. As only the $M_{P}$-even scalar fields get vevs, $M_{P}$ remains as an absolutely conserved residual symmetry, in such a way that the lightest amongst the $M_{P}$-odd particles is stable, see Table 1.

Taking into account the scalar fields in Table 1, we can write down the most general renormalisable potential as

	$\displaystyle V$	$\displaystyle=$	$\displaystyle\sum_{s}\left[\mu_{s}^{2}(s^{\dagger}s)+\frac{\lambda_{s}}{2}(s^{\dagger}s)^{2}\right]+\sum_{s_{1},s_{2}}^{s_{1}>s_{2}}\left[\lambda_{s_{1}s_{2}}(s_{1}^{\dagger}s_{1})(s_{2}^{\dagger}s_{2})\right]+\sum_{t_{1},t_{2}}^{t_{1}>t_{2}}\left[\lambda^{\prime}_{t_{1}t_{2}}(t_{1}^{\dagger}t_{2})(t_{2}^{\dagger}t_{1})\right]$
			$\displaystyle+\frac{\mu_{1}}{\sqrt{2}}\eta\rho\chi+\frac{\mu_{2}}{\sqrt{2}}(\zeta^{\dagger}\rho)\sigma+\lambda_{1}(\chi^{\dagger}\eta)(\zeta^{\dagger}\xi)+\lambda_{2}(\chi^{\dagger}\xi)(\zeta^{\dagger}\eta)+\lambda_{3}(\chi^{\dagger}\eta)(\xi^{\dagger}\rho)$
			$\displaystyle+\lambda_{4}(\chi^{\dagger}\rho)(\xi^{\dagger}\eta)+\lambda_{5}(\eta\zeta\chi)\sigma^{*}+h.c.~{},$

where $s=\eta,\rho,\chi,\sigma,\zeta,\xi$ (all scalars), and $t=\eta,\rho,\chi,\zeta,\xi$ (scalar triplets only).

We have proposed an $\mathrm{SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{N}}$ model where dark matter stability follows from the spontaneous breaking of the gauge symmetry. Neutrino masses are generated at loop level and mediated by DM in a scotogenic fashion. Our construction features a triplet scalar with anti-symmetric Yukawa couplings to neutrinos. This leads to the prediction of a massless neutrino and a lower bound for the $0\nu\beta\beta$ decay rate. Contrary to models where a massless neutrino arises from an ad hoc incomplete multiplet choice, here it is an unavoidable feature of the theory.