Scotogenic Cobimaximal Dirac Neutrino Mixing from $\Delta(27)$ and $U(1)_\chi$

In the context of $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_\chi$, where $U(1)_\chi$ comes from $SO(10) \to SU(5) \times U(1)_\chi$, supplemented by the non-Abelian discrete $\Delta(27)$ symmetry for three lepton families, Dirac neutrino masses and their mixing are radiatively generated through dark matter. The gauge $U(1)_\chi$ symmetry is broken spontaneously. The discrete $\Delta(27)$ symmetry is broken softly and spontaneously. Together, they result in two residual symmetries, a global $U(1)_L$ lepton number and a dark symmetry, which may be $Z_2$, $Z_3$, or $U(1)_D$ depending on what scalar breaks $U(1)_\chi$. Cobimaximal neutrino mixing, i.e. $\theta_{13} \neq 0$, $\theta_{23} = \pi/4$, and $\delta_{CP} = \pm \pi/2$, may also be obtained.


Introduction :
Whereas there exist powerful theoretical arguments that neutrinos are Majorana, there is yet no incontrovertible experimental evidence that they are so, i.e. no definitive measurement of a nonzero neutrinoless double beta decay. To make a case for neutrinos to be Dirac, the existence of a right-handed neutrino ν R must be justified, which is of course not required in the standard model (SM) of quarks and leptons. The canonical choice is to extend the SM gauge symmetry SU (3) C × SU (2) L × U (1) Y to the left-right symmetry SU (3) C × SU (2) L × SU (2) R × U (1) (B−L)/2 . In that case, the SU (2) R doublet (ν, e) R is required, and the charged W ± R gauge boson is predicted along with a neutral Z gauge boson.
A more recent choice is to consider U (1) χ which comes from SO(10) → SU (5) × U (1) χ , with SU (5) breaking to the SM at the same grand unified scale. Assuming that U (1) χ survives to an intermediate scale, the current experimental bound on the mass of Z χ being about 4.1 TeV [1,2], then ν R must exist for the cancellation of gauge anomalies. Now ν R is a singlet and W ± R is not predicted. In this context, new insights into dark matter [3,4] and Dirac neutrino masses [5,6] have emerged.
To make sure that ν R itself does not have a Majorana mass, the breaking of U (1) χ should not come from a scalar which couples to ν R ν R . This simple idea was first discussed [7] in 2013 in the case of singlet fermions charged under a gauge U (1) X . If the latter is broken by a scalar with three units of X charge, it is impossible for these fermions with one unit of X charge to acquire Majorana masses. Hence the residual symmetry is global U(1) in this case. It is straightforward then to apply this idea to lepton number [8].
A second issue regarding Dirac neutrinos is that the corresponding Yukawa couplings linking ν L to ν R through the SM Higgs boson must be very small. To avoid these tree-level couplings, it is often assumed that some additional symmetry exists which forbids these dimension-four couplings, but Dirac neutrino masses may be generated radiatively as this symmetry is broken softly by dimension-three terms. For a generic discussion, see Ref. [9], which is patterned after that for Majorana neutrinos [10]. In some applcations [11,12,13], the particles in the loop belong to the dark sector. This is called the scotogenic mechanism, from the Greek 'scotos' meaning darkness, the original one-loop example [14] of which was applied to Majorana neutrinos.

Outline of Model :
The particles of this model are shown in Table 1. Table 1: Particle content of model.
In the notation above, all fermion fields are left-handed. The usual right-handed fields are denoted by their charge conjugates. The SM particles transform under U (1) χ according to their SO(10) origin, as well as the particles of the dark sector (N, N c , η, σ). The input family symmetry is ∆ (27). The gauge U (1) χ is broken by ζ 2 or ζ 3 or ζ 4 . In each case, a residual U (1) L symmetry remains for lepton number whereas the dark symmetry becomes in all its terms, as well as ∆ (27)  The key feature of this model is the interplay between U (1) χ and ∆ (27) for restricting the interaction terms among the various fermions and scalars. The irreducible representations of ∆ (27) and their character table are given in Ref. [19]. The important point is that if a set of 3 complex fields transforms as the 3 representation of ∆ (27), then its conjugate transforms as 3 * , which is distinct from 3. The basic multiplication rules are From Table 1, the Yukawa term ee c φ 0 1 is allowed, but not νν c φ 0 2 because of ∆ (27). Furthermore, the usual dimension-five operator for Majorana neutrino mass, i.e. ννφ 0 2 φ 0 2 , is forbidden as well as the usual singlet Majorana mass term ν c ν c . Note that without U (1) χ , ν c ν c is a soft term breaking ∆(27) and would then have been allowed by itself. To obtain Dirac neutrino masses, the scalar doublet η and singlet σ with odd Q χ as well as the fermion singlets N, N c with even Q χ are added. Note that they belong to the dark sector because SM fermions have odd Q χ and the SM Higgs doublet has even Q χ , as pointed out in Ref. [3]. The dimension-three scalar trilinear couplings η 0 σφ 0 respect U (1) χ but not ∆ (27). The dimension-three N c N terms are also allowed to break ∆ (27).
Consider now the spontaneous breaking of U (1) χ . First, because ν, N ∼ 3 and ν c , N c ∼ 3 * under ∆(27), they cannot obtain Majorana masses through any scalar which is a singlet.
Hence U (1) L lepton number holds as indicated in Table 1. If ζ 2 is used, then the term ζ * 2 σ 2 is allowed, hence the residual dark symmetry is Z D 2 . Similarly, ζ 3 σ 3 yields Z D 3 with ω = exp(2πi/3), and ζ 4 yields U (1) D because ζ * 4 σ 4 is not allowed by renormalizability, and cannot be generated by the given particle content of the model. In the case of Z D 3 from ζ 3 , there is an equivalent assignment of lepton number and dark symmetry. Instead of the conventional thinking that N, N c must carry lepton number, the latter may be assigned [33] to the scalars η and σ. Hence lepton number becomes Z L 3 [34] with ν, σ ∼ ω; ν c , η ∼ ω 2 ; N, N c , φ ∼ 1; whereas dark symmetry remains Z D 3 with σ, N c ∼ ω; η, N ∼ ω 2 ; ν, ν c , φ ∼ 1.
Cobimaximal Neutrino Mixing : Using the decomposition 3 × 3 * and φ 0 i = v i , with 1 1 , 1 7 , 1 4 as defined in Ref. [19], instead of the usual 1 1 , 1 2 , 1 3 of the original A 4 model [35] of neutrino mixing, the charged-lepton mass matrix is given by where v 2 = v 3 = 0 has been assumed for the spontaneous breaking of φ 0 1,2,3 . This M l is diagonal and different from that of Ref. [35]. It allows also three independent masses for the charged leptons, and the emergence of lepton flavor triality [36,37] in the Yukawa interactions of the three charged leptons with the three Higgs doublets.
In the neutrino sector, the tree-level Yukawa couplings νν c φ 0 are forbidden by ∆ (27).
Hence the 3×3 Dirac neutrino mass matrix M ν is generated through dark matter (scotogenic) as shown in Fig. 1. Since η 0 σφ 0 is just one coupling, the flavor structure of M ν comes from the N c N mass terms which break ∆ (27) softly. Assuming the residual symmetry to be generalized N 2 − N 3 , N c 2 − N c 3 , and N 1,2,3 − N c 1,3,2 exchange with complex conjugation [24], where A, C are real. The above is exactly of the form [23,24,25] required for cobimaximal mixing [26,27,28,29,30,31], i.e. θ 13 = 0, θ 23 = π/4, and δ CP = ±π/2, because the neutrino basis is also the one where the charged leptons are diagonal. If B I = D I = 0, then θ 13 = 0 and θ 23 = π/4. If in addition, then tan 2 θ 12 = 1/2 as well, i.e. tribimaximal mixing is obtained. Since neutrino oscillation data are close to this limit, the above quantities may be considered small if not zero, hence the neutrino mass eigenvalues are approximately given by If B I = 0 or D I = 0 or both, cobimaximal mixing is obtained. However, θ 13 and θ 12 are not fixed. If Eq. (4) is valid together with B I = 2D I , then [38] in good agreement with data.
Dark Sector : To compute the Dirac neutrino mass matrix of Fig. 1, assume first that η 0 σ couples only toφ 0 1 , leading to the possibility of lepton flavor triality [36,37] which may be tested experimentally.
Note then that the one-loop calculation is equivalent to taking the difference of the exchanges of two scalar mass eigenstates where θ is the mixing angle due to theφ 0 η 0 σ term. Let the ν i N c k η 0 Yukawa coupling be h L U ik and the ν c j N k σ Yukawa coupling be h R U T kj , then the Dirac neutrino mass matrix is given by where m 1,2 are the masses of χ 1,2 and M k is the mass of N k . If |m 2 2 − m 2 1 | << m 2 2 + m 2 1 = 2m 2 0 << M 2 k , then the M k contribution reduces to  (7) to be very small, then it should be χ 1 . If its mass is greater than that of the SM Higgs boson, its annihilation to the latter is a well-known mechanism for generating the correct dark-matter relic abundance of the Universe.
If Eq. (10) holds, then the lightest N is dark matter, i.e. N 1 for the normal hierarchy of neutrino masses (m ν 1 < m ν 2 < m ν 3 ) or N 3 for the inverse hierarchy (m ν 3 < m ν 1 < m ν 2 ). In either case, the other two N s will decay to the lightest N plus a neutrino pair or charged lepton pair, through χ 1,2 or η ± . The annihilation of NN → νν through χ 1 exchange has a cross section × relative velocity given by assuming again that θ is very small. As a numerical example, let M N = 150 GeV, m 1 = 400 GeV, h R = 0.62, then this is about 1 pb, which is a typical value for obtaining the correct dark-matter relic abundance of the Universe, i.e. Ωh 2 = 0.12.
At the mass of 150 GeV, the constraint on the elastic scattering cross section of N off nuclei is about 1.5 × 10 −46 cm 2 from the latest XENON result [40]. This puts a lower limit on the mass of Z χ , i.e. where and Z = 54, A = 131 for xenon. In U (1) χ , the vector couplings are