Cobimaximal Mixing with Dirac Neutrinos

If neutrinos are Dirac, the conditions for cobimaximal mixing, i.e. $\theta_{23}=\pi/4$ and $\delta_{CP}=\pm \pi/2$ in the $3 \times 3$ neutrino mixing matrix, are derived. One example with $A_4$ symmetry and radiative Dirac neutrino masses is presented.

Introduction : Neutrinos are mostly assumed to be Majorana. The associated 3×3 mass matrix has been studied in numerous papers. One particularly interesting form was discovered in 2002 [1], i.e.
where A and D are real, which was shown subsequently [2] to be the result of a generalized CP transformation involving ν µ,τ exchange. This form predicts the so-called cobimaximal mixing pattern [3] of neutrinos, i.e. θ 23 = π/4 and δ CP = ±π/2, which is close to what is observed [4].
To understand the cobimaximal mixing matrix U CBM , consider its form in the PDG convention, i.e.
Note that the (2i) and (3i) entries for i = 1, 2, 3 are equal in magnitudes. It is easy then to obtain Eq. (1) as In Eq. (1), the neutrino basis is chosen for which the charged-lepton mass matrix M l is diagonal which links the left-handed (e, µ, τ ) to their right-handed counterparts. Suppose it is not, but rather that it is digonalized on the left by the special matrix [5,6] where ω = exp(2πi/3) = −1/2 + i √ 3/2. It was discovered in 2000 [7] that where O is an orthogonal matrix. The proof is very simple because the product U ω O enforces the equal magnitudes of the (2i) and (3i) entries.
This is the analog of Eq. (1) for Dirac neutrinos. An example was recently shown in Ref. [9].
However, Eq. (6) does not constrain M D uniquely because of the missing arbitrary U R .
Nevertheless, a possible form of M D is where a is real. It is then trivial to see that M D M † D yields exactly Eq. (6). The origin of this M D is a simple extension of the generalized CP transformation of Ref. [2], i.e.
together with complex conjugation. It is important to realize that whereas Eq. (7) guarantees Eq. (6), the former may be obtained without the latter, as shown already in Ref. [9] because of the missing arbitrary U R .
Scotogenic Dirac Neutrinos with Cobimaximal Mixing : The other approach to obtaining U CBM is through Eq. (5). Two previous models were constructed [10,11] this way for Majorana neutrinos. Their Dirac counterpart is presented here. It is actually simpler because a technical problem is naturally avoided in this case as shown below.
Following Ref. [8], the non-Abelian discrete symmetry A 4 is used, under which the three families of left-handed lepton doublets transform as the 3 representation, and the three charged-lepton singlets as 1, 1 , 1 . There are also three Higgs doublets Φ i = (φ + i , φ 0 i ) transforming as 3. The multiplication rules for two triplets a 1,2,3 and b 1,2,3 in this representation [8] are Assuming that φ 0 i is the same for i = 1, 2, 3, the 3 × 3 mass matrix linking (e, µ, τ ) L to (e, µ, τ ) R is then which is well-known since 2001.
To obtain Dirac neutrinos, three lepton singlets ν R transforming as 3 under A 4 are added to the SM. Since the products (a 1 b 2 c 3 + a 2 b 3 c 1 + a 3 b 1 c 2 ) and (a 1 b 3 c 2 + a 2 b 1 c 3 + a 3 b 2 c 1 ) are allowed, so that tree-level Dirac neutrino masses are obtained. To forbid this, a Z 2 symmetry is imposed, so that ν R are odd, and the SM fields are even as shown in Table 1. Note that all dimensionfour terms in the Lagrangian are required to obey A 4 × Z 2 which is only broken together softly by the ss mass terms. Added are dark scalars and fermions which are odd under an exactly conserved Z D 2 symmetry. Lepton number L is conserved as shown. Dirac neutrino masses are radiatively generated by dark matter [12] as shown in Fig. 1 in analogy with the original scotogenic model [13]. The key for cobimaximal mixing is that the s, s scalars are real fields [10,11]. The relevant Yukawa couplings are All respect A 4 ×Z 2 , with the latter two contributing to the 2×2 mass matrix linking (E 0 L , N L ) to (E 0 R , N R ), i.e.
As for the contribution of s and s , the mass-squared matrix for each is proportional to the identity, whereas the ss mixing is arbitrary, breaking both A 4 and Z 2 at the same time softly. Let it be denoted as M 2 ss and assuming that its entries are all much smaller then the invariant masses of s and s , then it is clear that the Dirac neutrino mass matrix in the basis of Fig. 1 is proportional to M 2 ss and is real up to an unobservable phase, i.e. the relative phase of f N and f E . This means that it is diagonalized by an orthogonal matrix. Combined with Eq. (10), cobimaximal mixing is assured.
The explicit expression for the scotogenic Dirac neutrino mass matrix is where x 1,2 = m 2 s /m 2 1,2 and y 1,2 = m 2 s /m 2 1,2 , and The A 4 → Z 3 Breaking : The breaking of A 4 by φ 0 i = v reduces this symmetry to Z 3 [14]. It must be maintained for U ω to be valid. However, the addition of s and s would allow the The key now is that both the quadratic mass terms s i s i and s i s i do not break Z 2 and are required also not to break A 4 . Only the s i s j terms break both A 4 and Z 2 softly together. Hence the one-loop correction to Φ † i Φ j is shown in Fig. 2. Two mass insertions are required, which render the diagram finite and suppressed so that the residual Z 3 symmetry is maintained to a good approximation. In Refs. [10,11], this option is not available for Majorana neutrinos because s is absent and s i s j breaks A 4 , which yields only one mass insertion in Fig. 2, thus making it logarithmically divergent.
Dark Sector : The dark sector fermions are (E 0 , E − ) and N . The two neutral ones have masses m 1,2 and the charged one m E . They are assumed greater than the masses of the scalars, m s and m s , with the small M 2 ss mixing between them. Let m s be the smaller, then the almost degenerate s 1,2,3 are dark-matter candidates. They interact with the SM Higgs boson h according to where This is a straightforward generation of the simplest model of dark matter, i.e. that of a real scalar. The comprehensive analysis of Ref. [15] is thus applicable.
Conclusion : It is shown how cobimaximal neutrino mixing, i.e. θ 23 = π/4 and δ CP = ±π/2, occurs for Dirac neutrinos. It is defined by Eq. (6) which is obtainable from Eq. (7) based on a generalized CP transformation. However, because of the missing arbitrary unitary matrix U R which diagonalizes M D on the right, there are certainly other solutions, one of which is discussed in Ref. [9].
Another approach is to use Eq. (5), which may be implemented with the non-Abelian discrete symmetry A 4 and a scotogenic Dirac neutrino mass matrix proportional to a real scalar mass-squared matrix. It is the analog of previous suggestions [10,11] for Majorana neutrinos, but in the case of Dirac neutrinos here, it is more technically natural.