UUCRHEP-T610Feb 2021
Cobimaximal Mixing with Dirac Neutrinos
Ernest Ma
Physics and Astronomy Department,University of California, Riverside, California 92521, USA
Abstract
If neutrinos are Dirac, the conditions for cobimaximal mixing, i.e. θ = π/ δ CP = ± π/ × A symmetry and radiative Dirac neutrino masses is presented. a r X i v : . [ h e p - ph ] F e b ntroduction : Neutrinos are mostly assumed to be Majorana. The associated 3 × M ν = A C C ∗ C B DC ∗ D B ∗ , (1)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 cobimaximalmixing pattern [3] of neutrinos, i.e. θ = π/ δ CP = ± π/
2, which is close to what isobserved [4].To understand the cobimaximal mixing matrix U CBM , consider its form in the PDGconvention, i.e. U CBM = c c s c ± is − (1 / √ s ± ic s ) (1 / √ c ∓ is s ) − c / √ / √ − s ± ic s ) (1 / √ c ± is s ) c / √ . (2)Note that the (2 i ) and (3 i ) entries for i = 1 , , M ν = U CBM M diag U TCBM . (3)In Eq. (1), the neutrino basis is chosen for which the charged-lepton mass matrix M l isdiagonal which links the left-handed ( e, µ, τ ) to their right-handed counterparts. Suppose itis not, but rather that it is digonalized on the left by the special matrix [5, 6] U ω = 1 √ ω ω ω ω , (4)where ω = exp(2 πi/
3) = − / i √ /
2. It was discovered in 2000 [7] that U CBM = U † ω O , (5)2here O is an orthogonal matrix. The proof is very simple because the product U ω O enforcesthe equal magnitudes of the (2 i ) and (3 i ) entries.The implications of these conditions regarding Dirac (instead of Majorana) neutrinosare the subject of this paper. A specific model of radiative Dirac neutrino masses with A symmetry [8] is also presented. Form of Dirac Neutrino Mass Matrix : Consider the 3 × ν L to ν R ,i.e. M D . It is diagonalized by two unitary matrices, U L on the left and U † R on the right. Toeliminate U R which is unobservable in the standard model (SM) of quarks and leptons, theproduct M D M † D should be studied. It is automatically Hermitian, and is diagonalized by U L on the left and U † L on the right. Using U CBM of Eq. (2), it is easily seen that M D M † D = A C C ∗ C ∗ B DC D ∗ B . (6)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 M D = a c c ∗ d b ed ∗ e ∗ b ∗ , (7)where a is real. It is then trivial to see that M D M † D yields exactly Eq. (6). The origin ofthis M D is a simple extension of the generalized CP transformation of Ref. [2], i.e. ν e ↔ ν e , ν µ ↔ ν τ , ν ce ↔ ν ce , ν cµ ↔ ν cτ , (8)together with complex conjugation. It is important to realize that whereas Eq. (7) guaranteesEq. (6), the former may be obtained without the latter, as shown already in Ref. [9] becauseof the missing arbitrary U R . 3 cotogenic 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 forMajorana neutrinos. Their Dirac counterpart is presented here. It is actually simpler becausea technical problem is naturally avoided in this case as shown below.Following Ref. [8], the non-Abelian discrete symmetry A is used, under which the threefamilies of left-handed lepton doublets transform as the 3 representation, and the threecharged-lepton singlets as 1 , (cid:48) , (cid:48)(cid:48) . There are also three Higgs doublets Φ i = ( φ + i , φ i ) trans-forming as 3. The multiplication rules for two triplets a , , and b , , in this representation [8]are a b + a b + a b = 1 , a b + ωa b + ω a b = 1 (cid:48) , a b + ω a b + ωa b = 1 (cid:48)(cid:48) . (9)Assuming that (cid:104) φ i (cid:105) is the same for i = 1 , ,
3, the 3 × e, µ, τ ) L to( e, µ, τ ) R is then M l = U ω m e m µ
00 0 m τ , (10)which is well-known since 2001.To obtain Dirac neutrinos, three lepton singlets ν R transforming as 3 under A are addedto the SM. Since 3 × (cid:48) + 1 (cid:48)(cid:48) + 3 + 3 , (11)the products ( a b c + a b c + a b c ) and ( a b c + a b c + a b c ) are allowed, so thattree-level Dirac neutrino masses are obtained. To forbid this, a Z (cid:48) symmetry is imposed, sothat ν R are odd, and the SM fields are even as shown in Table 1. Note that all dimension-four terms in the Lagrangian are required to obey A × Z (cid:48) which is only broken togethersoftly by the ss (cid:48) mass terms. Added are dark scalars and fermions which are odd under anexactly conserved Z D symmetry. Lepton number L is conserved as shown. Dirac neutrinomasses are radiatively generated by dark matter [12] as shown in Fig. 1 in analogy with the4ermion/scalar SU (2) L U (1) Y A Z (cid:48) Z D L ( ν, l ) iL − / l iR − , (cid:48) , (cid:48)(cid:48) + + 1 ν iR − + 1( φ + , φ ) i η + , η ) 2 1/2 1 + + 0 s i − s (cid:48) i − − E , E − ) L,R − / − N L,R − Z D and A × Z (cid:48) symmetries.original scotogenic model [13]. The key for cobimaximal mixing is that the s, s (cid:48) scalars arereal fields [10, 11]. × ν L ν R s s ′ E R N L η Figure 1: Scotogenic Dirac neutrino mass.The relevant Yukawa couplings are f E s ¯ E R ν L , f N s (cid:48) ¯ ν R N L , f NE η ¯ N L E R , f EN ¯ η ¯ E L N R . (12)All respect A × Z (cid:48) , with the latter two contributing to the 2 × E L , N L )to ( E R , N R ), i.e. M EN = (cid:32) m E m EN m NE m N (cid:33) = (cid:32) cos θ L − sin θ L sin θ L cos θ L (cid:33) (cid:32) m m (cid:33) (cid:32) cos θ R sin θ R sin θ R cos θ R (cid:33) . (13)5s for the contribution of s and s (cid:48) , the mass-squared matrix for each is proportional to theidentity, whereas the ss (cid:48) mixing is arbitrary, breaking both A and Z (cid:48) at the same timesoftly. Let it be denoted as M ss (cid:48) and assuming that its entries are all much smaller then theinvariant masses of s and s (cid:48) , then it is clear that the Dirac neutrino mass matrix in the basisof Fig. 1 is proportional to M ss (cid:48) and is real up to an unobservable phase, i.e. the relativephase of f N and f E . This means that it is diagonalized by an orthogonal matrix. Combinedwith Eq. (10), cobimaximal mixing is assured.The explicit expression for the scotogenic Dirac neutrino mass matrix is M ν = f ∗ N f E M ss (cid:48) π (cid:34) sin θ L cos θ R m [ F ( x ) − F ( y )][ x − y ] − cos θ L sin θ R m [ F ( x ) − F ( y )][ x − y ] (cid:35) , (14)where x , = m s /m , and y , = m s (cid:48) /m , , and F ( x ) = x ln xx − . (15) The A → Z Breaking : The breaking of A by (cid:104) φ i (cid:105) = v reduces this symmetry to Z [14].It must be maintained for U ω to be valid. However, the addition of s and s (cid:48) would allow thequartic terms s i s j Φ † i Φ j and s (cid:48) i s (cid:48) j Φ † i Φ j . The key now is that both the quadratic mass terms s i s i and s (cid:48) i s (cid:48) i do not break Z (cid:48) and are required also not to break A . Only the s i s (cid:48) j termsbreak both A and Z (cid:48) softly together. Hence the one-loop correction to Φ † i Φ j is shown inFig. 2. Two mass insertions are required, which render the diagram finite and suppressed Φ i Φ j s i , s ′ i s j , s ′ j s ′ k , s k × × Figure 2: Finite one-loop correction to Φ † j Φ j .6o that the residual Z symmetry is maintained to a good approximation. In Refs. [10, 11],this option is not available for Majorana neutrinos because s (cid:48) is absent and s i s j breaks A ,which yields only one mass insertion in Fig. 2, thus making it logarithmically divergent. Dark Sector : The dark sector fermions are ( E , E − ) and N . The two neutral ones havemasses m , and the charged one m E . They are assumed greater than the masses of thescalars, m s and m s (cid:48) , with the small M ss (cid:48) mixing between them. Let m s be the smaller, thenthe almost degenerate s , , are dark-matter candidates. They interact with the SM Higgsboson h according to L int = − λs i s i (cid:32) vh + h (cid:33) − λ (cid:48) s i s j (cid:32) vh + h (cid:33) , (16)where h = √ (cid:104) η (cid:105) Re ( η ) + (cid:104) φ i (cid:105) Re ( φ + φ + φ )] (cid:113) (cid:104) η (cid:105) + 3 (cid:104) φ i (cid:105) . (17)This is a straightforward generation of the simplest model of dark matter, i.e. that of a realscalar. The comprehensive analysis of Ref. [15] is thus applicable. Conclusion : It is shown how cobimaximal neutrino mixing, i.e. θ = π/ δ CP = ± π/ 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 isdiscussed in Ref. [9].Another approach is to use Eq. (5), which may be implemented with the non-Abeliandiscrete symmetry A and a scotogenic Dirac neutrino mass matrix proportional to a realscalar mass-squared matrix. It is the analog of previous suggestions [10, 11] for Majorananeutrinos, but in the case of Dirac neutrinos here, it is more technically natural. Acknowledgement : This work was supported in part by the U. S. Department of EnergyGrant No. DE-SC0008541. 7 eferences [1] K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett.
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