Neutrino Mixing and CP Phase Correlations

A special form of the $3 \times 3$ Majorana neutrino mass matrix derivable from $\mu - \tau$ interchange symmetry accompanied by a generalized $CP$ transformation was obtained many years ago. It predicts $\theta_{23} = \pi/4$ as well as $\delta_{CP} = \pm \pi/2$, with $\theta_{13} \neq 0$. Whereas this is consistent with present data, we explore a deviation of this result which occurs naturally in a recent proposed model of radiative inverse seesaw neutrino mass.

where A, B are real. It was shown that θ 13 = 0 and yet both θ 23 and the CP nonconserving phase δ CP are maximal, i.e. θ 23 = π/4 and δ CP = ±π/2. Subsequently, this pattern was shown [3] to be protected by a symmetry, i.e. e → e and µ ↔ τ exchange with CP conjugation. All three predictions are consistent with present experimental data. Recently, a radiative (scotogenic) model of inverse seesaw neutrino mass has been proposed [4] which naturally obtains where λ = f τ /f µ is the ratio of two real Yukawa couplings.
This model has three real singlet scalars s 1,2,3 and one Dirac fermion doublet (E 0 , E − ) and one Dirac fermion singlet N , all of which are odd under an exactly conserved (dark) Z 2 symmetry. As a result, the third one-loop radiative mechanism proposed in 1998 [5] for generating neutrino mass is realized, as shown below. The mass matrix linking (N L ,Ē 0 L ) to (N R , E 0 R ) is given by where m N , m E are invariant mass terms, and m D , m F come from the Higgs vacuum expectation value φ 0 = v/ √ 2. As a result, N and E 0 mix to form two Dirac fermions of masses m 1,2 , with mixing angles To connect the loop, Majorana mass terms (m L /2)N L N L and (m R /2)N R N R are assumed.
Since both E and N may be defined to carry lepton number, these new terms violate lepton number softly and may be naturally small, thus realizing the mechanism of inverse seesaw [6,7,8] as explained in Ref. [4]. Using the Yukawa interaction f sĒ 0 R ν L , the one-loop Majorana neutrino mass is given by . (6) It was also shown in Ref. [4] that the implementation of a discrete flavor Z 3 symmetry, which is softly broken by the 3 × 3 real scalar mass matrix spanning s 1,2,3 , leads to M λ ν of Eq. (2).
To explore how the predictions θ 23 = π/4 and δ CP = ±π/2 are changed for λ = 1, where We then have where We now diagonalize numerically where O is an orthogonal matrix, and M 2 new is diagonal with mass eigenvalues equal to the squares of the physical neutrino masses. Let us define then Since U is known with θ 23 = π/4 and δ = ±π/2, we know ∆ once λ is chosen. The orthogonal matrix O has three angles as parameters, so A has three parameters. In Eq. We then consider inverted ordering, using m 3 instead of m 1 . We plot in Figs. 5, 6, and 7 the corresponding results. Note that in our scheme, the effective neutrino mass m ee measured in neutrinoless double beta decay is very close to m 1 in normal ordering and m 3 + ∆m 2 32 in inverted ordering. We see similar constraints on sin 2 (2θ 23 ) and δ CP . In other words, our scheme is insensitive to whether normal or inverted ordering is chosen. Finally, we have checked numerically that θ 23 < π/4 if λ > 1, and θ 23 > π/4 if λ < 1. As we already mentioned, the two solutions are related by the mapping λ → λ −1 .  In conclusion, we have explored the possible deviation from the prediction of maximal θ 23 and maximal δ CP in a model of radiative inverse seesaw neutrino mass. We find that given the present 1σ bound of 0.98 on sin 2 (2θ 23 ), δ CP /(π/2) must be greater than about 0.95.
This work is supported in part by the U. S. Department of Energy under Grant No. de-sc0008541.