Soft $A_4 \to Z_3$ Symmetry Breaking and Cobimaximal Neutrino Mixing

I propose a model of radiative charged-lepton and neutrino masses with $A_4$ symmetry. The soft breaking of $A_4$ to $Z_3$ lepton triality is accomplished by dimension-three terms. The breaking of $Z_3$ by dimension-two terms allow cobimaximal neutrino mixing $(\theta_{13} \neq 0, \theta_{23} = \pi/4, \delta_{CP} = \pm \pi/2)$ to be realized with only very small finite calculable deviations from the residual lepton triality. This construction solves a long-standing technical problem inherent in renormalizable $A_4$ models since their inception.

For the past several years, some new things have been learned regarding the theory of neutrino flavor mixing. (1) Whereas the choice of symmetry, for example A 4 [1,2,3], and its representations are obviously important, the breaking of this symmetry into specific residual symmetries, for example A 4 → Z 3 lepton triality [4,5], is actually more important. (2) A mixing pattern may be obtained [6] independent of the masses of the charged leptons and neutrinos. (3) The clashing of residual symmetries between the charged-lepton, for example A 4 → Z 3 , and neutrino, for example A 4 → Z 2 , sectors is technically very difficult to maintain [7]. (4) The essential incorporation of CP transformations [8,9] may be the new approach [10,11,12,13,14,15] which will lead to an improved understanding of neutrino flavor mixing.
In this paper, a model of radiative charged-lepton and neutrino masses is proposed with the following properties. (1) The masses are generated in one loop through dark matter [16], i.e. particles distinguished from ordinary matter by an exactly conserved dark symmetry. This is the so-called scotogenic mechanism. (2) The symmetry A 4 × Z 2 is imposed on all dimension-four terms of the renormalizable Lagrangian with particle content given in Table 1. (3) Dimension-three terms break A 4 × Z 2 , but all such terms respect the residual Z 3 lepton triality. (4) Dimension-two terms break Z 3 , which is nevertheless retained in dimension-three (and dimension-four) terms with only finite calculable deviations. This solves the problem of clashing residual symmetries. (5) The proposed specific model results in cobimaximal [15] neutrino mixing (θ 13 = 0, θ 23 = π/4, δ CP = ±π/2), which is consistent with the present data [17,18]. It is also theoretically sound, because the residual Z 3 is protected, unlike previous proposals. Cobimaximal mixing becomes thus a genuine prediction, robustly supported in the context of a complete renormalizable theory of neutrino mass and mixing.
The dark U (1) D and Z 2 symmetries are assumed to be unbroken. The other Z 2 symmetry is used to forbid the dimension-four Yukawa couplingsl L l R φ 0 so that charged leptons only Table 1: acquire masses in one loop as shown in Fig. 1. Whereas this Z 2 is respected by the dimension- fourl R N L χ − terms, it is broken softly by the dimension-three trilinear η + χ − φ 0 term to complete the loop. This guarantees the one-loop charged-lepton mass to be finite. Note that a dark U (1) D symmetry [19,20] is supported here with χ + , (η + , η 0 ), and N L,R all transforming as 1 under U (1) D . The dimension-three soft termsN L N R are assumed to break A 4 to Z 3 through the well-known unitary matrix [1,21,22] U ω , i.e. where (2) In the A 4 limit, M N is proportional to the identity matrix. With three different mass eigenvalues, the residual symmetry is Z 3 lepton triality. Let the (η + , χ + ) mass eigenvalues be m 1,2 with mixing angle θ, then each lepton mass is given by [19] m where The dark U (1) D symmetry forbids the quartic scalar term (Φ † η) 2 , so that a neutrino mass is not generated as in Ref. [16]. It comes instead from Fig. 2, where the scalars s 1,2,3 are assumed real [10,23,24] to enable cobimaximal mixing, hence a separate dark Z 2 symmetry is required. Let theF L E R mass term be m D and assumed to be much smaller than m E , m F , where The dimension-two s i s j terms are allowed to break Z 3 arbitrarily. However, since this mass-squared matrix is real, it is diagonalized by an orthogonal matrix O, hence the neutrino mixing matrix is given by [10,25,26] resulting in U µi = U * τ i , thus guaranteeing cobimaximal mixing: θ 13 = 0, θ 23 = π/4, δ CP = ±π/2.
In a previous proposal [10], instead of Fig. 1 If the s i s j mass-squared terms break Z 3 as in Fig. 2, then the s 1 s 2 (x + 1 x − 2 + x + 2 x − 1 ) term from the above will induce a quadratic x 1 x 2 term as shown in Fig. 3. Whereas this diagram is Figure 3: One-loop generation of x 1 x 2 term from s 1 s 2 term.
not quadratically divergent, it is still logarithmically divergent. This means a counterterm is required for x + 1 x − 2 + x + 2 x − 1 , thereby invalidating the Z 3 residual symmetry necessary to derive U ω and thus Eq. (6).
In this proposal, the A 4 → Z 3 breaking comes fromN L N R , with the Dirac fermions N 1,2,3 distinguished from one another by the residual Z 3 lepton triality through U ω as shown in Eq. (1). The soft breaking of Z 3 by s 1 s 2 induces only a finite two-loop correction to the N 1 − N 2 wavefunction mixing as shown in Fig. 4. Therefore this construction solves a long- breaking in the dimension-three terms, justifying the use of U ω to obtain Eq. (6).
As for dark matter, there are in principle two stable components: the lightest N with U (1) D symmetry and the lightest s with Z 2 symmetry. Whereas N has only the allowed N R (ν L η 0 − l L η + ) interactions, s has others, i.e. s 2 Φ † Φ, s 2 η † η, s 2 χ + χ − , as well as s(ν L E 0 R + l L E − R ). Their interplay to make up the total correct dark-matter relic abundance of the Universe and how they may be detected in underground direct-search experiments require further study.
An immediate consequence of radiative charged-lepton mass is that the Higgs Yukawa coupling hll is no longer exactly m l /(246 GeV) as predicted by the standard model, as studied in detail already [27,28]. Because of the Z 3 lepton triality, large anomalous muon magnetic moment may be accommodated while µ → eγ is suppressed [28].
In conclusion, cobimaximal neutrino mixing (θ 13 = 0, θ 23 = π/4, δ CP = ±π/2) is achieved rigorously in a renormalizable model of radiative charged-lepton and neutrino masses. The key is the soft breaking of A 4 to Z 3 by dimension-three terms, so that the subsequent breaking of Z 3 by dimension-two terms only introduces very small finite corrections to the U ω transformation needed to obtain cobimaximal mixing as given by Eq. (6).
This work is supported in part by the U. S. Department of Energy under Grant No. de-sc0008541.