Low Scale Type I Seesaw Model for Lepton Masses and Mixings

PHYSICAL REVIEW D 101, 035005 (2020)

Low scale type I seesaw model for lepton masses and mixings

† ‡ A. E. Cárcamo Hernández,* Marcela González , and Nicolás A. Neill Universidad T´ecnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile

(Received 5 June 2019; accepted 15 January 2020; published 6 February 2020)

In contrast to the original type I seesaw mechanism that requires right-handed Majorana neutrinos at energies much higher than the electroweak scale, the so-called low scale seesaw models allow lighter masses for the additional neutrinos. Here we propose an alternative low scale type I seesaw model, where neither linear nor inverse seesaw mechanisms take place, but the spontaneous breaking of a discrete symmetry at an energy scale much lower than the model cutoff is responsible for the smallness of the light active neutrino masses. In this scenario, the model is defined with minimal particle content, where the right- handed Majorana neutrinos can have masses at the ∼50 GeV scale. The model is predictive in the neutrino sector having only four effective parameters that allow to successfully reproduce the experimental values of the six low energy neutrino observables.

DOI: 10.1103/PhysRevD.101.035005

I. INTRODUCTION three distinct one-dimensional irreducible representations where the three families of fermions can be accommodated After minimally extending the Standard Model (SM) to rather naturally. This group has been particularly promising include massive neutrinos, the observed fermion mass in providing a predictive description of the current pattern of hierarchy is extended over a range of 13 orders of magni- SM fermion masses and mixing angles [8–47]. Despite tude, from the lightest active neutrino mass scale up to the several models based on the A4 discrete symmetry have been top quark mass. In addition, the small quark mixing angles proposed, most of them have a nonminimal scalar sector, decrease from one generation to the next while in the lepton composed of several SUð2Þ Higgs doublets, even in their sector this hierarchy is not present since two of the mixing low energy limit, and have A4 scalar triplets in the scalar angles are large and the other one is small. Neither of these spectrum whose vacuum expectation value (VEV) configu- features in the flavor sector is explained in the SM. This is rations in the A4 direction are not the most natural solutions the so-called SM flavor puzzle, which has motivated the of the scalar potential minimization equations. Thus, it construction of theories with extended scalar and/or fermion would be desirable to build an A4 flavor model which at low sectors with additional continuous or discrete groups. energies reduces to the SM model and where the different In particular, extensions of the SM with non-Abelian gauge singlet scalars are accommodated into A4 singlets and discrete flavor symmetries are very attractive since they one A4 triplet [with VEV pattern in the (1,1,1) A4 direction] successfully describe the observed pattern of fermion which satisfies the minimization condition of the scalar masses and mixings (for recent reviews on discrete flavor potential for the whole range of values of the parameter groups see Refs. [1–6]), while they can naturally appear space. To this end, in this work we propose an extension of from the breaking of continuous non-Abelian gauge sym- the SM based on the A4 family symmetry, which is metries or from compactified extra dimensions (see Ref. [7] supplemented by a Z4 auxiliary symmetry, whose sponta- and references therein). Several discrete groups have been neous breaking at an energy scale (vS) much lower than the employed in extensions of the SM. In particular, A4 is the model cutoff (Λ) produces the small light active neutrino smallest discrete group with one three-dimensional and mass scale mν. As we will show in the next sections, in this scenario the masses for the active neutrinos are produced by a type I seesaw mechanism [48–51] mediated by *[email protected] † ∼50 [email protected] three GeV right-handed Majorana neutrinos, where ‡ 2 [email protected] mν ∝ ðvS=ΛÞ . Given the low mass scale of the right-handed neutrinos, this model can be classified as a low scale type I Published by the American Physical Society under the terms of seesaw, as it has been coined in the literature [52–62]. There the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to are different realizations of low scale seesaw models, as for the author(s) and the published article’s title, journal citation, example inverse or linear [35,46,63–78], where an addi- and DOI. Funded by SCOAP3. tional lepton number violating mass parameter is added.

2470-0010=2020=101(3)=035005(8) 035005-1 Published by the American Physical Society HERNÁNDEZ, GONZÁLEZ, and NEILL PHYS. REV. D 101, 035005 (2020)

In these models, the smallness of mν is related to the SM group. The lepton assignments under the group smallness of the additional parameter. In our case, however, A4 × Z4 are no extra small mass parameter has been included, and the smallness of the light neutrino masses is explained through LL ∼ ð3; 0Þ;NR ∼ ð3; 1Þ; the spontaneous breaking of the auxiliary discrete groups, 0 00 l1R ∼ ð1 ; 1Þ;l2R ∼ ð1; 1Þ;l3R ∼ ð1 ; 1Þ: ð2Þ which leads to a suppression in the Dirac neutrino mass matrix. A From the point of view of the low energy neutrino Here we specify the dimensions of the 4 irreducible observables, the model makes very particular predictions representations by the numbers in boldface and we write the different Z4 charges in additive notation. Regarding the for δCP and θ23, which are not aligned with the central lepton sector, note that the left- and right-handed leptonic values of current fits. Therefore, future improvements in the A A precision of neutrino measurements will provide an exper- fields are accommodated into 4 triplet and 4 singlet imental test of the model. Processes like (i) charged lepton irreducible representations, respectively, whereas the right- A flavor violating decays (l → l0γ) [11,12], (ii) flavor chang- handed Majorana neutrinos are unified into an 4 triplet. ing neutral currents, and (iii) rare top quark decays such as The scalar spectrum of the model includes the SM Higgs ϕ ρ ξ j ¼ 1 t → hc, t → cZ [15], are strongly suppressed, in contrast to doublet and the gauge singlet scalars , j ( , 2, 3). other A4 flavor models (that usually have several Higgs The scalar fields have the following transformation proper- A Z doublets), where these processes can have rates that are at ties under the flavor symmetry 4 × 4: the reach of forthcoming experiments. 00 0 The paper is organized as follows. In Sec. II we describe ϕ ∼ ð1; 0Þ; ρ1 ∼ ð1 ; −1Þ; ρ2 ∼ ð1 ; −1Þ; the model. In Sec. III we present a discussion on lepton ρ3 ∼ ð1; −2Þ; ξ ∼ ð3; −1Þ: ð3Þ masses and mixings and give the corresponding results. We draw our conclusions in Sec. IV. The Appendix provides a We assume the following vacuum configuration for the concise description of the A4 discrete group. A4-triplet gauge singlet scalar ξ:

II. MODEL DESCRIPTION vξ hξi¼pffiffiffi ð1; 1; 1Þ; ð4Þ 3 We propose an extension of the SM where the scalar sector is augmented by the inclusion of four gauge-singlet which satisfies the minimization condition of the scalar scalar fields and the SM gauge symmetry is supplemented potential for the whole range of values of the parameter by the A4 × Z4 discrete group. The symmetry G features the space, as shown in Ref. [31]. With the particle content following spontaneous symmetry breaking pattern: previously specified, we have the following relevant

vS Yukawa terms for the lepton sector, invariant under the G ¼ SUð2Þ Uð1Þ A Z →; L × Y × 4 × 4 symmetries of the model: v SUð2ÞL × Uð1ÞY → Uð1ÞQ; ð1Þ ðLÞ ðLÞ ¯ 1 ðLÞ ¯ 1 LY ¼ y1 ðLLϕξÞ100 l1R þ y2 ðLLϕξÞ1l2R where the symmetry-breaking scales satisfy the hierarchy Λ Λ 1 vS ∼ Oð1Þ TeV >v, vS is the scale of spontaneous break- ðLÞ ¯ ¯ C þ y3 ðLLϕξÞ10 l3R þ yNðNRNRÞ1ðρ3 þ cρ3Þ ing of the A4 × Z4 discrete group, and v ¼ 246 GeV is the Λ electroweak symmetry breaking scale. As mentioned ¯ ˜ ρ1 ¯ ˜ ρ2 þ y1νðLLϕNRÞ10 þ y2νðLLϕNRÞ100 before, the scalar sector of the SM is augmented by the Λ Λ ξ ξ inclusion of four SM gauge singlet scalars. We add these þ y ðL¯ ϕ˜ N Þ þ y ðL¯ ϕ˜ N Þ þ : :; ð Þ extra scalar fields for the following reasons: (i) to build 3ν L R 3s Λ 4ν L R 3a Λ H c 5 nonrenormalizable charged leptons and Dirac neutrino Yukawa terms invariant under the local and discrete groups, where the dimensionless couplings in Eq. (5) are Oð1Þ crucial to generate predictive textures for the lepton sector; parameters. (ii) to generate a renormalizable Yukawa term for the right- In what follows, we describe the role of each discrete handed Majorana neutrinos, that can give rise to ∼50 GeV group factor of our A4 flavor model. The A4 discrete group masses for these singlet fermions. As we will see below, the yields a reduction of the number of model parameters, observed pattern of SM charged lepton masses and leptonic giving rise to predictive textures for the lepton sector, which mixing angles will arise from the spontaneous breaking of are consistent with the lepton mass and mixing pattern, as the A4 × Z4 discrete group. In order to generate the masses will be shown in Sec. III. On the other hand, the Z4 discrete for the light active neutrinos via a type I seesaw mecha- group is the smallest cyclic symmetry allowing a renor- nism, we extend the fermion sector by including three right- malizable Yukawa term for the right-handed Majorana handed Majorana neutrinos, which are singlets under the neutrinos, giving rise to a diagonal Majorana neutrino

035005-2 LOW SCALE TYPE I SEESAW MODEL FOR LEPTON MASSES … PHYS. REV. D 101, 035005 (2020) 0 1 mass matrix that yields degenerate Majorana neutrinos with me mμ mτ electroweak scale masses. In addition, the spontaneous 1 B 2 C Ml ¼ pffiffiffi @ ω me mμ ωmτ A: ð9Þ breaking of the A4 × Z4 discrete group at an energy scale 3 2 much lower than the model cutoff is crucial to produce ωme mμ ω mτ small mixing mass terms between the active and sterile neutrinos, allowing the implementation of a low scale type I Regarding the neutrino sector, we find that the resulting seesaw mechanism. Finally, we assume that the VEVs of Dirac neutrino mass matrix reads ξ ρ i ¼ 1 the gauge singlet scalar fields , i ( , 2, 3) satisfy the 0 1 relation b þ caþ da− d D B 2 C vvS Mν ¼ @ a − d ωb þ cω a þ d A pffiffiffi ; ð10Þ vρ ∼ vξ ∼ Oð1Þ TeV ≪ Λ;j¼ 1; 2; 3; ð6Þ 2Λ j a þ da− d ω2b þ cω v ∼ v ∼ v where ξ ρ S is the discrete symmetry breaking scale 2πi ω ¼ e 3 a b c d and Λ is the model cutoff. where and , , , are effective parameters It is worth mentioning that this model at low energies related to the neutrino Yukawa couplings in Eq. (5). corresponds to a singlet-doublet model [79,80]. Con- The fermion sector is extended by including three right- m sequently, from a detailed analysis of the low energy handed Majorana neutrinos with masses N, where the M scalar potential (as done for example in Ref. [81]) one Majorana mass matrix N is proportional to the identity, M ¼ m 1ˆ m ≫ ðMDÞ i j ¼ 1 can show that the 125 GeV SM-like Higgs boson has N N 3×3. Given that N ν ij ( , , 2, 3), couplings close to the SM expectation, with small the light active neutrino mass matrix (Mν) arises from a 2 2 −2 deviations of order v =vS ∼ Oð10 Þ. The TeV-scale type I seesaw mechanism: 0 0 singlet s (s ¼ ξ, ρj) will mix with the CP-even neutral 0 M ¼ MDM−1ðMDÞT ð Þ component of the SM Higgs doublet, h , with a mixing ν ν N ν 11 angle γ ∼ Oðv=vSÞ. Thus, the couplings of the singlet 0 1 scalars to the SM particles will be equal to the SM Higgs b þ caþ da− d 1 B C couplings times the s0 − h0 mixing angle γ. The collider ¼ @ a − d ωb þ cω2 a þ d A m phenomenology of this scenario is well studied [82–86]. N a þ da− d ω2b þ cω For TeV-scale singlets, the most stringent limits at the 0 1 b þ ca− daþ d 8 TeV LHC come from indirect searches. A global fit B C v2v2 2 γ ≤ 0 23 @ a þ dcω2 þ bω a − d A S ; ð Þ to all SM signal strengths constrains sin . at × 2Λ2 12 95% C.L. [87,88], that assuming Oð1Þ couplings in the a − daþ dbω2 þ cω scalar potential translates to vS ≳ 500 GeV. For a summary of the sensitivity of future colliders see for example where we can read that the typical mass scale of the light Table 1 of Ref. [86]. As we will see in the next section, active neutrinos is v there is a broad range of values of S that are consistent 2 2 with the observed light neutrino masses and current limits v vS on singlet scalars. mν ∼ : ð13Þ 2mN Λ

III. NEUTRINO MASSES AND MIXINGS It is noteworthy that the smallness of the active neutrino masses is a consequence of their inverse scaling with the The lepton Yukawa terms in Eq. (5) imply that the mass square of the model cutoff, which is much larger than the matrix for charged leptons is given by breaking scale (vS) of the discrete symmetries. We can see from Eq. (13) that for heavy neutrinos with masses M ¼ V S ðm ; −m ;m Þ; ð Þ l lL ldiag e μ τ 7 mN ∼ Oð50 GeVÞ, there is a wide range of values of vS that produce the required suppression, depending on the where specific value of the model cutoff. To show that the model 0 1 0 1 is consistent with the neutrino oscillation experimental 11 1 0 −10 data, we fix mν ¼ 50 meV and vary the neutrino sector 1 B 2 C B C 2πi a b c d VlL ¼ pffiffiffi@1 ω ω A;Sl ¼ @100A; ω ¼ e 3 ; parameters , , and , to adjust the neutrino mass 3 Δm2 Δm2 1 ωω2 001 squared splittings 21, 31, the leptonic mixing angles θ12, θ13, θ23, and the leptonic Dirac CP violating phase δCP ð8Þ to their experimental values. Tables I and II show the average model values and so experimental values of the neutrino observables for both

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TABLE I. Normal mass hierarchy.—Model and experimental values of the neutrino mass squared splittings, leptonic mixing angles, and CP-violating phase. The second column shows the average model value for each observable, calculated from the model solutions that reproduce the neutrino observables at the 90% C.L. The experimental values are taken from Refs. [89,90].

Neutrino oscillation global fit values (NH) Average Observable model value Best fit 1σ [89] Best fit 1σ [90] 3σ range [89] 3σ range [90] Δm2 10−5 2 7 55þ0.20 7 39þ0.21 – – 21 [ eV ] 7.61 . −0.16 . −0.20 7.05 8.14 6.79 8.01 Δm2 10−3 2 2 50 0 03 2 525þ0.033 – – 31 [ eV ] 2.51 . . . −0.032 2.41 2.60 2.427 2.625 θ ð Þ 34 5þ1.2 33 82þ0.78 – – 12 ° 34.1 . −1.0 . −0.76 31.5 38.0 31.61 36.27 þ0 16 θ13ð Þ 8 45 . 8 61 0 13 – – ° 8.45 . −0.14 . . 8.0 8.9 8.22 8.99 θ ð Þ 47 7þ1.2 49 6þ1.0 – – 23 ° 42.8 . −1.7 . −1.2 41.8 50.7 40.3 52.4 δ ð Þ þ38 þ40 – – CP ° 313 218−27 215−29 157 349 125 392

TABLE II. Inverted mass hierarchy.—Model and experimental values of the neutrino mass squared splittings, leptonic mixing angles, and CP-violating phase. The second column shows the average model value for each observable, calculated from the model solutions that reproduce the neutrino observables at the 90% C.L. The experimental values are taken from Refs. [89,90].

Neutrino oscillation global fit values (IH) Average Observable model value Best fit 1σ [89] Best fit 1σ [90] 3σ range [89] 3σ range [90] Δm2 10−5 2 7 55þ0.20 7 39þ0.21 – – 21 [ eV ] 7.61 . −0.16 . −0.20 7.05 8.14 6.79 8.01 Δm2 10−3 2 2 42þ0.03 2 512þ0.034 – – 13 [ eV ] 2.41 . −0.04 . −0.032 2.31 2.51 2.412 2.611 þ1 2 þ0 78 θ12ð Þ 34 5 . 33 82 . – – ° 34.7 . −1.0 . −0.76 31.5 38.0 31.61 36.27 þ0 14 θ13ð Þ 8 53 . 8 65 0 13 – – ° 8.56 . −0.15 . . 8.1 9.0 8.27 9.03 θ ð Þ 47 9þ1.0 49 8þ1.0 – – 23 ° 48.7 . −1.7 . −1.1 42.3 50.7 40.6 52.5 δ ð Þ þ23 þ27 – – CP ° 297 281−27 284−29 202 349 196 360 normal hierarchy (NH) and inverted hierarchy (IH). for comparison are plotted over the experimental values Figure 1 shows several solutions consistent with the taken from Ref. [89]. To give an example, for each global fits for the NH (the IH has the same behavior). hierarchy we choose a representative value of the neutrino The dots in orange correspond to the model values, which sector parameters:

2 2 2 2 2 FIG. 1. Correlation between the observables predicted by the model: sin θ12, sin θ13, sin θ23, δCP, Δm21, and Δm31 for the normal hierarchy, superimposed on the global fits from Ref. [89]. Model predictions are shown in orange, while the 90%, 95%, and 99% C.L. contours of the global fit are in purple, blue, and light blue, respectively.

035005-4 LOW SCALE TYPE I SEESAW MODEL FOR LEPTON MASSES … PHYS. REV. D 101, 035005 (2020) a≃−0.474;b≃−0.367;c≃0.487;d≃−0.0590 ðNHÞ discrete group. This mechanism for inducing CP violation in the fermion sector via the spontaneous breaking of a≃−0.254;b≃0.352;c≃−0.795;d≃0.0174 ðIHÞ discrete groups is the so-called geometrical CP violation ð14Þ [46,75,76,91–97]. Now we determine the effective Majorana neutrino which produces the following mass spectrum: mass parameter, which is proportional to the neutrinoless double beta decay (0νββ) amplitude. The effective Majorana m1 ≃5.84½meV;m2 ≃10.5½meV;m3 ≃50.4½meVðNHÞ neutrino mass parameter is given by m ≃50 4½ ;m≃51 1½ ;m≃11 7½ ð Þ: 1 . meV 2 . meV 3 . meV IH X ð15Þ m ¼ m U2 ; ð Þ ββ νj ej 16 j Thus, with four effective parameters, the model reproduces the experimental values of the six physical observables of U2 m the neutrino sector, i.e., the neutrino mass squared splittings where ej and νj are the Pontecorvo-Maki-Nakagawa- 2 2 Δm21 and Δm31, the leptonic mixing angles θ12, θ23, θ13, Sakata mixing matrix elements and the Majorana neutrino and the leptonic Dirac CP violating phase δCP. The model masses, respectively. As we can see from Fig. 2,the values are consistent with the current neutrino oscillation predicted effective Majorana neutrino mass parameter is experimental data, for both normal and inverted mass within the range: ordering, as shown in Tables I and II. From Fig. 1,we 2 2 2 can see that for the normal hierarchy, Δm31, Δm21, sin θ12, 2 mββ ¼ð5.0 − 7.4Þ meV ðNHÞ; and sin θ13 are evenly distributed in the allowed range. On 2 m ¼ð24 7 − 35 3Þ ð Þ; ð Þ the other hand, for sin θ23 and δCP, the model features ββ . . meV IH 17 more definite predictions. The same behavior is found for the inverted hierarchy. which is below the sensitivity of present 0νββ-decay experi- It is worth mentioning that in a generic scenario, the ments. The current experimental sensitivity on the Majorana neutrino Yukawa couplings are complex, thus the light neutrino mass parameter is obtained from the KamLAND- active neutrino sector has eight parameters. However, not 136 0νββ T0νββð136 Þ ≥ all of them are physical. Considering the case of real VEVs Zen limit on the Xe decay half-life, 1=2 Xe 26 for the gauge-singlet scalars ρ1, ρ2, and ξ, the phase 1.07 × 10 yr [98], which yields the corresponding upper redefinition of the leptonic fields LL and NR allows to limit on the Majorana mass, jmββj ≤ ð61 − 165Þ meV at rotate away the phase of one of the neutrino Yukawa 90% C.L. For other 0νββ-decay experiments see Refs. [99– couplings, leading to seven physical parameters. On the 104]. The experimental sensitivity of neutrinoless double other hand, if we consider complex VEVs for the gauge- beta decay searches is expected to improve in the near future. singlet scalars ρ1, ρ2, and ξ, we can use their phases to set Note that the model predicts a range of values for neutrino- three of the four neutrino Yukawa couplings real. less double beta decay rates that can be tested by the next- Therefore, in this case we are left with five effective generation bolometric CUORE experiment [102],aswellas parameters in the neutrino sector. However, for the sake the next-to-next-generation ton-scale 0νββ-decay experi- of simplicity, we are considering a particular benchmark ments [98,101,105,106]. scenario with real neutrino Yukawa couplings, i.e., four Finally, we briefly comment on the prospects of observ- effective parameters. In this simplified benchmark scenario, ing heavy neutrinos with masses around 50 GeV in collider the complex phase responsible for CP violation in neutrino experiments. In the type I seesaw model, the heavy light 2 oscillation arises from the spontaneous breaking of the A4 mixing squared, jUj is given by

FIG. 2. Model predictions for the Dirac CP violating phase versus the effective Majorana mass parameter.

035005-5 HERNÁNDEZ, GONZÁLEZ, and NEILL PHYS. REV. D 101, 035005 (2020) 2 MD mν The effective Majorana neutrino mass parameter is pre- jUj2 ∼ ∼ ; ð18Þ MN mN dicted to be which in our case (for mN ≈ 50 GeV and mν ≈ 50 meV) ð6 2 0 5Þ ; ð Þ 2 −12 . . meV NH gives jUj ∼ Oð10 Þ. Even though this is a very small mββ ¼ ð21Þ ð31 1 2 6Þ ; ð Þ: mixing, typical of the type I seesaw model, for masses . . meV IH mN ∼ 50 GeV it might be within the reach of future colliders such as the FCC-ee [107]. The most sensitive The scalar sector of the model corresponds to the SM Higgs channel at the FCC-ee would be Z → νN, when the decays 0 doublet supplemented with additional singlet scalars. The of N are fully reconstructible, i.e., N → lW → lqq¯ . phenomenology of this kind of extended Higgs sectors is According to the analysis in Ref. [107], most of the well studied, and many direct and indirect searches have backgrounds for this decay can be reduced if one takes been proposed in the literature. For masses of the additional into consideration (i) the displaced vertex topology pro- scalar singlets (mS) in the range 1 TeV ≲ mS ≲ 11 TeV, N duced by the long-lived (expected for these small the scalar sector would be within the reach of future couplings) and (ii) the full reconstruction of the heavy colliders. neutrino mass, allowed by its visible decay. For ∼ 1013 Z decays, this would allow reaching sensitivities down to a heavy-light mixing jUj2 ∼ 10−12, for heavy neutrino ACKNOWLEDGMENTS masses between 40 and 80 GeV. This work was supported by FONDECYT (Chile) under Grants No. 1170803, No. 3170906 and No. 11180873, and IV. SUMMARY AND CONCLUSIONS in part by Conicyt PIA/Basal FB0821. We have proposed a low scale seesaw model based on the A4×Z4 discrete symmetry, where the masses for the A light active neutrinos are produced by a type I seesaw APPENDIX: THE PRODUCT RULES FOR 4 ∼50 mechanism mediated by three GeV scale right-handed The A4 group has one three-dimensional 3 and three Majorana neutrinos. Contrary to the original type I seesaw, distinct one-dimensional 1, 10, and 100 irreducible repre- where the right-handed neutrinos are required to have sentations, satisfying the following product rules: masses much larger than the electroweak scale to reproduce mν mν the light active neutrino mass scale , in this case is 0 00 suppressed by the ratio between the discrete symmetry 3 ⊗ 3 ¼ 3s ⊕ 3a ⊕ 1 ⊕ 1 ⊕ 1 ; breaking scale (vS) and the cutoff (Λ) of the model: 1 ⊗ 1 ¼ 1; 10 ⊗ 100 ¼ 1; 2 mν ∝ ðvS=ΛÞ . That is, the large Λ=vS ratio plays the role 10 ⊗ 10 ¼ 100; 100 ⊗ 100 ¼ 10: ð Þ of the heavy mass scale in the original seesaw. This allows A1 lighter Majorana masses that might be eventually tested at future colliders such as the FCC-ee. Considering ðx1;y1;z1Þ and ðx2;y2;z2Þ as the basis vectors The model is predictive in the sense that it reproduces for two A4-triplets 3, the following relations are fulfilled: the experimental values of the six low energy neutrino observables with only four effective parameters. Two of the predicted observables (being within the 90% C.L. global-fit ð3 ⊗ 3Þ1 ¼ x1y1 þ x2y2 þ x3y3; regions) are not aligned with the central values of the global ð3 ⊗ 3Þ3 ¼ðx2y3 þ x3y2;x3y1 þ x1y3;x1y2 þ x2y1Þ; fits, so are distinctive predictions of the model. These are s 2 the CP-violating angle, predicted to be ð3 ⊗ 3Þ10 ¼ x1y1 þ ωx2y2 þ ω x3y3;

ð3 ⊗ 3Þ ¼ðx2y3 − x3y2;x3y1 − x1y3;x1y2 − x2y1Þ; 312.9° 2.4°; ðNHÞ 3a δCP ¼ ð19Þ ð3 ⊗ 3Þ ¼ x y þ ω2x y þ ωx y ; ð Þ 297.2° 2.7°; ðIHÞ; 100 1 1 2 2 3 3 A2 and the “atmospheric” neutrino mixing angle i2π where ω ¼ e 3 . The representation 1 is trivial, while the 10 100 0 465 0 004; ð Þ nontrivial and are complex conjugate to each other. 2 . . NH sin θ23 ¼ ð20Þ Reviews of discrete symmetries in particle physics can be 0.565 0.001; ðIHÞ: found in Refs. [1–4].

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