Based Neutrino Masses with Majoron Decaying Dark Matter

Chapter 14 A4-based neutrino masses with Majoron decaying dark matter Joao˜ N. Esteves Abstract We propose an A4 flavor-symmetric SU(3) SU(2) U(1) seesaw model where lepton number is broken spontaneously. A consistent two-zero texture⊗ pattern of⊗ neutrino masses and mixing emerges from the inter- play of type-I and type-II seesaw contributions, with important phenomenological predictions. We also discuss the possibility of the decaying but long-lived Majoron to be a good candidate for dark matter. We suggest [1] a version of the seesaw mechanism containing both type-I [2,4,3,5,5,7,8,9] and type-II contributions [8,9,10,11,12,13] in which we implement an A4 flavor symmetry with spontaneous violation of lepton number [7,9]. We study the resulting pattern of vacuum expectation values (vevs) and show that the model reproduces the phenomenologically consistent and predictive two-zero texture proposed in Ref. [14]. In the presence of explicit global symmetry breaking effects, as might follow from gravitational interactions, the resulting pseudo-Goldstone boson - Majoron - may constitute a viable candidate for decaying dark matter if it acquires mass in the keV-MeV range. Indeed, this is not in conflict with the lifetime constraints which follow from current cosmic microwave background (CMB) observations provided by the Wilkinson Microwave Anisotropy Probe (WMAP) [15]. We also show how the corresponding mono-energetic emission line arising from the sub-leading one-loop induced electromagnetic decay of the Majoron may be observed in future X-ray missions [16]. 14.1. model Our model is described by the multiplet content specified in Table 14.1 where the transformation properties under the SM and A4 groups are shown (as well as the corresponding lepton number L). The Li and lRi fields are the usual SM lepton doublets and singlets and νR the right-handed neutrinos. The scalar sector contains an SU(2) triplet ∆, three Higgs doublets Φi (which transform as a triplet of A4) and a scalar singlet σ. Three additional fermion singlets Si are also included. 96 97 Table 14.1 Lepton multiplet structure (Q = T3 + Y=2) L1 L2 L3 lRi νiR Φi ∆ σ Si SU(2) 2 2 2 1 1 2 3 1 1 U(1)Y 1 1 1 2 0 1 2 0 0 −0 − −00 − − 00 00 A4 1 1 1 3 3 3 1 1 3 L 1 1 1 1 1 0 2 2 1 − − Taking into account the information displayed in Table 14.1, and imposing lepton number conservation, the Lagrangian responsible for neutrino masses reads 0 00 T T = h1L1 (ν Φ) + h2L2 (ν Φ) + h3L3 (ν Φ) + λL C∆L2 + λL C∆L1 −LL R 1 R 1 R 1 1 2 0 T T 0 + λ L3 C∆L3 + MR SLνR 1 + h SL CSL 1 σ + h.c. ; (14.1) where h and λ are adimensional couplings, MR is a mass scale and + 0 ∆0 ∆ =p2 φi ∆ = − ; Φi = − : (14.2) ∆+=p2 ∆++ φ − i T 0 Note that the term (νR CνR)1σ is allowed by the imposed symmetry. This term however does not contribute to the light neutrino masses to the leading order in the seesaw expansion and we omit it. Alternatively, such term may be forbidden by holomorphy in a supersymmetric framework with the following superpotential terms 1 = + λ hν Lâν^cH^ b + M ν^cS^ + hS^S^σ^ W ··· ab i i u R 2 where the hats denote superfields and the last term replaces the corresponding bilinear employed in Ref. [17, 0 18]. Assuming that the Higgs bosons Φi, ∆ and σ acquire the following vevs 0 0 0 v 0 φ1 = φ2 = φ3 = ; ∆ = u∆; σ = uσ ; (14.3) p3 h i we obtain an extended seesaw neutrino mass matrix [17,18,19] in the (ν , νc, S) basis M L 0 1 0 1 1 !2 ! 0 mD 0 T 1 B C = @ m 0 M A ; mD = v diag(h1; h2; h3) U; U = B 1 1 1 C ; (14.4) D p M 0 M T µ 3 @ A 1 !!2 2πi=3 2 with ! = e , M = MR diag(1; 1; 1) and µ = uσh diag(1; w ; w). This leads to an effective light neutrino mass matrix I given by Mν 0 2 1 h1 0 0 2 I T −1 −1 T hv uσ B C ν = mDM µM mD = 2 B 0 0 h2h3 C : (14.5) M MR @ A 0 h2h3 0 98 On the other hand the vev of the triplet, u∆, will induce an effective mass matrix for the light neutrinos from type-II seesaw mechanism 0 0 λ 0 1 II B C = 2u∆ B λ 0 0 C ; (14.6) Mν @ A 0 0 λ0 and the total effective light neutrino mass matrix will then be = I + II : (14.7) Mν Mν Mν In Ref.[14] it was shown that the neutrino mass matrix given by Eq. (14.7) could explain the currently available neutrino data. In section 14.2 we will present an update of that analysis taking into account the latest neutrino oscillation data. In [1] we show that the minimization of the Higgs potential is consistent with the A4 symmetry. 14.2. neutrino parameter analysis Given the two contributions to the light neutrino mass matrix discussed in Eqs. (14.5) and (14.6) one finds that the total neutrino mass matrix has the following structure: 0 a b 0 1 B C ν = B b 0 c C : (14.8) M @ A 0 c d This matrix with two-zero texture has been classified as B1 in [8]. One can show that considering the 0 00 00 0 (L1;L2;L3) transformation properties under A4 as being (1 ; 1 ; 1) or (1 ; 1 ; 1) an effective neutrino mass matrix with (1; 2) = (3; 3) = 0 is obtained (type B2 in [8]). Moreover, by choosing ∆; σ 10 and Mν Mν ∼ appropriate transformation properties of the Li doublets, we could obtain the textures B1 and B2 as well. Still, the configuration ∆; σ 1 would lead to textures which are incompatible with neutrino data since, in this case, both type I and type∼ II contributions to the effective neutrino mass matrix would have the same form. Since the textures of the type B1 and B2 are very similar in what concerns to neutrino parameter predictions, we will restrict our analysis to B1, shown in (14.8). In general, the neutrino mass matrix is described by nine parameters: three masses, three mixing angles and three phases (one Dirac + two Majorana). From neutrino oscillation experiments we have good determi- nations for two of the mass parameters (mass squared differences) and for two of the mixing angles (θ12 and θ23) as well as an upper-bound on the third mixing angle θ13. Using the 3σ allowed ranges for these five parameters and the structure of the mass matrix in Eq. (14.8) we can determine the remaining four parameters. The phenomenological implications of this kind of mass matrix have been analysed in Refs. [14] and [21]. Here we will update the results in light of the recently determined neutrino oscillation parameters [2]. The main results are shown in 14.1. On the left of figure 14.1 we plot the correlation of the mass parameter characterizing the neutrinoless double beta decay amplitude: 2 2 2 2 2iα 2 2iβ mee = c c m1 + c s m2e + s m3e ; (14.9) j j 13 12 13 12 13 with the atmospheric mixing angle θ23. Here cij and sij stand for cos θij and sin θij respectively. At the 2 ◦ zeroth order approximation m1=m3 = tan θ23, and therefore θ23 < 45 for normal hierarchy (NH), while ◦ θ23 > 45 for inverted hierarchy (IH). The main result from this plot is a lower bound on the effective neutrino 99 Figure 14.1. Left panel: correlation between the neutrinoless double beta decay amplitude parameter mee and the atmospheric mixing parameter. Experimental sensitivities are also given for comparison. Centerj j and right panels: CP violating phase δ and CP-invariant J in terms of the reactor mixing parameter. The 2 3 σ-excluded range for sin θij is given for comparison. mass: mee > 0:03 eV. For comparison the range of sensitivities of planned experiments as well as current boundsj isj also given. Note that the lower bound we obtain lies within reach of the future generation of neutrinoless double beta decay experiments. The center and right panels in Fig. 14.1 show the CP-violating phase δ and the corresponding CP- 2 violating invariant J in neutrino oscillations versus sin θ13. Note that these hold both for normal and in- 2 verted hierarchy spectra. In the middle panel one sees that cos δ < 0 since, at first order in sin θ13, cos θ23 m1=m2 = 1 + 2 sin θ13 cos δ, and the ratio of masses should satisfy: m1=m2 < 1. Moreover, cos θ12 sin θ12 sin θ23 for large θ13 values, where CP violation is likely to be probed in neutrino oscillations, one can see that our model predicts maximal violation of CP. Quantitatively, from the right panel one sees that the 3σ bound on 2 θ13: sin θ13 < 0:053 implies an upper bound: J . 0:06 on the CP-invariant. In addition, the two-zero texture structure ofj j our neutrino mass matrix may have other implications, for example for the expected pattern of lepton flavor violating decays. In fact, thanks to the strong renormalization effects due to the presence of the triplet states, the latter are quite sizeable in sypersymmetric models [23,24, 25].

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